![2 Problem of Spacecraft Trajectory Optimization
not be known. The VEEGA trajectory for Galileo [1] is an example; this was not the
only feasible trajectory but was determined to be the optimal flyby sequence.
There are many types of spacecraft trajectories. Until 1998 (and the very suc-
cessful Deep Space 1 mission), spacecraft were propelled only impulsively, using
chemical rockets whose burn duration is so brief in comparison to the total flight time
that it is reasonable to model it as instantaneous. Between impulses, the spacecraft
motion, as a reasonable first approximation, can be considered Keplerian. Interplan-
etary cases add the possibility of planetary flyby maneuvers, which again, as a first
approximation, may be modeled as nearly instantaneous velocity changes, preceded
andfollowedbyKeplerianmotion. Theimpulsivetransfercase, evenincludingflybys,
is thus a parameter optimization problem with the parameters being such quantities
as the timing, magnitude, and direction of the impulsive V’s and the timing and
altitude of gravity assist maneuvers. Of course, for extremely accurate spacecraft
trajectory optimization, the resulting approximate trajectories must be reconsidered
with the perturbations of other solar system bodies, the effect of solar radiation
pressure, and other small but not insignificant effects included.
While the potential benefits of low-thrust electric propulsion have been known
for many years, it has only been relatively recently that spacecraft missions have
been flown using this technology, for example in the NEAR and Deep Space 1
missions. Electric propulsion produces very small thrust, so that typical spacecraft
acceleration is on the order of 10−5 g, and thus thrust is used either continuously or
nearly so. The continuous thrust optimal control problem is qualitatively different
from the impulsive case as there are now no integrable arcs and the control itself,
for example the thrust magnitude and direction, have continuous time histories that
must be modeled and determined. If the electric power is provided by solar cells,
the variation of power available with distance from the sun must also be taken into
consideration. A qualitatively similar continuous thrust case is that of solar sail-
powered spacecraft, which of course are also subject to variation in effectiveness as
they move away from the sun.
While orbit transfer, for example LEO-GEO transfer, and interplanetary tra-
jectories have been the focus of the bulk of research into spacecraft trajectory
optimization, there are certainly many other applications of optimal control theory
and numerical optimization to astrodynamics. Recent interesting problems include:
(1) multi-vehicle navigation and maneuver optimization for cooperative vehicles,
for example a fleet of small satellites in a specified formation [2]; (2) multi-vehicle
noncooperative maneuver optimization, for example pursuit-evasion problems such
as the interception of a maneuvering ICBM warhead by an intercepting spacecraft
or missile [3]; (3) so-called “low-energy” transfer using invariant manifolds of the
three-body problem, alone [4] or in combination with conventional or low-thrust
propulsion [5] and; (4) trajectory optimization for a spacecraft sent to collide with a
threatening Earth-approaching asteroid, with the objective of maximizing the subse-
quent miss distance of the asteroid at its closest approach to Earth [6] [7]. These are
only a few of many examples that could be drawn from recent literature and from
the programs of the principal conferences in the subject.](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-20-320.jpg)
![1.2 Solution Methods 3
Necessary conditions for optimality for every one of these types of spacecraft tra-
jectoryoptimizationproblemsmaybederivedusingthecalculusofvariations(COV).
Unfortunately the solution of the resulting system of equations and boundary condi-
tions is either difficult or impossible. For certain simplified but still very useful cases
of either impulsive-thrust or continuous-thrust orbit transfer, the analytical neces-
sary conditions may described using the “Primer Vector” theory of Lawden [8], as
will be described briefly in Section 1.2.1 of this Chapter and then in much greater
detail in Chapter 2. Analytical solutions for the optimal trajectory (i.e. solutions sat-
isfying the necessary conditions) can be obtained in special cases, for example for
very-low-thrust orbit raising [9], even in the presence of some perturbations [10].
However, the vast majority of researchers and analysts today use numerical opti-
mization. Numerical optimization methods for continuous optimal control problems
are generally divided into two types. Indirect solutions are those using the analytical
necessary conditions from the calculus of variations [11]. This requires the addition
of the costate variables (or adjoint variables or Lagrange multipliers) of the prob-
lem, equal in number to the state variables, and their governing equations. This
instantly doubles the size of the dynamical system (which alone, of course, makes it
more difficult to solve). Direct solutions, of which there are many types, transcribe
the continuous optimal control problem into a parameter optimization problem [12]
[13] [14]. Satisfaction of the system equations is accomplished by integrating them
stepwise using either implicit or explicit (for example Runge-Kutta) rules; in either
case, the effect is to generate nonlinear constraint equations that must be satisfied by
the parameters, which are the discrete representations of the state and control time
histories. The problem is thus converted into a nonlinear programming problem.
There is a comprehensive survey paper by Betts [15] that describes direct and indi-
rect optimization, the relation between these two approaches, and the development
of these two approaches.
1.2 Solution Methods
In just the decade since the publication of Betts’ survey paper [15], there has been
considerable advancement of direct numerical solutions for optimal control prob-
lems. There also has been even more development and improvement, in relative
terms, of a qualitatively different approach to solving such problems, one using evo-
lutionary algorithms. The best known of these are genetic algorithms (GA) [16].
Another evolutionary algorithm, the Particle Swarm Optimizer (PSD) will be dis-
cussed in Chapter 10. The evolutionary algorithms have two principal advantages
over other extant methods; they are comparatively simple and thus easy to learn to
use, and they are generally more likely, in comparison to conventional optimizers,
to locate global minima. In addition, there has been progress in analytical solutions
such as those using primer vector theory [8] [17], “shape based” trajectories [18] [19],
or Hamilton-Jacobi theory.
All of the solutions may be broadly categorized as being either analytical or
numerical, though of course the analytical solutions (with only a few exceptions](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-21-320.jpg)
![4 Problem of Spacecraft Trajectory Optimization
such as the Hohmann transfer) use numerical methods and the numerical solutions
include some methods that explicitly use the analytical necessary conditions for opti-
mality. In the following sections, the analytical and numerical solution methods will
be defined and various examples, some historical and some very recent, will be pre-
sented for many of the methods that fall within these categories. This is not intended
to be a survey and will be unapologetically incomplete, as the subject is a vast one with
a large literature. Rather, the intention in this introductory chapter is to describe the
problem of spacecraft trajectory optimization, categorize the solution approaches,
provide a small amount of history, and describe the “state of the art” so that the work
of the various book chapter authors describing their approaches to the problem will
be in context.
1.2.1 Analytical Solutions
This is the original approach for space trajectory optimization, the oldest example
of which (1925) is due to Hohmann’s conjecture [20] regarding the optimal circular
orbit to circular orbit transfer. (The proof of the optimality of the Hohmann transfer
came much later [21] [22].) Most of the analytical solutions are based on the necessary
conditions of the problem that come from the calculus of variations (COV). Suppose
that the system equations may be written in form
ẋ = f(x, u, t) (1.1)
where x represents an n-dimensional state (vector) and u represents the m-
dimensional control (vector). The state vector is problem dependent; there are many
choices available. Typically, conventional elliptic elements, equinoctial variables, or
Delaunay variables are used for problems that are Keplerian or nearly-Keplerian,
for example, very low-thrust orbit raising. Another common choice is spherical polar
coordinates. Cartesian coordinates are typically used for three-body problems. The
control u is typically a control of thrust magnitude and direction or its equivalent,
for example the orientation of a solar-sail spacecraft with respect to the Sun. The
problem has some initial conditions specified, that is,
xi(0) given for i = 1, 2, …, k with k ≤ n (1.2)
and some terminal conditions, or functions of the terminal conditions, specified as
the vector
[x(T), T] = 0. (1.3)
The objective may be written in the Bolza form as
J = φ [x(T), T] +
T
0
L [x, u, t] dt (1.4)
where φ is a terminal cost function while the integral expresses a cost incurred during
the entire trajectory.](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-22-320.jpg)
![1.2 Solution Methods 5
The first step in deriving the conditions for an extremum of (1.4) subject to the
system (1.1) and the boundary conditions (1.3) is to define a system Hamiltonian
H = L + λT
f
Then, in terms of H and the other quantities introduced, the necessary conditions
become [11]
λ̇ = −
∂H
∂x
T
with boundary condition λ(T) =
∂φ
∂x
+ νT
∂
∂x
T
t=T
(1.5)
∂H
∂u
= 0. (1.6)
The system of equations (1.1)–(1.6) constitutes a two-point-boundary-value problem
(TPBVP); some boundary conditions on the states are specified at the initial time
and some boundary conditions on the states and adjoints are specified at the terminal
time. In addition, if the terminal time is unspecified (that is free to be optimized), as
is often the case, an additional scalar equation obtains
∂φ
∂t
+ νT
∂
∂t
+
∂φ
∂x
+ νT
∂
∂x
f + L
t=T
= 0. (1.7)
For all but the most elementary optimal control problems, the solution of
this TPBVP is challenging and numerical solutions are required. Despite this, it
is interesting that when this set of necessary conditions is applied to the optimal
space trajectory problem, which is by no means elementary, several very useful
observations may be made.
The system equations of motion (1.1) may be written in the form
˙
x̄ = f̄ =
˙
r̄
˙
v̄
=
v̄
ḡ(r̄) + û
(1.8)
where g(r) is the gravitational acceleration, is the thrust acceleration magnitude,
and û is a unit vector indicating the thrust direction.
To minimize the velocity change required, one chooses the integrand in the cost
function (1.4) to be L = the acceleration provided by the motor; then the integral
will represent the V provided by the motor. The Hamiltonian then becomes
H = + λ̄T
r v̄ + λ̄T
v [ḡ(r̄) + û] = 1 + λ̄T
v û + λ̄T
r v̄ + λ̄T
v ḡ(r̄). (1.9)
Because H is linear in u, equation (1.6) does not obtain. The optimal control is
instead chosen according to Pontryagin’s Minimum Principle, stating that at any
time on the optimal trajectory, the control variables are chosen in order to minimize
the Hamiltonian. Thus the first simple observation is that the thrust pointing unit
vector is chosen to be parallel to the opposite of the adjoint (to the velocity) vector,](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-23-320.jpg)
![6 Problem of Spacecraft Trajectory Optimization
i.e. −λ̄v(t). Because of its physical significance to the problem, this (adjoint) vector
is referred to as the primer vector [8]. A second simple observation is that with this
choice of thrust direction, it is then optimal in this case to choose the thrust magnitude
at its maximum possible value if the “switching function”
1 + λ̄T
v û (1.10)
is negative and choose = 0 if the switching function is positive. The adjoint vector
λ̄v(t) is governed by the system equations (1.5) with the Hamiltonian (1.9).
In addition, it is straightforward to show that if the Hamiltonian H is not explic-
itly time dependent, then H is a constant on the optimal trajectory. This result is
not necessarily useful for obtaining the optimal control but can be of great use in
determining, by its use a posteriori, the accuracy of the numerical solution of the
TPBVP, that is, a good solution will have H the same, to several significant figures,
when evaluated at any point on the numerical solution [14] [23].
Finally, while the necessary conditions guarantee only that the trajectory repre-
sents an extremum of the cost, by the nature of the space trajectory problem, there
is clearly no upper bound to the fuel that could be consumed on a feasible trajectory
(other than consuming all the fuel available). So one may be confident that a solution
is a local minimum and not a local maximum.
Further results can be obtained from a description of the necessary conditions
in terms of the primer vector, and these will be described in Chapter 2. It will suf-
fice to say here that while the primer vector is defined, and has the significance
with regard to optimal thrust direction found above, this is of course true only on
the optimal trajectory. The improvement of a known, nonoptimal trajectory via
primer vector theory was first discussed by Lion and Handelsman [17]. Jezewski and
Rozendaal [24] showed under what conditions an optimal N impulse trajectory could
beimprovedbytheadditionofanotherimpulse, andwhereandwithwhatdirectionto
apply it.
Solution of the analytical necessary conditions is possible for some special
cases. One useful example is the case of very-low-thrust orbit raising. With certain
assumptions, it is possible to find approximate solutions of the analytical neces-
sary conditions. Many of these are found in a survey paper of the subject by
Petropoulos and Sims [25]. The most common simplifications include: assuming
that the thrust direction is always tangential; assuming that the thrust pointing is
always in the direction of the velocity vector; or assuming that the orbit is always
circular. Surprisingly, exact solutions also exist in certain cases, including this low-
thrust orbit raising, even in the presence of nonspherical Earth perturbations [10].
This will be discussed in Chapter 7. The mathematics and analysis become very
involved.
The solution of the TPBVP resulting from (or constituting) the necessary con-
ditions becomes quite difficult for other problems, particularly those with path
constraints (typically on the state variables or on functions of the state variables)
or constraints on total fuel available.](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-24-320.jpg)
![1.2 Solution Methods 7
Many methods have been developed to solve the TPBVP numerically. The most
obvious and well known is probably shooting (an archetype of shooting applied
to spacecraft trajectory optimization may be found in the paper by Breakwell and
Redding [26]) but there are other methods including finite-difference methods [27]
[28] and collocation [12] [13] [14]. The long-recognized difficulty of the “indirect”
approach to determining the optimal trajectory is that the initial costate variables
of the TPBVP are unknown and further that the nonlinearity of the problem means
that the vector flow is very sensitive to some or all of these initial costate variables.
A further difficulty is that the costate variables lack the physical significance of the
state variables so that estimating the order of magnitude or even the sign of the initial
costates is very difficult. For problems with constrained arcs, another difficulty that
arises is discontinuity of controls and costate variables at the junctions of constrained
and unconstrained arcs. This also increases the difficulty of solving the associated
TPBVP.
Another solution method that satisfies both the necessary and sufficient condi-
tions for optimality is the method of Static/Dynamic control (SDC) of Whiffen [29]
[30]. The term static refers to decision variables that are discrete, such as launch
dates or planetary flyby dates, while the term dynamic refers to controls that have
a continuous variation in time, such as thrust pointing angle time histories. SDC is
a general nonlinear optimal control algorithm based on Bellman’s principle of opti-
mality [11]. The implementation of SDC in the program Mystic is a very capable
low-thrust spacecraft trajectory optimizer.
A recent, qualitatively different approach to the determination of optimal space
trajectoriesisthatofGuiboutandScheeres[31]. Inthiswork, thedynamicalsystemof
state and costate variables (the vector field) is solved for specified terminal conditions
andfinaltimebysolvingtheassociatedHamilton-Jacobi(H-J)equation. Thesolution
of the H-J equation is a generating function for a canonical transformation. Once
this solution is determined, the initial value of the costate vector may be found; the
optimal trajectory and the optimal control may then be found by forward integration
of the flow field. Scheeres et al. show an example of an optimal rendezvous in the
vicinity of a nominal circular orbit [32].
1.2.2 Numerical Solutions via Discretization
Many recent methods for solving optimal control problems seek to reduce them to
parameter optimization problems that can then be solved by a NLP problem solver.
One principal way in which such methods are distinguished is with regard to what
quantities are parameterized. In one popular method, the collocation method that
will be discussed in Chapter 3, it is possible to parameterize the state variables and
the costate variables (that is, to solve the TPBVP). It is also possible in collocation to
parameterize only the state variables and the control variables, as will be discussed in
the next section. A third possibility, yielding the smallest number of parameters for a
given problem, is to parameterize only the control variables and some free terminal
states, but then the system equations must be numerically integrated (as opposed](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-25-320.jpg)
![8 Problem of Spacecraft Trajectory Optimization
to the implicit integration that occurs in collocation). This is referred to as “control
parameterization” and will be discussed in Chapter 5.
Of course all of the solutions described in the previous section are obtained
numerically, that is, they will employ methods such as numerical integration, solving
TPBVP problems using “shooting” methods, or solving boundary value problems
by converting them into nonlinear programming (NLP) problems. What is meant
in this section by “numerical solution” is solutions that do not explicitly employ
the analytical necessary conditions of the COV, for example, solutions that do not
employ the costate (adjoint) variables of the problem or solutions that satisfy the
H-J-B equation or Bellman’s principle for discrete systems.
Why would one want to avoid the use of the necessary conditions, particularly
when the resulting trajectory has a “guarantee” of being a local extremum (that one
loses in a numerical solution) and has other benefits previously discussed, such as
information about sensitivity to terminal conditions and guidance toward improving
a solution by for example, adding/subtracting thrust arcs? The principal reason is the
lack of robustness of the various methods for solving the Euler-Lagrange TPBVP
stemming, as previously mentioned, from the nonlinearity of the problem and a
lack, in the general case, of a systematic means for determining a sufficiently good
approximation to the initial adjoint variables of the problem.
A variety of direct solution methods have been developed. They are best catego-
rized by the way in which they handle the discretization of the equations of motion,
which appear as function-space constraints in the original optimal control problem.
A more complete survey will be presented in Chapter 3. In the last two decades,
however, the most successful approach is arguably one in which the continuous
problem is discretized and state and control variables are known only at discrete
times. Satisfaction of the equations of motion is achieved by employing an explicit
or implicit numerical integration rule that needs to be satisfied at each step; this
results in a large NLP problem with a large number of nonlinear constraints. This
approach was termed “direct transcription” by Canon et al. [33]. While known to
mathematicians in the 1960s and 1970s, it became known in the aerospace commu-
nity principally through two papers. Dickmanns and Well [34] used the collocation
scheme to solve the TPBVP of the indirect method. This approach is significantly
more robust than shooting methods because it eliminates the sequential nature of
the shooting solution, with its forward numerical integration, in favor of a solution
in which simultaneous changes in all of the discrete state and costate parameters
are made in order to satisfy algebraic constraints (while minimizing the objective
of course).
However, the most useful development for space trajectory optimization was
the observation in 1987 by Hargraves and Paris [12] that it was not necessary to use
this approach to solve the indirect TPBVP, that in fact the adjoint variables (which
had been used to determine the optimal control from Pontryagin’s principle) could
be removed from the solution provided that discrete control variables were intro-
duced as additional NLP parameters. This significantly improved the robustness of
the method; by eliminating the adjoint variables, the problem size is reduced almost](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-26-320.jpg)
![1.2 Solution Methods 9
by half, and there is no longer a need to provide the NLP problem solver with an
estimate of the adjoint variables, something that is always problematic. A fortunate
coincidence is that at about the same time (1980s), the NLP technology required to
efficiently and robustly solve large problems became available (and has been continu-
ouslyimprovedsincethen)[35][36]. Theastrodynamicscommunityswiftlyembraced
this method. Many optimal spacecraft trajectories have since been determined using
direct methods. The direct method has also been significantly developed in the last
two decades. There are now many approaches, differing primarily (for collocation
methods) on how the implicit integration rules are constructed [37]. The most com-
mon approaches are to use trapezoidal [38] or Hermite-Simpson [12] integration
rule constraints, or higher-degree rules from the same Gauss-Lobatto family [13]
or a Gauss-pseudospectral method [39]. There also exist commercial software pack-
ages implementing direct methods for general optimal control problems, for example
DIDO [40] and SOCS [38], and even solvers specifically for space and launch vehicle
trajectory optimization, for example OTIS [41] and ALTOS [42] [43].
It would be accurate to say that the great majority of optimal space trajecto-
ries are now determined numerically, with methods that do not make explicit use
of the analytical necessary conditions of the problem, as will be described briefly
below and in detail in Chapter 2. However, that does not mean that the necessary
conditions are no longer useful. On the contrary, they provide useful information
that many numerical solutions naturally lack. For example, primer vector theory
can provide important information on how a solution may be improved, for exam-
ple by adding thrust arcs or coast arcs or by adding impulses for an impulsive
trajectory. The solution of the TPBVP of the necessary conditions also provides
information on the sensitivity of the solution to changes in terminal conditions and
constraints.
Fortunately, without solving the TPBVP, it is possible to make use of some
of these beneficial features of the solution of the necessary conditions, as will be
described in Chapter 3. This occurs because of a correspondence between the final
adjoint variables of the continuous TPBVP and some Kuhn-Tucker multipliers gen-
erated in (some) numerical solutions of the trajectory optimization problem [13] [14].
With these multiplier variables available, it is possible, for example, to compute the
value of the system Hamiltonian over the entire trajectory time history. For many
problems in which H should be a constant, this can provide a check on the accuracy
of the numerical solution. Or, knowing the final adjoints and final states from, for
example, a direct solution using collocation and NLP, one can integrate the E-L
equations backward to the initial time. If the initial states are recovered, one can
then say that the numerical solution satisfies the analytical necessary conditions and
thus represents an extremal path.
1.2.3 Evolutionary Algorithms
A qualitatively different approach, recently applied to spacecraft trajectory opti-
mization, is the use of “evolutionary” algorithms (EA). The best known of the EAs](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-27-320.jpg)
![10 Problem of Spacecraft Trajectory Optimization
is the genetic algorithm (GA). EA’s are numerical optimizers that determine, using
methods similar to those found in nature, an optimal set of discrete parameters that
has been used to characterize the problem solution. The EA’s have two principal
advantages over all of the direct and indirect solution methods previously described
in this chapter: they require no initial “guess” of the solution (in fact they gener-
ate a population of initial solutions randomly), and they are more likely than other
methods to locate a global minimum in the search space rather than be attracted to
a local minimum.
All of the EAs require that the problem solution be capable of being described
by a relatively small, in comparison to the vector of parameters of a nonlinear pro-
gram, set of discrete parameters. This can be accomplished, for spacecraft trajectory
optimization problems, in a number of ways:
(1) If the trajectory can naturally be described by a finite set, for example an impul-
sive thrust trajectory, the parameters will be such things as times, magnitudes,
and directions of impulses. Between impulses the trajectories may be determined
by solving Lambert’s problem. In this case a small number of parameters will
suffice to completely describe the solution.
(2) If the trajectory contains non-integrable arcs, for example low-thrust arcs, it is
still the case that much of the trajectory can be described with a small number
of parameters such as departure and arrival dates and times for the beginning
and end of thrust arcs. Quantities that must be described continuously, such
as thrust magnitude or pointing time history, can be parameterized using, for
example, polynomial equations in time. Then the additional parameters are a
small number of polynomial coefficients [44]
(3) Low-thrust arcs can also be described using “shape-based” methods [18] [19]. In
this approach, a shape, which is an analytical expression for the trajectory, can be
generated from a small number of parameters such that the resulting trajectory
will actually be a solution of the system equations of motion. Unfortunately the
thrust time history that allows this beneficial result can only be determined a
posteriori. An EA is then used to choose the parameters defining the shape to
satisfy the boundary conditions of the problem and to minimize the cost. The
resulting trajectory may not be realizable, as it may require greater thrust than is
available. However the trajectory may well be satisfactory as an initial guess for
a more accurate method, for example a direct method such as collocation [12]
[13] [14].
In the simplest form of the genetic algorithm, the set of parameters describ-
ing the solution is written as a string or sequence of numbers. Suppose that this
sequence is converted to binary form; it is then similar to a chromosome but con-
sisting only of two possible variables, a 1 or a 0. Every sequence can be “decoded”
to yield a trajectory whose cost or objective value can be determined. The first step
in the GA is the generation of a “population” of sequences using a random pro-
cess. The great majority of these randomly generated sequences will have very large](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-28-320.jpg)
![1.2 Solution Methods 11
costs; many may even be infeasible. The population is then improved using three
natural processes: selection, combination, and mutation. Selection removes the worst
sequences and may also, via elitism, guarantee that the best sequence survives into
the next generation unchanged. Following selection, remaining sequences are used
as “parents,” that is partial sequences from two parents are combined to form new
individuals. Finally, mutation changes a randomly chosen bit in a small fraction of
the population.
The process is then repeated; the cost of every individual in the new generation
is determined. Since the best individual from the previous generation was retained,
the objective may improve but cannot worsen. In practice, there is generally rapid
improvement in the early generations; if the process locates the global minimum
then, of course, improvement will cease. Termination of the algorithm is usually done
either after a fixed number of generations or after the objective has reached a plateau.
Of course neither of these termination conditions guarantees that a minimum has
been found, nor are there necessary conditions for optimality with this method.
Additional shortcomings are that there is no way to enforce satisfaction of boundary
conditions; normally a “penalty function” approach is taken in which unsatisfied
boundary conditions are added to the cost, and that the solution will be less accurate
than a typical direct solution (and even less accurate than an indirect solution).
Nevertheless, the method has been very useful when applied to optimizing space
trajectories, either for finding approximate extremals [44] or when used to provide
an initial guess for more accurate methods, for example collocation with NLP.
Betts [15] notes that one significant advantage of the GA in comparison to all
other solution methods is how straightforward it is to use. There are many GA
routines available (a commonly used one is found in MATLAB) so the user need
only provide a subroutine for decoding the sequence to evaluate the cost (which for
space trajectory problems can be as simple as a routine that integrates the system
equations of motion) provide bounds on the parameters, and then provide values
for certain constant parameters that control the evolutionary processes.
