![4 J.-P. Merlet and J. Alexandre-dit-Sandretto
Fig. 1 Cable driven parallel
robots with sagging cables
A1 A2
A3
A4
A5
A6
A7
B7
B3
B1
B2
B4
B5
B6
• real-time FK: used for control purposes with the objective of determining the
current pose of the robot. It need to be fast (running within a sampling time of the
robot controller) and uses the knowledge that the solution is “close” to a known
pose. It may also be used for simulation purposes
• full FK: the purpose is to determine all solutions of the FK. It is used off-line for
determining the possible initial states of the robot when running a simulation
We will see later on that the real time FK is not a problem and we will address only
the full FK. This problem has been addressed for CDPRDs in a very preliminary
stage only very recently and under restrictive assumptions on the behavior of the
cables. Indeed we may consider:
• non-deformable cables: they are aligned along the direction Ai Bi
• elastic cables: they are also aligned along Ai Bi but their lengths depend upon the
tension to which they are submitted
• catenary cables: they exhibit elasticity and their own mass leads to a deformed
shape
Most of the kinematic works have assumed non-deformable cables. For robots having
at least 6 cables the FK is equivalent to the one of classical parallel robots (for more
than 6 cables at most 6 will be under tension simultaneously [1]). With less than 6
cables the problem is still open as the geometrical constraints relating the length of
the cables to the pose leads to less equations than unknowns, which imposes to add
the 6 additional statics equations and the cable tensions as additional unknowns. For
a CDPR with m cables the minimal system has 6 + m equations in 6 + m unknowns,
to be compared with the system of 6 equations for the Gough platform. Although
there has been progress recently to solve these problems [2–4] there are still a lot of
progress to be made in order to determine the maximal number of solutions according
to m, solutions that should have only positive tensions and are stable.](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-18-320.jpg)
![The Forward Kinematics of Cable-Driven Parallel Robots . . . 5
If we assume elasticity in the cables there has been some works for the IK [5, 6]
but to the best of our knowledge the FK has never been addressed in the general case.
This paper addresses the most general case of FK for CDPR having catenary type
cables. Very few works have addressed the IK and FK of such a robot [7–9] and none
have considered the general case.
2 Problem Statement
2.1 Cable Model
We will assume cables with linear density μ, cross-section A0 and E will denote
the Young modulus of the cable material. A reference frame O, x, y, z will be used
and the coordinates of the Ai points are known in this frame. In the vertical plane of
the cable we may assume that the cable is attached at point A with coordinates (0,0)
while the other extremity is attached at point B with coordinates (xb, yb) (Fig.2).
The vertical and horizontal forces Fz, Fx are exerted on the cable at point B and the
cable length at rest is L0. With this notation the coordinates of B are related to the
forces Fx , Fz [10] by:
xb = Fx (
L0
E A0
+
sinh−1(Fz) − sinh−1(Fz − μgL0
Fx
)
μg
) (1)
zb =
F2
x + F2
z −
F2
x + (Fz − μgL0)2
μg
+
Fz L0
E A0
−
μgL2
0
2E A0
(2)
2.2 The FK Problem
We consider a spatial robot with m cables whose lengths will be denoted L1
0, . . . , Lm
0 .
Without loss of generality we will assume that the Young modulus, linear density and
cross-section of the cables are identical. The problem we have to solve is to determine
all the possible poses of the platform being given the L
j
0 and the location of the Ai ,
together with the external forces/torques F that act on the platform. In terms of
unknowns we will assume a minimal representation of the pose with 6 parameters](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-19-320.jpg)
![The Forward Kinematics of Cable-Driven Parallel Robots . . . 7
3 Solving the FK
As a theoretical solving appears to be difficult to be used we will have to resort
to a numerical solving method, that has to provide all the solutions. We will use an
interval analysis (IA) approach, which guarantees to find all the solutions lying within
some given ranges. The basis of IA is the interval evaluation: being given a function
f (x1, . . . , xn) in n variables and assuming that each variable xi lies in the range
[xi , xi ] the interval evaluation of f is a range [A, B] such that ∀xi ∈ [xi , xi ], i ∈
[1, n] we have A ≤ f (x1, . . . , xn) ≤ B. There are multiple ways to define an interval
evaluation but the most simple is the natural evaluation: each mathematical operator
has an interval equivalent (for example the addition interval operator + is defined
as [a, b] + [c, d] = [a + c, b + d]) and transforming any function by using the
interval operators allows to calculate the interval evaluation. One of the property
of the interval evaluation [A, B] is that if A 0 of B 0, then f cannot cancel
whatever is the value of the variables in their ranges. Note that an interval evaluation
may be overestimated: there may not be value of the variables in their respective
range such that f (x1, . . . , xn) = A or f (x1, . . . , xn) = B. Indeed overestimation
may occur because of multiple occurrences of a given variable that are considered
as independent: for example the evaluation of x − x when x ∈ [−1, 1] is [−2, 2].
But the size of the overestimation decreases with the width of the variable ranges.
The second key ingredient of IA is the branch and bound algorithm. A box B is
defined as a set of ranges for the variables. If for a given box we have f (B = [A, B]
with A 0, B 0, then we select one of the variable, xi bisect its range in two
and create two new boxes B1, B2 that are identical to B except for the range for xi
which result from the bisection. These boxes are stored in a list and will be processed
later on. We will see in the next section that if a box is tiny enough we may determine
if it includes a single solution and compute this solution with an arbitrary accuracy.
However we will not use the minimal set of equations for the FK. Indeed the
pose of the platform will not be represented by the coordinates of C and three
orientation angles. The motivation is that coordinates xb, zb in (1, 2) will be obtained
after using the rotation matrix and will include several occurrences of the rotation
angles, possibly leading to large overestimation. We prefer to represent the pose
of the platform by the 12 coordinates in the reference frame of 4 of the points
Bi , which are not coplanar (we assume here that B1, B2, B3, B4 are chosen). With
this choice the coordinates of any point M on the platform may be obtained as
OM = α1OB1 + α2OB2 + α3OB3 + α4OB4 where the α are known constants.
Such a representation allows one to obtain the coordinates of the Bj , j 4 and
of the center of mass C. As we have now 12 unknowns for representing the pose
of the platform instead of 6 with the minimal representation we need 6 additional
equations that are obtained by stating that the distance between a pair of points in
the set B1, . . . B4 is a known constant. Note that these equations are not sufficient
to fully describe the geometry of the platform (e.g. the equations does not allow to
differentiate if a Bj point is over or under the plane that includes the three other
points). Another test is needed and we will use the fact that for any point M of the
platform there exist constants βj such that](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-21-320.jpg)
![8 J.-P. Merlet and J. Alexandre-dit-Sandretto
BjM = β1BjBk + β2BjBl + β3BjBk × BjBl, j, k,l ∈ [1, 4] (5)
It remains to manage the angles ui : formally they can be obtained using the equation
x
j
b sin(u j ) + y
j
b cos(u j ) = 0 (6)
where x
j
b , y
j
b are the coordinates of Bj in the reference frame. But by so doing
their interval evaluation will be relatively large even for small width for the x
j
b , y
j
b
intervals, especially if j 4. Hence we prefer to add them as additional unknowns
and to use (6) as additional equations. Hence we end up with 12 + 3m unknowns
for 12 + 3m equations. Note that we have checked that solving this system in the
context of a real-time FK is not difficult as soon as a certified strategy is used [11]:
this strategy allows one to determine the current pose of the robot or eventually that
the pose is too close to a singularity (in which case the FK has an infinite number of
solutions).
An IA approach impose to determine a domain in which are located all the solu-
tions. This can be easily done for the x
j
b , y
j
b variable that are restricted to lie within
the convex hull of the Ai points. This can also be done for z
j
b, the z coordinate of
Bj that cannot be greater than the highest z coordinate of the Ai points and cannot
be lower than the length obtained if we assume that the cable is vertical and bears
the platform. As for the ui as the cable have to lie within the convex hull of the Ai
we can also get bounds for these variables. It remains the variables Fx , Fz which
have no natural bounds except that Fx cannot be negative. We will first define m new
variables λj such that F
j
z = λj Fx (which allows one to have simpler expressions
for (1, 2). We take then as upper bound for Fx 10 times the value of mg and for the λ
a range of [−10, 10] (at the extremities of this range the cables are almost vertical).
