Automated Patch Point Placement for Spacecraft Trajectory Targeting

Automated Patch Point Placement for
Spacecraft Trajectory Targeting
Galen Harden
Amanda Haapala
Kathleen Howell
Belinda Marchand
2014 AAS/AIAA Space Flight Mechanics Meeting
Image compliments of www.nasa.gov

Introduction
 Problem Summary
 Targeting in dynamically sensitive regimes benefits from multi-
phase algorithms, which simultaneously operate on a startup arc
divided into multiple patch states (e.g. nodes)
 Gradient-based targeting algorithms (optimal or sub-optimal) are
sensitive to the quality of the initial guess.
 Arbitrary placement of patch states (e.g. nodes) can negatively
impact algorithm response in dynamically sensitive regimes
 Solution Approach
 Develop an automated patch state / node selection strategy,
suitable for onboard guidance processes, that can intelligently
select patch state sets.
 Seek algorithm that reduces impact of dynamical sensitivities to
improve overall algorithm response.
January 27th, 2014 Harden, Haapala, Howell, & Marchand 2

Lyapunov Exponents
January 27th, 2014 Harden, Haapala, Howell, & Marchand
 Characterize the rate of separation of two infinitesimally
close trajectories as they evolve in time, and given by:
where
.
 If , the full Lyapunov exponent spectrum is
characterized by , one for each linearly
independent fundamental direction.
 In general, there is no analytical means of identifying the
Lyapunov exponents. They must be approximated
numerically over a finite horizon.
3

Finite-Time Lyapunov Exponents
and Local Lyapunov Exponents
 The term Finite Time Lyapunov Exponents is often used
to refer to the full spectrum of Lyapunov exponents over
a specific finite time horizon.
 A reasonable approximation of the local growth rate is
determined by considering only the largest exponent in
the set, or Local Lyapunov Exponent (LLE):
 Here, t denotes the selected time horizon.
 Note that if the trajectory spans over , then:
4

Visualization of LLE Contours
as a Function of Normalization Time
 The relation between the time
along the trajectory and the
horizon (or normalization time) is :
 A large LLE value is indicative of
a high degree of dynamical
sensitivity at that specific location
along the arc.
 The dark regions in the contour
are associated with local minima
of the LLE value, while the
highest intensity corresponds to
local maxima. (Clearly sometimes
embracing the dark side is a good
thing  )
5

LLE Contour
Dependence on Horizon Time
Note that the LLE contour for a given
arc will change according to the
normalizing factor selected

Patch Point Placement
on an LLE Surface

on an LLE Surface

Automated Patch State Selection:
Motivation
 Previous research reveals that patch states placed at
local minima along the LLE contour offer the best
convergence for targeting and optimization algorithms.
 This observation suggests a patch state selection
strategy that automatically identify candidate points,
based on the LLE criteria, is desirable.
 To develop an automated patch state placement
algorithm, it is useful to establish a simple metric by
which to systematically and autonomously compare the
“quality” of a given patch state set against another.

Evaluation Metric (1)
 All multi-phase targeting algorithms presented operate
on the initial guess by modifying a set of control
parameters:
in an effort to satisfy a set of linearized constraint
equations:
 The vector of control parameters varies depending on
the exact targeting algorithm selected.

 To evaluate the impact of varying a specific
patch state set, on the constraint error, we seek
a simple scalar expression that
 Relates the norm of the constraint vector as a
function of the norm of the patch state errors.
 Captures the impact of our “confidence” on the
quality of the patch states.

 By leveraging the properties of the expected value, and
some properties of the trace, this expression reduces to:
 For the specific targeting algorithm selected, this
reduces to:
and ultimately to

Computational Process
 Having established the metric for comparison,
the computation of a patch state set proceeds as
follows:
 Start with one patch state at the beginning of the trajectory, and
at any scheduled maneuver points, iteratively.
 For a specific segment, identify a set of candidate states, any
one of which could represent the new patch state.
 Each candidate must satisfy the constraint between duration and
normalization (i.e. select points along diagonals of the LLE contour).
 Select a reasonably representative number of candidates to properly
characterize the options along that diagonal.
 For each candidate state, evaluate the approximate error metric
and identify which is associated with the smallest error.

Motivating Example #1:
Altitude Targeting Near Earth
 Fix initial position, target final position.
 Target final position vector aligned with initial guess, but
seeks change in altitude.
 Compare candidate multi-phase targeter performance
using evenly-spaced vs. automatically-selected nodes.

Two-Stage Corrector:
Performance Comparison Across
Patch State Selection Strategies

Orion trans-Earth Trajectory
2x2BP Initial Guess in Earth-Moon 3BP
 One Earth-centered arc
 One moon-centered of arcs
 LLO to Apogee raise seg.
 Apogee to Inc. change seg.
 Inc. change to Trans-Earth
seg.
 Segments  patch points.
 2BP patch points  3BP
 Discontinuities between
segments and @ interface
20

Orion trans-Earth Trajectory
Converged Solution in Earth-Moon 3BP
 Target entry altitude via
3-maneuver sequence
 Poor initial guess quality
degrades performance of
Linear targeting
 Targeting performance
 Equally spaced patch
states: DNC
 Automated patch state
selection: 8 iterations
21

Conclusions
 Preliminary results indicate automated patch
point placement algorithm improves response of
multi-phase targeting algorithms.
 The initial error prediction model considered
offers a simple effective metric by which to
compare the quality of candidate patch state
sets.

Automated Patch Point Placement for Spacecraft Trajectory Targeting

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