There are other EAs that have begun to prove very useful in the determination of
optimal space trajectories. One qualitatively different method is particle swarm opti-
mization (PSO). In PSO, some number (say 100) of particles are randomly distributed
in a N-dimensional decision parameter space. The objective value is determined for
the solution vector corresponding to each particle. Taking an anthropomorphic view,
it is then assumed that the particles can communicate so that all know the objective
value for all the others. Let xi(n) denote the position of particle i at the nth time step.
At the next iteration, the particles take a step vi(n + 1) in the parameter space so
that the new position of particle i becomes
xi(n + 1) = xi(n) + vi(n + 1) (1.11)
with (in one form of the PSO)
vij(n + 1) = vij(n) + c1r1j(n) yij(n) − xij(n) + c2r2j(n) ŷj(n) − xij(n) (1.12)](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-29-320.jpg)
![12 Problem of Spacecraft Trajectory Optimization
where vij(n) is the velocity (step) for component j of particle i at time step n,
xij(n) is the jth component of the position of particle i at the nth time step,
r1j(n) and r2j(n) ⊂ U(0, 1) are random values in the range [0, 1] sampled from a
uniform distribution. yi(n) is the “personal best” position, the best position located
by the ith particle since the first time step; ŷj(n) is the “global best” position, the
best position located by the any particle of the swarm since the first time step. The
step described in equation (1.12) thus has three components. The first is an “inertia”
that causes the particle to move in the direction it had previously been moving, the
second “nostalgia” component reflects a tendency for the particle to move toward its
own most satisfactory position, and the third “social” component draws the particle
toward the best position found by any of its colleagues. The c’s are constants that
weight the importance of the three components and the r’s provide stochasticity to
the system.
As with the GA, the process can be terminated after a fixed number of iterations
or when the “best” solution has not changed for several iterations. This method has
proven quite robust, is also very simple to use, and is particularly good in locating
globalminimawhenthesolutionspacecontainsmanylocalminima. Amorethorough
description of the PSO method and its application to space trajectory problems will
be provided in Chapter 10.
There are many other EAs, for example ant colony optimization (ACO) or
differential evolution (DE). The interested reader can easily find information on the
use of these methods [45].
1.3 The Situation Today with Regard to Solving Optimal Control Problems
One can safely say, for example by considering papers published recently in astrody-
namics journals, that solutions using analytical methods, that is analytical solutions
of the first-order necessary conditions, are seldom found. This is due, as previously
mentioned, to the complexity of the problem when realistic terminal boundary condi-
tions and when bounds on the controls are present. Also, solutions found numerically
using indirect methods, for example with shooting methods, are also becoming less
common. This is almost certainly due to the success that has been achieved with direct
methods, particularly those using collocation via low-degree rules such as trapezoid
or Hermite-Simpson [11] [13], via the pseudospectral method [39], or by higher
degree G-L implicit integration [13]. (These collocation methods are all derivable
from the same source, as will be seen in Chapter 3.) These methods have proven
particularly robust and efficient and have been used to solve many types of prob-
lems including low-thrust orbit raising [46], Earth-Moon transfer [47] [5] [48], and
interplanetary transfers [49].
An early difficulty faced by users of these methods was that, while robust, it was
still necessary to supply a reasonable initial guess of the solution parameters, that is
a discretized form of the state and control time-histories on the optimal trajectory,
to the NLP problem solver. This, of course, is not always a simple matter. For
some cases, for example for low-thrust orbit raising, approximate analytical solutions
such as a Lawden spiral, as described in Section 1.2.1, are available and make a](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-30-320.jpg)
![References 13
very satisfactory initial guess. For other problems, such as the optimal low-thrust
Earth-Moon transfer, obtaining a satisfactory initial guess is much more difficult.
Today, however, the situation is much improved since evolutionary algorithms such
as the GA, which can provide a solution to the problem in their own right, can also
be used as “pre-processors” to provide an initial guess of the solution from which a
method such as direct collocation with NLP can converge to a much more accurate
solution. An additional advantage of this approach is that some of the EAs are better
suited to locating the global minimum than are the methods using NLP, as the NLP
solver will tend to converge to a local minimum in the neighborhood of the initial
guess it is given. Thus starting from a guess provided by an EA is more likely to
enable the direct solver to find a global minimum. (Of course there is no guarantee
in any case.)
John Betts’ observation in 1988 [15] that “one may expect many of the best
features of seemingly disparate techniques to merge, forming still more powerful
methods” was clearly very prescient.
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[2] Wang, P. K. C., and Hadaegh, F. Y. (1998) Optimal Formation-Reconfiguration for Mul-
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[3] Pontani, M., and Conway, B. A. (2008) Optimal Interception of Evasive Missile War-
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[4] Marsden, J. E., and Ross, S. D. (2005) New Methods in Celestial Mechanics and Mission
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[7] Conway, B. A. (1997) Optimal Low-Thrust Interception of Earth-Crossing Asteroids, J.
of Guidance, Control, and Dynamics, 20, 995–1002.
[8] Lawden, D. F. (1963) Optimal Trajectories for Space Navigation, Butterworths, London.
[9] Kechichian, J. A. (1997) Reformulation of Edelbaum’s Low-Thrust Transfer Prob-
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[10] Kechichian, J. A. (2000) Minimum-Time Constant Acceleration Orbit Transfer With
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[12] Hargraves, C. R., and Paris, S. W. (1987) Direct Trajectory Optimization Using Non-
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[13] Herman, A. L., and Conway, B. A. (1996) Direct Optimization Using Collocation Based
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![14 Problem of Spacecraft Trajectory Optimization
[14] Enright, P.J., andConway, B.A.(1992)DiscreteApproximationstoOptimalTrajectories
Using Direct Transcription and Nonlinear Programming, Journal of Guidance, Control,
and Dynamics, 15, 994–1002.
[15] Betts, J. T. (1998) Survey of Numerical Methods for Trajectory Optimization, Journal of
Guidance, Control and Dynamics, 21, 193–207.
[16] Goldberg, D. (1989) Genetic Algorithms in Search, Optimization, and Machine Learning,
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[17] Lion, P. M., and Handelsman, M. (1968) The Primer Vector on Fixed-Time Impulsive
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[18] Petropoulos, A. E., and Longuski, J. M. (2004) Shape-Based Algorithm for Automated
Design of Low-Thrust, Gravity-Assist Trajectories, Journal of Spacecraft and Rockets,
41, 787–796.
[19] Wall, B. J., and Conway, B. A. (2009) Shape-Based Approach to Low-Thrust Trajectory
Design, Journal of Guidance, Control and Dynamics, 32, 95–101.
[20] Hohmann, W. (1925). Die Erreichbarkeit der Himmelskörper. Verlag Oldenbourg,
München.
[21] Prussing, J. E. (1991) Simple Proof of the Global Optimality of the Hohmann Transfer,
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[22] Barrar, R. B., (1963) An Analytic Proof that the Hohmann-Type Transfer is the True
Minimum Two-Impulse Transfer, Astronautica Acta, 9, 1–11.
[23] Herman, A. L. (1995) Improved Collocation Methods Used for Direct Trajectory
Optimization, Ph.D. Thesis, University of Illinois at Urbana-Champaign.
[24] Jezewski, D. J., and Rozendaal, H. L. (1968) An Efficient Method for Calculating Optimal
Free-Space N-Impulse Trajectories, AIAA Journal, 6, 2160–2165.
[25] Petropoulos, A. E., and Sims, J. A. (2004) A Review of Some Exact Solutions to the
Planar Equations of Motion of a Thrusting Spacecraft, DSpace at JPL, http://hdl.handle.
net/2014/8673
[26] Breakwell, J. V., and Redding, D. C. (1984) Optimal Low-Thrust Transfers to Syn-
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[27] Fox, L. (1957) The Numerical Solution of Two-Point Boundary Problems, Oxford Press,
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[28] Keller, H.B. (1968) Numerical Methods for Two-Point Boundary Value Problems,
Blaisdell, New York.
[29] Whiffen, G. (2006) Mystic: Implementation of the Static Dynamic Optimal Control
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gov/dspace/bitstream/2014/40782/1/06-2356.pdf
[30] Whiffen, G. J., and Sims, J. A. (2002) Application of the SDC Optimal Control Algorithm
to Low-Thrust Escape and Capture Including Fourth Body Effects, Second International
Symposium on Low Thrust Trajectories, Toulouse, France.
[31] Guibout, V. M., and Scheeres, D. J. (2006) Solving Two-Point Boundary Value Problems
Using Generating Functions: Theory and Applications to Astrodynamics, in Modern
Astrodynamics, Gurfil, P. (ed.), Elsevier, Amsterdam, 53–105.
[32] Scheeres, D. J., Park, C., and Guibout, V. (2003) Solving Optimal Control Problems with
Generating Functions, Paper AAS 03-575.
[33] Canon, M. D., Cullum, C. D., and Polak, E. (1970) Theory of Optimal Control and
Mathematical Programming, McGraw-Hill, New York.
[34] Dickmanns, E. D., and Well, K. H. (1975) Approximate Solution of Optimal Con-
trol Problems Using Third-Order Hermite Polynomial Functions, Proceedings of the
6th Technical Conference on Optimization Techniques, IFIP-TC7, Springer–Verlag,
New York.
[35] Gill, P., Murray, W., and Saunders, M. A. (2005) SNOPT: An SQP Algorithm for Large-
Scale Constrained Optimization. SIAM Review, 47, 99–131.
[36] Gill, P. E. et al. (1993) User’s Guide for NZOPT 1.0: A Fortran Package For Nonlinear
Programming, McDonnell Douglas Aerospace, Huntington Beach, CA.](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-32-320.jpg)
![References 15
[37] Paris, S. W., Riehl, J. P., and Sjauw, W. K. (2006) Enhanced Procedures for Direct Trajec-
tory Optimization Using Nonlinear Programming and Implicit Integration, Paper AIAA-
2006-6309, AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Keystone,
Colorado.
[38] Betts, J. T., and Huffman, W. P. (1997) Sparse Optimal Control Software SOCS,
Mathematics and Engineering Analysis Technical Document, MEA-LR-085, Boeing
Information and Support Services, The Boeing Company, Seattle.
[39] Ross, I. M., and Fahroo, F. (2003) Legendre Pseudospectral Approximations of Optimal
Control Problems, Lecture Notes in Control and Information Sciences, 295, Springer–
Verlag, New York, 327–342.
[40] Ross, I. M. (2004) User’s Manual for DIDO: A MATLAB Application Package for Solving
Optimal Control Problems, TOMLAB Optimization, Sweden.
[41] Paris, S. (1992) OTIS-Optimal Trajectories by Implicit Simulation, User’s Manual, The
Boeing Company.
[42] Well, K. H., Markl, A., and Mehlem, K. (1997) ALTOS-A Trajectory Analysis and
Optimization Software for Launch and Reentry Vehicles, International Astronautical
Federation Paper IAF-97-V4.04.
[43] Wiegand, A., Well, K. H., Mehlem, K., Steinkopf, M., and Ortega, G. (1999) ALTOS-
ESA’s Trajectory Optimization Tool Applied to Reentry Vehicle Trajectory Design,
International Astronautical Federation Paper IAF-99-A.6.09
[44] Wall, B. J., and Conway, B. A. (2005) Near-Optimal Low-Thrust Earth-Mars Trajectories
Found Via a Genetic Algorithm,” Journal of Guidance, Control, and Dynamics, 28,
1027–1031.
[45] Engelbrecht, A. P. (2007) Computational Intelligence, 2nd ed., John Wiley Sons, West
Sussex, England.
[46] Scheel, W. A., and Conway, B. A. (1994) Optimization of Very-Low-Thrust, Many
Revolution Spacecraft Trajectories, Journal of Guidance, Control, and Dynamics, 17,
1185–1192.
[47] Herman, A. L., and Conway, B. A. (1998) Optimal Low-Thrust, Earth-Moon Orbit
Transfer, Journal of Guidance, Control, and Dynamics, 21, 141–147.
[48] Betts, J. T., and Erb, S. O. (2003) Optimal Low Thrust Trajectories to the Moon, SIAM
Journal on Applied Dynamical Systems, 2, 144–170.
[49] Tang, S., and Conway, B. A. (1995) Optimization of Low-Thrust Interplanetary Trajec-
tories Using Collocation and Nonlinear Programming, Journal of Guidance, Control, and
Dynamics, 18, 599–604.](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-33-320.jpg)
![2 Primer Vector Theory and Applications
John E. Prussing
Department of Aerospace Engineering, University of Illinois
at Urbana-Champaign, Urbana, IL
2.1 Introduction
In this chapter, the theory and a resulting indirect method of trajectory optimization
are derived and illustrated. In an indirect method, an optimal trajectory is deter-
mined by satisfying a set of necessary conditions (NC), and sufficient conditions
(SC) if available. By contrast, a direct method uses the cost itself to determine an
optimal solution.
Even when a direct method is used, these conditions are useful to determine
whether the solution satisfies the NC for an optimal solution. If it does not, it is not
an optimal solution. As an example, the best two-impulse solution obtained by a
direct method is not the optimal solution if the NC indicate that three impulses are
required. Thus, post-processing a direct solution using the NC (and SC if available)
is essential to verify optimality.
Optimal Control [1], a generalization of the calculus of variations, is used to
derive a set of necessary conditions for an optimal trajectory. The primer vector is a
term coined by D. F. Lawden [2] in his pioneering work in optimal trajectories. [This
terminology is explained after Equation (2.24).] First-order necessary conditions for
both impulsive and continuous-thrust trajectories can be expressed in terms of the
primer vector. For impulsive trajectories, the primer vector determines the times and
positions of the thrust impulses that minimize the propellant cost. For continuous-
thrusttrajectories, boththeoptimalthrustdirectionandtheoptimalthrustmagnitude
as functions of time are determined by the primer vector. As is standard practice,
the word “optimal” is loosely used as shorthand for “satisfies the first-order NC.”
The most completely developed primer vector theory is for impulsive trajec-
tories. Terminal coasting periods for fixed-time trajectories and the addition of
midcourse impulses can sometimes lower the cost. The primer vector indicates when
these modifications should be made. Gradients of the cost with respect to termi-
nal impulse times and midcourse impulse times and positions were first derived by
Lion and Handelsman [3]. These gradients were then implemented in a nonlinear
Figures 2.2 and 2.4–2.8 were generated using the MATLAB computer code written by
Suzannah L. Sandrik [13].
16](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-34-320.jpg)
![2.2 First-Order Necessary Conditions 17
programming algorithm to iteratively improve a nonoptimal solution and converge
to an optimal trajectory by Jezewski and Rozendaal [4].
2.2 First-Order Necessary Conditions
2.2.1 Optimal Constant-Specific-Impulse Trajectory
For a constant specific impulse (CSI) engine, the thrust is bounded by 0 ≤ T ≤
Tmax (where Tmax is a constant), corresponding to bounds on the mass flow rate:
0 ≤ b ≤ bmax (where bmax is a constant). Note that one can also prescribe bounds
on the thrust acceleration (thrust per unit mass) ≡ T/m as 0 ≤ ≤ max, where
max is achieved by running the engine at Tmax. However, max is not constant but
increases due to the decreasing mass. One must keep track of the changing mass in
order to compute for a given thrust level. This is easy to do, especially if the thrust
is held constant, for example, at its maximum value. However, if the propellant mass
required is a small fraction of the total mass because of being optimized, a constant
max approximation can be made.
The cost functional representing minimum propellant consumption for the CSI
case is
J =
tf
to
(t)dt. (2.1)
The state vector is defined as
x(t) =
r(t)
v(t)
(2.2)
where r(t) is the spacecraft position vector and v(t) is its velocity vector. The mass
m can be kept track of without defining it to be a state variable by noting that
m(t) = moe−F(t)/c
(2.3)
where c is the exhaust velocity and
F(t) =
t
to
(ξ)dξ. (2.4)
Note that from Equation (2.4), F(tf ) is equal to the cost J. In the constant thrust case,
varies according to ˙
= 1
c 2, which is consistent with the mass decreasing linearly
with time.
The equation of motion is
ẋ =
ṙ
v̇
=
v
g(r) + u
(2.5)
with the initial state x(to) specified.](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-35-320.jpg)
![18 Primer Vector Theory and Applications
In Equation (2.5), g(r) is the gravitational acceleration and u represents a
unit vector in the thrust direction. An example gravitational field is the inverse-
square field:
g(r) = −
μ
r2
r
r
= −
μ
r3
r. (2.6)
The first-order necessary conditions for an optimal CSI trajectory were first derived
by Lawden [2] using classical calculus of variations. In the derivation that follows,
an optimal control theory formulation is used, but the derivation is similar to that of
Lawden. One difference is that the mass is not considered a state variable but is kept
track of separately.
In order to minimize the cost in Equation (2.1), one forms the Hamiltonian using
Equation (2.5) as
H = + λT
r v + λT
v [g(r) + u]. (2.7)
The adjoint equations are then
λ̇
T
r = −
∂H
∂r
= −λT
v G(r) (2.8)
λ̇
T
v = −
∂H
∂v
= −λT
r (2.9)
where
G(r) ≡
∂g(r)
∂r
(2.10)
is the symmetric 3 × 3 gravity gradient matrix.
For terminal constraints of the form
ψ[r(tf ), v(tf ), tf ] = 0, (2.11)
which may describe an orbital intercept, rendezvous, etc., the boundary conditions
on Equations (2.8–2.9) are given in terms of
≡ vT
ψ[r(tf ), v(tf ), tf] (2.12)
as
λT
r (tf ) =
∂
∂r(tf )
= vT ∂ψ
∂r(tf )
(2.13)
λT
v (tf ) =
∂
∂v(tf )
= vT ∂ψ
∂v(tf )
. (2.14)
There are two control variables, the thrust direction u and the thrust acceleration
magnitude , that must be chosen to satisfy the minimum principle [1], that is, to min-
imize the instantaneous value of the Hamiltonian H. By inspection, the Hamiltonian
of Equation (2.7) is minimized over the choice of thrust direction by aligning the unit](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-36-320.jpg)
![2.2 First-Order Necessary Conditions 19
vector u(t) opposite to the adjoint vector λv(t). Because of the significance of the
vector −λv(t), Lawden [2] termed it the primer vector p(t):
p(t) ≡ −λv(t). (2.15)
The optimal thrust unit vector is then in the direction of the primer vector, specifically
u(t) =
p(t)
p(t)
(2.16)
and
λT
v u = −λv = −p (2.17)
in the Hamiltonian of Equation (2.7).
From Equations (2.9) and (2.15), it is evident that
λr(t) = ṗ(t). (2.18)
Equations (2.8), (2.9), (2.15), and (2.18) combine to yield the primer vector equation
p̈ = G(r)p. (2.19)
The boundary conditions on the solution to Equation (2.19) are obtained from
Equations (2.13) (2.14)
p(tf ) = −vT ∂ψ
∂v(tf )
(2.20)
ṗ(tf ) = vT ∂ψ
∂r(tf )
. (2.21)
Note that in Equation (2.20), the final value of the primer vector for an optimal
intercept is the zero vector, because the terminal constraint ψ does not depend
on v(tf ).
Using Equations (2.15)–(2.18), the Hamiltonian of Equation (2.7) can be
rewritten as
H = −(p − 1) + ṗT
v − pT
g. (2.22)
To minimize the Hamiltonian over the choice of the thrust acceleration magnitude
, one notes that the Hamiltonian is a linear function of , and thus the minimizing
value for 0 ≤ ≤ max will depend on the algebraic sign of the coefficient of in
Equation (2.22). It is convenient to define the switching function
S(t) ≡ p − 1. (2.23)
The choice of the thrust acceleration magnitude that minimizes H is then given by
the “bang-bang” control law
=
max for S 0 (p 1)
0 for S 0 (p 1)
. (2.24)](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-37-320.jpg)
![20 Primer Vector Theory and Applications
S (t )
MT MT MT
NT NT
0
t
Γmax
Figure 2.1. Three-burn CSI switching function and thrust profile.
That is, the thrust magnitude switches between its limiting values of 0 (an NT null-
thrust arc) and Tmax (an MT maximum-thrust arc) each time S(t) passes through 0
[p(t) passes through 1] according to Equation (2.24). Figure 2.1 shows an example
switching function for a three-burn trajectory.
The possibility also exists that S(t) ≡ 0 [p(t) ≡ 1] on an interval of finite duration.
From Equation (2.22), it is evident that in this case the thrust acceleration magnitude
is not determined by the minimum principle and may take on intermediate values
between 0 and max. This IT “intermediate thrust arc” [2] is referred to as a singular
arc in optimal control [1].
Lawden explained the origin of the term primer vector in a personal letter in 1990:
“In regard to the term ‘primer vector’, you are quite correct in your supposition. I
served in the artillery during the war [World War II] and became familiar with the
initiation of the burning of cordite by means of a primer charge. Thus, p = 1 is the
signal for the rocket motor to be ignited.”
It follows then from Equation (2.3) that if T = Tmax and the engine is on for a
total of t time units,
max(t) = eF(t)/c
Tmax/mo = Tmax/(mo − bmaxt). (2.25)
Other necessary conditions are that the variables p and ṗ must be continuous
everywhere. Equation (2.23) then indicates that the switching function S(t) is also
continuous everywhere.
Even though the gravitational field is time-invariant, the Hamiltonian in this
formulation does not provide a first integral (constant of the motion) on an MT
arc, because is an explicit function of time as shown in Equation (2.25). From
Equation (2.22)
H = −S + ṗT
v − pT
g. (2.26)
Note that the Hamiltonian is continuous everywhere because S = 0 at the
discontinuities in the thrust acceleration magnitude.](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-38-320.jpg)
![2.2 First-Order Necessary Conditions 21
2.2.2 Optimal Impulsive Trajectory
For a high-thrust CSI engine the thrust durations are very small compared with the
times between thrusts. Because of this, one can approximate each MT arc as an
impulse (Dirac delta function) having unbounded magnitude (max → ∞) and zero
duration. The primer vector then determines both the optimal times and directions
of the thrust impulses with p ≤ 1 corresponding to S ≤ 0. The impulses can occur
only at those instants at which S = 0 (p = 1). These impulses are separated by NT
arcs along which S 0 (p 1). At the impulse times the primer vector is then a unit
vector in the optimal thrust direction.
The necessary conditions (NC) for an optimal impulsive trajectory, first derived
by Lawden [2], are shown in Table 2.1.
For a linear system, these NC are also sufficient conditions for an optimal tra-
jectory [5]. Also in [5], an upper bound on the number of impulses required for an
optimal solution is given.
Figure 2.2 shows a trajectory (at top) and a primer vector magnitude (at bottom)
for an optimal three-impulse solution. (In all of the trajectory plots in this chapter,
the direction of orbital motion is counterclockwise.) Canonical units are used. The
canonical time unit is the orbital period of the circular orbit that has a radius of
one canonical distance unit. The initial orbit is a unit radius circular orbit, shown as
the topmost orbit going counterclockwise from the symbol ⊕ at (1,0) to (−1,0). The
transfer time is 0.5 original (initial) orbit periods (OOP). The target is in a coplanar
circular orbit of radius 2, with an initial lead angle (ila) of 270◦ and shown by the
symbol at (0,−2). The spacecraft departs and intercepts at approximately
(1.8,−0.8) as shown. The + signs at the initial and final points indicate thrust impulses
and the + sign on the transfer orbit very near (0,0) indicates the location of the
midcourse impulse. The magnitudes of the three Vs are shown at the left, with the
total V equal to 1.3681 in units of circular orbit speed in the initial orbit.
The examples shown in this chapter are coplanar, but the theory and applica-
tions apply to three-dimensional trajectories as well, for example, see Prussing and
Chiu [6].
The bottom graph in Figure 2.2 displays the time history of the primer vec-
tor magnitude. Note that it satisfies the necessary conditions of Table 2.1 for an
optimal transfer.
Table 2.1. Impulsive necessary conditions
1. The primer vector and its first derivative are continuous everywhere.
2. The magnitude of the primer vector satisfies p(t) ≤ 1 with the impulses occurring at those instants
at which p = 1.
3. At the impulse times the primer vector is a unit vector in the optimal thrust direction.
4. As a consequence of the above conditions, dp/dt = ṗ = ṗT p = 0 at an intermediate impulse
(not at the initial or final time).](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-39-320.jpg)
![22 Primer Vector Theory and Applications
−4 −3 −2 −1 0 1 2 3 4
−2
−1.5
−1
−0.5
0
0.5
0.5585
0.14239
0.66717
ila = 270°
Canonical Distance
Canonical
Distance
tf = 0.5 OOP, ΔV = 1.3681
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0.2
0.4
0.6
0.8
1
1.2
Time [OOP]
Primer
Magnitude
Figure 2.2. Optimal three-impulse trajectory and primer magnitude.