3.1 Determining Exact Solutions
The classical IA branch and bound algorithm assume that if the width of a box is
smaller than a small value and the interval evaluations of all equations include 0, then
we have found an approximate solution of the system. Here we proceed in another
way: for each box of the algorithm we run a few iterations of the Newton-Raphson
scheme with as estimate of the solution the center of the box H. Note that even if
the NR algorithm converges there is no guarantee (1) that the result is indeed an
approximate solution of the system, (2) that the solution lie within the box or even
within the search space, (3) that the result satisfies the constraint (5). In order to check
if the result is really a solution of the system we use Kantorovitch theorem [11] that
allows one to verify that there is indeed a single solution of the system in a ball
centered at H with a known radius. If this test succeed we have furthermore the
property that the NR scheme, initialized with H as guess point, converges toward](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-22-320.jpg)
![The Forward Kinematics of Cable-Driven Parallel Robots . . . 9
the solution. We will see in the implementation section that this property will allow
us to compute an approximation of the solution with an arbitrary accuracy.
As soon as a solution H0 is found it is stored and in a first step we will assume
that there is no other solution in a ball centered at H0 with a given radius, this being
applied only on the 12 coordinates of the point B1, . . . , B4. Any box that is fully
included in this ball will be eliminated and if a box has an intersection with the ball,
then the intersection part will be removed from the box. Our purpose in this first
step is to determine balls that include a solution and possibly others. In a second
step we will run the algorithm on this ball and this check will be faster because the
search domain will be drastically reduced. With this approach the IA algorithm is
guaranteed to complete.
3.2 Heuristics
A drawback of the usual IA branch and bound algorithm that eliminates boxes only
according to the interval evaluation of the equations is that is not efficient as soon as
we have complex equations with multiple occurrences of the variables. But several
heuristics allows one to drastically improve the efficiency of the algorithm. A first
set of heuristics is called the consistency approach, which is based on a rewriting
of the equations. Consider for example the equation that described that the distance
between the points of a pair of Bi point is constant. This equation is written as
(xi
b − x
j
b )2
+ (yi
b − y
j
b )2
+ (zi
b − z
j
b)2
= d2
i j (7)
which may be rewritten as
(xi
b − x
j
b )2
= d2
i j − (yi
b − y
j
b )2
− (zi
b − z
j
b)2
Let [A, B] denote the interval evaluation of the right hand-side of this equation. We
deduce that
• if B 0 the equation has no solution
• if A ≤ 0, B ≥ 0 then −
√
B + x
j
b ≤ xi
b ≤ −x
j
b +
√
B
• if A ≥ 0 then xi
b belongs to [−
√
B + x
j
b , −
√
A + x
j
b ] ∪ [
√
A + x
j
b ,
√
B + x
j
b ]
With this approach we may improve the range for any variable in the equation or even
eliminate a box without having to use the bisection process. It is important to note that
if the set of variable is denoted x and we are able to write an equation under the form
g(xi ) = G(x) the consistency requires an inverse operator of g in order to be able to
update xi . This also motivate our choice not to use the minimal representation of the
pose but a more algebraic formulation whose inverse is trivial. In our implementation
the consistency is applied on all equations of the system and for all variables. It is](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-23-320.jpg)
![10 J.-P. Merlet and J. Alexandre-dit-Sandretto
also used on Eq.(5) and on the equations BiBj.BiBk = di j dik cos(θ) where θ is the
known angle between the lines going through (Bi , Bj ) and (Bi , Bk).
Another efficient heuristic is the 3B method. Assume that we have a box and select
one of the variable xi whose range is [xi , xi ]. We change this range to [xi , [xi + ε],
where ε has a small value. Interval evaluation of the equations and the consistency
are used to determine if this new box may include a solution. If the answer is negative
we can safely modify the initial box by setting the range for xi to [xi + ε, xi ]. This
process is applied for all variables but also on the right side of the interval for xi .
Another approach is used for the equations that have multiple occurrences of the
same variable. We calculate the gradient of these equations and it interval evaluation
for the current box. If this evaluation has a constant sign we set the variable to the
appropriate lower or upper bound of the range to improve the interval evaluation.
This process has to be recursive: as soon as a variable is set to an extremity of its
range, then the interval evaluation of the gradient for another variable may become
of constant sign.
Another important issue is the choice of the variable that will be selected for
the bisection process. Our strategy is to bisect in priority the 12 coordinates of the
B1, . . . B4 until the width of their interval is lower than a given threshold. Indeed
these variables play an important role in the equations: if they are fixed the Eqs.(1),
(2) admit a single solution that correspond to the minimal potential energy of the
cables.
3.3 Implementation
The algorithm is implemented using the interval arithmetics of BIAS/PROFIL1 while
the higher level uses the functions of our ALIAS library2 that mixes a C++ library
and a Maple interface. The Maple interface has allowed to generate most of the C++
code for the algorithm and includes an arbitrary accuracy Newton scheme which
allow us to calculate an approximation of the solution with n digits, the n-th digit
being guaranteed to be exact, n being a number given by the end-user.
Another property of the Maple interface is that it allows one to implement the
algorithm in a distributed manner, i.e. running the algorithm on several computers.
Indeed it must be noticed that in the solving scheme the treatment of a given box
is independent from the treatment of the other boxes to be processed. This allow to
have a master program that manages the list of boxes to be processed and the list
o solutions and an arbitrary number m of slave computers. The master computer
process the initial box until it has a fixed number of boxes in its list. Then it sends
the top boxes to the slave computer that a few iterations of the solving algorithm
and send back to the master the eventual solution and the boxes that remain to be
1 http://www.ti3.tuhh.de/keil/profil/index_e.html.
2 http://www-sop.inria.fr/coprin/developpements/main.html.](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-24-320.jpg)
![The Forward Kinematics of Cable-Driven Parallel Robots . . . 11
Fig. 3 The robot developed for the ANR Cogiro project. Although the robot is real we present a
CAD drawing that allows one to better figure out the CDPR
Table 1 Coordinates of the attachment points on the base (in meters)
x y z x y z
−7.175 −5.244 5.462 −7.316 −5.1 5.47
−7.3 5.2 5.476 −7.161 5.3 5.485
7.182 5.3 5.488 7.323 5.2 5.499
7.3 −5.1 5.489 7.161 −5.27 5.497
treated. As the communication overhead is small compared to the computing time
of the algorithm the distributed version allows to divide the processing time by m.
4 Example
We consider the large scale robot developed by LIRMM and Tecnalia as part of
the ANR project Cogiro [7] (Fig.3). This robot is a suspended CDPR (i.e. there is
no cable pulling the platform downward) with 8 cables, whose Ai coordinates are
given in Table1. The cables characteristics are E = 1009 N/m2, μ = 0.346kg/m
and their diameter is 10mm. The mass of the platform of 10kg. The value of the L0s
(Table2) are the non-deformed cable lengths for the pose (1, 0, 2) in meters and for
an orientation such that the reference frame and the mobile frame are aligned.
Table 2 Lengths of the cable
at rest (in meters)
1 2 3 4
10.48215 9.838952 10.16035 8.96827
5 6 7 8
10.310003 8.421629 8.663245 8.655556](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-25-320.jpg)
![12 J.-P. Merlet and J. Alexandre-dit-Sandretto
Table 3 The coordinates of B1, B2, B3 for the 19 solutions (meter)
Sol x1 y1 z1 x2 y2 z2 x3 y3 z3
1 1.55 0.43 4.26 0.04 0.21 4.89 1.05 −0.35 4.71
2 0.57 −0.08 3.53 1.47 −0.80 4.71 0.44 −0.89 4.15
3 1.45 −0.47 2.37 0.42 0.34 3.39 0.43 −0.27 2.38
4 1.27 0.65 4.41 −0.04 −0.31 4.11 1.10 −0.37 4.39
5 0.37 0.47 3.96 1.27 −0.33 5.10 1.34 0.11 4.01
6 0.91 0.98 3.45 0.94 0.04 4.81 1.64 0.64 4.09
7 1.53 −0.74 3.40 0.47 0.53 3.49 1.13 −0.18 4.16
8 1.46 −0.42 4.29 0.52 0.33 3.16 0.94 0.46 4.25
9 1.06 0.64 4.06 1.00 −0.60 2.97 0.93 −0.38 4.12
10 1.66 −0.07 3.45 0.46 0.77 4.20 0.94 −0.30 4.16
11 0.71 0.64 4.10 1.08 −0.64 3.11 0.58 −0.39 4.14
12 1.65 −0.32 4.99 0.73 −0.07 3.64 0.83 −0.78 4.57
13 0.29 0.69 3.64 1.11 −0.67 3.17 0.12 −0.32 3.71
14 1.39 0.45 3.81 0.55 −0.23 2.55 0.88 0.81 2.99
15 1.28 0.53 3.78 0.11 0.01 4.82 1.22 −0.20 4.50
16 1.04 0.90 3.63 0.54 −0.12 4.82 1.52 0.33 4.34
17 1.37 0.52 3.59 0.19 −0.64 3.63 0.44 0.40 3.15
18 0.71 0.92 3.69 0.53 −0.69 4.01 −0.08 0.31 3.94
19 1.86 −0.15 4.12 0.27 −0.60 4.19 1.18 −0.31 4.88
With 8 cables we have to solve a system of 36 equations and this is probably
the most challenging FK task that has even been considered. The solving algorithm
has been implemented using 10 computers and nineteen solutions were found in
the search domain in a computation of about 24h. They are presented in Table3,
while the cable tensions are given in Table4. The solutions are depicted in Fig.4.