Note also that at a thrust impulse at time tk
(t) = vkδ(t − tk) (2.27)
and from Equation (2.4)
vk =
t+
k
t−
k
(t)dt = F(t+
k ) − F(t−
k ) (2.28)
where t+
k and t−
k are times immediately after and before the impulse time, respec-
tively. Equation (2.3) then becomes the familiar solution to the rocket equation:
m(t+
k ) = m(t−
k )e−vk/c
. (2.29)
2.2.3 Optimal Variable-Specific-Impulse Trajectory
A variable-specific-impulse (VSI) engine is also known as a power-limited (PL)
engine, because the power source is separate from the engine itself, for example,
solar panels, and radioisotope thermoelectric generator. The power delivered to
the engine is bounded between 0 and a maximum value Pmax, with the optimal
value being constant and equal to the maximum. The cost functional representing
minimum propellant consumption for the VSI case is
J =
1
2
tf
to
2
(t)dt. (2.30)](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-40-320.jpg)
![2.3 Solution to the Primer Vector Equation 23
Writing 2 as T , the corresponding Hamiltonian function can be written as
H =
1
2
T
+ λT
r v + λT
v [g(r) + ]. (2.31)
For the VSI case, there is no need to consider the thrust acceleration magnitude and
direction separately, so the vector is used in place of the term u that appears in
Equation (2.7).
Because H is a nonlinear function of , the minimum principle is applied by
setting
∂H
∂
= T
+ λT
v = 0T
(2.32)
or
(t) = −λv(t) = p(t) (2.33)
using the definition of the primer vector in Equation (2.15). Thus for a VSI engine,
the optimal thrust acceleration vector is equal to the primer vector: (t) = p(t).
Because of this, Equation (2.5), written as r̈ = g(r) + , can be combined with
Equation (2.19), as in [7] to yield a fourth-order differential equation in r:
riv
− Ġṙ + G(g − 2r̈) = 0. (2.34)
Every solution to Equation (2.34) is an optimal VSI trajectory through the gravity
field g(r). But desired boundary conditions, such as specified position and velocity
vectors at the initial and final times, must be satisfied.
Note also that from Equation (2.32)
∂2H
∂2
=
∂
∂
∂H
∂
T
= I3 (2.35)
where I3 is the 3 × 3 identity matrix. Equation (2.35) shows that the (Hessian) matrix
of second partial derivatives is positive definite, verifying that H is minimized.
Because the VSI thrust acceleration of Equation (2.33) is continuous, a recently
developed procedure [8] to test whether second-order NC and SC are satisfied can be
applied. Equation (2.35) shows that an NC for minimum cost (Hessian matrix posi-
tive semidefinite) and part of the SC (Hessian matrix positive definite) are satisfied.
The other condition that is both an NC and an SC is the Jacobi no-conjugate-point
condition. Reference [8] details the recently developed test for that.
2.3 Solution to the Primer Vector Equation
The primer vector equation, Equation (2.19), can be written in first-order form as
the linear system
d
dt
p
ṗ
=
O3 I3
G O3
p
ṗ
(2.36)
where O3 is the 3 × 3 zero matrix.](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-41-320.jpg)
![24 Primer Vector Theory and Applications
Equation (2.36) is of the form ẏ = A(t)y, and its solution can be written in terms
of a transition matrix (t, to) as
y(t) = (t, to)y(to) (2.37)
for a specified initial condition y(to).
Glandorf [9] presents a form of the transition matrix for an inverse-square grav-
itational field. [In that Technical Note, the missing Equation (2.33) is (t, to) =
P(t)P−1(to).]
Note that on an NT (no-thrust or coast) arc, the variational (linearized) state
equation is, from Equation (2.5),
δẋ =
δṙ
δv̇
=
O3 I3
G O3
δr
δv
, (2.38)
which is the same as Equation (2.36). So the transition matrix in Equation (2.37)
is also the transition matrix for the state variation, that is, the state transition
matrix [10].
This state transition matrix has the usual properties from linear system theory
and is also symplectic [10], which has the useful property that
−1
(t, to) = −JT
(t, to)J (2.39)
where
J =
O3 I3
−I3 O3
. (2.40)
Note that J2 = −I6, indicating that J is a matrix analog of the imaginary number i.
Equation (2.39) is useful when the state transition matrix is determined numer-
ically because the inverse matrix −1(t, to) = (to, t) can be computed without
explicitly inverting a 6 × 6 matrix.
2.4 Application of Primer Vector Theory to an Optimal Impulsive Trajectory
If the primer vector evaluated along an impulsive trajectory fails to satisfy the nec-
essary conditions of Table 2.1 for an optimal solution, the way in which the NC are
violated provides information that can lead to a solution that does satisfy the NC.
This process was first derived by Lion and Handelsman [3]. For given boundary con-
ditions and a fixed transfer time, an impulsive trajectory can be modified either by
allowing a terminal coast or by adding a midcourse impulse. A terminal coast can
be either an initial coast, in which the first impulse occurs after the initial time, or a
final coast, in which the final impulse occurs before the final time. In the former case,
the spacecraft coasts along the initial orbit after the initial time until the first impulse](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-42-320.jpg)
![2.4 Application of Primer Vector Theory 25
F
ro rf
Initial Orbit
Fnal Orbit
vo
vo
vf
+
_
Δvo
Δvf
vf
_
+
Transfer Orbit
to tf
Figure 2.3. A fixed-time impulsive rendezvous trajectory.
occurs. In the latter case, the rendezvous actually occurs before the final time, and
the spacecraft coasts along the final orbit until the final time is reached.
To determine when a terminal coast will result in a trajectory that has a lower fuel
cost, consider the two-impulse fixed-time rendezvous trajectory shown in Figure 2.3.
In the two-body problem, if the terminal radii ro and rf are specified along with
the transfer time τ ≡ tf − to, the solution to Lambert’s Problem [10] [11] provides
the terminal velocity vectors v+
o (after the initial impulse) and v−
f (before the final
impulse) on the transfer orbit. Because the velocity vectors are known on the initial
orbit (v−
o before the first impulse) and on the final orbit (v+
f after the final impulse),
the required velocity changes can be determined as
vo = v+
o − v−
o (2.41)
and
vf = v+
f − v−
f . (2.42)
Once the vector velocity changes are known, the primer vector can be evaluated
along the trajectory to determine if the NC are satisfied. In order to satisfy the NC
that on an optimal trajectory the primer vector at an impulse time is a unit vector in
the direction of the impulse, one imposes the following boundary conditions on the
primer vector
p(to) ≡ po =
vo
vo
(2.43)
p(tf ) ≡ pf =
vf
vf
. (2.44)](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-43-320.jpg)
![26 Primer Vector Theory and Applications
The primer vector can then be evaluated along the transfer orbit using the 6 × 6
transition matrix solution of Equation (2.37)
p(t)
ṗ(t)
= (t, to)
p(to)
ṗ(to)
(2.45)
where the 3 × 3 partitions of the 6 × 6 transition matrix are designated as
(t, to) ≡
M(t, to) N(t, to)
S(t, to) T(t, to)
. (2.46)
Equation (2.45) can then be evaluated for the fixed terminal times to and tf to yield
pf = Mfopo + Nfoṗo (2.47)
and
ṗf = Sfopo + Tfoṗo (2.48)
where the abbreviated notation is used that pf ≡ p(tf ), Mfo ≡ M(tf , to), and so on.
Equation (2.47) can be solved for the initial primer vector rate
ṗo = N−1
fo [pf − Mfopo] (2.49)
where the inverse matrix N−1
fo exists except for isolated values of τ = tf −to. With both
the primer vector and the primer vector rate known at the initial time, the primer
vector along the transfer orbit for to ≤ t ≤ tf can be calculated as using Equations
(2.43–2.46, 2.49) as
p(t) = NtoN−1
fo
vf
vf
+ [Mto − NtoN−1
fo Mfo]
vo
vo
. (2.50)
2.4.1 Criterion for a Terminal Coast
One of the options available to modify a two-impulse solution that does not satisfy
the NC for an optimal transfer is to include a terminal coast period in the form of
either an initial coast, a final coast, or both. To do this, one allows the possibility that
the initial impulse occurs at time to +dto due to a coast in the initial orbit of duration
dto 0 and that the final impulse occurs at a time tf +dtf . In the case of a final coast,
dtf 0 in order that the final impulse occur prior to the nominal final time, allowing
a coast in the final orbit until the nominal final time. A negative value of dto or a
positive value of dtf also has a physical interpretation as will be seen.
To determine whether a terminal coast will lower the cost of the trajectory,
an expression for the difference in cost between the perturbed trajectory (with the
terminal coasts) and the nominal trajectory (without the coasts) must be derived. The](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-44-320.jpg)
![2.4 Application of Primer Vector Theory 27
discussion that follows summarizes and interprets results by Lion and Handelsman
[3]. The cost on the nominal trajectory is simply
J = vo + vf (2.51)
for the two-impulse solution. In order to determine the differential change in the
cost due to the differential coast periods the concept of a noncontemporaneous,
or “skew” variation is needed. This variation combines two effects: the variation
due to being on a perturbed trajectory and the variation due to a difference in the
time of the impulse. The variable d will be used to denote a noncontemporaneous
variation in contrast to the variable δ that represents a contemporaneous variation,
as in Equation (2.38). The rule for relating the two types of variations is given by
dx(to) = δx(to) + ẋ∗
odto (2.52)
where ẋ∗
o is the derivative on the nominal (unperturbed) trajectory at the nominal
final time and the variation in the initial state has been used as an example.
Next, the noncontemporaneous variation in the cost must be determined.
Because the coast periods result in changes in the vector velocity changes, the
variation in the cost can be expressed, from Equation (2.51) as
dJ =
∂vo
∂vo
dvo +
∂vf
∂vf
dvf . (2.53)
Using the fact that for any vector a having magnitude a
∂a
∂a
=
aT
a
(2.54)
the variation in the cost in Equation (2.53) can be expressed as
dJ =
vT
o
vo
dvo +
vT
f
vf
dvf . (2.55)
Finally, Equation (2.55) can be rewritten in terms of the initial and final primer vector
using the conditions of Equations (2.43–2.44) as
dJ = pT
o dvo + pT
f dvf . (2.56)
The analysis in [3] leads to the result that
dJ = −ṗT
o vodto − ṗT
f vf dtf (2.57)
The final form of the expression for the variation in cost is obtained by expressing the
vector velocity changes in terms of the primer vector using Equations (2.43–2.44) as
dJ = −voṗT
o podto − vf ṗT
f pf dtf . (2.58)](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-45-320.jpg)
![2.4 Application of Primer Vector Theory 31
impulse is more complicated than including terminal coasts because, in the general
case, four new parameters are introduced: three components of the position of the
impulse and the time of the impulse. One must first derive a criterion that indicates
that the addition of an impulse will lower the cost and then determine where in space
and when in time the impulse should occur. The where and when will be done in
two steps. The first step is to determine initial values of position and time of the
added impulse that will lower the cost. The second step is to iterate on the values of
position and time using gradients that will be developed, until a minimum of the cost
is achieved. Note that this procedure is more complicated than for terminal coasts,
because the starting value of the coast time for the iteration was simply taken to be
zero, that is, no coast.
When considering the addition of a midcourse impulse, let us assume dto =
dtf = 0, that is, there are no terminal coasts. Because we are doing a first-order
perturbation analysis, superposition applies and we can combine the previous results
for terminal coasts easily with our new results for a midcourse impulse. Also, we will
discuss the case of adding a third impulse to a two-impulse trajectory, but the same
theory applies to the case of adding a midcourse impulse to any two-impulse segment
of an n-impulse trajectory. The cost on the nominal, two-impulse trajectory is given
by Equation (2.50)
J = vo + vf .
The variation in the cost due to adding an impulse is given by adding the midcourse
velocity change magnitude vm to Equation (2.56)
dJ = pT
o dvo + vm + pT
f dvf . (2.62)
The analysis in [3] results in
dJ = vm
1 − pT
m
vm
vm
. (2.63)
In Equation (2.63), the expression for dJ involves a dot product between the primer
vector and a unit vector. If the numerical value of this dot product is greater than one,
dJ 0 and the perturbed trajectory has a lower cost than the nominal trajectory.
In order for the value of the dot product to be greater than one, it is necessary that
pm 1. Here again we have an alternative derivation of the necessary condition
that p ≤ 1 on an optimal trajectory. We also have the criterion that tells us when the
addition of a midcourse impulse will lower the cost.
If the value of p(t) exceeds unity along the trajectory, the addition of a midcourse impulse
at a time for which p 1 will lower the cost.
Figure 2.7 shows a primer magnitude time history that indicates the need for a mid-
course impulse (but not for a terminal coast). The final radius is 2, the ila is 270◦, and
the transfer time is relatively small, equal to 0.5 OOP.
The first step is to determine initial values for the position and time of the mid-
course impulse. From Equation (2.63) it is evident that for a given pm, the largest](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-49-320.jpg)
![32 Primer Vector Theory and Applications
−4 −3 −2 −1 0 1 2 3 4
−2
−1.5
−1
−0.5
0
0.5
1.1758
0.57976
ila = 270°
Canonical Distance
Canonical
Distance
tf = 0.5 OOP, ΔV = 1.7555
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
1
2
3
4
5
6
Time (OOP)
Primer
Magnitude
Figure 2.7. Primer magnitude indicating a need for a midcourse impulse.
decrease in the cost is obtained by maximizing the value of the dot product, that is,
by choosing a position for the impulse that causes vm to be parallel to the vector
pm and by choosing the time tm to be the time at which the primer magnitude has
a maximum value. Choosing the position of the impulse so that the velocity change
is in the direction of the primer vector sounds familiar because it is one of the nec-
essary conditions derived previously, but how to determine this position is not at all
obvious, and we will have to derive an expression for this. Choosing the time tm to
be the time of maximum primer magnitude does not guarantee that the decrease in
cost is maximized, because the value of vm in the expression for dJ depends on the
value of tm. However, all we are doing is obtaining an initial position and time of the
midcourse impulse to begin an iteration process. As long as our initial choice repre-
sents a decrease in the cost, we will opt for the simple device of choosing the time of
maximum primer magnitude as our initial estimate of tm. In Figure 2.7, tm is 0.1.
Having determined an initial value for tm, the initial position of the impulse,
namely the value δrm to be added to rm, must also be determined. Obviously δrm
must be nonzero, otherwise the midcourse impulse would have zero magnitude. The
property that must be satisfied in determining δrm is that vm be parallel to pm. The
analysis of [3] results in
vm = Aδrm (2.64)](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-50-320.jpg)
![2.4 Application of Primer Vector Theory 33
where the matrix A is defined as
A ≡ −(MT
fmN−T
fm + TmoN−1
mo). (2.65)
Next, in order to have vm parallel to pm, it is necessary that vm = εpm with scalar
ε 0. Combining this fact with Equation (2.64) yields
Aδrm = vm = εpm (2.66)
which yields the solution for δrm as
δrm = εA−1
pm (2.67)
assuming A is invertible.
The question then arises how to select a value for the scalar ε. Obviously too
large a value will violate the linearity assumptions of the perturbation analysis. This
is not addressed in [3], but one can maintain a small change by specifying
δrm
rm
= β (2.68)
where β is a specified small positive number such as 0.05. Equation (2.67) then yields
a value for ε
ε
A−1pm
rm
= β ⇒ ε =
βrm
A−1pm
. (2.69)
If the resulting dJ ≥ 0, then decrease ε and repeat Equation (2.67). One should
never accept a midcourse impulse position that does not decrease the cost, because
a sufficiently small ε will always provide a lower cost.
The initial values of midcourse impulse position and time are now determined.
One adds the δrm of Equation (2.67) to the value of rm on the nominal trajectory
at the time tm at which the primer magnitude achieves its maximum value (greater
than one).
The primer history after the addition of the initial midcourse impulse is shown
in Figure 2.8. Note that pm = 1 but ṗm is discontinuous and the primer magnitude
exceeds unity, both of which violate the NC. However, the addition of the midcourse
impulse has decreased the cost slightly, from 1.7555 to 1.7549.
2.4.3 Iteration on a Midcourse Impulse Position and Time
To determine how to efficiently iterate on the components of position of the mid-
course impulse and its time, one needs to derive expressions for the gradients of the
cost with respect to these variables. To do this, one must compare the three-impulse
trajectory (or three-impulse segment of an n-impulse trajectory) that resulted from
the addition of the midcourse impulse with a perturbed three-impulse trajectory.](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-51-320.jpg)
![34 Primer Vector Theory and Applications
−4 −3 −2 −1 0 1 2 3 4
−2
−1.5
−1
−0.5
0
0.5
1.1749
0.00026491
0.57973
ila = 270°
Canonical Distance
Canonical
Distance
tf = 0.5 OOP, ΔV = 1.7549
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0.5
1
1.5
2
2.5
3
Time (OOP)
Primer
Magnitude
Figure 2.8. Initial (nonoptimal) three-impulse primer magnitude.
Note that, unlike a terminal coast, the values of drm and dtm are independent. (By
contrast, on an initial coast dro = v−
o dto and on a final coast drf = v+
f dtf .)
The cost on the nominal three-impulse trajectory is
J = vo + vm + vf (2.70)
and the variation in the cost due to perturbing the midcourse time and position is
dJ =
∂vo
∂vo
dvo +
∂vm
∂vm
dvm +
∂vf
∂vf
dvf (2.71)
which, analogous to Equation (2.56), can be written as
dJ = pT
o dvo + pT
mdvm + pT
f dvf . (2.72)
The analysis of [3] leads to the result that
dJ =
ṗ+
m − ṗ−
m
T
drm −
ṗT+
m v+
m − ṗT−
m v−
m
dtm. (2.73)](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-52-320.jpg)
![2.4 Application of Primer Vector Theory 35
In Equation (2.73), a discontinuity in ṗm has been allowed because there is no guar-
antee that it will be continuous at the inserted midcourse impulse, as demonstrated
in Figure 2.8.
Equation (2.73) can be written more simply in terms of the Hamiltonian function
Equation (2.22) for pm = 1: Hm = ṗT
mvm − pT
mgm (for which the second term pT
mgm
is continuous because pm = vm/vm) and gm(rm) are continuous).
dJ = (ṗ+
m − ṗ−
m)T
drm − (H+
m − H−
m)dtm. (2.74)
Equation (2.74) provides the gradients of the cost with respect to the independent
variations in the position and time of the midcourse impulse for use in a nonlinear
programming algorithm:
∂J
∂rm
= (ṗ+
m − ṗ−
m) (2.75)
and
∂J
∂tm
= −(H+
m − H−
m). (2.76)
AsasolutionsatisfyingtheNCisapproached, thegradientstendtozero, inwhichcase
both the primer rate vector ṗm and the Hamiltonian function Hm become continuous
at the midcourse impulse.
Note that when the NC are satisfied, the gradient with respect to tm in Equation
(2.76) being zero implies that
H+
m − H−
m = 0 = ṗT
m(v+
m − v−
m) = ṗT
mvm = vmṗT
mpm = 0 (2.77)
which, in turn, implies that ṗm = 0, indicating that the primer magnitude attains a
local maximum value of unity. This is consistent with the NC that p ≤ 1 and that ṗ
be continuous.
Figure 2.2 shows the converged, optimal three-impulse trajectory that results
from improving the primer histories shown in Figures 2.7 and 2.8. Note that the
final cost of 1.3681 is significantly less that the value of 1.7555 prior to adding the
midcourse impulse. Also, the time of the midcourse impulse changed during the
iteration from its initial value of 0.1 to a final value of approximately 0.17.
The absolute minimum cost solution for the final radius and ila value of Figure 2.2
is, of course, the Hohmann transfer shown in Figure 2.4. Its cost is significantly less
at 0.28446, but the transfer time is nearly three times as long at 1.8077 OOP. Of
this, 0.889 OOP is an initial coast to achieve the correct target phase angle for the
Hohmann transfer. Depending on the specific application, the total time required
may be unacceptably long.
(As a side note, a simple proof of the global optimality of the Hohmann transfer
using ordinary calculus rather than primer vector theory is given in [12].)](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-53-320.jpg)
![36 Primer Vector Theory and Applications
R E F E R E N C E S
[1] Bryson, A. E., and Ho, Y-C. (1975) Applied Optimal Control, Hemisphere Publishing
Co., Washington DC.
[2] Lawden, D. F. (1963) Optimal Trajectories for Space Navigation, Butterworths, London.
[3] Lion, P. M., and Handelsman, M. (1968) Primer Vector on Fixed-Time Impulsive
Trajectories. AIAA Journal, 6, No. 1, 127–132.
[4] Jezewski, D. J., and Rozendaal, H. L. (1968) An Efficient Method for Calculating Optimal
Free-Space n-impulse Trajectories. AIAA Journal, 6, No. 11, 2160–2165.
[5] Prussing, J. E. (1995) Optimal Impulsive Linear Systems: Sufficient Conditions and
Maximum Number of Impulses, The Journal of the Astronautical Sciences, 43, No. 2,
195–206.
[6] Prussing, J. E., and Chiu, J-H. (1986) Optimal Multiple-Impulse Time-Fixed Rendezvous
between Circular Orbits, Journal of Guidance, Control, and Dynamics, 9, No. 1, 17–22.
also Errata, 9, No. 2, 255.
[7] Prussing, J. E. (1993) Equation for Optimal Power-Limited Spacecraft Trajectories,
Journal of Guidance, Control, and Dynamics, 16, No. 2, 391–393.
[8] Prussing, J. E., and Sandrik, S. L. (2005) Second-Order Necessary Conditions and Suf-
ficient Conditions Applied to Continuous-Thrust Trajectories, Journal of Guidance,
Control, and Dynamics, 28, No. 4, 812–816.
[9] Glandorf, D. R. (1969) Lagrange Multipliers and the State Transition Matrix for Coasting
Arcs, AIAA Journal, 7, Vol. 2, 363–365.
[10] Battin, R. H. (1999) An Introduction to the Mathematics and Methods of Astrodynamics,
Revised Edition, AIAA Education Series, New York.
[11] Prussing, J. E., and Conway, B. A. (1993) Orbital Mechanics, Oxford University Press,
New York.
[12] Prussing, J. E. (1992) Simple Proof of the Global Optimality of the Hohmann Transfer,
Journal of Guidance, Control, and Dynamics, 15, No. 4, 1037–1038.
[13] Sandrik, S. (2006) Primer-Optimized Results and Trends for Circular Phasing and Other
Circle-to-Circle Impulsive Coplanar Rendezvous. Ph.D. Thesis, University of Illinois at
Urbana-Champaign.](https://image.slidesharecdn.com/1314345-250531020221-ca0ab4e7/85/Spacecraft-Trajectory-Optimization-Conway-B-Ed-54-320.jpg)
Spacecraft Trajectory Optimization Conway B Ed
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- 7. S P A C E C R A F T T R A J E C T O R Y O P T I M I Z A T I O N This is a long-overdue volume dedicated to space trajectory optimization. Interest in the subject has grown, as space missions of increasing levels of sophistication, complexity, and scientific return – hardly imaginable in the 1960s – have been designed and flown. Although the basic tools of optimization theory remain an accepted canon, there has been a revolution in the manner in which they are applied and in the development of numerical optimization. This vol- ume purposely includes a variety of both analytical and numerical approaches to trajectory optimization. The choice of authors has been guided by the editor’s intention to assemble the most expert and active researchers in the various specialties presented. Bruce A. Conway is a Professor of Aeronautical and Astronauti- cal Engineering at the University of Illinois, Urbana-Champaign. He earned his Ph.D. in aeronautics and astronautics at Stanford Uni- versity in 1981. Professor Conway’s research interests include orbital mechanics, optimal control, and improved methods for the numerical solution of problems in optimization. He is the author of numerous refereed journal articles and (with John Prussing) the textbook Orbital Mechanics.
- 9. Cambridge Aerospace Series Editors: Wei Shyy and Michael J. Rycroft 1. J.M. Rolfe and K.J. Staples (eds.): Flight Simulation 2. P. Berlin: The Geostationary Applications Satellite 3. M.J.T. Smith: Aircraft Noise 4. N.X. Vinh: Flight Mechanics of High-Performance Aircraft 5. W.A. Mair and D.L. Birdsall: Aircraft Performance 6. M.J. Abzug and E.E. Larrabee: Airplane Stability and Control 7. M.J. Sidi: Spacecraft Dynamics and Control 8. J.D. Anderson: A History of Aerodynamics 9. A.M. Cruise, J.A. Bowles, C.V. Goodall, and T.J. Patrick: Principles of Space Instrument Design 10. G.A. Khoury and J.D. Gillett (eds.): Airship Technology 11. J. Fielding: Introduction to Aircraft Design 12. J.G. Leishman: Principles of Helicopter Aerodynamics, 2nd Edition 13. J. Katz and A. Plotkin: Low Speed Aerodynamics, 2nd Edition 14. M.J. Abzug and E.E. Larrabee: Airplane Stability and Control: A History of the Technologies that made Aviation Possible, 2nd Edition 15. D.H. Hodges and G.A. Pierce: Introduction to Structural Dynamics and Aeroelasticity 16. W. Fehse: Automatic Rendezvous and Docking of Spacecraft 17. R.D. Flack: Fundamentals of Jet Propulsion with Applications 18. E.A. Baskharone: Principles of Turbomachinery in Air-Breathing Engines 19. D.D. Knight: Numerical Methods for High-Speed Flows 20. C. Wagner, T. Hüttl, P. Sagaut (eds.): Large-Eddy Simulation for Acoustics 21. D. Joseph, T. Funada, and J. Wang: Potential Flows of Viscous and Viscoelastic Fluids 22. W. Shyy, Y. Lian, H. Liu, J. Tang, D. Viieru: Aerodynamics of Low Reynolds Number Flyers 23. J.H. Saleh: Analyses for Durability and System Design Lifetime 24. B.K. Donaldson: Analysis of Aircraft Structures, 2/e 25. C. Segal: The Scramjet Engine: Processes and Characteristics 26. J. Doyle: Guided Explorations of the Mechanics of Solids and Structures 27. A. Kundu: Aircraft Design 28. M. Friswell, J. Penny, S. Garvey, A. Lees: Fundamentals of Rotor Dynamics 29. B.A. Conway (ed.): Spacecraft Trajectory Optimization 30. R.J. Adrian and J. Westerweel: Particle Image Velocimetry 31. S. Ching, Y. Eun, C. Gokcek, P.T. Kabamba, and S.M. Meerkov: Quasilinear Control Theory: Performance Analysis and Design of Feedback Systems with Nonlinear Actuators and Sensors
- 11. Spacecraft Trajectory Optimization Edited by Bruce A. Conway University of Illinois at Urbana-Champaign
- 12. CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK First published in print format ISBN-13 978-0-521-51850-5 ISBN-13 978-0-511-90945-0 © Cambridge University Press 2010 2010 Information on this title: www.cambridge.org/9780521518505 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Published in the United States of America by Cambridge University Press, New York www.cambridge.org eBook (NetLibrary) Hardback
- 13. Contents Preface page xi 1 The Problem of Spacecraft Trajectory Optimization . . . . . 1 Bruce A. Conway 1.1 Introduction 1 1.2 Solution Methods 3 1.3 The Situation Today with Regard to Solving Optimal Control Problems 12 References 13 2 Primer Vector Theory and Applications . . . . . . . . . . . 16 John E. Prussing 2.1 Introduction 16 2.2 First-Order Necessary Conditions 17 2.3 Solution to the Primer Vector Equation 23 2.4 Application of Primer Vector Theory to an Optimal Impulsive Trajectory 24 References 36 3 Spacecraft Trajectory Optimization Using Direct Transcription and Nonlinear Programming . . . . . . . . . . . . . . . . 37 Bruce A. Conway and Stephen W. Paris 3.1 Introduction 37 3.2 Transcription Methods 40 3.3 Selection of Coordinates 52 3.4 Modeling Propulsion Systems 60 3.5 Generating an Initial Guess 62 3.6 Computational Considerations 65 3.7 Verifying Optimality 71 References 76 vii
- 14. viii Contents 4 Elements of a Software System for Spacecraft Trajectory Optimization . . . . . . . . . . . . . . . . . . . . . . . . 79 Cesar Ocampo 4.1 Introduction 79 4.2 Trajectory Model 80 4.3 Equations of Motion 85 4.4 Finite Burn Control Models 85 4.5 Solution Methods 90 4.6 Trajectory Design and Optimization Examples 93 4.7 Concluding Remarks 110 References 110 5 Low-Thrust Trajectory Optimization Using Orbital Averaging and Control Parameterization . . . . . . . . . . . . . . . . 112 Craig A. Kluever 5.1 Introduction and Background 112 5.2 Low-Thrust Trajectory Optimization 113 5.3 Numerical Results 125 5.4 Conclusions 136 Nomenclature 136 References 138 6 Analytic Representations of Optimal Low-Thrust Transfer in Circular Orbit . . . . . . . . . . . . . . . . . . . . . . 139 Jean A. Kéchichian 6.1 Introduction 139 6.2 The Optimal Unconstrained Transfer 141 6.3 The Optimal Transfer with Altitude Constraints 145 6.4 The Split-Sequence Transfers 157 References 177 7 Global Optimization and Space Pruning for Spacecraft Trajectory Design . . . . . . . . . . . . . . . . . . . . . 178 Dario Izzo 7.1 Introduction 178 7.2 Notation 179 7.3 Problem Transcription 179 7.4 The MGA Problem 181 7.5 The MGA-1DSM Problem 183 7.6 Benchmark Problems 186 7.7 Global Optimization 190 7.8 Space Pruning 194 7.9 Concluding Remarks 197
- 15. Contents ix Appendix 7A 198 Appendix 7B 199 References 200 8 Incremental Techniques for Global Space Trajectory Design . . . . . . . . . . . . . . . . . . . . . 202 Massimiliano Vasile and Matteo Ceriotti 8.1 Introduction 202 8.2 Modeling MGA Trajectories 203 8.3 The Incremental Approach 209 8.4 Testing Procedure and Performance Indicators 216 8.5 Case Studies 221 8.6 Conclusions 234 References 235 9 Optimal Low-Thrust Trajectories Using Stable Manifolds . . . 238 Christopher Martin and Bruce A. Conway 9.1 Introduction 238 9.2 System Dynamics 240 9.3 Basics of Trajectory Optimization 247 9.4 Generation of Periodic Orbit Constructed as an Optimization Problem 250 9.5 Optimal Earth Orbit to Lunar Orbit Transfer: Part 1—GTO to Periodic Orbit 253 9.6 Optimal Earth Orbit to Lunar Orbit Transfer: Part 2—Periodic Orbit to Low-Lunar Orbit 256 9.7 Extension of the Work to Interplanetary Flight 259 9.8 Conclusions 260 References 261 10 Swarming Theory Applied to Space Trajectory Optimization . 263 Mauro Pontani and Bruce A. Conway 10.1 Introduction 263 10.2 Description of the Method 266 10.3 Lyapunov Periodic Orbits 269 10.4 Lunar Periodic Orbits 274 10.5 Optimal Four-Impulse Orbital Rendezvous 277 10.6 Optimal Low-Thrust Orbital Transfers 284 10.7 Concluding Remarks 290 References 291 Index 295
- 17. Preface It has been a very long time since the publication of any volume dedicated solely to space trajectory optimization. The last such work may be Jean-Pierre Marec’s Optimal Space Trajectories. That book followed, after 16 years, Derek Lawden’s pio- neering OptimalTrajectoriesforSpaceNavigationof1963. Ifeitherofthesebookscan be found now, it is only at a specialized used-book seller, for “astronomical” prices. In the intervening several decades, interest in the subject has only grown, with space missions of sophistication, complexity, and scientific return hardly possible to imagine in the 1960s having been designed and flown. While the basic tools of opti- mization theory – such things as the calculus of variations, Pontryagin’s principle, Hamilton-Jacobi theory, or Bellman’s principle, all of which are useful tools for the mission designer – have not changed in this time, there has been a revolution in the manner in which they are applied and in the development of numerical optimization. The scientists and engineers responsible have thus learned what they know about spacecraft trajectory optimization from their teachers or colleagues, with the assis- tance, primarily, of journal and conference articles, some of which are now “classics” in the field. This volume is thus long overdue. Of course one book of ten chapters cannot hope to comprehensively describe this complex subject or summarize the advances of three decades. While it purposely includes a variety of both analytical and numerical approaches to trajectory optimization, it is bound to omit solution methods preferred by some researchers. It is also the case that a solution method espoused by one author and shown to be successful for his examples may prove completely unsatisfactory when applied by a reader to his own problems. Even very experienced practitioners of optimal control theory cannot be certain a priori of success with any particular method applied to any particular challenging problem. The choice of authors has been guided by the editor’s intention to assemble the most expert and active researchers in the various specialties presented. The authors were given considerable freedom to choose their subjects, and while this may yield a somewhat eclectic volume, it also yields chapters written with palpable enthusiasm that are relevant to contemporary problems. Bruce Conway Urbana, Illinois xi
- 19. 1 The Problem of Spacecraft Trajectory Optimization Bruce A. Conway Dept. of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 1.1 Introduction The subject of spacecraft trajectory optimization has a long and interesting history. The problem can be simply stated as the determination of a trajectory for a spacecraft that satisfies specified initial and terminal conditions, that is conducting a required mission, while minimizing some quantity of importance. The most common objective is to minimize the propellant required or equivalently to maximize the fraction of the spacecraft that is not devoted to propellant. Of course, as is common in the optimization of continuous dynamical systems, it is usually necessary to provide some practical upper bound for the final time or the optimizer will trade time for propellant. There are also spacecraft trajectory problems where minimizing flight time is the important thing, or problems, for example those using continuous thrust, where minimizing flight time and minimizing propellant use are synonymous. Except in very special (integrable) cases, which reduce naturally to parameter optimization problems, the problem is a continuous optimization problem of an espe- cially complicated kind. The complications include the following: (1) the dynamical system is nonlinear; (2) many practical trajectories include discontinuities in the state variables, for example, there may be instantaneous velocity changes (also known as “V’s”) from use of rocket motors or from planetary flyby (or “gravity assist”) maneuvers, there may be instantaneous changes in spacecraft mass from staging or from using the rocket motor, or there may be sudden changes due to coordinate transformations necessary as the spacecraft moves from the gravitational sphere of influence of one body to that of another; (3) the terminal conditions, initial, final or both, may not be known explicitly, for example, for an interplanetary trajectory, the positions of the departure and arrival planets depend on the terminal times, which are often optimization variables; (4) there may be time-dependent forces, for exam- ple, the perturbations from other planets during an interplanetary trajectory can only be determined after the positions of the planets are determined using an ephemeris; (5) the basic structure of the optimal trajectory may not be a priori specified but is instead subject to optimization. For example, the optimal number of impulses or the optimal number of planetary flybys (or even the planets to use for the flybys) may 1
- 20. 2 Problem of Spacecraft Trajectory Optimization not be known. The VEEGA trajectory for Galileo [1] is an example; this was not the only feasible trajectory but was determined to be the optimal flyby sequence. There are many types of spacecraft trajectories. Until 1998 (and the very suc- cessful Deep Space 1 mission), spacecraft were propelled only impulsively, using chemical rockets whose burn duration is so brief in comparison to the total flight time that it is reasonable to model it as instantaneous. Between impulses, the spacecraft motion, as a reasonable first approximation, can be considered Keplerian. Interplan- etary cases add the possibility of planetary flyby maneuvers, which again, as a first approximation, may be modeled as nearly instantaneous velocity changes, preceded andfollowedbyKeplerianmotion. Theimpulsivetransfercase, evenincludingflybys, is thus a parameter optimization problem with the parameters being such quantities as the timing, magnitude, and direction of the impulsive V’s and the timing and altitude of gravity assist maneuvers. Of course, for extremely accurate spacecraft trajectory optimization, the resulting approximate trajectories must be reconsidered with the perturbations of other solar system bodies, the effect of solar radiation pressure, and other small but not insignificant effects included. While the potential benefits of low-thrust electric propulsion have been known for many years, it has only been relatively recently that spacecraft missions have been flown using this technology, for example in the NEAR and Deep Space 1 missions. Electric propulsion produces very small thrust, so that typical spacecraft acceleration is on the order of 10−5 g, and thus thrust is used either continuously or nearly so. The continuous thrust optimal control problem is qualitatively different from the impulsive case as there are now no integrable arcs and the control itself, for example the thrust magnitude and direction, have continuous time histories that must be modeled and determined. If the electric power is provided by solar cells, the variation of power available with distance from the sun must also be taken into consideration. A qualitatively similar continuous thrust case is that of solar sail- powered spacecraft, which of course are also subject to variation in effectiveness as they move away from the sun. While orbit transfer, for example LEO-GEO transfer, and interplanetary tra- jectories have been the focus of the bulk of research into spacecraft trajectory optimization, there are certainly many other applications of optimal control theory and numerical optimization to astrodynamics. Recent interesting problems include: (1) multi-vehicle navigation and maneuver optimization for cooperative vehicles, for example a fleet of small satellites in a specified formation [2]; (2) multi-vehicle noncooperative maneuver optimization, for example pursuit-evasion problems such as the interception of a maneuvering ICBM warhead by an intercepting spacecraft or missile [3]; (3) so-called “low-energy” transfer using invariant manifolds of the three-body problem, alone [4] or in combination with conventional or low-thrust propulsion [5] and; (4) trajectory optimization for a spacecraft sent to collide with a threatening Earth-approaching asteroid, with the objective of maximizing the subse- quent miss distance of the asteroid at its closest approach to Earth [6] [7]. These are only a few of many examples that could be drawn from recent literature and from the programs of the principal conferences in the subject.
- 21. 1.2 Solution Methods 3 Necessary conditions for optimality for every one of these types of spacecraft tra- jectoryoptimizationproblemsmaybederivedusingthecalculusofvariations(COV). Unfortunately the solution of the resulting system of equations and boundary condi- tions is either difficult or impossible. For certain simplified but still very useful cases of either impulsive-thrust or continuous-thrust orbit transfer, the analytical neces- sary conditions may described using the “Primer Vector” theory of Lawden [8], as will be described briefly in Section 1.2.1 of this Chapter and then in much greater detail in Chapter 2. Analytical solutions for the optimal trajectory (i.e. solutions sat- isfying the necessary conditions) can be obtained in special cases, for example for very-low-thrust orbit raising [9], even in the presence of some perturbations [10]. However, the vast majority of researchers and analysts today use numerical opti- mization. Numerical optimization methods for continuous optimal control problems are generally divided into two types. Indirect solutions are those using the analytical necessary conditions from the calculus of variations [11]. This requires the addition of the costate variables (or adjoint variables or Lagrange multipliers) of the prob- lem, equal in number to the state variables, and their governing equations. This instantly doubles the size of the dynamical system (which alone, of course, makes it more difficult to solve). Direct solutions, of which there are many types, transcribe the continuous optimal control problem into a parameter optimization problem [12] [13] [14]. Satisfaction of the system equations is accomplished by integrating them stepwise using either implicit or explicit (for example Runge-Kutta) rules; in either case, the effect is to generate nonlinear constraint equations that must be satisfied by the parameters, which are the discrete representations of the state and control time histories. The problem is thus converted into a nonlinear programming problem. There is a comprehensive survey paper by Betts [15] that describes direct and indi- rect optimization, the relation between these two approaches, and the development of these two approaches. 1.2 Solution Methods In just the decade since the publication of Betts’ survey paper [15], there has been considerable advancement of direct numerical solutions for optimal control prob- lems. There also has been even more development and improvement, in relative terms, of a qualitatively different approach to solving such problems, one using evo- lutionary algorithms. The best known of these are genetic algorithms (GA) [16]. Another evolutionary algorithm, the Particle Swarm Optimizer (PSD) will be dis- cussed in Chapter 10. The evolutionary algorithms have two principal advantages over other extant methods; they are comparatively simple and thus easy to learn to use, and they are generally more likely, in comparison to conventional optimizers, to locate global minima. In addition, there has been progress in analytical solutions such as those using primer vector theory [8] [17], “shape based” trajectories [18] [19], or Hamilton-Jacobi theory. All of the solutions may be broadly categorized as being either analytical or numerical, though of course the analytical solutions (with only a few exceptions
- 22. 4 Problem of Spacecraft Trajectory Optimization such as the Hohmann transfer) use numerical methods and the numerical solutions include some methods that explicitly use the analytical necessary conditions for opti- mality. In the following sections, the analytical and numerical solution methods will be defined and various examples, some historical and some very recent, will be pre- sented for many of the methods that fall within these categories. This is not intended to be a survey and will be unapologetically incomplete, as the subject is a vast one with a large literature. Rather, the intention in this introductory chapter is to describe the problem of spacecraft trajectory optimization, categorize the solution approaches, provide a small amount of history, and describe the “state of the art” so that the work of the various book chapter authors describing their approaches to the problem will be in context. 1.2.1 Analytical Solutions This is the original approach for space trajectory optimization, the oldest example of which (1925) is due to Hohmann’s conjecture [20] regarding the optimal circular orbit to circular orbit transfer. (The proof of the optimality of the Hohmann transfer came much later [21] [22].) Most of the analytical solutions are based on the necessary conditions of the problem that come from the calculus of variations (COV). Suppose that the system equations may be written in form ẋ = f(x, u, t) (1.1) where x represents an n-dimensional state (vector) and u represents the m- dimensional control (vector). The state vector is problem dependent; there are many choices available. Typically, conventional elliptic elements, equinoctial variables, or Delaunay variables are used for problems that are Keplerian or nearly-Keplerian, for example, very low-thrust orbit raising. Another common choice is spherical polar coordinates. Cartesian coordinates are typically used for three-body problems. The control u is typically a control of thrust magnitude and direction or its equivalent, for example the orientation of a solar-sail spacecraft with respect to the Sun. The problem has some initial conditions specified, that is, xi(0) given for i = 1, 2, …, k with k ≤ n (1.2) and some terminal conditions, or functions of the terminal conditions, specified as the vector [x(T), T] = 0. (1.3) The objective may be written in the Bolza form as J = φ [x(T), T] + T 0 L [x, u, t] dt (1.4) where φ is a terminal cost function while the integral expresses a cost incurred during the entire trajectory.
- 23. 1.2 Solution Methods 5 The first step in deriving the conditions for an extremum of (1.4) subject to the system (1.1) and the boundary conditions (1.3) is to define a system Hamiltonian H = L + λT f Then, in terms of H and the other quantities introduced, the necessary conditions become [11] λ̇ = − ∂H ∂x T with boundary condition λ(T) = ∂φ ∂x + νT ∂ ∂x T t=T (1.5) ∂H ∂u = 0. (1.6) The system of equations (1.1)–(1.6) constitutes a two-point-boundary-value problem (TPBVP); some boundary conditions on the states are specified at the initial time and some boundary conditions on the states and adjoints are specified at the terminal time. In addition, if the terminal time is unspecified (that is free to be optimized), as is often the case, an additional scalar equation obtains ∂φ ∂t + νT ∂ ∂t + ∂φ ∂x + νT ∂ ∂x f + L t=T = 0. (1.7) For all but the most elementary optimal control problems, the solution of this TPBVP is challenging and numerical solutions are required. Despite this, it is interesting that when this set of necessary conditions is applied to the optimal space trajectory problem, which is by no means elementary, several very useful observations may be made. The system equations of motion (1.1) may be written in the form ˙ x̄ = f̄ = ˙ r̄ ˙ v̄ = v̄ ḡ(r̄) + û (1.8) where g(r) is the gravitational acceleration, is the thrust acceleration magnitude, and û is a unit vector indicating the thrust direction. To minimize the velocity change required, one chooses the integrand in the cost function (1.4) to be L = the acceleration provided by the motor; then the integral will represent the V provided by the motor. The Hamiltonian then becomes H = + λ̄T r v̄ + λ̄T v [ḡ(r̄) + û] = 1 + λ̄T v û + λ̄T r v̄ + λ̄T v ḡ(r̄). (1.9) Because H is linear in u, equation (1.6) does not obtain. The optimal control is instead chosen according to Pontryagin’s Minimum Principle, stating that at any time on the optimal trajectory, the control variables are chosen in order to minimize the Hamiltonian. Thus the first simple observation is that the thrust pointing unit vector is chosen to be parallel to the opposite of the adjoint (to the velocity) vector,
- 24. 6 Problem of Spacecraft Trajectory Optimization i.e. −λ̄v(t). Because of its physical significance to the problem, this (adjoint) vector is referred to as the primer vector [8]. A second simple observation is that with this choice of thrust direction, it is then optimal in this case to choose the thrust magnitude at its maximum possible value if the “switching function” 1 + λ̄T v û (1.10) is negative and choose = 0 if the switching function is positive. The adjoint vector λ̄v(t) is governed by the system equations (1.5) with the Hamiltonian (1.9). In addition, it is straightforward to show that if the Hamiltonian H is not explic- itly time dependent, then H is a constant on the optimal trajectory. This result is not necessarily useful for obtaining the optimal control but can be of great use in determining, by its use a posteriori, the accuracy of the numerical solution of the TPBVP, that is, a good solution will have H the same, to several significant figures, when evaluated at any point on the numerical solution [14] [23]. Finally, while the necessary conditions guarantee only that the trajectory repre- sents an extremum of the cost, by the nature of the space trajectory problem, there is clearly no upper bound to the fuel that could be consumed on a feasible trajectory (other than consuming all the fuel available). So one may be confident that a solution is a local minimum and not a local maximum. Further results can be obtained from a description of the necessary conditions in terms of the primer vector, and these will be described in Chapter 2. It will suf- fice to say here that while the primer vector is defined, and has the significance with regard to optimal thrust direction found above, this is of course true only on the optimal trajectory. The improvement of a known, nonoptimal trajectory via primer vector theory was first discussed by Lion and Handelsman [17]. Jezewski and Rozendaal [24] showed under what conditions an optimal N impulse trajectory could beimprovedbytheadditionofanotherimpulse, andwhereandwithwhatdirectionto apply it. Solution of the analytical necessary conditions is possible for some special cases. One useful example is the case of very-low-thrust orbit raising. With certain assumptions, it is possible to find approximate solutions of the analytical neces- sary conditions. Many of these are found in a survey paper of the subject by Petropoulos and Sims [25]. The most common simplifications include: assuming that the thrust direction is always tangential; assuming that the thrust pointing is always in the direction of the velocity vector; or assuming that the orbit is always circular. Surprisingly, exact solutions also exist in certain cases, including this low- thrust orbit raising, even in the presence of nonspherical Earth perturbations [10]. This will be discussed in Chapter 7. The mathematics and analysis become very involved. The solution of the TPBVP resulting from (or constituting) the necessary con- ditions becomes quite difficult for other problems, particularly those with path constraints (typically on the state variables or on functions of the state variables) or constraints on total fuel available.