It is interesting to note that the solution poses are distributed all over the possible
workspace: for example the x, y, z coordinates of B1 are in the ranges [0.29, 1.86],
[−0.74, 0.98], [2.37, 4.99]. The Fx forces exhibit also a very large range. For example
for cable 1 this force ranges from 20.08 to 417N. We observe the same variation for
the Fz force: for the same cable its ranges from −30.25 to 15.91N. In 15 cases on
19 the FZ tension in cable 1 is positive, meaning that the cable exert a downward
force on the platform. The number of cables that exert an upward force to support
the load is either 2, 3 or 4, meaning that only a small subset of cables contributes to
this support: this may be an useful information for dimensioning the cable.](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-26-320.jpg)
![An Elastic Cable Model for Cable-Driven
Parallel Robots Including Hysteresis Effects
Philipp Miermeister, Werner Kraus, Tian Lan and Andreas Pott
Abstract Experimental results indicate that time invariant linear elastic models for
cable-driven parallel robots show a significant error in the force prediction during
operation. This paper proposes the use of an extended model for polymer cables
which allows to regard the hysteresis effects depending on the excitation amplitude,
frequency, and initial tension level. The experimental design as well as the parameter
identification are regarded.
1 Introduction
A cable-driven parallel robot, in the following simply called cable robot, is a parallel
kinematic machine mainly consisting of a platform, cables, and winches as shown
in Fig.1. The cables connect the platform to the winches which in turn control the
platform pose by changing the cable length. The control inputs for the winches
usually are computed using a simplified kinematic robot model which regards the
platform and frame geometry under the assumption that the attachment points at the
platform and the contact points at the winch are time invariant. Methods for design
and workspace computation of such systems can be found in [3, 4, 10, 12]. Extended
kinematic models also include the pulley geometry at the winches and result in higher
positioning accuracy [9]. The cable robot IPAnema at Fraunhofer IPA uses Dyneema
cables instead of conventional steel cables which brings the advantage of the lower
weight but at the same time introduces a more complex elastic behavior in the most
relevant force transition element of the cable robot. It showed that the Dyneema
polyethylene cables have a changing elastic behavior over time, are subject to settling
effects, are sensitive to overload, and show hysteresis effects. Since it is very difficult
tobuildthemodelsandidentifyingtherelatedparametersusingmodelsfromdifferent
fields such as tribology, viscoelasticity, and multibody systems, here we propose a
black box approach to model the drive chain. While white box modeling demands a
P. Miermeister (B) · W. Kraus · T. Lan · A. Pott
Fraunhofer IPA, Stuttgart, Germany
e-mail: philipp.miermeister@ipa.fraunhofer.de
© Springer International Publishing Switzerland 2015
A. Pott and T. Bruckmann (eds.), Cable-Driven Parallel Robots,
Mechanisms and Machine Science 32, DOI 10.1007/978-3-319-09489-2_2
17](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-30-320.jpg)
![An Elastic Cable Model for Cable-Driven Parallel Robots . . . 19
Fig. 2 Kinematic loop for one cable
li = ai − (r + Rbi ) (1)
can be used to compute the cable length li = li 2 from a given platform posi-
tion r and rotation R. More complex models deal with pulleys by introducing time
variant anchorage points and handle non negligible cable mass by utilizing catenary
equations to model the cable sagging [5, 11].
Since the mass of the cables of the IPAnema robot is small in comparison to the
applied tension level it is not necessary to regard the cable sagging in the model. For
cable force control, admittance control, and parameter identification it is necessary
to predict the cable force very accurately. The force and torque equilibrium of a cable
robot is described by the structure equation
u1 · · · um
b1 × u1 · · · bm × um
AT
⎡
⎢
⎣
f1
.
.
.
fm
⎤
⎥
⎦
f
+
fp
τp
w
= 0 (2)
where AT is the well known structure matrix [12], ui = li li −1
2 is a unit vector in
direction of the cables, f is the vector of cable forces and w is the external wrench.
Going from the kinematic model to a linear elastic model of type](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-32-320.jpg)
![20 P. Miermeister et al.
fi =
fi = ci Δli for Δli 0
0 for Δli 0
(3)
with Δli = lSP,i − li where ci is the cable stiffness and lSP,i is the set point for
the cable length while li is the actual cable length in the workspace one can achieve
a more accurate prediction. Modeling the cable forces in this way gives acceptable
results for positive cable tensions, but leads to numerical problems in case of Δli 0
due to the discontinuity and the zero area.
Comparing the linear elastic model with experimental data showed that cables also
come along with a hysteresis behavior which depends on the actuation frequency,
amplitude, tension level and amplitude as well as the current angle of the cables with
regards to the redirection pulleys of the robot. Different models such as the Bouc–
Wen-Model, Bilinear Model, Bingham Model, or Polynomial model can be used to
describe the hysteresis behavior [1, 2, 6]. A single tensioned cable of the robot can
be seen as single degree of freedom system with an elastic–plastic behavior whose
hysteresis by means of the cable elongation x can be described by two polynomial
functions, one for the upper part and one for the lower part
FU(x) =
nP
i=0
ai xi
, ẋ 0 (4)
and
FL(x) =
nP
i=0
(−1)i+1
ai xi
, ẋ 0, (5)
respectively. Under the assumption of displacement anti-symmetry one can compute
the hysteresis function by combining Eqs.(4) and (5) to
F(x, ẋ) = g(x) + h(x)sgn(ẋ) (6)
with the polynomials
g(x) = a1 + a3x3 + · · · + am xng
h(x) = a0 + a2x2 + · · · + anxnh
(7)
where ng is an odd number and nh = ng − 1. The degree of the polynomials can
be chosen according to the expected accuracy. The sum of the polynomials g(x)
and h(x) can be interpreted as the superposition of an anhysteretic nonlinear elastic
part and hysteretic nonlinear damping part as shown in Fig.3. The expression in
Eq.(6) allows to describe the hysteresis for a certain amplitude A independent of
the current velocity state. Experiments showed that the hysteretic behavior of the
cable is different comparing very small an very large amplitudes. Extending the
model in order to deal with variable amplitudes and replacing the solely polynomial
description by a velocity depended damping expression one can write](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-33-320.jpg)
![On the Improvement of Cable Collision
Detection Algorithms
Dinh Quan Nguyen and Marc Gouttefarde
Abstract This paper presents several algorithms to detect the cable interferences for
a general spatial Cable-Driven Parallel Robot (CDPR). Two types of cable interfer-
ences are considered. The first type is the collisions between cables and cables. The
second type is the interferences between cables and the CDPR mobile platform. In
each case, an algorithm is proposed to efficiently verify the cable interferences. The
useoftheproposedalgorithmsisthenillustratedbyaverificationprocedureofthecol-
lision free condition over a given Cartesian workspace and orientation workspace of
a CDPR. These tools can be used in the design or planning phase of a general CDPR.
1 Introduction
For cable-driven parallel robots, collision detections happen in several cases:
• Interferences between cables and cables
• Interferences between cables and mobile platform
• Interferences between mobile platform and surrounding environment
• Interferences between cables and surrounding environment.
Efficientmethodstodetectsuchcollisionsbecomenecessary,especiallyforspatial
CDPR having a large number of cables (e.g. m ≥ 6) such as the NIST robot crane
[1], the Marionet CDPR [2, 3] and the CoGiRo prototype [4]. These methods could
be used in two main situations:
• Design/planning: required to check the capability of CDPR (e.g. compute the
bounds on the orientation and Cartesian spaces within which there is no cable
interference).
• Control: required to guarantee safety issues in operating CDPR in real-time.
D.Q. Nguyen (B) · M. Gouttefarde
Laboratoire d’Informatique, de Robotique et de Micro-électronique de Montpellier
(LIRMM-CNRS-UM2),161 rue Ada, 34392 Montpellier Cedex 5, France
e-mail: dinhquan.nguyen@lirmm.fr
M. Gouttefarde
e-mail: marc.gouttefarde@lirmm.fr
© Springer International Publishing Switzerland 2015
A. Pott and T. Bruckmann (eds.), Cable-Driven Parallel Robots,
Mechanisms and Machine Science 32, DOI 10.1007/978-3-319-09489-2_3
29](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-42-320.jpg)
![30 D.Q. Nguyen and M. Gouttefarde
In most situations, the latter case can be avoided if all the safety constraints are dealt
with from checking the capability of CDPR over a desired workspace. In this paper,
we mainly discuss the verification of collision free conditions for a CDPR in the
design or offline planning phases.