- 25. 1.2 Solution Methods 7 Many methods have been developed to solve the TPBVP numerically. The most obvious and well known is probably shooting (an archetype of shooting applied to spacecraft trajectory optimization may be found in the paper by Breakwell and Redding [26]) but there are other methods including finite-difference methods [27] [28] and collocation [12] [13] [14]. The long-recognized difficulty of the “indirect” approach to determining the optimal trajectory is that the initial costate variables of the TPBVP are unknown and further that the nonlinearity of the problem means that the vector flow is very sensitive to some or all of these initial costate variables. A further difficulty is that the costate variables lack the physical significance of the state variables so that estimating the order of magnitude or even the sign of the initial costates is very difficult. For problems with constrained arcs, another difficulty that arises is discontinuity of controls and costate variables at the junctions of constrained and unconstrained arcs. This also increases the difficulty of solving the associated TPBVP. Another solution method that satisfies both the necessary and sufficient condi- tions for optimality is the method of Static/Dynamic control (SDC) of Whiffen [29] [30]. The term static refers to decision variables that are discrete, such as launch dates or planetary flyby dates, while the term dynamic refers to controls that have a continuous variation in time, such as thrust pointing angle time histories. SDC is a general nonlinear optimal control algorithm based on Bellman’s principle of opti- mality [11]. The implementation of SDC in the program Mystic is a very capable low-thrust spacecraft trajectory optimizer. A recent, qualitatively different approach to the determination of optimal space trajectoriesisthatofGuiboutandScheeres[31]. Inthiswork, thedynamicalsystemof state and costate variables (the vector field) is solved for specified terminal conditions andfinaltimebysolvingtheassociatedHamilton-Jacobi(H-J)equation. Thesolution of the H-J equation is a generating function for a canonical transformation. Once this solution is determined, the initial value of the costate vector may be found; the optimal trajectory and the optimal control may then be found by forward integration of the flow field. Scheeres et al. show an example of an optimal rendezvous in the vicinity of a nominal circular orbit [32]. 1.2.2 Numerical Solutions via Discretization Many recent methods for solving optimal control problems seek to reduce them to parameter optimization problems that can then be solved by a NLP problem solver. One principal way in which such methods are distinguished is with regard to what quantities are parameterized. In one popular method, the collocation method that will be discussed in Chapter 3, it is possible to parameterize the state variables and the costate variables (that is, to solve the TPBVP). It is also possible in collocation to parameterize only the state variables and the control variables, as will be discussed in the next section. A third possibility, yielding the smallest number of parameters for a given problem, is to parameterize only the control variables and some free terminal states, but then the system equations must be numerically integrated (as opposed
- 26. 8 Problem of Spacecraft Trajectory Optimization to the implicit integration that occurs in collocation). This is referred to as “control parameterization” and will be discussed in Chapter 5. Of course all of the solutions described in the previous section are obtained numerically, that is, they will employ methods such as numerical integration, solving TPBVP problems using “shooting” methods, or solving boundary value problems by converting them into nonlinear programming (NLP) problems. What is meant in this section by “numerical solution” is solutions that do not explicitly employ the analytical necessary conditions of the COV, for example, solutions that do not employ the costate (adjoint) variables of the problem or solutions that satisfy the H-J-B equation or Bellman’s principle for discrete systems. Why would one want to avoid the use of the necessary conditions, particularly when the resulting trajectory has a “guarantee” of being a local extremum (that one loses in a numerical solution) and has other benefits previously discussed, such as information about sensitivity to terminal conditions and guidance toward improving a solution by for example, adding/subtracting thrust arcs? The principal reason is the lack of robustness of the various methods for solving the Euler-Lagrange TPBVP stemming, as previously mentioned, from the nonlinearity of the problem and a lack, in the general case, of a systematic means for determining a sufficiently good approximation to the initial adjoint variables of the problem. A variety of direct solution methods have been developed. They are best catego- rized by the way in which they handle the discretization of the equations of motion, which appear as function-space constraints in the original optimal control problem. A more complete survey will be presented in Chapter 3. In the last two decades, however, the most successful approach is arguably one in which the continuous problem is discretized and state and control variables are known only at discrete times. Satisfaction of the equations of motion is achieved by employing an explicit or implicit numerical integration rule that needs to be satisfied at each step; this results in a large NLP problem with a large number of nonlinear constraints. This approach was termed “direct transcription” by Canon et al. [33]. While known to mathematicians in the 1960s and 1970s, it became known in the aerospace commu- nity principally through two papers. Dickmanns and Well [34] used the collocation scheme to solve the TPBVP of the indirect method. This approach is significantly more robust than shooting methods because it eliminates the sequential nature of the shooting solution, with its forward numerical integration, in favor of a solution in which simultaneous changes in all of the discrete state and costate parameters are made in order to satisfy algebraic constraints (while minimizing the objective of course). However, the most useful development for space trajectory optimization was the observation in 1987 by Hargraves and Paris [12] that it was not necessary to use this approach to solve the indirect TPBVP, that in fact the adjoint variables (which had been used to determine the optimal control from Pontryagin’s principle) could be removed from the solution provided that discrete control variables were intro- duced as additional NLP parameters. This significantly improved the robustness of the method; by eliminating the adjoint variables, the problem size is reduced almost
- 27. 1.2 Solution Methods 9 by half, and there is no longer a need to provide the NLP problem solver with an estimate of the adjoint variables, something that is always problematic. A fortunate coincidence is that at about the same time (1980s), the NLP technology required to efficiently and robustly solve large problems became available (and has been continu- ouslyimprovedsincethen)[35][36]. Theastrodynamicscommunityswiftlyembraced this method. Many optimal spacecraft trajectories have since been determined using direct methods. The direct method has also been significantly developed in the last two decades. There are now many approaches, differing primarily (for collocation methods) on how the implicit integration rules are constructed [37]. The most com- mon approaches are to use trapezoidal [38] or Hermite-Simpson [12] integration rule constraints, or higher-degree rules from the same Gauss-Lobatto family [13] or a Gauss-pseudospectral method [39]. There also exist commercial software pack- ages implementing direct methods for general optimal control problems, for example DIDO [40] and SOCS [38], and even solvers specifically for space and launch vehicle trajectory optimization, for example OTIS [41] and ALTOS [42] [43]. It would be accurate to say that the great majority of optimal space trajecto- ries are now determined numerically, with methods that do not make explicit use of the analytical necessary conditions of the problem, as will be described briefly below and in detail in Chapter 2. However, that does not mean that the necessary conditions are no longer useful. On the contrary, they provide useful information that many numerical solutions naturally lack. For example, primer vector theory can provide important information on how a solution may be improved, for exam- ple by adding thrust arcs or coast arcs or by adding impulses for an impulsive trajectory. The solution of the TPBVP of the necessary conditions also provides information on the sensitivity of the solution to changes in terminal conditions and constraints. Fortunately, without solving the TPBVP, it is possible to make use of some of these beneficial features of the solution of the necessary conditions, as will be described in Chapter 3. This occurs because of a correspondence between the final adjoint variables of the continuous TPBVP and some Kuhn-Tucker multipliers gen- erated in (some) numerical solutions of the trajectory optimization problem [13] [14]. With these multiplier variables available, it is possible, for example, to compute the value of the system Hamiltonian over the entire trajectory time history. For many problems in which H should be a constant, this can provide a check on the accuracy of the numerical solution. Or, knowing the final adjoints and final states from, for example, a direct solution using collocation and NLP, one can integrate the E-L equations backward to the initial time. If the initial states are recovered, one can then say that the numerical solution satisfies the analytical necessary conditions and thus represents an extremal path. 1.2.3 Evolutionary Algorithms A qualitatively different approach, recently applied to spacecraft trajectory opti- mization, is the use of “evolutionary” algorithms (EA). The best known of the EAs
- 28. 10 Problem of Spacecraft Trajectory Optimization is the genetic algorithm (GA). EA’s are numerical optimizers that determine, using methods similar to those found in nature, an optimal set of discrete parameters that has been used to characterize the problem solution. The EA’s have two principal advantages over all of the direct and indirect solution methods previously described in this chapter: they require no initial “guess” of the solution (in fact they gener- ate a population of initial solutions randomly), and they are more likely than other methods to locate a global minimum in the search space rather than be attracted to a local minimum. All of the EAs require that the problem solution be capable of being described by a relatively small, in comparison to the vector of parameters of a nonlinear pro- gram, set of discrete parameters. This can be accomplished, for spacecraft trajectory optimization problems, in a number of ways: (1) If the trajectory can naturally be described by a finite set, for example an impul- sive thrust trajectory, the parameters will be such things as times, magnitudes, and directions of impulses. Between impulses the trajectories may be determined by solving Lambert’s problem. In this case a small number of parameters will suffice to completely describe the solution. (2) If the trajectory contains non-integrable arcs, for example low-thrust arcs, it is still the case that much of the trajectory can be described with a small number of parameters such as departure and arrival dates and times for the beginning and end of thrust arcs. Quantities that must be described continuously, such as thrust magnitude or pointing time history, can be parameterized using, for example, polynomial equations in time. Then the additional parameters are a small number of polynomial coefficients [44] (3) Low-thrust arcs can also be described using “shape-based” methods [18] [19]. In this approach, a shape, which is an analytical expression for the trajectory, can be generated from a small number of parameters such that the resulting trajectory will actually be a solution of the system equations of motion. Unfortunately the thrust time history that allows this beneficial result can only be determined a posteriori. An EA is then used to choose the parameters defining the shape to satisfy the boundary conditions of the problem and to minimize the cost. The resulting trajectory may not be realizable, as it may require greater thrust than is available. However the trajectory may well be satisfactory as an initial guess for a more accurate method, for example a direct method such as collocation [12] [13] [14]. In the simplest form of the genetic algorithm, the set of parameters describ- ing the solution is written as a string or sequence of numbers. Suppose that this sequence is converted to binary form; it is then similar to a chromosome but con- sisting only of two possible variables, a 1 or a 0. Every sequence can be “decoded” to yield a trajectory whose cost or objective value can be determined. The first step in the GA is the generation of a “population” of sequences using a random pro- cess. The great majority of these randomly generated sequences will have very large
- 29. 1.2 Solution Methods 11 costs; many may even be infeasible. The population is then improved using three natural processes: selection, combination, and mutation. Selection removes the worst sequences and may also, via elitism, guarantee that the best sequence survives into the next generation unchanged. Following selection, remaining sequences are used as “parents,” that is partial sequences from two parents are combined to form new individuals. Finally, mutation changes a randomly chosen bit in a small fraction of the population. The process is then repeated; the cost of every individual in the new generation is determined. Since the best individual from the previous generation was retained, the objective may improve but cannot worsen. In practice, there is generally rapid improvement in the early generations; if the process locates the global minimum then, of course, improvement will cease. Termination of the algorithm is usually done either after a fixed number of generations or after the objective has reached a plateau. Of course neither of these termination conditions guarantees that a minimum has been found, nor are there necessary conditions for optimality with this method. Additional shortcomings are that there is no way to enforce satisfaction of boundary conditions; normally a “penalty function” approach is taken in which unsatisfied boundary conditions are added to the cost, and that the solution will be less accurate than a typical direct solution (and even less accurate than an indirect solution). Nevertheless, the method has been very useful when applied to optimizing space trajectories, either for finding approximate extremals [44] or when used to provide an initial guess for more accurate methods, for example collocation with NLP. Betts [15] notes that one significant advantage of the GA in comparison to all other solution methods is how straightforward it is to use. There are many GA routines available (a commonly used one is found in MATLAB) so the user need only provide a subroutine for decoding the sequence to evaluate the cost (which for space trajectory problems can be as simple as a routine that integrates the system equations of motion) provide bounds on the parameters, and then provide values for certain constant parameters that control the evolutionary processes. There are other EAs that have begun to prove very useful in the determination of optimal space trajectories. One qualitatively different method is particle swarm opti- mization (PSO). In PSO, some number (say 100) of particles are randomly distributed in a N-dimensional decision parameter space. The objective value is determined for the solution vector corresponding to each particle. Taking an anthropomorphic view, it is then assumed that the particles can communicate so that all know the objective value for all the others. Let xi(n) denote the position of particle i at the nth time step. At the next iteration, the particles take a step vi(n + 1) in the parameter space so that the new position of particle i becomes xi(n + 1) = xi(n) + vi(n + 1) (1.11) with (in one form of the PSO) vij(n + 1) = vij(n) + c1r1j(n) yij(n) − xij(n) + c2r2j(n) ŷj(n) − xij(n) (1.12)
- 30. 12 Problem of Spacecraft Trajectory Optimization where vij(n) is the velocity (step) for component j of particle i at time step n, xij(n) is the jth component of the position of particle i at the nth time step, r1j(n) and r2j(n) ⊂ U(0, 1) are random values in the range [0, 1] sampled from a uniform distribution. yi(n) is the “personal best” position, the best position located by the ith particle since the first time step; ŷj(n) is the “global best” position, the best position located by the any particle of the swarm since the first time step. The step described in equation (1.12) thus has three components. The first is an “inertia” that causes the particle to move in the direction it had previously been moving, the second “nostalgia” component reflects a tendency for the particle to move toward its own most satisfactory position, and the third “social” component draws the particle toward the best position found by any of its colleagues. The c’s are constants that weight the importance of the three components and the r’s provide stochasticity to the system. As with the GA, the process can be terminated after a fixed number of iterations or when the “best” solution has not changed for several iterations. This method has proven quite robust, is also very simple to use, and is particularly good in locating globalminimawhenthesolutionspacecontainsmanylocalminima. Amorethorough description of the PSO method and its application to space trajectory problems will be provided in Chapter 10. There are many other EAs, for example ant colony optimization (ACO) or differential evolution (DE). The interested reader can easily find information on the use of these methods [45]. 1.3 The Situation Today with Regard to Solving Optimal Control Problems One can safely say, for example by considering papers published recently in astrody- namics journals, that solutions using analytical methods, that is analytical solutions of the first-order necessary conditions, are seldom found. This is due, as previously mentioned, to the complexity of the problem when realistic terminal boundary condi- tions and when bounds on the controls are present. Also, solutions found numerically using indirect methods, for example with shooting methods, are also becoming less common. This is almost certainly due to the success that has been achieved with direct methods, particularly those using collocation via low-degree rules such as trapezoid or Hermite-Simpson [11] [13], via the pseudospectral method [39], or by higher degree G-L implicit integration [13]. (These collocation methods are all derivable from the same source, as will be seen in Chapter 3.) These methods have proven particularly robust and efficient and have been used to solve many types of prob- lems including low-thrust orbit raising [46], Earth-Moon transfer [47] [5] [48], and interplanetary transfers [49]. An early difficulty faced by users of these methods was that, while robust, it was still necessary to supply a reasonable initial guess of the solution parameters, that is a discretized form of the state and control time-histories on the optimal trajectory, to the NLP problem solver. This, of course, is not always a simple matter. For some cases, for example for low-thrust orbit raising, approximate analytical solutions such as a Lawden spiral, as described in Section 1.2.1, are available and make a
- 31. References 13 very satisfactory initial guess. For other problems, such as the optimal low-thrust Earth-Moon transfer, obtaining a satisfactory initial guess is much more difficult. Today, however, the situation is much improved since evolutionary algorithms such as the GA, which can provide a solution to the problem in their own right, can also be used as “pre-processors” to provide an initial guess of the solution from which a method such as direct collocation with NLP can converge to a much more accurate solution. An additional advantage of this approach is that some of the EAs are better suited to locating the global minimum than are the methods using NLP, as the NLP solver will tend to converge to a local minimum in the neighborhood of the initial guess it is given. Thus starting from a guess provided by an EA is more likely to enable the direct solver to find a global minimum. (Of course there is no guarantee in any case.) John Betts’ observation in 1988 [15] that “one may expect many of the best features of seemingly disparate techniques to merge, forming still more powerful methods” was clearly very prescient. R E F E R E N C E S [1] D’Amario, L. et al. (1989) Galileo 1989 VEEGA Trajectory Design, Journal of the Astronautical Sciences, 37, 281–306. [2] Wang, P. K. C., and Hadaegh, F. Y. (1998) Optimal Formation-Reconfiguration for Mul- tiple Spacecraft, AIAA-1998-4226, AIAA Guidance, Navigation, and Control Conference and Exhibit, Boston, MA, Collection of Technical Papers. Pt. 2 (A98-37001 10–63) [3] Pontani, M., and Conway, B. A. (2008) Optimal Interception of Evasive Missile War- heads: Numerical Solution of the Differential Game, Journal of Guidance, Control and Dynamics, 31, 1111–1122. [4] Marsden, J. E., and Ross, S. D. (2005) New Methods in Celestial Mechanics and Mission Design, Bulletin of the American Mathematical Society, 43, 43–73. [5] Mingotti, G., Topputo, F., and Bernelli-Zazzera, F. (2007) Combined Optimal Low- Thrust and Stable-Manifold Trajectories to the Earth-Moon Halo Orbits, New Trends in Astrodynamics and Applications III, E. Belbruno (ed.) 100–110. [6] Englander, J., and Conway, B. A. (2009) Optimal Strategies Found Using Genetic Algo- rithms for Deflecting Hazardous Near-Earth Objects, IEEE Congress on Evolutionary Computation, Trondheim, Norway. [7] Conway, B. A. (1997) Optimal Low-Thrust Interception of Earth-Crossing Asteroids, J. of Guidance, Control, and Dynamics, 20, 995–1002. [8] Lawden, D. F. (1963) Optimal Trajectories for Space Navigation, Butterworths, London. [9] Kechichian, J. A. (1997) Reformulation of Edelbaum’s Low-Thrust Transfer Prob- lem Using Optimal Control Theory, J. of Guidance, Control and Dynamics, 20, 988–994. [10] Kechichian, J. A. (2000) Minimum-Time Constant Acceleration Orbit Transfer With First-Order Oblateness Effect, Journal of Guidance, Control, and Dynamics, 23, 595–603. [11] Bryson, A. E., and Ho, Y-C. (1975) Applied Optimal Control, Revised Printing, Hemisphere Publ., Washington, DC. [12] Hargraves, C. R., and Paris, S. W. (1987) Direct Trajectory Optimization Using Non- linear Programming and Collocation, Journal of Guidance, Control, and Dynamics, 10, 338–342. [13] Herman, A. L., and Conway, B. A. (1996) Direct Optimization Using Collocation Based on High-Order Gauss-Lobatto Quadrature Rules, Journal of Guidance, Control, and Dynamics, 19, 592–599.
- 32. 14 Problem of Spacecraft Trajectory Optimization [14] Enright, P.J., andConway, B.A.(1992)DiscreteApproximationstoOptimalTrajectories Using Direct Transcription and Nonlinear Programming, Journal of Guidance, Control, and Dynamics, 15, 994–1002. [15] Betts, J. T. (1998) Survey of Numerical Methods for Trajectory Optimization, Journal of Guidance, Control and Dynamics, 21, 193–207. [16] Goldberg, D. (1989) Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley Publ., Reading, MA. [17] Lion, P. M., and Handelsman, M. (1968) The Primer Vector on Fixed-Time Impulsive trajectories, AIAA Journal, 6, 127–132. [18] Petropoulos, A. E., and Longuski, J. M. (2004) Shape-Based Algorithm for Automated Design of Low-Thrust, Gravity-Assist Trajectories, Journal of Spacecraft and Rockets, 41, 787–796. [19] Wall, B. J., and Conway, B. A. (2009) Shape-Based Approach to Low-Thrust Trajectory Design, Journal of Guidance, Control and Dynamics, 32, 95–101. [20] Hohmann, W. (1925). Die Erreichbarkeit der Himmelskörper. Verlag Oldenbourg, München. [21] Prussing, J. E. (1991) Simple Proof of the Global Optimality of the Hohmann Transfer, Journal of Guidance, Control, and Dynamics, 15, 1037–1038. [22] Barrar, R. B., (1963) An Analytic Proof that the Hohmann-Type Transfer is the True Minimum Two-Impulse Transfer, Astronautica Acta, 9, 1–11. [23] Herman, A. L. (1995) Improved Collocation Methods Used for Direct Trajectory Optimization, Ph.D. Thesis, University of Illinois at Urbana-Champaign. [24] Jezewski, D. J., and Rozendaal, H. L. (1968) An Efficient Method for Calculating Optimal Free-Space N-Impulse Trajectories, AIAA Journal, 6, 2160–2165. [25] Petropoulos, A. E., and Sims, J. A. (2004) A Review of Some Exact Solutions to the Planar Equations of Motion of a Thrusting Spacecraft, DSpace at JPL, http://hdl.handle. net/2014/8673 [26] Breakwell, J. V., and Redding, D. C. (1984) Optimal Low-Thrust Transfers to Syn- chronous Orbit, Journal of Guidance, Control, and Dynamics, 7, 148–155. [27] Fox, L. (1957) The Numerical Solution of Two-Point Boundary Problems, Oxford Press, New York. [28] Keller, H.B. (1968) Numerical Methods for Two-Point Boundary Value Problems, Blaisdell, New York. [29] Whiffen, G. (2006) Mystic: Implementation of the Static Dynamic Optimal Control Algorithm for High-Fidelity, Low-Thrust Trajectory Design, http://trs-new.jpl.nasa. gov/dspace/bitstream/2014/40782/1/06-2356.pdf [30] Whiffen, G. J., and Sims, J. A. (2002) Application of the SDC Optimal Control Algorithm to Low-Thrust Escape and Capture Including Fourth Body Effects, Second International Symposium on Low Thrust Trajectories, Toulouse, France. [31] Guibout, V. M., and Scheeres, D. J. (2006) Solving Two-Point Boundary Value Problems Using Generating Functions: Theory and Applications to Astrodynamics, in Modern Astrodynamics, Gurfil, P. (ed.), Elsevier, Amsterdam, 53–105. [32] Scheeres, D. J., Park, C., and Guibout, V. (2003) Solving Optimal Control Problems with Generating Functions, Paper AAS 03-575. [33] Canon, M. D., Cullum, C. D., and Polak, E. (1970) Theory of Optimal Control and Mathematical Programming, McGraw-Hill, New York. [34] Dickmanns, E. D., and Well, K. H. (1975) Approximate Solution of Optimal Con- trol Problems Using Third-Order Hermite Polynomial Functions, Proceedings of the 6th Technical Conference on Optimization Techniques, IFIP-TC7, Springer–Verlag, New York. [35] Gill, P., Murray, W., and Saunders, M. A. (2005) SNOPT: An SQP Algorithm for Large- Scale Constrained Optimization. SIAM Review, 47, 99–131. [36] Gill, P. E. et al. (1993) User’s Guide for NZOPT 1.0: A Fortran Package For Nonlinear Programming, McDonnell Douglas Aerospace, Huntington Beach, CA.
- 33. References 15 [37] Paris, S. W., Riehl, J. P., and Sjauw, W. K. (2006) Enhanced Procedures for Direct Trajec- tory Optimization Using Nonlinear Programming and Implicit Integration, Paper AIAA- 2006-6309, AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Keystone, Colorado. [38] Betts, J. T., and Huffman, W. P. (1997) Sparse Optimal Control Software SOCS, Mathematics and Engineering Analysis Technical Document, MEA-LR-085, Boeing Information and Support Services, The Boeing Company, Seattle. [39] Ross, I. M., and Fahroo, F. (2003) Legendre Pseudospectral Approximations of Optimal Control Problems, Lecture Notes in Control and Information Sciences, 295, Springer– Verlag, New York, 327–342. [40] Ross, I. M. (2004) User’s Manual for DIDO: A MATLAB Application Package for Solving Optimal Control Problems, TOMLAB Optimization, Sweden. [41] Paris, S. (1992) OTIS-Optimal Trajectories by Implicit Simulation, User’s Manual, The Boeing Company. [42] Well, K. H., Markl, A., and Mehlem, K. (1997) ALTOS-A Trajectory Analysis and Optimization Software for Launch and Reentry Vehicles, International Astronautical Federation Paper IAF-97-V4.04. [43] Wiegand, A., Well, K. H., Mehlem, K., Steinkopf, M., and Ortega, G. (1999) ALTOS- ESA’s Trajectory Optimization Tool Applied to Reentry Vehicle Trajectory Design, International Astronautical Federation Paper IAF-99-A.6.09 [44] Wall, B. J., and Conway, B. A. (2005) Near-Optimal Low-Thrust Earth-Mars Trajectories Found Via a Genetic Algorithm,” Journal of Guidance, Control, and Dynamics, 28, 1027–1031. [45] Engelbrecht, A. P. (2007) Computational Intelligence, 2nd ed., John Wiley Sons, West Sussex, England. [46] Scheel, W. A., and Conway, B. A. (1994) Optimization of Very-Low-Thrust, Many Revolution Spacecraft Trajectories, Journal of Guidance, Control, and Dynamics, 17, 1185–1192. [47] Herman, A. L., and Conway, B. A. (1998) Optimal Low-Thrust, Earth-Moon Orbit Transfer, Journal of Guidance, Control, and Dynamics, 21, 141–147. [48] Betts, J. T., and Erb, S. O. (2003) Optimal Low Thrust Trajectories to the Moon, SIAM Journal on Applied Dynamical Systems, 2, 144–170. [49] Tang, S., and Conway, B. A. (1995) Optimization of Low-Thrust Interplanetary Trajec- tories Using Collocation and Nonlinear Programming, Journal of Guidance, Control, and Dynamics, 18, 599–604.