In fact, CDPR cable interference problem has not extensively been addressed.
Studies on this aspect can be listed as [5–10].
In term of the collisions between cables or CDPR mobile platform with surround-
ing environment, one can use AABB or OOBB tree methods [11]. These methods are
fast and effective for large and complex shape objects (triangulations of the mobile
platform and obstacles may consist of a lot of vertices).
For CDPR with light-weight cables, the cables can be considered as straight
lines. For CDPR using hefty cables, the mobile platform weight is expectedly large,
the cable sagging effects may not really affect the algorithms, thus it may also be
sufficient enough to consider the cables as straight lines. In either cases, the inter-
ferences between cables and cables can be treated as interferences between straight
line segments. The interferences between cables and CDPR mobile platform can
be considered as collisions between straight line segments and triangles (the latter
triangulating the surface of the CDPR mobile platform). Usual methods [11] can
be applied to detect cable interferences. However, these methods are only suited
for real-time situations and are not satisfactory enough to our application of interest
which is verifying the cable interference free conditions for a given desired Cartesian
workspace and set of orientation ranges. In [5], Merlet discussed algorithms to detect
interferences between cables and cables as well as between cables and mobile plat-
form. However, the proposed methods were only applied to fixed orientation cases
and did not applied for a range of orientations. In [9], Perreault presented an analysis
of the cable interference-free workspace of CDPR. The analysis was also mainly
applied to the cases of CDPR with constant orientation.
In this paper, we aim to develop algorithms that could improve the efficiency of
the verification of collision free operation with respect to given CDPR Cartesian
workspace and orientation workspace. These algorithms concern only the interfer-
ences between cables and cables, and cables and CDPR mobile platform.
The paper is organized as follows. Section 2 presents the algorithm to detect
interferences between cables and cables. The collision detection algorithm for the
interferences between cables and CDPR mobile platform is discussed in Sect.3. An
illustrating example of using the proposed algorithms to verify the collision free
operation of a CDPR over a given Cartesian workspace and orientation workspace
is given in Sect.4. Finally, Sect.5 gives an illustration of the performance of the
presented methods.
2 Interferences Between Cables and Cables
Let us consider a m-cable CDPR as shown in Fig.1. We will consider the interference
between two cables AiBi and AjBj (i = j).](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-43-320.jpg)
![On the Improvement of Cable Collision Detection Algorithms 31
Ai
Aj Am
Bi
Bj
Bm
Fig. 1 A general m-cable CDPR
i
B
i
A
i
B
i
A
j
A
j
B
ij
d ε
i
B
j
A
i
A
j
B
ij
d ε
j
B
j
A
ij
d ε
(a) (b) (c)
Fig. 2 Usual method of checking interference between two cables. a Not colliding, b (Going to)
collide, c Colliding
Figure2 illustrates a general method to detect the collision between two cables.
In the first case (Fig.2a), the two cables i and j are not colliding since the distance
between the two cables dij is greater than a given small value ε (this value can be
chosen as the cable diameter). In the second case (Fig.2b) when the cable i moves
toward the cable j, according to the collision condition dij ε, the two cables
collide. Note that, in this state, the cable j is “behind” the plane (AiBiBj). In this
case, one can say that the two cables are going to collide but a real collision has
not yet happened. In the third case (Fig.2c), the same collision condition is valid.
However, the cable j is “in front of” the plane (AiBiBj). In this case, a real collision
between the two cables i and j has occurred. This algorithm can be formalized as
follows [6, 7, 9]:](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-44-320.jpg)
![32 D.Q. Nguyen and M. Gouttefarde
A real collision between the two cables i and j will occur if the two following condi-
tions are met
(i) the distance between the two cables is very small: dij ε
(ii) the position of the cable j (or the cable exit point Aj) with respect to the plane
(AiBiBj) changes sign (e.g. switch from “behind” position to “in front of” posi-
tion)
It is enough to use this algorithm for real-time collision detection where the positions
of cable exit point Ai and cable anchor point Bi are updated online in each sample
time (while the mobile platform is following a trajectory). The computation of dij
can be found in [11].
However, in the design phase, where the collision free conditions need to be
verified with respect to a range of orientations and a volume of Cartesian space, this
usual method may not be really effective.
In fact, to check the cable interferences, it could be enough to consider the
second condition (ii) (in the usual algorithm) while neglecting the first condition
(i) (dij ε). The two cables i and j can be far away (the distance dij can be large)
but their relative positions will tell us whether or not there was a collision when the
mobile platform “moved” from an arbitrary pose Xp to another pose Xq in the CDPR
workspace.
Figure3 illustrates the method proposed in this paper to detect the interference
between the two cables i and j. Suppose that the CDPR mobile platform moves from
an initial pose Xp to an arbitrary pose Xq where a rotation and/or a translation occur.
In the first case (Fig.3a), when the mobile platform “moves” from pose Xp to pose
Xq, the cable j is always “behind” the plane (AiBiBj). There should be no collision
between the two cables i and j. In the second case (Fig.3b), the position of cable j
with respect to the plane (AiBiBj) changed sign so that a collision probably occurred
between the two cables.
i
B
i
A
i
B
i
A
j
A
j
B
i
B
j
A
i
A
j
B
ij
d
j
B
j
A
ij
d ij
d
i
B
i
A
j
B
j
A
ij
d
(a) (b)
Fig. 3 Checking interference between two cables. a Not colliding, b Colliding](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-45-320.jpg)
![34 D.Q. Nguyen and M. Gouttefarde
3 Interferences Between Cables and CDPR Mobile
Platform
3.1 Method 1
Suppose that the CDPR mobile platform is triangulated into N triangles. The first
approach to detect the interferences between cables and CDPR mobile platform is
quite straightforward using the method to detect collision between line segments and
triangles [11].
At pose Xp, for each cable i, we check for the interferences between cable i and
all the triangles that do not belong to the planes which contain the cable end point
Bi. If there is a collision then we stop the checking process and give out a warning.
Figure4 shows an example of the collision between cables i and the mobile plat-
form.
The computational time in this case depends on the number of vertices of the
mobile platform as well as the number of cables. This method is quite “heavy” and
not really effective if the mobile platform has a complex shape (triangulated with
a large number of triangles N 1). To avoid excessive computational time, we
can approximate the mobile platform shape by a more simple convex shape whose
number of triangles is reduced considerably, e.g. in Fig.5 (the simplified shape should
enclose the CDPR mobile platform).
Fig. 4 Interferences between
cables and CDPR mobile
platform. a Not colliding, b
Colliding
i
B
i
A
i
B
i
A
(a) (b)
Fig. 5 Simplification of the
mobile platform shape. a Real
shape, b Simplified shape
(a) (b)](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-47-320.jpg)
![On the Improvement of Cable Collision Detection Algorithms 35
Fig. 6 Detecting collision
between cable i and the mobile
platform. a Not colliding,
b Colliding
i
B
i
A
i
B
i
A
( 1)
i k
D +
Bi
iN
D
ik
D
M
( 1)
i k
D +
M
Bi
iN
D
ik
D
(a) (b)
3.2 Method 2
Although the first method (Sect.3.1) to detect the interferences between cables and
mobile platform is simple, the issue of heavy computational time may remain if
the mobile platform has a complex shape and the simplification procedure cannot
significantly reduce the number of its vertices.
We propose a second heuristic method which consists in checking whether or not
the cable AiBi belongs to the subspace (convex cone) spanned by its nearest edges.
Fig.6 shows an illustrating example of this approach.
The algorithm is given in the following steps:
• Step 1. Perform a simplification of the mobile platform to transform it into a
simpler convex shape while keeping important vertices. This simplified convex
shape should enclose the CDPR mobile platform.
• Step 2. Determine the nearest neighbor vertices Dik (k = 1, NBi) of anchor points
Bi in such a way that the convex cone spanned by the vectors (
−
−
→
BiDi1,
−
−
→
BiDi2, ...,
−
−
→
BiDiNBi ) includes the CDPR mobile platform. NBi should be the minimum number
of such nearest neighbor vertices of Bi.
• Step 3. Compute the positions (or the signs) of an arbitrary point M lying within
the mobile platform shape with respect to the planes (BiDikDi(k+1)) and store them
into vector SBi of size [NBi × 1]:
SBi(k) = sign
−
−
→
BiDik ×
−
−
−
−
−
→
BiDi(k+1)
T
·
−
−
−
→
DikM (5)
For instance, the point M can be chosen as the origin of the local frame attached
to the mobile platform or as its center of mass. Note that all the cases where
SBi(k) = 0 in (5) are considered invalid (the point M must lie strictly inside the
mobile platform shape).
• Step 4. At an arbitrary pose Xp, compute the signs SAi of the cable exit point Ai
with respect to the NBi planes (BiDikDi(k+1)) (vector SAi is of size [NBi × 1]):
SAi(k) = sign
−
−
→
BiDik ×
−
−
−
−
−
→
BiDi(k+1)
T
·
−
−
→
DikAi (6)](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-48-320.jpg)
![36 D.Q. Nguyen and M. Gouttefarde
If ∃ i and ∃ k (i = 1, m, k = 1, NBi) such that SAi(k) = 0, then the cable AiBi is
considered to be colliding with the mobile platform.
If SAi ≡ SBi then the cable AiBi is considered to be colliding with the mobile platform.
If SAi = SBi and SAi(k) = 0 (∀ k) then there is no collision.
Note that the steps 1–3 should only be done at the initial step of an optimization
or operation process. Then, step 4 will be used to check the interferences between
the cables and the mobile platform for each considered robot configuration.
This approach utilizes the fact that the positions of a point lying within the mobile
platform with respect to the planes (BiDikDi(k+1)) never change. One only need to
evaluate (6) Ncp times to check the collision, where:
Ncp =
m
i=1
NBi (7)
This method is fast and reliable. However, there are still a few limitations to this
approach. The algorithm only works under the condition that the mobile platform
has a convex shape. In case the mobile platform shape is concave, a pre-process
(Step 1) is needed to convert it into a convex object (with a number of vertices as
small as possible) in order to apply the algorithm. Currently, we are not aware of an
efficient (fast) method of selecting the right number of the nearest neighbor vertices
Dik of anchor point Bi. One still has to manually select the vertices Dik. The process
of simplifying the mobile platform shape to reduce its complexity can be done with
available CAD softwares e.g. [12].
4 Verification of Collision Free Condition for a Given
Workspace
Let us consider an application where one want to verify the cable interferences of a
CDPR with respect to a given Cartesian workspace and orientation range. The CDPR
workspace is given as follows:
xmin ≤ x ≤ xmax
ymin ≤ y ≤ ymax
zmin ≤ z ≤ zmax
θx min ≤ θx ≤ θx max
θy min ≤ θy ≤ θy max
θz min ≤ θz ≤ θz max
where X = (x y z, θx θy θz) denotes the mobile platform pose. Assume that the
Cartesian workspace is discretized into a finite set of Np points and the orientation
workspace is discretized into a finite set of Nq points (these points can be chosen as
extreme points which lie on the workspace boundaries). Let us take an arbitrary pose](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-49-320.jpg)
![[161]
[162]
“Yes, but our expenses are high,” said Tom. “We’ve got
to look ahead all the time. Dad will have used all of the
money he took with him in a little less than six months.
After that it will be up to us to have the cash in the
bank. Right now we’ve just a little under a hundred
dollars in the bank. Current bills will take more than
that, and our own living expenses, that is for mother
and we two, will run at least $100 a month. With our
total income from the paper only slightly more than
$200 a month on the basis of the present amount of
advertising, you see we’re not going to be able to save
much toward helping Dad.”
“Then we’ll have to find ways of increasing our volume
of business,” said Helen.
“That won’t be easy to do in a town this size,” replied
Tom, “and I won’t go out and beg for advertising.”
“No one is going to ask you to,” said Helen. “We’ll make
the Herald such a bright, outstanding paper that all of
the business men will want to advertise.”
“We’ll do the best we can,” agreed Tom.
“Then let’s start right now by putting in a farm page,”
suggested Helen.
“But there won’t be many farm sales from now on,”
argued Tom.
“No,” conceded his sister, “but there is haying, threshing
and then corn picking and all of the stores have supplies
to sell to the farmers.”
“I believe you’re right. If you’ll do the collecting this
afternoon, I’ll go down to Gladbrook and see if we can](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-57-320.jpg)
![[163]
get the cooperation of the county agent. Lots of the
townships near here have farm bureaus and I’ll get the
names of all of their leaders and we’ll write and tell
them what we plan to do.”
After lunch Tom teased the family flivver into motion
and set out for Gladbrook while Helen took the sheaf of
bills and started the rounds of the business houses. She
had no trouble getting her money from all of the regular
advertisers and in every store in which she stopped she
took care to ask the owner about news of the store and
of his family. She noticed that it flattered each one and
she resolved to call on them at least once a week.
Tom returned from Gladbrook late in the afternoon. He
was enthusiastic over the success of his talk with the
county agent.
“He’s a fine chap,” Tom explained. “Had a course in
agricultural journalism in college and knows news and
how to write it. The Gladbrook papers, the News and
the Times, don’t come up in this section of the county
and he’ll be only too glad to send us a column each
week.”
“When will he start?”
“Next week will be the first one. He’ll mail his column
every Tuesday evening and we’ll have it on the
Wednesday morning mail. Now, here’s even better
news. I went to several of the department stores at
Gladbrook and told them we were going to put out a
real farm page. They’re actually anxious to buy space
and by driving down there once a week I can get two or
three good ads.”
“How will the local merchants feel?” asked Helen.](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-58-320.jpg)
![[164]
“They won’t object,” replied Tom, “for I was careful to
stress that I would only accept copy which would not
conflict with that used by our local stores.”
“That was a wise thing to do,” Helen said. “We can’t
afford to antagonize our local advertisers. I made the
rounds and collected all of the regular accounts. There’s
only about eighteen dollars outstanding on this month’s
bills and I’ll get all but about five dollars of that before
the week is over.”
“Want to go to Cranston Friday or Saturday?” asked
Tom.
“I surely do,” Helen replied. “But what for, Tom, and can
we afford it?”
“One of us will have to make the trip,” her brother said.
“Putting on this farm page means we’ll have to print two
more pages at home, six altogether, and will need only
two pages of ready-print a week from the World Printing
Company. We’ll go down and talk with their manager at
Cranston and select the features we want for the two
pages they will continue to print for us.”
“Our most important features in the ready-print now are
the comics, the serial story and the fashion news for
women,” said Helen.
“Then we’ll have one page of comics,” said Tom, “and fill
the other page with features of special interest to our
women readers.”
The next three days found the young Blairs so busy
getting out the current edition of the paper that they
had little time to talk about their plans.](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-59-320.jpg)
![[165]
They had decided to go to Cranston Friday but when
Helen found that there were special rates for Saturday,
they postponed the trip one day. When the Friday
morning mail arrived, Helen was glad they had changed
their plans. While sorting the handful of letters, most of
them circulars destined for the wastepaper basket, she
came upon the letter she had been looking forward to
for days. The words in the upper left hand corner
thrilled her. It was from the Cranston bureau of the
Associated Press.
With fingers that trembled slightly, she tore it open.
Would she get the job as Rolfe correspondent? A green
slip dropped out of the envelope and Tom, who had
come in from the composing room, reached down and
picked it up.
“Ten dollars!” he whistled.
“What’s that?” demanded Helen, incredulously.
“It’s your check from the Associated Press for covering
the tornado,” explained Tom. “Look!”
Helen took the slip of crisp, green paper. She wasn’t
dreaming. It was a check, made out in her name and
for $10.
“But there must be some mistake,” she protested. “They
didn’t mean to pay me that much.”
“If you think there’s a mistake,” grinned Tom, “you can
go and see them when we reach Cranston tomorrow.
However, if I were you, I’d tuck it in my pocket, invite
my brother across the street to the drug store, and buy
him a big ice cream soda.”](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-60-320.jpg)
![[166]
[167]
“Wait until I see what the letter says,” replied Helen.
She pulled it out of the envelope and Tom leaned over
to read it with her.
“Dear Miss Blair,” it started, “enclosed you will find check
for your fine work in reporting the tornado near Rolfe.
Please consider this letter as your appointment as Rolfe
correspondent for the Associated Press. Serious
accidents, fires of more than $5,000 damage and deaths
of prominent people should be sent as soon as possible.
Telegraph or telephone, sending all your messages
collect. In using the telegraph, send messages by press
rate collect when the story is filed in the daytime. If at
night, send them night press collect. And remember,
speed counts but accuracy must come first. Stories of a
feature or time nature should be mailed. We are
counting on you to protect us on all news that breaks in
and near Rolfe. Very truly yours, Alva McClintock,
Correspondent in charge of the Cranston Bureau.”
“He certainly said a lot in a few words,” was Tom’s
comment. “Now you’re one up on me. You’re editor of
the Herald and Associated Press correspondent and I’m
only business manager.”
“Don’t get discouraged,” laughed Helen, “I’ll let you
write some of the Associated Press stories.”