- 34. 2 Primer Vector Theory and Applications John E. Prussing Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 2.1 Introduction In this chapter, the theory and a resulting indirect method of trajectory optimization are derived and illustrated. In an indirect method, an optimal trajectory is deter- mined by satisfying a set of necessary conditions (NC), and sufficient conditions (SC) if available. By contrast, a direct method uses the cost itself to determine an optimal solution. Even when a direct method is used, these conditions are useful to determine whether the solution satisfies the NC for an optimal solution. If it does not, it is not an optimal solution. As an example, the best two-impulse solution obtained by a direct method is not the optimal solution if the NC indicate that three impulses are required. Thus, post-processing a direct solution using the NC (and SC if available) is essential to verify optimality. Optimal Control [1], a generalization of the calculus of variations, is used to derive a set of necessary conditions for an optimal trajectory. The primer vector is a term coined by D. F. Lawden [2] in his pioneering work in optimal trajectories. [This terminology is explained after Equation (2.24).] First-order necessary conditions for both impulsive and continuous-thrust trajectories can be expressed in terms of the primer vector. For impulsive trajectories, the primer vector determines the times and positions of the thrust impulses that minimize the propellant cost. For continuous- thrusttrajectories, boththeoptimalthrustdirectionandtheoptimalthrustmagnitude as functions of time are determined by the primer vector. As is standard practice, the word “optimal” is loosely used as shorthand for “satisfies the first-order NC.” The most completely developed primer vector theory is for impulsive trajec- tories. Terminal coasting periods for fixed-time trajectories and the addition of midcourse impulses can sometimes lower the cost. The primer vector indicates when these modifications should be made. Gradients of the cost with respect to termi- nal impulse times and midcourse impulse times and positions were first derived by Lion and Handelsman [3]. These gradients were then implemented in a nonlinear Figures 2.2 and 2.4–2.8 were generated using the MATLAB computer code written by Suzannah L. Sandrik [13]. 16
- 35. 2.2 First-Order Necessary Conditions 17 programming algorithm to iteratively improve a nonoptimal solution and converge to an optimal trajectory by Jezewski and Rozendaal [4]. 2.2 First-Order Necessary Conditions 2.2.1 Optimal Constant-Specific-Impulse Trajectory For a constant specific impulse (CSI) engine, the thrust is bounded by 0 ≤ T ≤ Tmax (where Tmax is a constant), corresponding to bounds on the mass flow rate: 0 ≤ b ≤ bmax (where bmax is a constant). Note that one can also prescribe bounds on the thrust acceleration (thrust per unit mass) ≡ T/m as 0 ≤ ≤ max, where max is achieved by running the engine at Tmax. However, max is not constant but increases due to the decreasing mass. One must keep track of the changing mass in order to compute for a given thrust level. This is easy to do, especially if the thrust is held constant, for example, at its maximum value. However, if the propellant mass required is a small fraction of the total mass because of being optimized, a constant max approximation can be made. The cost functional representing minimum propellant consumption for the CSI case is J = tf to (t)dt. (2.1) The state vector is defined as x(t) = r(t) v(t) (2.2) where r(t) is the spacecraft position vector and v(t) is its velocity vector. The mass m can be kept track of without defining it to be a state variable by noting that m(t) = moe−F(t)/c (2.3) where c is the exhaust velocity and F(t) = t to (ξ)dξ. (2.4) Note that from Equation (2.4), F(tf ) is equal to the cost J. In the constant thrust case, varies according to ˙ = 1 c 2, which is consistent with the mass decreasing linearly with time. The equation of motion is ẋ = ṙ v̇ = v g(r) + u (2.5) with the initial state x(to) specified.
- 36. 18 Primer Vector Theory and Applications In Equation (2.5), g(r) is the gravitational acceleration and u represents a unit vector in the thrust direction. An example gravitational field is the inverse- square field: g(r) = − μ r2 r r = − μ r3 r. (2.6) The first-order necessary conditions for an optimal CSI trajectory were first derived by Lawden [2] using classical calculus of variations. In the derivation that follows, an optimal control theory formulation is used, but the derivation is similar to that of Lawden. One difference is that the mass is not considered a state variable but is kept track of separately. In order to minimize the cost in Equation (2.1), one forms the Hamiltonian using Equation (2.5) as H = + λT r v + λT v [g(r) + u]. (2.7) The adjoint equations are then λ̇ T r = − ∂H ∂r = −λT v G(r) (2.8) λ̇ T v = − ∂H ∂v = −λT r (2.9) where G(r) ≡ ∂g(r) ∂r (2.10) is the symmetric 3 × 3 gravity gradient matrix. For terminal constraints of the form ψ[r(tf ), v(tf ), tf ] = 0, (2.11) which may describe an orbital intercept, rendezvous, etc., the boundary conditions on Equations (2.8–2.9) are given in terms of ≡ vT ψ[r(tf ), v(tf ), tf] (2.12) as λT r (tf ) = ∂ ∂r(tf ) = vT ∂ψ ∂r(tf ) (2.13) λT v (tf ) = ∂ ∂v(tf ) = vT ∂ψ ∂v(tf ) . (2.14) There are two control variables, the thrust direction u and the thrust acceleration magnitude , that must be chosen to satisfy the minimum principle [1], that is, to min- imize the instantaneous value of the Hamiltonian H. By inspection, the Hamiltonian of Equation (2.7) is minimized over the choice of thrust direction by aligning the unit
- 37. 2.2 First-Order Necessary Conditions 19 vector u(t) opposite to the adjoint vector λv(t). Because of the significance of the vector −λv(t), Lawden [2] termed it the primer vector p(t): p(t) ≡ −λv(t). (2.15) The optimal thrust unit vector is then in the direction of the primer vector, specifically u(t) = p(t) p(t) (2.16) and λT v u = −λv = −p (2.17) in the Hamiltonian of Equation (2.7). From Equations (2.9) and (2.15), it is evident that λr(t) = ṗ(t). (2.18) Equations (2.8), (2.9), (2.15), and (2.18) combine to yield the primer vector equation p̈ = G(r)p. (2.19) The boundary conditions on the solution to Equation (2.19) are obtained from Equations (2.13) (2.14) p(tf ) = −vT ∂ψ ∂v(tf ) (2.20) ṗ(tf ) = vT ∂ψ ∂r(tf ) . (2.21) Note that in Equation (2.20), the final value of the primer vector for an optimal intercept is the zero vector, because the terminal constraint ψ does not depend on v(tf ). Using Equations (2.15)–(2.18), the Hamiltonian of Equation (2.7) can be rewritten as H = −(p − 1) + ṗT v − pT g. (2.22) To minimize the Hamiltonian over the choice of the thrust acceleration magnitude , one notes that the Hamiltonian is a linear function of , and thus the minimizing value for 0 ≤ ≤ max will depend on the algebraic sign of the coefficient of in Equation (2.22). It is convenient to define the switching function S(t) ≡ p − 1. (2.23) The choice of the thrust acceleration magnitude that minimizes H is then given by the “bang-bang” control law = max for S 0 (p 1) 0 for S 0 (p 1) . (2.24)
- 38. 20 Primer Vector Theory and Applications S (t ) MT MT MT NT NT 0 t Γmax Figure 2.1. Three-burn CSI switching function and thrust profile. That is, the thrust magnitude switches between its limiting values of 0 (an NT null- thrust arc) and Tmax (an MT maximum-thrust arc) each time S(t) passes through 0 [p(t) passes through 1] according to Equation (2.24). Figure 2.1 shows an example switching function for a three-burn trajectory. The possibility also exists that S(t) ≡ 0 [p(t) ≡ 1] on an interval of finite duration. From Equation (2.22), it is evident that in this case the thrust acceleration magnitude is not determined by the minimum principle and may take on intermediate values between 0 and max. This IT “intermediate thrust arc” [2] is referred to as a singular arc in optimal control [1]. Lawden explained the origin of the term primer vector in a personal letter in 1990: “In regard to the term ‘primer vector’, you are quite correct in your supposition. I served in the artillery during the war [World War II] and became familiar with the initiation of the burning of cordite by means of a primer charge. Thus, p = 1 is the signal for the rocket motor to be ignited.” It follows then from Equation (2.3) that if T = Tmax and the engine is on for a total of t time units, max(t) = eF(t)/c Tmax/mo = Tmax/(mo − bmaxt). (2.25) Other necessary conditions are that the variables p and ṗ must be continuous everywhere. Equation (2.23) then indicates that the switching function S(t) is also continuous everywhere. Even though the gravitational field is time-invariant, the Hamiltonian in this formulation does not provide a first integral (constant of the motion) on an MT arc, because is an explicit function of time as shown in Equation (2.25). From Equation (2.22) H = −S + ṗT v − pT g. (2.26) Note that the Hamiltonian is continuous everywhere because S = 0 at the discontinuities in the thrust acceleration magnitude.
- 39. 2.2 First-Order Necessary Conditions 21 2.2.2 Optimal Impulsive Trajectory For a high-thrust CSI engine the thrust durations are very small compared with the times between thrusts. Because of this, one can approximate each MT arc as an impulse (Dirac delta function) having unbounded magnitude (max → ∞) and zero duration. The primer vector then determines both the optimal times and directions of the thrust impulses with p ≤ 1 corresponding to S ≤ 0. The impulses can occur only at those instants at which S = 0 (p = 1). These impulses are separated by NT arcs along which S 0 (p 1). At the impulse times the primer vector is then a unit vector in the optimal thrust direction. The necessary conditions (NC) for an optimal impulsive trajectory, first derived by Lawden [2], are shown in Table 2.1. For a linear system, these NC are also sufficient conditions for an optimal tra- jectory [5]. Also in [5], an upper bound on the number of impulses required for an optimal solution is given. Figure 2.2 shows a trajectory (at top) and a primer vector magnitude (at bottom) for an optimal three-impulse solution. (In all of the trajectory plots in this chapter, the direction of orbital motion is counterclockwise.) Canonical units are used. The canonical time unit is the orbital period of the circular orbit that has a radius of one canonical distance unit. The initial orbit is a unit radius circular orbit, shown as the topmost orbit going counterclockwise from the symbol ⊕ at (1,0) to (−1,0). The transfer time is 0.5 original (initial) orbit periods (OOP). The target is in a coplanar circular orbit of radius 2, with an initial lead angle (ila) of 270◦ and shown by the symbol at (0,−2). The spacecraft departs and intercepts at approximately (1.8,−0.8) as shown. The + signs at the initial and final points indicate thrust impulses and the + sign on the transfer orbit very near (0,0) indicates the location of the midcourse impulse. The magnitudes of the three Vs are shown at the left, with the total V equal to 1.3681 in units of circular orbit speed in the initial orbit. The examples shown in this chapter are coplanar, but the theory and applica- tions apply to three-dimensional trajectories as well, for example, see Prussing and Chiu [6]. The bottom graph in Figure 2.2 displays the time history of the primer vec- tor magnitude. Note that it satisfies the necessary conditions of Table 2.1 for an optimal transfer. Table 2.1. Impulsive necessary conditions 1. The primer vector and its first derivative are continuous everywhere. 2. The magnitude of the primer vector satisfies p(t) ≤ 1 with the impulses occurring at those instants at which p = 1. 3. At the impulse times the primer vector is a unit vector in the optimal thrust direction. 4. As a consequence of the above conditions, dp/dt = ṗ = ṗT p = 0 at an intermediate impulse (not at the initial or final time).
- 40. 22 Primer Vector Theory and Applications −4 −3 −2 −1 0 1 2 3 4 −2 −1.5 −1 −0.5 0 0.5 0.5585 0.14239 0.66717 ila = 270° Canonical Distance Canonical Distance tf = 0.5 OOP, ΔV = 1.3681 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.2 0.4 0.6 0.8 1 1.2 Time [OOP] Primer Magnitude Figure 2.2. Optimal three-impulse trajectory and primer magnitude. Note also that at a thrust impulse at time tk (t) = vkδ(t − tk) (2.27) and from Equation (2.4) vk = t+ k t− k (t)dt = F(t+ k ) − F(t− k ) (2.28) where t+ k and t− k are times immediately after and before the impulse time, respec- tively. Equation (2.3) then becomes the familiar solution to the rocket equation: m(t+ k ) = m(t− k )e−vk/c . (2.29) 2.2.3 Optimal Variable-Specific-Impulse Trajectory A variable-specific-impulse (VSI) engine is also known as a power-limited (PL) engine, because the power source is separate from the engine itself, for example, solar panels, and radioisotope thermoelectric generator. The power delivered to the engine is bounded between 0 and a maximum value Pmax, with the optimal value being constant and equal to the maximum. The cost functional representing minimum propellant consumption for the VSI case is J = 1 2 tf to 2 (t)dt. (2.30)
- 41. 2.3 Solution to the Primer Vector Equation 23 Writing 2 as T , the corresponding Hamiltonian function can be written as H = 1 2 T + λT r v + λT v [g(r) + ]. (2.31) For the VSI case, there is no need to consider the thrust acceleration magnitude and direction separately, so the vector is used in place of the term u that appears in Equation (2.7). Because H is a nonlinear function of , the minimum principle is applied by setting ∂H ∂ = T + λT v = 0T (2.32) or (t) = −λv(t) = p(t) (2.33) using the definition of the primer vector in Equation (2.15). Thus for a VSI engine, the optimal thrust acceleration vector is equal to the primer vector: (t) = p(t). Because of this, Equation (2.5), written as r̈ = g(r) + , can be combined with Equation (2.19), as in [7] to yield a fourth-order differential equation in r: riv − Ġṙ + G(g − 2r̈) = 0. (2.34) Every solution to Equation (2.34) is an optimal VSI trajectory through the gravity field g(r). But desired boundary conditions, such as specified position and velocity vectors at the initial and final times, must be satisfied. Note also that from Equation (2.32) ∂2H ∂2 = ∂ ∂ ∂H ∂ T = I3 (2.35) where I3 is the 3 × 3 identity matrix. Equation (2.35) shows that the (Hessian) matrix of second partial derivatives is positive definite, verifying that H is minimized. Because the VSI thrust acceleration of Equation (2.33) is continuous, a recently developed procedure [8] to test whether second-order NC and SC are satisfied can be applied. Equation (2.35) shows that an NC for minimum cost (Hessian matrix posi- tive semidefinite) and part of the SC (Hessian matrix positive definite) are satisfied. The other condition that is both an NC and an SC is the Jacobi no-conjugate-point condition. Reference [8] details the recently developed test for that. 2.3 Solution to the Primer Vector Equation The primer vector equation, Equation (2.19), can be written in first-order form as the linear system d dt p ṗ = O3 I3 G O3 p ṗ (2.36) where O3 is the 3 × 3 zero matrix.
- 42. 24 Primer Vector Theory and Applications Equation (2.36) is of the form ẏ = A(t)y, and its solution can be written in terms of a transition matrix (t, to) as y(t) = (t, to)y(to) (2.37) for a specified initial condition y(to). Glandorf [9] presents a form of the transition matrix for an inverse-square grav- itational field. [In that Technical Note, the missing Equation (2.33) is (t, to) = P(t)P−1(to).] Note that on an NT (no-thrust or coast) arc, the variational (linearized) state equation is, from Equation (2.5), δẋ = δṙ δv̇ = O3 I3 G O3 δr δv , (2.38) which is the same as Equation (2.36). So the transition matrix in Equation (2.37) is also the transition matrix for the state variation, that is, the state transition matrix [10]. This state transition matrix has the usual properties from linear system theory and is also symplectic [10], which has the useful property that −1 (t, to) = −JT (t, to)J (2.39) where J = O3 I3 −I3 O3 . (2.40) Note that J2 = −I6, indicating that J is a matrix analog of the imaginary number i. Equation (2.39) is useful when the state transition matrix is determined numer- ically because the inverse matrix −1(t, to) = (to, t) can be computed without explicitly inverting a 6 × 6 matrix. 2.4 Application of Primer Vector Theory to an Optimal Impulsive Trajectory If the primer vector evaluated along an impulsive trajectory fails to satisfy the nec- essary conditions of Table 2.1 for an optimal solution, the way in which the NC are violated provides information that can lead to a solution that does satisfy the NC. This process was first derived by Lion and Handelsman [3]. For given boundary con- ditions and a fixed transfer time, an impulsive trajectory can be modified either by allowing a terminal coast or by adding a midcourse impulse. A terminal coast can be either an initial coast, in which the first impulse occurs after the initial time, or a final coast, in which the final impulse occurs before the final time. In the former case, the spacecraft coasts along the initial orbit after the initial time until the first impulse
- 43. 2.4 Application of Primer Vector Theory 25 F ro rf Initial Orbit Fnal Orbit vo vo vf + _ Δvo Δvf vf _ + Transfer Orbit to tf Figure 2.3. A fixed-time impulsive rendezvous trajectory. occurs. In the latter case, the rendezvous actually occurs before the final time, and the spacecraft coasts along the final orbit until the final time is reached. To determine when a terminal coast will result in a trajectory that has a lower fuel cost, consider the two-impulse fixed-time rendezvous trajectory shown in Figure 2.3. In the two-body problem, if the terminal radii ro and rf are specified along with the transfer time τ ≡ tf − to, the solution to Lambert’s Problem [10] [11] provides the terminal velocity vectors v+ o (after the initial impulse) and v− f (before the final impulse) on the transfer orbit. Because the velocity vectors are known on the initial orbit (v− o before the first impulse) and on the final orbit (v+ f after the final impulse), the required velocity changes can be determined as vo = v+ o − v− o (2.41) and vf = v+ f − v− f . (2.42) Once the vector velocity changes are known, the primer vector can be evaluated along the trajectory to determine if the NC are satisfied. In order to satisfy the NC that on an optimal trajectory the primer vector at an impulse time is a unit vector in the direction of the impulse, one imposes the following boundary conditions on the primer vector p(to) ≡ po = vo vo (2.43) p(tf ) ≡ pf = vf vf . (2.44)
- 44. 26 Primer Vector Theory and Applications The primer vector can then be evaluated along the transfer orbit using the 6 × 6 transition matrix solution of Equation (2.37) p(t) ṗ(t) = (t, to) p(to) ṗ(to) (2.45) where the 3 × 3 partitions of the 6 × 6 transition matrix are designated as (t, to) ≡ M(t, to) N(t, to) S(t, to) T(t, to) . (2.46) Equation (2.45) can then be evaluated for the fixed terminal times to and tf to yield pf = Mfopo + Nfoṗo (2.47) and ṗf = Sfopo + Tfoṗo (2.48) where the abbreviated notation is used that pf ≡ p(tf ), Mfo ≡ M(tf , to), and so on. Equation (2.47) can be solved for the initial primer vector rate ṗo = N−1 fo [pf − Mfopo] (2.49) where the inverse matrix N−1 fo exists except for isolated values of τ = tf −to. With both the primer vector and the primer vector rate known at the initial time, the primer vector along the transfer orbit for to ≤ t ≤ tf can be calculated as using Equations (2.43–2.46, 2.49) as p(t) = NtoN−1 fo vf vf + [Mto − NtoN−1 fo Mfo] vo vo . (2.50) 2.4.1 Criterion for a Terminal Coast One of the options available to modify a two-impulse solution that does not satisfy the NC for an optimal transfer is to include a terminal coast period in the form of either an initial coast, a final coast, or both. To do this, one allows the possibility that the initial impulse occurs at time to +dto due to a coast in the initial orbit of duration dto 0 and that the final impulse occurs at a time tf +dtf . In the case of a final coast, dtf 0 in order that the final impulse occur prior to the nominal final time, allowing a coast in the final orbit until the nominal final time. A negative value of dto or a positive value of dtf also has a physical interpretation as will be seen. To determine whether a terminal coast will lower the cost of the trajectory, an expression for the difference in cost between the perturbed trajectory (with the terminal coasts) and the nominal trajectory (without the coasts) must be derived. The
- 45. 2.4 Application of Primer Vector Theory 27 discussion that follows summarizes and interprets results by Lion and Handelsman [3]. The cost on the nominal trajectory is simply J = vo + vf (2.51) for the two-impulse solution. In order to determine the differential change in the cost due to the differential coast periods the concept of a noncontemporaneous, or “skew” variation is needed. This variation combines two effects: the variation due to being on a perturbed trajectory and the variation due to a difference in the time of the impulse. The variable d will be used to denote a noncontemporaneous variation in contrast to the variable δ that represents a contemporaneous variation, as in Equation (2.38). The rule for relating the two types of variations is given by dx(to) = δx(to) + ẋ∗ odto (2.52) where ẋ∗ o is the derivative on the nominal (unperturbed) trajectory at the nominal final time and the variation in the initial state has been used as an example. Next, the noncontemporaneous variation in the cost must be determined. Because the coast periods result in changes in the vector velocity changes, the variation in the cost can be expressed, from Equation (2.51) as dJ = ∂vo ∂vo dvo + ∂vf ∂vf dvf . (2.53) Using the fact that for any vector a having magnitude a ∂a ∂a = aT a (2.54) the variation in the cost in Equation (2.53) can be expressed as dJ = vT o vo dvo + vT f vf dvf . (2.55) Finally, Equation (2.55) can be rewritten in terms of the initial and final primer vector using the conditions of Equations (2.43–2.44) as dJ = pT o dvo + pT f dvf . (2.56) The analysis in [3] leads to the result that dJ = −ṗT o vodto − ṗT f vf dtf (2.57) The final form of the expression for the variation in cost is obtained by expressing the vector velocity changes in terms of the primer vector using Equations (2.43–2.44) as dJ = −voṗT o podto − vf ṗT f pf dtf . (2.58)
- 46. 28 Primer Vector Theory and Applications In Equation (2.58), one can identify the gradients of the cost with respect to the terminal impulse times to and tf as ∂J ∂to = −voṗT o po (2.59) and ∂J ∂tf = −vf ṗT f pf . (2.60) One notes that the dot products in Equations (2.59–2.60) are simply the slopes of the primer magnitude time history at the terminal times, due to the fact that p2 = pT p and, after differentiation with respect to time, 2pṗ = 2ṗT p. Because p = 1 at the impulse times, ṗT p = ṗ. (2.61) The criteria for adding an initial or final coast can now be summarized by examining the algebraic signs of the gradients in Equations (2.59–2.60): If ṗo 0, an initial coast (represented by dto 0) will lower the cost. Similarly, if ṗf 0, a final coast (represented by dtf 0) will lower the cost. It is worth noting that, conversely, if ṗo ≤ 0, an initial coast will not lower the cost. This is consistent with the NC for an optimal solution and represents an alternate proof of the NC that p ≤ 1 on an optimal solution. Similarly, if ṗf ≥ 0, a final coast will not lower the cost. However, one can interpret these results even further. If ṗo 0, a value of dto 0 yields dJ 0, indicating that an earlier initial impulse time would lower the cost. This is the opposite of an initial coast and simply means that the cost can be lowered by increasing the transfer time by starting the transfer earlier. Similarly, a value of ṗf 0 implies that a dtf 0 will yield dJ 0. In this case, the cost can be lowered by increasing the transfer time by increasing the final time. From these observations, one can conclude that for a time-open optimal solution, such as the Hohmann transfer, the slopes of the primer magnitude time history must be zero at the terminal times, indicating that no improvement in the cost can be made by slightly increasing or decreasing the times of the terminal impulses. Figure 2.4 shows the primer time history for a Hohmann transfer rendezvous trajectory. An initial coast of 0.889 OOP is required to obtain the correct phase angle of the target body for the given ila and there is no final coast. Figure 2.5 shows an example of a primer history that violates the NC in a manner indicating that an initial coast or final coast or both will lower the cost. The final radius is 1.6, the ila is 90◦, and the transfer time is 0.9 OOP. In this case, the choice is made to add an initial coast, and the gradient of Equation (2.59) is used in a nonlinear programming (NLP) algorithm to iterate on the time of the first impulse. This is a one-dimensional search in which small changes in the time of the first impulse are made using the gradient of Equation (2.59) until the gradient is driven to zero. On each iteration, new values for the terminal velocity
- 47. 2.4 Application of Primer Vector Theory 29 −4 −2 0 2 4 6 −2 −1 0 1 β0 = 270° Canonical Distance Canonical Distance Hohmann Trajectory: ΔVH = 0.28446 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.8 0.85 0.9 0.95 1 Time (OOP) Primer Magnitude tfH = 1.8077 OOP twait = 0.88911 OOP tellipse = 0.91856 OOP Figure 2.4. Hohmann transfer orbit and primer magnitude. −5 −4 −3 −2 −1 0 1 2 3 4 −1.5 −1 −0.5 0 0.5 1 1.5 0.24379 0.13087 ila = 90° Canonical Distance Canonical Distance tf = 0.9 OOP, ΔV = 0.37466 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.5 2 2.5 3 3.5 Time (OOP) Primer Magnitude Figure 2.5. Primer magnitude indicating initial/final coast.