“Thanks of the compliment,” grinned Tom. “I’m still
waiting for that ice cream soda, Miss Plutocrat.”
“You’ll grumble until I buy it, I suppose, so I might as
well give in right now,” said Helen. “Come on. I’m
hungry for one myself.”
Tom and Helen boarded the nine forty-five Saturday
morning and arrived at the state capital shortly after](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-61-320.jpg)
![[168]
noon. It was Helen’s first trip to Cranston and she
enjoyed every minute of it, the noise and confusion of
the great railroad terminal, the endless bobbing about
of the red caps, the cries of news boys heralding
noonday editions and the ceaseless roar of the city.
They went into the large restaurant at the station for
lunch and after that Tom inquired at the information
desk for directions on how to reach the plant of the
World Printing Company. He copied the information on a
slip of paper and the two young newspaper people
boarded a street car.
Half an hour later they were on the outskirts of the
industrial district and even before the conductor called
their stop, Tom heard the steady roar of great presses.
“Here we are,” he told Helen as they stepped down from
the car and looked up at a hulking ten story building
that towered above them.
“The Cranston plant of the Rolfe Herald,” chuckled
Helen. “Lead on.”
They walked up the steps into the office, gave their
names and indicated their business to the office girl.
After waiting a few minutes they were ushered into an
adjoining office where an energetic, middle aged man
who introduced himself as Henry Walker, service
manager, greeted them.
“Let’s see, you’re from the Rolfe Herald?” he asked.
“My sister and I are running the paper while Dad is in
the southwest regaining his health,” explained Tom.
“We’ve got to expand the paper to increase our](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-62-320.jpg)
![[169]
advertising space and the only thing we can see to do is
cut down our ready-print to two pages.”
“Explain just what you mean,” suggested the service
manager.
Tom outlined their advertising field and how they hoped
to increase business by adding two more pages of home
print, one of which would be devoted to farm
advertising and news and the other to be available for
whatever additional advertising they could produce.
“We’ll be sorry to have you drop two pages of ready-
print,” said Mr. Walker, “but I believe you’re doing the
right thing. Now let’s see what you want on the two
pages you’ll retain.”
“Helen is editor,” Tom explained, “and it’s up to her to
pick out what she wants.”
“You’re doing a splendid job on the Herald,” the service
manager told Helen. “I get copies of every paper we
serve and I’ve been noticing the changes in make-up
and the lively stories. However, I am sorry to hear about
your father but with you two youngsters to give him pep
and courage he ought to be back on the job in a few
months.”
“We’re sure he will,” smiled Helen as she unfolded a
copy of their last edition of the Herald. “I’ve pasted up
two pages of the features I want to retain,” she
explained as she placed them in front of the service
manager.
“I see,” he said. “You’re going to be quite metropolitan
with a full page of comics and a page devoted to
women. I’m glad of that. Too many editors of weeklies](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-63-320.jpg)
![[170]
fail to realize that the women and not the men are the
real readers of their papers. If you run a paper which
appeals to women and children you’ll have a winner.
Comics for the youngsters and a serial story with a
strong love element and fashions and style news for the
women.”
“How about cost?” asked Tom.
“Dropping the two pages won’t quite cut your bill with
us in half,” explained Mr. Walker, “for you’re retaining all
of our most expensive features. However, this new plan
of yours will reduce your weekly bill about 40 per cent.”
“That’s satisfactory,” agreed Tom, “and we’d like to have
it effective at once. Helen has written the headings she
wants for each page.”
“We’ll send the pages, made up in the new way, down
at the usual time next week,” promised the service
manager, “and when there is anything else we can do,
don’t hesitate to let us know.”
When they were out of the building, they paused to
decide what to do next.
“I liked Mr. Walker,” said Helen. “He didn’t attempt to
keep us from making the change. It means less money
for his company yet he didn’t object.”
“It was good business on his part,” replied Tom. “Now
we feel kindly toward him and although he has lost
temporarily he will gain in the end for we’ll give him
every bit of business we can in the way of ordering
supplies for job printing and extra stock for the paper.”](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-64-320.jpg)
![[171]
[172]
“If we have time,” suggested Helen, “I’d like to go down
to the Associated Press office.”
“Good idea,” agreed Tom. “I’d like to see how they
handle all of the news.”
They boarded the first down town street car and got off
fifteen minutes later in the heart of Cranston’s loop
district. Across the street was the building which housed
the Cranston Chronicle, the largest daily newspaper in
the state. They consulted the directory in the lobby of
the building and took the elevator to the fifth floor
where the Associated Press offices were located.
They stepped out of the elevator and into a large room,
filled with the clatter of many machines. A boy, his face
smeared with blue smudges off carbon paper, rushed up
to them and inquired their business.
“I’m Helen Blair, a new correspondent at Rolfe,”
explained the editor of the Herald, “and I’d like to see
Mr. McClintock, the chief correspondent.”
“Okay,” grinned the boy. “I’ll tell him. You wait here.”
The youngster hurried across the room to a large table,
shaped like a half moon and behind which sat a
touseled haired chap of indeterminate age. He might be
30 and he might be 40, decided Helen.
“Glad to know you, Miss Blair,” he said. “You did a nice
piece of work on the storm.”
“Thank you, Mr. McClintock,” replied Helen. “But my
brother, Tom, deserves all of the credit. He suggested
calling the story to you.”](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-65-320.jpg)
![[173]
“Then I’ll thank Tom, too,” laughed the head of the
Cranston bureau of the Associated Press.
“We’re here today on business for our paper,” explained
Helen, “and with a few minutes to spare before train
time hoped you wouldn’t mind if we came in and saw
how the ‘wheels go round’ here.”
“I’ll be happy to show you the ‘works’,” replied Mr.
McClintock, and he took them over to a battery of
electric printers.
“These,” he explained, “bring us news from every part
of the country, east, south and far west. In reality, they
are electric typewriters controlled from the sending
station in some other city. We take the news which
comes in here, sift it out and decide what will interest
people in our own state, and send it on to daily papers
in our territory.”
“Do these electric printers run all day?” asked Tom.
“Some of them go day and night,” continued Mr.
McClintock, “for the A.P. never sleeps. Whenever news
breaks, we’ve got to be ready to cover it. That’s why we
appreciated your calling us on the storm. We knew
there was trouble in your part of the state but we didn’t
have a correspondent at Rolfe. It was a mighty pleasant
surprise when you phoned.”
They visited with the Associated Press man for another
fifteen minutes and would have continued longer if Tom
had not realized that they had less than twenty minutes
to make their train. The last two blocks to the terminal
were covered at a run and they raced through the train
gates just before they clanged shut.](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-66-320.jpg)
![[174]
“Close call,” panted Tom as they swung onto the steps
of the local and it slid out of the train shed.
“Too close,” agreed Helen, who was breathless from
their dash.
“Had to make it, though,” added Tom, “or we’d have
been stranded here flat broke with the next train for
home Monday night.”
“Don’t worry about something that didn’t happen,”
Helen said. “I’ve enjoyed every minute of our trip and
we’re all ready now to start our expansion program for
the Herald in earnest.”
Adding two more pages of home print to the paper
meant more work than either Tom or Helen had
realized. There was more news to be written and more
ads to be set and another run to be made on the press.
With early June at hand the summer season at the
resorts on the lower end of Lake Dubar got under way
and Helen resolved to make a trip at least once a week
and run a column or two of personals about people
coming and going. She also gave liberal space to the
good roads election in July, stressing the value the
paved scenic highway would be to Rolfe.
The two pages of ready-print arrived on Tuesday and
Tom and Helen were delighted with the appearance of
the comic page and the feature page for women
readers.
“We’ll have the snappiest looking paper in the county,”
chuckled Tom. “Dad won’t know the old paper when he
sees this week’s issue.”](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-67-320.jpg)
![[175]
The county agent kept his promise to send them at least
a column of farm news and Helen made it a point to
gather all she could while Tom went to the county seat
Tuesday morning and solicited ads for the page. The
result was a well-balanced page, half ads and half news.
Careful solicitation of home town merchants also
brought additional ads and when they made up the last
two pages Thursday noon they felt the extra work which
increasing the size of the paper meant was more than
repaid in extra advertising.
“I’m printing a number of extra copies this week,”
explained Tom. “There are lots of people around here
who ought to take the Herald. With our expansion
program we may pick up some extra subscriptions and
we might get a chance at the county printing.”
“Tom!” exclaimed Helen. “Do you really think we might
get to be an official county paper.”
“I don’t see why not,” said Tom. “Of course the two
Gladbrook papers will always be on the county list but
there are always three who print the legal news and the
third one is the Auburn Advocate. Auburn isn’t any
larger than Rolfe and I know darned well we have
almost as many subscriptions as they do.”