- 48. 30 Primer Vector Theory and Applications −5 −4 −3 −2 −1 0 1 2 3 4 −1.5 −1 −0.5 0 0.5 1 1.5 0.11059 0.104 ila = 90° Canonical Distance Canonical Distance tf = 0.9 OOP, ΔV = 0.21459 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Time (OOP) Primer Magnitude Figure 2.6. Optimal initial coast trajectory and primer magnitude. changes are calculated by re-solving Lambert’s Problem and a new primer vector solution is obtained. Note that once the iteration begins, the time of the first impulse is no longer to, but a later value denoted by t1. In a similar way, if the final impulse time becomes an iteration variable, it is denoted by tn where the last impulse is con- sidered to be the nth impulse. For a two-impulse trajectory, n = 2, but as will be seen shortly, optimal solutions can require more than two impulses. When the times of the first and last impulse become iteration variables, in all the formulas in the preceding analysis, the subscript o is replaced by 1 everywhere and f is replaced by n. Figure 2.6 shows the converged result of an iteration on the time of the initial impulse. Note that the necessary condition p ≤ 1 is satisfied and the gradient of the cost with respect to t1, the time of the first impulse (at approximately t1 = 0.22), is zero because ṗ1 = 0, making the gradient of Equation (2.59) equal to zero. This simply means that a small change in t1 will cause no change in the cost, that is, the cost has achieved a stationary value and satisfies the first-order necessary conditions. Com- paring Figures 2.5 and 2.6, one notes that the cost has decreased significantly from 0.37466 to 0.21459, and that an initial coast is required but no final coast is required. 2.4.2 Criterion for Addition of a Midcourse Impulse Besides terminal coasts, the addition of one or more midcourse impulses is another potential way of lowering the cost of an impulsive trajectory. The addition of an
- 49. 2.4 Application of Primer Vector Theory 31 impulse is more complicated than including terminal coasts because, in the general case, four new parameters are introduced: three components of the position of the impulse and the time of the impulse. One must first derive a criterion that indicates that the addition of an impulse will lower the cost and then determine where in space and when in time the impulse should occur. The where and when will be done in two steps. The first step is to determine initial values of position and time of the added impulse that will lower the cost. The second step is to iterate on the values of position and time using gradients that will be developed, until a minimum of the cost is achieved. Note that this procedure is more complicated than for terminal coasts, because the starting value of the coast time for the iteration was simply taken to be zero, that is, no coast. When considering the addition of a midcourse impulse, let us assume dto = dtf = 0, that is, there are no terminal coasts. Because we are doing a first-order perturbation analysis, superposition applies and we can combine the previous results for terminal coasts easily with our new results for a midcourse impulse. Also, we will discuss the case of adding a third impulse to a two-impulse trajectory, but the same theory applies to the case of adding a midcourse impulse to any two-impulse segment of an n-impulse trajectory. The cost on the nominal, two-impulse trajectory is given by Equation (2.50) J = vo + vf . The variation in the cost due to adding an impulse is given by adding the midcourse velocity change magnitude vm to Equation (2.56) dJ = pT o dvo + vm + pT f dvf . (2.62) The analysis in [3] results in dJ = vm 1 − pT m vm vm . (2.63) In Equation (2.63), the expression for dJ involves a dot product between the primer vector and a unit vector. If the numerical value of this dot product is greater than one, dJ 0 and the perturbed trajectory has a lower cost than the nominal trajectory. In order for the value of the dot product to be greater than one, it is necessary that pm 1. Here again we have an alternative derivation of the necessary condition that p ≤ 1 on an optimal trajectory. We also have the criterion that tells us when the addition of a midcourse impulse will lower the cost. If the value of p(t) exceeds unity along the trajectory, the addition of a midcourse impulse at a time for which p 1 will lower the cost. Figure 2.7 shows a primer magnitude time history that indicates the need for a mid- course impulse (but not for a terminal coast). The final radius is 2, the ila is 270◦, and the transfer time is relatively small, equal to 0.5 OOP. The first step is to determine initial values for the position and time of the mid- course impulse. From Equation (2.63) it is evident that for a given pm, the largest
- 50. 32 Primer Vector Theory and Applications −4 −3 −2 −1 0 1 2 3 4 −2 −1.5 −1 −0.5 0 0.5 1.1758 0.57976 ila = 270° Canonical Distance Canonical Distance tf = 0.5 OOP, ΔV = 1.7555 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 1 2 3 4 5 6 Time (OOP) Primer Magnitude Figure 2.7. Primer magnitude indicating a need for a midcourse impulse. decrease in the cost is obtained by maximizing the value of the dot product, that is, by choosing a position for the impulse that causes vm to be parallel to the vector pm and by choosing the time tm to be the time at which the primer magnitude has a maximum value. Choosing the position of the impulse so that the velocity change is in the direction of the primer vector sounds familiar because it is one of the nec- essary conditions derived previously, but how to determine this position is not at all obvious, and we will have to derive an expression for this. Choosing the time tm to be the time of maximum primer magnitude does not guarantee that the decrease in cost is maximized, because the value of vm in the expression for dJ depends on the value of tm. However, all we are doing is obtaining an initial position and time of the midcourse impulse to begin an iteration process. As long as our initial choice repre- sents a decrease in the cost, we will opt for the simple device of choosing the time of maximum primer magnitude as our initial estimate of tm. In Figure 2.7, tm is 0.1. Having determined an initial value for tm, the initial position of the impulse, namely the value δrm to be added to rm, must also be determined. Obviously δrm must be nonzero, otherwise the midcourse impulse would have zero magnitude. The property that must be satisfied in determining δrm is that vm be parallel to pm. The analysis of [3] results in vm = Aδrm (2.64)
- 51. 2.4 Application of Primer Vector Theory 33 where the matrix A is defined as A ≡ −(MT fmN−T fm + TmoN−1 mo). (2.65) Next, in order to have vm parallel to pm, it is necessary that vm = εpm with scalar ε 0. Combining this fact with Equation (2.64) yields Aδrm = vm = εpm (2.66) which yields the solution for δrm as δrm = εA−1 pm (2.67) assuming A is invertible. The question then arises how to select a value for the scalar ε. Obviously too large a value will violate the linearity assumptions of the perturbation analysis. This is not addressed in [3], but one can maintain a small change by specifying δrm rm = β (2.68) where β is a specified small positive number such as 0.05. Equation (2.67) then yields a value for ε ε A−1pm rm = β ⇒ ε = βrm A−1pm . (2.69) If the resulting dJ ≥ 0, then decrease ε and repeat Equation (2.67). One should never accept a midcourse impulse position that does not decrease the cost, because a sufficiently small ε will always provide a lower cost. The initial values of midcourse impulse position and time are now determined. One adds the δrm of Equation (2.67) to the value of rm on the nominal trajectory at the time tm at which the primer magnitude achieves its maximum value (greater than one). The primer history after the addition of the initial midcourse impulse is shown in Figure 2.8. Note that pm = 1 but ṗm is discontinuous and the primer magnitude exceeds unity, both of which violate the NC. However, the addition of the midcourse impulse has decreased the cost slightly, from 1.7555 to 1.7549. 2.4.3 Iteration on a Midcourse Impulse Position and Time To determine how to efficiently iterate on the components of position of the mid- course impulse and its time, one needs to derive expressions for the gradients of the cost with respect to these variables. To do this, one must compare the three-impulse trajectory (or three-impulse segment of an n-impulse trajectory) that resulted from the addition of the midcourse impulse with a perturbed three-impulse trajectory.
- 52. 34 Primer Vector Theory and Applications −4 −3 −2 −1 0 1 2 3 4 −2 −1.5 −1 −0.5 0 0.5 1.1749 0.00026491 0.57973 ila = 270° Canonical Distance Canonical Distance tf = 0.5 OOP, ΔV = 1.7549 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.5 1 1.5 2 2.5 3 Time (OOP) Primer Magnitude Figure 2.8. Initial (nonoptimal) three-impulse primer magnitude. Note that, unlike a terminal coast, the values of drm and dtm are independent. (By contrast, on an initial coast dro = v− o dto and on a final coast drf = v+ f dtf .) The cost on the nominal three-impulse trajectory is J = vo + vm + vf (2.70) and the variation in the cost due to perturbing the midcourse time and position is dJ = ∂vo ∂vo dvo + ∂vm ∂vm dvm + ∂vf ∂vf dvf (2.71) which, analogous to Equation (2.56), can be written as dJ = pT o dvo + pT mdvm + pT f dvf . (2.72) The analysis of [3] leads to the result that dJ = ṗ+ m − ṗ− m T drm − ṗT+ m v+ m − ṗT− m v− m dtm. (2.73)
- 53. 2.4 Application of Primer Vector Theory 35 In Equation (2.73), a discontinuity in ṗm has been allowed because there is no guar- antee that it will be continuous at the inserted midcourse impulse, as demonstrated in Figure 2.8. Equation (2.73) can be written more simply in terms of the Hamiltonian function Equation (2.22) for pm = 1: Hm = ṗT mvm − pT mgm (for which the second term pT mgm is continuous because pm = vm/vm) and gm(rm) are continuous). dJ = (ṗ+ m − ṗ− m)T drm − (H+ m − H− m)dtm. (2.74) Equation (2.74) provides the gradients of the cost with respect to the independent variations in the position and time of the midcourse impulse for use in a nonlinear programming algorithm: ∂J ∂rm = (ṗ+ m − ṗ− m) (2.75) and ∂J ∂tm = −(H+ m − H− m). (2.76) AsasolutionsatisfyingtheNCisapproached, thegradientstendtozero, inwhichcase both the primer rate vector ṗm and the Hamiltonian function Hm become continuous at the midcourse impulse. Note that when the NC are satisfied, the gradient with respect to tm in Equation (2.76) being zero implies that H+ m − H− m = 0 = ṗT m(v+ m − v− m) = ṗT mvm = vmṗT mpm = 0 (2.77) which, in turn, implies that ṗm = 0, indicating that the primer magnitude attains a local maximum value of unity. This is consistent with the NC that p ≤ 1 and that ṗ be continuous. Figure 2.2 shows the converged, optimal three-impulse trajectory that results from improving the primer histories shown in Figures 2.7 and 2.8. Note that the final cost of 1.3681 is significantly less that the value of 1.7555 prior to adding the midcourse impulse. Also, the time of the midcourse impulse changed during the iteration from its initial value of 0.1 to a final value of approximately 0.17. The absolute minimum cost solution for the final radius and ila value of Figure 2.2 is, of course, the Hohmann transfer shown in Figure 2.4. Its cost is significantly less at 0.28446, but the transfer time is nearly three times as long at 1.8077 OOP. Of this, 0.889 OOP is an initial coast to achieve the correct target phase angle for the Hohmann transfer. Depending on the specific application, the total time required may be unacceptably long. (As a side note, a simple proof of the global optimality of the Hohmann transfer using ordinary calculus rather than primer vector theory is given in [12].)
- 54. 36 Primer Vector Theory and Applications R E F E R E N C E S [1] Bryson, A. E., and Ho, Y-C. (1975) Applied Optimal Control, Hemisphere Publishing Co., Washington DC. [2] Lawden, D. F. (1963) Optimal Trajectories for Space Navigation, Butterworths, London. [3] Lion, P. M., and Handelsman, M. (1968) Primer Vector on Fixed-Time Impulsive Trajectories. AIAA Journal, 6, No. 1, 127–132. [4] Jezewski, D. J., and Rozendaal, H. L. (1968) An Efficient Method for Calculating Optimal Free-Space n-impulse Trajectories. AIAA Journal, 6, No. 11, 2160–2165. [5] Prussing, J. E. (1995) Optimal Impulsive Linear Systems: Sufficient Conditions and Maximum Number of Impulses, The Journal of the Astronautical Sciences, 43, No. 2, 195–206. [6] Prussing, J. E., and Chiu, J-H. (1986) Optimal Multiple-Impulse Time-Fixed Rendezvous between Circular Orbits, Journal of Guidance, Control, and Dynamics, 9, No. 1, 17–22. also Errata, 9, No. 2, 255. [7] Prussing, J. E. (1993) Equation for Optimal Power-Limited Spacecraft Trajectories, Journal of Guidance, Control, and Dynamics, 16, No. 2, 391–393. [8] Prussing, J. E., and Sandrik, S. L. (2005) Second-Order Necessary Conditions and Suf- ficient Conditions Applied to Continuous-Thrust Trajectories, Journal of Guidance, Control, and Dynamics, 28, No. 4, 812–816. [9] Glandorf, D. R. (1969) Lagrange Multipliers and the State Transition Matrix for Coasting Arcs, AIAA Journal, 7, Vol. 2, 363–365. [10] Battin, R. H. (1999) An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition, AIAA Education Series, New York. [11] Prussing, J. E., and Conway, B. A. (1993) Orbital Mechanics, Oxford University Press, New York. [12] Prussing, J. E. (1992) Simple Proof of the Global Optimality of the Hohmann Transfer, Journal of Guidance, Control, and Dynamics, 15, No. 4, 1037–1038. [13] Sandrik, S. (2006) Primer-Optimized Results and Trends for Circular Phasing and Other Circle-to-Circle Impulsive Coplanar Rendezvous. Ph.D. Thesis, University of Illinois at Urbana-Champaign.
- 55. Other documents randomly have different content
- 56. the development of the French Revolution up to the spring of 1791, and though the excesses of the revolutionists compelled him a few years after to express his entire agreement with the opinions of Burke, its defence of the “rights of man” is a valuable statement of the cultured Whig’s point of view at the time. The History of the Revolution in England, breaking off at the point where William of Orange is preparing to intervene in the affairs of England, is chiefly interesting because of Macaulay’s admiring essay on it and its author. A Life, by his son R. J. Mackintosh, was published in 1836. MACKLIN, CHARLES (c. 1699-1797), Irish actor and playwright, whose real name was McLaughlin, was born in Ireland, and had an adventurous youth before coming to Bristol, where he made his first appearance on the stage as Richmond in Richard III. He was at Lincoln’s Inn Fields about 1725, and by 1733 was at Drury Lane, where the quarrel between the manager and the principal actors resulted in his getting better parts. When the trouble was over and these were taken from him, he went to the Haymarket, but he returned in 1734 to Drury Lane and acted there almost continuously until 1748. Then for two seasons he and his wife (d. c. 1758), an excellent actress, were in Dublin under Sheridan, then back in London at Covent Garden. He played a great number of
- 57. characters, principally in comedy, although Shylock was his greatest part, and Iago and the Ghost in Hamlet were in his repertory. At the end of 1753 Macklin bade farewell to the stage to open a tavern, near the theatre, where he personally supervised the serving of dinner. He also delivered an evening lecture, followed by a debate, which was soon a hopeless subject of ridicule. The tavern failed, and Macklin returned to the stage, and played for a number of years in London and Dublin. His quick temper got him into constant trouble. In a foolish quarrel over a wig in 1735 he killed a fellow actor in the green-room at Drury Lane, and he was constantly at law over his various contracts and quarrels. The bitterest of these arose on account of his appearing as Macbeth at Covent Garden in 1772. The part was usually played there by William Smith, and the public would not brook a change. A few nights later the audience refused to hear Macklin as Shylock, and shouted their wish, in response to the manager’s question, to have him discharged. This was done in order to quell the riot. His lawsuit, well conducted by himself, against the leaders of the disturbance resulted in an award of £600 and costs, but Macklin magnanimously elected instead that the defendants should take £100 in tickets at three benefits—for himself, his daughter and the management. He returned to Covent Garden, but his appearances thereafter were less frequent, ending in 1789, when as Shylock, at his benefit, he was only able to begin the play, apologize for his wandering memory, and retire. He lived until the 11th of July 1797, and his last years were provided for by a subscription edition of two of his best plays, The Man of the World and Love in a Maze. Macklin’s daughter, Mary Macklin (c. 1734- 1781), was a well-known actress in her day. See Edward A. Parry, Charles Macklin (1891).
- 58. MACK VON LEIBERICH, KARL, Freiherr (1752-1828), Austrian soldier, was born at Nenslingen, in Bavaria, on the 25th of August 1752. In 1770 he joined an Austrian cavalry regiment, in which his uncle, Leiberich, was a squadron commander, becoming an officer seven years later. During the brief war of the Bavarian Succession he was selected for service on the staff of Count Kinsky, under whom, and subsequently under the commander-in-chief Field Marshal Count Lacy, he did excellent work. He was promoted first lieutenant in 1778, and captain on the quartermaster-general’s staff in 1783. Count Lacy, then the foremost soldier of the Austrian army, had the highest opinion of his young assistant. In 1785 Mack married Katherine Gabrieul, and was ennobled under the name of Mack von Leiberich. In the Turkish war he was employed on the headquarter staff, becoming in 1788 major and personal aide-de- camp to the emperor, and in 1789 lieutenant-colonel. He distinguished himself greatly in the storming of Belgrade. Shortly after this, disagreements between Mack and Loudon, now commander-in-chief, led to the former’s demanding a court-martial and leaving the front. He was, however, given a colonelcy (1789) and the order of Maria Theresa, and in 1790 Loudon and Mack, having become reconciled, were again on the field together. During these campaigns Mack received a severe injury to his head, from which he never fully recovered. In 1793 he was made
- 59. quartermaster-general (chief of staff) to Prince Josias of Saxe- Coburg, commanding in the Netherlands; and he enhanced his reputation by the ensuing campaign. The young Archduke Charles, who won his own first laurels in the action of the 1st of March 1793, wrote after the battle, “Above all we have to thank Colonel Mack for these successes.” Mack distinguished himself again on the field of Neerwinden; and had a leading part in the negotiations between Coburg and Dumouriez. He continued to serve as quartermaster- general, and was now made titular chief (Inhaber) of a cuirassier regiment. He received a wound at Famars, but in 1794 was once more engaged, having at last been made a major-general. But the failure of the allies, due though it was to political and military factors and ideas, over which Mack had no control, was ascribed to him, as their successes of March-April 1793 had been, and he fell into disfavour in consequence. In 1797 he was promoted lieutenant field marshal, and in the following year he accepted, at the personal request of the emperor, the command of the Neapolitan army. But with the unpromising material of his new command he could do nothing against the French revolutionary troops, and before long, being in actual danger of being murdered by his men, he took refuge in the French camp. He was promised a free pass to his own country, but Napoleon ordered that he should be sent to France as a prisoner of war. Two years later he escaped from Paris in disguise. The allegation that he broke his parole is false. He was not employed for some years, but in 1804, when the war party in the Austrian court needed a general to oppose the peace policy of the Archduke Charles, Mack was made quartermaster-general of the army, with instructions to prepare for a war with France. He did all that was possible within the available time to reform the army, and on the opening of the war of 1805 he was made quartermaster-
- 60. general to the titular commander-in-chief in Germany, the Archduke Ferdinand. He was the real responsible commander of the army which opposed Napoleon in Bavaria, but his position was ill-defined and his authority treated with slight respect by the other general officers. For the events of the Ulm campaign and an estimate of Mack’s responsibility for the disaster, see Napoleonic Campaigns. After Austerlitz, Mack was tried by a court-martial, sitting from February 1806 to June 1807, and sentenced to be deprived of his rank, his regiment, and the order of Maria Theresa, and to be imprisoned for two years. He was released in 1808, and in 1819, when the ultimate victory of the allies had obliterated the memory of earlier disasters, he was, at the request of Prince Schwarzenberg, reinstated in the army as lieutenant field marshal and a member of the order of Maria Theresa. He died on the 22nd of October 1828 at S. Pölten. See Schweigerd, Oesterreichs Helden (Vienna, 1854); Würzbach, Biogr. Lexikon d. Kaiserthums Oesterr. (Vienna, 1867); Ritter von Rittersberg, Biogr. d. ausgezeichneten Feldherren d. oest. Armee (Prague, 1828); Raumer’s Hist. Taschenbuch (1873) contains Mack’s vindication. A short critical memoir will be found in Streffleur for January 1907. McLANE, LOUIS (1786-1857), American political leader, was born in Smyrna, Delaware, on the 28th of May 1786, son of Allan
- 61. McLane (1746-1829), a well-known Revolutionary soldier. He was admitted to the bar in 1807. He entered politics as a Democrat, and served in the Federal House of Representatives in 1817-1827 and in the Senate in 1827-1829. He was minister to England in 1829-1831, and secretary of the treasury in Jackson’s cabinet from 1831 (when in his annual report he argued for the United States Bank) until May 1833, when he was transferred to the state department. He retired from the cabinet in June 1834. He was president of the Baltimore Ohio railway in 1837-1847, minister to England in 1845-1846, and delegate to the Maryland constitutional convention of 1850-1851. He died in Baltimore, Maryland, on the 7th of October 1857. His son, Robert Milligan McLane (1815-1898), graduated at West Point in 1837, resigned from the army in 1843, and practised law in Baltimore. He was a Democratic representative in Congress in 1847- 1851 and again in 1879-1883, governor of Maryland in 1884-1885, U.S. commissioner to China in 1853-1854, and minister to Mexico in 1859-1860 and to France in 1885-1889. See R. M. McLane’s Reminiscences, 1827-1897 (privately printed, 1897). MACLAREN, CHARLES (1782-1866), Scottish editor, was born at Ormiston, Haddingtonshire, on the 7th of October 1782, the son of a farmer and cattle-dealer. He was almost entirely self-
- 62. educated, and when a young man became a clerk in Edinburgh. In 1817, with others, he established the Scotsman newspaper in Edinburgh and at first acted as its editor. Offered a post as clerk in the custom house, he resigned his editorial position, resuming it in 1820, and resigning it again in 1845. In 1820 Maclaren was made editor of the sixth edition of the Encyclopaedia Britannica. From 1864-1866 he was president of the Geological Society of Edinburgh, in which city he died on the 10th of September 1866. MACLAREN, IAN, the pseudonym of John Watson (1850- 1907), Scottish author and divine. The son of John Watson, a civil servant, he was born at Manningtree, Essex, on the 3rd of November 1850, and was educated at Stirling and at Edinburgh University, afterwards studying theology at New College, Edinburgh, and at Tübingen. In 1874 he entered the ministry of the Free Church of Scotland and became assistant minister of Barclay Church, Edinburgh. Subsequently he was minister at Logiealmond in Perthshire and at Glasgow, and in 1880 he became minister of Sefton Park Presbyterian church, Liverpool, from which he retired in 1905. In 1896 he was Lyman Beecher lecturer at Yale University, and in 1900 he was moderator of the synod of the English Presbyterian church. While travelling in America he died at Mount Pleasant, Iowa, on the 6th of May 1907. Ian Maclaren’s first sketches of rural Scottish life, Beside the Bonnie Briar Bush (1894), achieved
- 63. extraordinary popularity and were followed by other successful books, The Days of Auld Lang Syne (1895), Kate Carnegie and those Ministers (1896) and Afterwards and other Stories (1898). Under his own name Watson published several volumes of sermons, among them being The Upper Room (1895); The Mind of the Master (1896) and The Potter’s Wheel (1897). See Sir W. Robertson Nicoll, Ian Maclaren (1908). MACLAURIN, COLIN (1698-1746), Scottish mathematician, was the son of a clergyman, and born at Kilmodan, Argyllshire. In 1709 he entered the university of Glasgow, where he exhibited a decided genius for mathematics, more especially for geometry; it is said that before the end of his sixteenth year he had discovered many of the theorems afterwards published in his Geometria organica. In 1717 he was elected professor of mathematics in Marischal College, Aberdeen, as the result of a competitive examination. Two years later he was admitted F.R.S. and made the acquaintance of Sir Isaac Newton. In 1719 he published his Geometria organica, sive descriptio linearum curvarum universalis. In it Maclaurin developed several theorems due to Newton, and introduced the method of generating conics which bears his name, and showed that many curves of the third and fourth degrees can be described by the intersection of two movable angles. In 1721 he
- 64. wrote a supplement to the Geometria organica, which he afterwards published, with extensions, in the Philosophical Transactions for 1735. This paper is principally based on the following general theorem, which is a remarkable extension of Pascal’s hexagram: “If a polygon move so that each of its sides passes through a fixed point, and if all its summits except one describe curves of the degrees m, n, p, c., respectively, then the free summit moves on a curve of the degree 2mnp... which reduces to mnp ... when the fixed points all lie on a right line.” In 1722 Maclaurin travelled as tutor and companion to the eldest son of Lord Polwarth, and after a short stay in Paris resided for some time in Lorraine, where he wrote an essay on the percussion of bodies, which obtained the prize of the French Academy of Sciences for the year 1724. The following year he was elected professor of mathematics in the university of Edinburgh on the urgent recommendation of Newton. After the death of Newton, in 1728, his nephew, John Conduitt, applied to Maclaurin for his assistance in publishing an account of Newton’s life and discoveries. This Maclaurin gladly undertook, but the death of Conduitt put a stop to the project. In 1740 Maclaurin divided with Leonhard Euler and Daniel Bernoulli the prize offered by the French Academy of Sciences for an essay on tides. His Treatise on Fluxions was published at Edinburgh in 1742, in two volumes. In the preface he states that the work was undertaken in consequence of the attack on the method of fluxions made by George Berkeley in 1734. Maclaurin’s object was to found the doctrine of fluxions on geometrical demonstration, and thus to answer all objections to its method as being founded on false reasoning and full of mystery. The most valuable part of the work is that devoted to physical applications, in which he embodied his essay on the tides. In this he showed that a homogeneous fluid
- 65. mass revolving uniformly round an axis under the action of gravity ought to assume the form of an ellipsoid of revolution. The importance of this investigation in connexion with the theory of the tides, the figure of the earth, and other kindred questions, has always caused it to be regarded as one of the great problems of mathematical physics. Maclaurin was the first to introduce into mechanics, in this discussion, the important conception of surfaces of level; namely, surfaces at each of whose points the total force acts in the normal direction. He also gave in his Fluxions, for the first time, the correct theory for distinguishing between maxima and minima in general, and pointed out the importance of the distinction in the theory of the multiple points of curves. In 1745, when the rebels were marching on Edinburgh, Maclaurin took a most prominent part in preparing trenches and barricades for its defence. The anxiety, fatigue and cold to which he was thus exposed, affecting a constitution naturally weak, laid the foundation of the disease to which he afterwards succumbed. As soon as the rebel army got possession of Edinburgh Maclaurin fled to England, to avoid making submission to the Pretender. He accepted the invitation of T. Herring, then archbishop of York, with whom he remained until it was safe to return to Edinburgh. He died of dropsy on the 14th of June 1746, at Edinburgh. Maclaurin was married in 1733 to Anne, daughter of Walter Stewart, solicitor-general for Scotland. His eldest son John, born in 1734, was distinguished as an advocate, and appointed one of the judges of the Scottish court of session, with the title of Lord Dreghorn. He inherited an attachment to scientific discovery, and was one of the founders of the Royal Society of Edinburgh, in 1782. After Maclaurin’s death his account of Newton’s philosophical discoveries was published by Patrick Murdoch, and also his
- 66. algebra in 1748. As an appendix to the latter appeared his De linearum geometricarum proprietatibus generalibus tractatus, a treatise of remarkable elegance. Of the more immediate successors of Newton in Great Britain Maclaurin is probably the only one who can be placed in competition with the great mathematicians of the continent of Europe at the time. (B. W.) M‘LENNAN, JOHN FERGUSON (1827-1881), Scottish ethnologist, was born at Inverness on the 14th of October 1827. He studied at King’s college, Aberdeen, where he graduated with distinction in 1849, thence proceeding to Cambridge, where he remained till 1855 without taking a degree. He was called to the Scottish bar in 1857, and in 1871 was appointed parliamentary draughtsman for Scotland. In 1865 he published Primitive Marriage, in which, arguing from the prevalence of the symbolical form of capture in the marriage ceremonies of primitive races, he developed an intelligible picture of the growth of the marriage relation and of systems of kinship (see Family) according to natural laws. In 1866 he wrote in the Fortnightly Review (April and May) an essay on “Kinship in Ancient Greece,” in which he proposed to test by early Greek facts the theory of the history of kinship set forth in Primitive Marriage; and three years later appeared a series of essays on “Totemism” in the same periodical for 1869-1870 (the germ of which had been contained in the paper just named), which mark the second great
- 67. step in his systematic study of early society. A reprint of Primitive Marriage, with “Kinship in Ancient Greece” and some other essays not previously published, appeared in 1876, under the title of Studies in Ancient History. The new essays in this volume were mostly critical, but one of them, in which perhaps his guessing talent is seen at its best, “The Divisions of the Irish Family,” is an elaborate discussion of a problem which has long puzzled both Celtic scholars and jurists; and in another, “On the Classificatory System of Relationship,” he propounded a new explanation of a series of facts which, he thought, might throw light upon the early history of society, at the same time putting to the test of those facts the theories he had set forth in Primitive Marriage. Papers on “The Levirate and Polyandry,” following up the line of his previous investigations (Fortnightly Review, 1877), were the last work he was able to publish. He died of consumption on the 14th of June 1881 at Hayes Common, Kent. Besides the works already cited, M‘Lennan wrote a Life of Thomas Drummond (1867). The vast materials which he had accumulated on kinship were edited by his widow and A. Platt, under the title Studies in Ancient History: Second Series (1896). MACLEOD, HENRY DUNNING (1821-1902), Scottish economist, was born in Edinburgh, and educated at Eton, Edinburgh
- 68. University, and Trinity College, Cambridge, where he graduated in 1843. He travelled in Europe, and in 1849 was called to the English bar. He was employed in Scotland on the work of poor-law reform, and devoted himself to the study of economics. In 1856 he published his Theory and Practice of Banking, in 1858 Elements of Political Economy, and in 1859 A Dictionary of Political Economy. In 1873 appeared his Principles of Economist Philosophy, and other books on economics and banking were published later. Between 1868 and 1870 he was employed by the government in digesting and codifying the law of bills of exchange. He died on the 16th of July 1902. Macleod’s principal contribution to the study of economics consists in his work on the theory of credit, to which he was the first to give due prominence. For a judicious discussion of the value of Macleod’s writings, see an article on “The Revolt against Orthodox Economics” in the Quarterly Review for October 1901 (No. 388). MACLEOD, NORMAN (1812-1872), Scottish divine, son of Rev. Norman Macleod (1783-1862), and grandson of Rev. Norman Macleod, minister of Morven, Argyllshire, was born at Campbeltown on the 3rd of June 1812. In 1827 he became a student at Glasgow University, and in 1831 went to Edinburgh to study divinity under Dr Thomas Chalmers. On the 18th of March 1838 he became parish
- 69. minister at Loudoun, Ayrshire. At this time the troubles in the Scottish Church were already gathering to a head (see Free Church of Scotland). Macleod, although he had no love for lay patronage, and wished the Church to be free to do its proper work, clung firmly to the idea of a national Established Church, and therefore remained in the Establishment when the disruption took place. He was one of those who took a middle course in the non-intrusion controversy, holding that the fitness of those who were presented to parishes should be judged by the presbyteries—the principle of Lord Aberdeen’s Bill. On the secession of 1843 he was offered many different parishes, and having finally settled at Dalkeith, devoted himself to parish work and to questions affecting the Church as a whole. He was largely instrumental in the work of strengthening the Church. In 1847 he became one of the founders of the Evangelical Alliance, and from 1849 edited the Christian Instructor (Edinburgh). In 1851 he was called to the Barony church, Glasgow, in which city the rest of his days were passed. There the more liberal theology rapidly made way among a people who judged it more by its fruits than its arguments, and Macleod won many adherents by his practical schemes for the social improvement of the people. He instituted temperance refreshment rooms, a congregational penny savings bank, and held services specially for the poor. In 1860 Macleod was appointed editor of the new monthly magazine Good Words. Under his control the magazine, which was mainly of a religious character, became widely popular. His own literary work, nearly all of which originally appeared in its pages—sermons, stories, travels, poems—was only a by-product of a busy life. By far his best work was the spontaneous and delightful Reminiscences of a Highland Parish (1867). While Good Words made his name known, and helped the cause he had so deeply at heart, his relations with
- 70. the queen and the royal family strengthened yet further his position in the country. Never since Principal Carstairs had any Scottish clergyman been on such terms with his sovereign. In 1865 he risked an encounter with Scottish Sabbatarian ideas. The presbytery of Glasgow issued a pastoral letter on the subject of Sunday trains and other infringements of the Sabbath. Macleod protested against the grounds on which its strictures were based. For a time, owing partly to a misleading report of his statement, he became “the man in all Scotland most profoundly distrusted.” But four years later the Church accorded him the highest honour in her power by choosing him as moderator of her general assembly. In 1867, along with Dr Archibald Watson, he was sent to India, to inquire into the state of the missions. He undertook the journey in spite of failing health, and seems never to have recovered from its effects. He returned resolved to devote the rest of his days to rousing the Church to her duty in the sphere of foreign missions, but his health was now broken, and his old energy flagged. He died on the 16th of June 1872, and was buried at Campsie. He was one of the greatest of Scottish religious leaders, a man of wide sympathy and high ideals. His Glasgow church was named after him the “Macleod Parish Church,” and the “Macleod Missionary Institute” was erected by the Barony church in Glasgow. Queen Victoria gave two memorial windows to Crathie church as a testimony of her admiration for his work. See Memoir of Norman Macleod, by his brother, Donald Macleod (1876).
- 71. MACLISE, DANIEL (1806-1870), Irish painter, was born at Cork, the son of a Highland soldier. His education was of the plainest kind, but he was eager for culture, fond of reading, and anxious to become an artist. His father, however, placed him, in 1820, in Newenham’s Bank, where he remained for two years, and then left to study in the Cork school of art. In 1825 it happened that Sir Walter Scott was travelling in Ireland, and young Maclise, having seen him in a bookseller’s shop, made a surreptitious sketch of the great man, which he afterwards lithographed. It was exceedingly popular, and the artist became celebrated enough to receive many commissions for portraits, which he executed, in pencil, with very careful treatment of detail and accessory. Various influential friends perceived the genius and promise of the lad, and were anxious to furnish him with the means of studying in the metropolis; but with rare independence he refused all aid, and by careful economy saved a sufficient sum to enable him to leave for London. There he made a lucky hit by a sketch of the younger Kean, which, like his portrait of Scott, was lithographed and published. He entered the Academy schools in 1828, and carried off the highest prizes open to the students. In 1829 he exhibited for the first time in the Royal Academy. Gradually he began to confine himself more exclusively to subject and historical pictures, varied occasionally by portraits of Campbell, Miss Landon, Dickens, and other of his literary friends. In
- 72. 1833 he exhibited two pictures which greatly increased his reputation, and in 1835 the “Chivalric Vow of the Ladies and the Peacock” procured his election as associate of the Academy, of which he became full member in 1840. The years that followed were occupied with a long series of figure pictures, deriving their subjects from history and tradition and from the works of Shakespeare, Goldsmith and Le Sage. He also designed illustrations for several of Dickens’s Christmas books and other works. Between the years 1830 and 1836 he contributed to Fraser’s Magazine, under the pseudonym of Alfred Croquis, a remarkable series of portraits of the literary and other celebrities of the time—character studies, etched or lithographed in outline, and touched more or less with the emphasis of the caricaturist, which were afterwards published as the Maclise Portrait Gallery (1871). In 1858 Maclise commenced one of the two great monumental works of his life, the “Meeting of Wellington and Blücher,” on the walls of Westminster Palace. It was begun in fresco, a process which proved unmanageable. The artist wished to resign the task; but, encouraged by Prince Albert, he studied in Berlin the new method of “water-glass” painting, and carried out the subject and its companion, the “Death of Nelson,” in that medium, completing the latter painting in 1864. The intense application which he gave to these great historic works, and various circumstances connected with the commission, had a serious effect on the artist’s health. He began to shun the company in which he formerly delighted; his old buoyancy of spirits was gone; and when, in 1865, the presidentship of the Academy was offered to him he declined the honour. He died of acute pneumonia on the 25th of April 1870. His works are distinguished by powerful intellectual and imaginative qualities, but most of them are marred by harsh and dull colouring, by metallic hardness of surface and texture, and by frequent touches
- 73. of the theatrical in the action and attitudes of the figures. His fame rests most securely on his two greatest works at Westminster. A memoir of Maclise, by his friend W. J. O’Driscoll, was published in 1871. MACLURE, WILLIAM (1763-1840), American geologist, was born at Ayr in Scotland in 1763. After a brief visit to New York in 1782 he began active life as a partner in a London firm of American merchants. In 1796 business affairs took him to Virginia, U.S.A., which he thereafter made his home. In 1803 he visited France as one of the commissioners appointed to settle the claims of American citizens on the French government; and during the few years then spent in Europe he applied himself with enthusiasm to the study of geology. On his return home in 1807 he commenced the self- imposed task of making a geological survey of the United States. Almost every state in the Union was traversed and mapped by him, the Alleghany Mountains being crossed and recrossed some fifty times. The results of his unaided labours were submitted to the American Philosophical Society in a memoir entitled Observations on the Geology of the United States explanatory of a Geological Map, and published in the Society’s Transactions (vol. iv. 1809, p. 91) together with the first geological map of that country. This antedates William Smith’s geological map of England by six years. In 1817
- 74. Maclure brought before the same society a revised edition of his map, and his great geological memoir was issued separately, with some additional matter, under the title Observations on the Geology of the United States of America. Subsequent survey has corroborated the general accuracy of Maclure’s observations. In 1819 he visited Spain, and attempted, unsuccessfully, to establish an agricultural college near the city of Alicante. Returning to America in 1824, he settled for some years at New Harmony, Indiana, and sought to develop his scheme of the agricultural college. Failing health ultimately constrained him to relinquish the attempt, and to seek (in 1827) a more congenial climate in Mexico. There, at San Angel, he died on the 23rd of March 1840. See S. G. Morton, “Memoir of William Maclure,” Amer. Journ. Sci., vol. xlvii. (1844), p. 1. MacMAHON, MARIE EDMÉ PATRICE MAURICE DE, duke of Magenta (1808-1893), French marshal and president of the French republic, was born on the 13th of July 1808 at the château of Sully, near Autun. He was descended from an Irish family which went into exile with James II. Educated at the military school of St Cyr, in 1827 he entered the army, and soon saw active service in the first French campaign in Algeria, where his ability and bravery became conspicuous. Being recalled to France, he gained renewed
- 75. distinction in the expedition to Antwerp in 1832. He became captain in 1833, and in that year returned to Algeria. He led daring cavalry raids across plains infested with Bedouin, and especially distinguished himself at the siege of Constantine in 1837. From then until 1855 he was almost constantly in Algeria, and rose to the rank of general of division. During the Crimean War MacMahon was given the command of a division, and in September 1855 he successfully conducted the assault upon the Malakoff works, which led to the fall of Sebastopol. After his return to France honours were showered upon him, and he was made a senator. Desiring a more active life, however, and declining the highest command in France, he was once more sent out, at his own request, to Algeria, where he completely defeated the Kabyles. After his return to France he voted as a senator against the unconstitutional law for general safety, which was brought forward in consequence of Orsini’s abortive attempt on the emperor’s life. MacMahon greatly distinguished himself in the Italian campaign of 1859. Partly by good luck and partly by his boldness and sagacity in pushing forward without orders at a critical moment at the battle of Magenta, he enabled the French to secure the victory. For his brilliant services MacMahon received his marshal’s baton and was created duke of Magenta. In 1861 he represented France at the coronation of William I. of Prussia, and in 1864 he was nominated governor-general of Algeria. MacMahon’s action in this capacity formed the least successful episode of his career. Although he did institute some reforms in the colonies, complaints were so numerous that twice in the early part of 1870 he sent in his resignation to the emperor. When the ill-fated Ollivier cabinet was formed the emperor abandoned his Algerian schemes and MacMahon was recalled.
- 76. War being declared between France and Prussia in July 1870, MacMahon was appointed to the command of the Alsace army detachment (see Franco-German War). On the 6th of August MacMahon fought the battle of Wörth (q.v.). His courage was always conspicuous on the field, but the two-to-one numerical superiority of the Germans triumphed. MacMahon was compelled to fall back upon Saverne, and thence to Toul. Though he suffered further losses in the course of his retreat, his movements were so ably conducted that the emperor confided to him the supreme command of the new levies which he was mustering at Châlons, and he was directed to effect a junction with Bazaine. This operation he undertook against his will. He had an army of 120,000 men, with 324 guns; but large numbers of the troops were disorganized and demoralized. Early on the 1st of September the decisive battle of Sedan began. MacMahon was dangerously wounded in the thigh, whereupon General Ducrot, and soon afterwards General de Wimpffen, took command. MacMahon shared the captivity of his comrades, and resided at Wiesbaden until the conclusion of peace. In March 1871 MacMahon was appointed by Thiers commander-in- chief of the army of Versailles; and in that capacity he suppressed the Communist insurrection, and successfully conducted the second siege of Paris. In the following December he was invited to become a candidate for Paris in the elections to the National Assembly, but declined nomination. On the resignation of Thiers as president of the Republic, on the 24th of May 1873, MacMahon was elected to the vacant office by an almost unanimous vote, being supported by 390 members out of 392. The duc de Broglie was empowered to form a Conservative administration, but the president also took an early opportunity of showing that he intended to uphold the sovereignty of the National Assembly. On the 5th of November 1873 General
- 77. Changarnier presented a motion in the Assembly to confirm MacMahon’s powers for a period of ten years, and to provide for a commission of thirty to draw up a form of constitutional law. The president consented, but in a message to the Assembly he declared in favour of a confirmation of his own powers for seven years, and expressed his determination to use all his influence in the maintenance of Conservative principles. After prolonged debates the Septennate was adopted on the 19th of November by 378 votes to 310. There was no coup d’état in favour of “Henri V.,” as had been expected, and the president resolved to abide by “existing institutions.” One of his earliest acts was to receive the finding of the court-martial upon his old comrade in arms, Marshal Bazaine, whose death sentence he commuted to one of twenty years’ imprisonment in a fortress. Though MacMahon’s life as president of the Republic was of the simplest possible character, his term of office was marked by many brilliant displays, while his wife was a leader in all works of charity and benevolence. The president was very popular in the rural districts of France, through which he made a successful tour shortly after the declaration of the Septennate. But in Paris and other large cities his policy soon caused great dissatisfaction, the Republican party especially being alienated by press prosecutions and the attempted suppression of Republican ideas. Matters were at a comparative deadlock in the National Assembly, until the accession of some Orleanists to the Moderate Republican party in 1875 made it possible to pass various constitutional laws. In May 1877, however, the constitutional crisis became once more acute. A peremptory letter of censure from MacMahon to Jules Simon caused the latter to resign with his colleagues. The duc de Broglie formed a ministry, but Gambetta carried a resolution in the Chamber of Deputies in favour
- 78. of parliamentary government. The president declined to yield, and being supported by the Senate, he dissolved the Chamber, by decree, on the 25th of June. The prosecution of Gambetta followed for a speech at Lille, in which he had said “the marshal must, if the elections be against him, se soumettre ou se démettre.” In a manifesto respecting the elections, the president referred to his successful government and observed, “I cannot obey the injunctions of the demagogy; I can neither become the instrument of Radicalism nor abandon the post in which the constitution has placed me.” His confidence in the result of the elections was misplaced. Notwithstanding the great pressure put upon the constituencies by the government, the elections in October resulted in the return of 335 Republicans and only 198 anti-Republicans, the latter including 30 MacMahonists, 89 Bonapartists, 41 Legitimists, and 38 Orleanists. The president endeavoured to ignore the significance of the elections, and continued his reactionary policy. As a last resort he called to power an extra-parliamentary cabinet under General Rochebouet, but the Republican majority refused to vote supplies, and after a brief interval the president was compelled to yield, and to accept a new Republican ministry under Dufaure. The prolonged crisis terminated on the 14th of December 1877, and no further constitutional difficulties arose in 1878. But as the senatorial elections, held early in 1879, gave the Republicans an effective working majority in the Upper Chamber, they now called for the removal of the most conspicuous anti-Republicans among the generals and officials. The president refused to supersede them, and declined to sanction the law brought in with this object. Perceiving further resistance to be useless, however, MacMahon resigned the presidency on the 30th of January 1879, and Jules Grévy was elected as his successor.
- 79. MacMahon now retired into private life. Relieved from the cares of state, his simple and unostentatious mode of existence enabled him to pass many years of dignified repose. He died at Paris on the 17th of October 1893, in his eighty-sixth year. A fine, tall, soldierly man, of a thoroughly Irish type, in private life MacMahon was universally esteemed as generous and honourable; as a soldier he was brave and able, without decided military genius; as a politician he was patriotic and well-intentioned, but devoid of any real capacity for statecraft. (G. B. S.) McMASTER, JOHN BACH (1852- ), American historian, was born in Brooklyn, New York, on the 29th of June 1852. He graduated from the college of the City of New York in 1872, worked as a civil engineer in 1873-1877, was instructor in civil engineering at Princeton University in 1877-1883, and in 1883 became professor of American history in the university of Pennsylvania. He is best known for his History of the People of the United States from the Revolution to the Civil War (1883 sqq.), a valuable supplement to the more purely political writings of Schouler, Von Holst and Henry Adams.
- 80. MACMILLAN, the name of a family of English publishers. The founders of the firm were two Scotsmen, Daniel Macmillan (1813- 1857) and his younger brother Alexander (1818-1896). Daniel was a native of the Isle of Arran, and Alexander was born in Irvine on the 3rd of October 1818. Daniel was for some time assistant to the bookseller Johnson at Cambridge, but entered the employ of Messrs Seeley in London in 1837; in 1843 he began business in Aldersgate Street, and in the same year the two brothers purchased the business of Newby in Cambridge. They did not confine themselves to bookselling, but published educational works as early as 1844. In 1845 they became the proprietors of the more important business of Stevenson, in Cambridge, the firm being styled Macmillan, Barclay Macmillan. In 1850 Barclay retired and the firm resumed the name of Macmillan Co. Daniel Macmillan died at Cambridge on the 27th of June 1857. In that year an impetus was given to the business by the publication of Kingsley’s Two Years Ago. A branch office was opened in 1858 in Henrietta Street, London, which led to a great extension of trade. These premises were surrendered for larger ones in Bedford Street, and in 1897 the buildings in St Martin’s Street were opened. Alexander Macmillan died in January 1896. By his great energy and literary associations, and with the aid of his partners, there had been built up in little over half a century one of the most important publishing houses in the world. Besides the issue
- 81. of many important series of educational and scientific works, they published the works of Kingsley, Huxley, Maurice, Tennyson, Lightfoot, Westcott, J. R. Green, Lord Roberts, Lewis Carroll, and of many other well-known authors. In 1898 they took over the old- established publishing house of R. Bentley Son, and with it the works of Mrs Henry Wood, Miss Rhoda Broughton, The Ingoldsby Legends, and also Temple Bar and the Argosy. In 1893 the firm was converted into a limited liability company, its chairman being Frederick Macmillan (b. 1851), who was knighted in 1909. The American firm of the Macmillan Company, of which he was also a director, is a separate business. See Thomas Hughes, Memoir of Daniel Macmillan (1882); A Bibliographical Catalogue of Macmillan Co’s Publications from 1843 to 1889 (1891), with portraits of the brothers Daniel and Alexander after Lowes Dickinson and Hubert Herkomer; also articles in Le Livre (September 1886), Publishers’ Circular (January 14, 1893), the Bookman (May 1901), c. MACMONNIES, FREDERICK WILLIAM (1863- ) American sculptor and painter, was born at Brooklyn, New York, on the 20th of September 1863. His mother was a niece of Benjamin West. At the age of sixteen MacMonnies was received as an apprentice in the studio of Augustus St Gaudens, the sculptor, where
- 82. he remained for five years. In 1884 he went to Paris and thence to Munich, where he painted for some months. Returning to Paris next year he became the most prominent pupil of Falguière. His “Diana” brought him a mention at the Salon of 1889. Three life-sized figures of angels for the church of St Paul, New York, were followed by his “Nathan Hale,” in the City Hall Park, New York, and a portrait of James S. T. Stranahan, for Brooklyn. This last brought him a “second medal” in the Salon of 1891, the first time an American sculptor had been so honoured. In 1893 he was chosen to design and carry out the Columbian Fountain for the Chicago World’s Fair, which placed him instantly in the front rank. His largest work is a decoration for the Memorial Arch to Soldiers and Sailors, in Prospect Park, Brooklyn, consisting of three enormous groups in bronze. In Prospect Park, Brooklyn, MacMonnies has also a large “Horse Tamer,” a work of much distinction. A “Winged Victory” at the U.S. military academy at West Point, New York, is of importance; and his “Bacchante,” an extraordinary combination of realism and imagination, rejected by the Boston Public Library, is now at the Metropolitan Museum of Art, New York. He also became well known as a painter, mainly of portraits. In 1888 he married Mary Fairchild, a figure painter of distinction, but in 1909 they were divorced and she married Will H. Low.
- 83. MACNAGHTEN, SIR WILLIAM HAY, Bart. (1793- 1841), Anglo-Indian diplomatist, was the second son of Sir Francis Macnaghten, Bart., judge of the supreme courts of Madras and Calcutta. He was born in August 1793, and educated at Charterhouse. He went out to Madras as a cadet in 1809, but was appointed in 1816 to the Bengal Civil Service. He early displayed a great talent for languages, and also published several treatises on Hindu and Mahommedan law. His political career began in 1830 as secretary to Lord William Bentinck; and in 1837 he became one of the most trusted advisers of the governor-general, Lord Auckland, with whose policy of supporting Shah Shuja against Dost Mahommed, the reigning amir of Kabul, Macnaghten was closely identified. As political agent at Kabul he came into conflict with the military authorities and subsequently with his subordinate Sir Alexander Burnes. Macnaghten attempted to placate the Afghan chiefs with heavy subsidies, but when the drain on the Indian exchequer became too great, and the allowances were reduced, this policy led to an outbreak. Burnes was murdered on the 2nd of November 1841; and owing to the incapacity of the aged General Elphinstone the British army in Kabul degenerated into a leaderless mob. Macnaghten tried to save the situation by negotiating with the Afghan chiefs and, independently of them, with Dost Mahommed’s son, Akbar Khan, by whom he was assassinated on the 23rd of December 1841; the disastrous retreat from Kabul and the massacre of the British army in the Kurd Kabul pass followed. These events threw doubt on Macnaghten’s capacity for dealing with the problems of Indian diplomacy, though his fearlessness and integrity were unquestioned. He had been created a baronet in 1840, and four months before his death was nominated to the governorship of Bombay.
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