“How do they decide the official papers?” Helen wanted
to know.
“The county board of supervisors meets once a year to
select the three official papers,” Tom explained, “and the
three showing the largest circulation are selected. It
would mean at least $2,000 extra revenue to us, most
of which would be profit.”
“Then why didn’t Dad try for it?” Helen asked.](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-68-320.jpg)
![[176]
[177]
“I’m not sure,” said Tom slowly. “There are probably
several reasons, the principal one being that he wasn’t
strong enough to make the additional effort to build up
the circulation list. The other is probably Burr Atwell,
owner and publisher of the Auburn Advocate. I’ve heard
Dad often remark that Atwell is the crookedest
newspaperman in the state.”
“How much circulation do you think the Advocate has
now?” Helen asked.
“Their last postoffice statement showed only 108 more
than ours,” replied Tom.
“And when do the supervisors have their annual
meeting?”
“About the 15th of December,” said Tom. “Now what’s
up?”
“Nothing much,” smiled Helen. “Only, when the
supervisors meet next the Rolfe Herald is going to have
enough circulation to be named an official county paper.
“Why Tom,” she went on enthusiastically, “think what it
would mean to Dad?”
“I’m thinking of that,” nodded her brother, “but I’m also
thinking of what Burr Atwell might do to the Herald.”](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-69-320.jpg)
![[178]
[179]
extra afternoon this summer I’ll drive out in the country
and see if I can’t get some people who haven’t been
subscribers to take our paper.”
Tom agreed with Helen’s suggestion and that very
afternoon they took the old family touring car, filled it
with gas and oil, and ambled through the countryside.
Tom had a list of farmers who were non-subscribers and
before the afternoon was over they had added half a
dozen new names to the Herald’s circulation list. In
addition, they had obtained at least one item of farm
news at every place they stopped.
“I call that a good afternoon’s work,” Helen commented
when they drove the ancient flivver into the garage at
home.
“Not bad at all,” Tom agreed. “Only, we’ll keep quiet
about our circulation activities. No use to stir up Burr
Atwell until he finds it out for himself, which will be soon
enough.”
The remaining weeks of June passed uneventfully. The
days were bright and warm with the softness of early
summer and the countryside was green with a richness
that only the middle west knows. Helen devoted the
first part of each week to getting news in Rolfe and on
Fridays and Saturdays took the old car and rambled
through the countryside, stopping at farmhouses to
make new friends for the Herald and gather news for
the farm page. The revenue of the paper was increasing
rapidly and they rejoiced at the encouraging news which
was coming from their father.
The Fourth of July that year came on Saturday, which
meant a two day celebration for Rolfe and the summer](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-71-320.jpg)
![[180]
resorts on Lake Dubar. Special trains would be routed in
over the railroad and the boats on the lake would do a
rushing business.
The managers of Crescent Beach and Sandy Point
planned big programs for their resorts and ordered full
page bills to be distributed throughout that section of
the state. The county seat papers had usually obtained
these large job printing orders but by carefully figuring,
Tom put in the lowest bids.
Kirk Foster, the manager of Crescent Beach, ordered five
thousand posters while Art Provost, the owner of Sandy
Point, ordered twenty thousand. Crescent Beach catered
to a smaller and more exclusive type of summer visitors
while Sandy Point welcomed everyone to its large and
hospitable beach.
There was not much composition for the posters but the
printing required hours and it seemed to Helen that the
old press rattled continuously for the better part of
three days as Tom fed sheet after sheet of paper into
the ancient machine. The wonder of it was that they
had no breakdowns and the bills were printed and
delivered on time.
“All of which means,” said Tom when he had finished,
“that we’ve added a clear profit of $65 to our bank
account.”
“If we keep on at this rate,” Helen added, “we’ll have
ample to take care of Dad when he needs more money.”
“And he’ll be needing it sometime this fall,” Tom said
slowly. “Gee whizz, but it sure does cost to be in one of
those sanitariums. Lucky we could step in and take hold
here for Dad.”](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-72-320.jpg)
![[181]
“We owe him more than we’ll ever repay,” said Helen,
“and the experience we’re getting now will be
invaluable. We’re working hard but we find time to do
the things we like.”
Helen planned special stories for the edition just before
the Fourth and visited the managers of both resorts to
get their complete programs for the day.
Kirk Foster at Crescent Beach explained that there
would be nothing unusual there except the special
display of night fireworks but Art Provost over at Sandy
Point had engaged a line of free attractions that would
rival any small circus. Besides the usual boating and
bathing, there would be free acts by aerialists, a high
dive by a girl into a small tank of water, half a dozen
clowns to entertain the children, a free band concert
both afternoon and evening, two ball games and in
addition to the merry-go-round on the grounds there
would be a ferris wheel and several other “thrill” rides
brought in for the Fourth.
“You ought to have a great crowd,” said Helen.
“Goin’ to be mighty disappointed if I don’t,” said the old
resort manager. “Plannin’ a regular rip-snorter of a day.
No admission to the grounds, but Boy! it’ll cost by the
time they leave.”
“Going to double the prices of everything?” asked
Helen.
“Nope. Goin’ to have so many things for folks to do
they’ll spend everything they got before they leave.”
“In that case,” replied Helen, “I see where I stay at
home. I’m a notorious spendthrift when it comes to](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-73-320.jpg)
![[182]
celebrating the Fourth.”
“I should say you’re not goin’ to stay home,” said Mr.
Provost. “You and your mother and Tom are goin’ to be
my guests. I’ve got your passes all filled out. Swim, ride
in the boats, dance, roller skate, see the ball games,
enjoy any of the ‘thrill rides’ you want to. Won’t cost you
a cent.”
“But I can’t accept them,” protested Helen. “We’ll pay if
we come down. Besides, we didn’t give you all of those
bills for nothing.”
“Seemed mighty near nothin’ compared with the prices
all the other printers in the county wanted,” smiled Mr.
Provost. “You’ve been down every week writin’ items
about the folks who come here and, believe me, I
appreciate it. These passes are just a little return of the
courtesy you’ve shown me this summer.”
“When you put it that way, I can scarcely refuse them,”
laughed Helen.
“As a matter of fact,” she added, “I wanted them terribly
for we honestly couldn’t afford to come otherwise.”
When Helen returned to the office she told Tom about
the passes and he agreed that acceptance of them
would not place the Herald under obligation to the
resort owner.
“I always thought old man Provost a pretty good scout,”
he said, “but I hardly expected him to do this. And say,
these passes are good for both Saturday and Sunday.
What a break!”](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-74-320.jpg)
![[183]
[184]
“If we see everything Saturday we’ll be so tired we
won’t want to go back Sunday,” Helen said. “Besides,
Mother has some pretty strong ideas on Sunday
celebrations.”
The telephone rang and Helen hastened into the
editorial office to answer.
She talked rapidly for several minutes, jotting down
notes on a pad of scratch paper. When she had finished,
she hurried back into the composing room.
“Tom,” she cried, “that was Mr. Provost calling.”
“Did he cancel the passes?”
“I should say not. He called to say he had just received
a telegram from the Ace Flying Circus saying it would be
at Sandy Point to do stunt flying and carry passengers
for the Fourth of July celebration.”
“Why so excited about that? We’ve had flying circuses
here before.”
“Yes, I know, Tom, but ‘Speed’ Rand is in charge of the
Ace outfit this year.”
“‘Speed’ Rand!” whistled Tom. “Well, I should say that
was different. That’s news. Why Rand’s the man who
flew from Tokyo to Seattle all alone. Other fellows had
done it in teams but Rand is the only one to go solo.
He’s big news in all of the dailies right now. Everyone is
wondering what daredevil stunt he’ll do next.”
“He’s very good looking and awfully rich,” smiled Helen.](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-75-320.jpg)
![[185]
“Flies just for fun,” added Tom. “With all of the oil land
he’s got he doesn’t have to worry about work. Tell you
what, I’ll write to the Cranston Chronicle and see if
they’ll send us a cut of Rand. It would look fine on the
front page of this week’s issue.”
“Oh,” exclaimed Helen “I almost forgot the most
important part of Mr. Provost’s call. He wants you to get
out 10,000 half page bills on the Ace Flying Circus. Here
are the notes. He said for you to write the bill and run
them off as soon as you can.”
The order for the bills put Tom behind on his work with
the paper and it was late Thursday afternoon before
Helen started folding that week’s issue. But they didn’t
mind being late. The bill order from Sandy Point had
meant another piece of profitable job work and Mr.
Provost had also taken a half page in the Herald to
advertise the coming of his main attraction for the
Fourth. Mrs. Blair came down to help with the folding
and Margaret Stevens, just back from a vacation in the
north woods with her father, arrived in time to lend a
hand.
“Nice trip?” Helen asked as she deftly folded the printed
sheets.
“Wonderful,” smiled Margaret, “but I’m glad to get back.
I missed helping you and Tom. Honestly, I get a terrific
thrill out of reporting.”
“We’re glad to have you back,” replied Helen, “and I
think Mr. Provost down at Sandy Point will be glad to
give me an extra pass for the Fourth. I’ll tell him you’re
our star reporter.”](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-76-320.jpg)
![[186]
“I’d rather go to Crescent Beach for the Fourth,” said
Margaret. “It’s newer and much more ritzy than Sandy
Point.”
“You’d better stop and look at the front page carefully,”
warned Tom, who had shut off the press just in time to
hear Margaret’s words.
She stopped folding papers long enough to read the
type under the two column picture on the front page.
“What!” she exclaimed, “‘Speed’ Rand coming here?”
“None other and none such,” laughed Tom. “Guaranteed
to be the one and only ‘Speed’ Rand. Step right this way
folks for your airplane tickets. Five dollars for five
minutes. See the beauty of Lake Dubar from the air.
Don’t crowd, please.”
“Do you still want me to get a pass?” Helen asked. “It
will be honored any place at Sandy Point during the
celebration and Mr. Provost says we can all have rides
with the air circus ‘Speed’ Rand is running.”
“I should say I do want a pass,” said Margaret. “At least
it’s some advantage to being a newspaper woman
besides just the fun of it.”
The famous Ace air circus of half a dozen planes roared
over Rolfe just before sunset Friday night and the whole
town turned out to see them and try to identify the
plane which “Speed” Rand was flying.
The air circus was flying in two sections, three fast, trim
little biplanes that led the way, followed by three large
cabin planes used for passenger carrying. Every ship
was painted a brilliant scarlet and they looked like](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-77-320.jpg)
![[187]
tongues of flames darting through the sky, the
afternoon sun glinting on their wings.
The air circus swung over Rolfe in a wide circle and the
leading plane dropped down out of the sky, its motor
roaring so loud the windows in the houses rattled in
their frames.
“He’s going to crash!” cried Margaret.
“Nothing of the kind,” shouted Tom, who had read
widely of planes and pilots and flying maneuvers.
“That’s just a power dive—fancy flying.”
Tom was right. When the scarlet biplane seemed
headed for certain destruction the pilot pulled its nose
up, levelled off, shot over Rolfe at dizzying speed and
then climbed his craft back toward the fleecy, lazy white
clouds.
“That’s Rand,” announced Tom with a certainty that left
no room for argument. “He’s always up to stunts like
that.”
“It must be awfully dangerous,” said Helen as she
watched the plane, now a mere speck in the sky.
“It is,” agreed Tom. “Everything depends on the motor
in a dive like that. If it started to miss some editor
would have to write that particular flyer’s obituary.”
The morning of Saturday, the Fourth, dawned clear and
bright. Small boys whose idea of fun was to arise at four
o’clock and spend the next two hours throwing cannon
crackers under windows had their usual good time and
Tom and Helen, unable to sleep, were up at six o’clock.
Half an hour later Margaret Stevens, also awakened by](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-78-320.jpg)
![[188]
[189]
the almost continuous cannonading of firecrackers,
came across the street.
“Jim Preston is going to take us down the lake on his
seven-thirty trip before the special trains and the big
crowds start coming in,” said Tom.
“But I’d like to see the trains come in,” protested Helen.
“If we wait until then,” explained Tom, “we’ll be caught
in the thick of the rush for the boats and we may never
get to Sandy Point. We’d better take the seven-thirty
boat.”
From the hill on which the Blair home stood they looked
down on the shore of Lake Dubar with its half dozen
boat landings, each with two or three motorboats
awaiting the arrival of the first special excursion train.
Mrs. Blair called them to breakfast and they were
getting up to go inside when Margaret’s exclamation
drew their attention back to the lake.
“Am I seeing things or is that the old Queen?” she
asked, pointing down the lake.
Tom and Helen looked in the direction she pointed. An
old, double decked boat, smoke rolling from its lofty,
twin funnels, was churning its way up the lake.
“We may all be seeing things,” cried Tom, “but it looks
like the Queen. I thought she had been condemned by
the steamboat inspectors as unfit for further service.”
“The news that ‘Speed’ Rand is going to be at Sandy
Point is bringing hundreds more than the railroad
expected,” said Helen. “I talked with the station agent](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-79-320.jpg)
![[190]
last night and they have four specials scheduled in this
morning and they usually only have two.”
“If they vote the paved roads at the special election
next week,” commented Tom, “the railroad will lose a lot
of summer travel. As it is now, folks almost have to
come by train for the slightest rain turns the roads
around here into swamps and they can’t run the risk of
being marooned here for several days.”
The Queen puffed sedately toward shore. They heard
the clang of bells in the engine room and the steady
chouf-chouf of the exhaust cease. The smoke drifted
lazily from the funnels. Bells clanged again and the
paddle wheel at the stern went into the back motion,
churning the water into white froth. The forward speed
of the Queen was checked and the big double-decker
nosed into its pier.
“There’s old Capt. Billy Tucker sticking his white head
out of the pilot house,” said Tom. “He’s probably put a
few new planks in the Queen’s rotten old hull and
gotten another O. K. from the boat inspectors. But if
that old tub ever hits anything, the whole bottom will
cave in and she’ll sink in five minutes.”
“That’s not a very cheerful Fourth of July idea,” said
Margaret. “Come on, let’s eat. Your mother called us
hours ago.”
They had finished breakfast and were leaving the table
when Mrs. Blair spoke.
“I’ve decided not to go down to Sandy Point with you,”
she said. “The crowd will be so large I’m afraid I
wouldn’t enjoy it very much.”](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-80-320.jpg)
![[191]
“But we’ve planned on your going, Mother,” said Helen.
“I’m sorry to disappoint you,” smiled her mother, “but
Margaret’s mother and I will spend the day on the hill
here. We’ll be able to see the aerial circus perform and
really we’ll enjoy a quiet day here at home more than
being in the crowd.”
“It won’t be very quiet if those kids keep on shooting
giant crackers,” said Tom.
“They’ll be going to the celebration in another hour or
two and then things will quiet down,” said Mrs. Blair.
“How about a plane ride if the circus has time to take
us?” asked Tom.
Helen saw her mother tremble at Tom’s question, but
she replied quickly.
“That’s up to you, Tom. You know more about planes
than I do and if you’re convinced the flying circus is
safe, I have no objection.” But Helen made a mental
reservation that the planes would have to look mighty
safe before any of them went aloft.
They hurried down the hill to the pier which Jim Preston
used. The boatman and his helpers had just finished
polishing the three speed boats Preston owned, the
Argosy, the Liberty and the Flyer, which had been raised
from the bottom of the lake and partially rebuilt.
“All ready for the big day?” asked the genial boatman.
“We’re shy a few hours sleep,” grinned Tom. “Those
cannon crackers started about four o’clock but outside
of that we’re all pepped up and ready to go.”](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-81-320.jpg)
![[192]
“About three or four years ago,” reminded the boatman,
“you used to be gallivantin’ around town with a
pocketful of those big, red crackers at sun-up. Guess
you can’t complain a whole lot now.”
Tom admitted that he really couldn’t complain and they
climbed into the Liberty.
“I’m takin’ some last minute supplies down to the hotel
at Sandy Point,” said the boatman, “so we won’t wait for
anyone else.”
He switched on the starter and the boat quivered as the
powerful motor took hold. They were backing away
from the pier when the pilot of one of the other boats
shouted for them to stop.
A boy was running down Main Street, waving a yellow
envelope in his hand.
Jim Preston nosed the Liberty back to the pier and the
boy ran onto the dock.
“Telegram for you,” he told Helen. “It’s a rush message
and I just had to get it to you.”
“Thanks a lot,” replied Helen. “Are there any charges?”
“Nope. Message is prepaid.”
Helen ripped open the envelope with nervous fingers.
Who could be sending her a telegram? Was there
anything wrong with her father? No, that couldn’t be it
for her mother would have received the message.
She unfolded the single sheet of yellow paper and read
the telegraph operator’s bold scrawl.](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-82-320.jpg)
![[193]
[194]
“To: Helen Blair, The Herald, Rolfe. Understand ‘Speed’
Rand is at Rolfe for two days. Have rumor his next flight
will be an attempted non-stop refueling flight around
the world. See Rand at once and try for confirmation of
rumor. Telephone as soon as possible. McClintock, The
AP.”
Helen turned to Tom and Margaret.
“I’m to interview ‘Speed’ Rand for the Associated Press,”
she exclaimed. “Let’s go!”](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-83-320.jpg)
![[195]
[196]
“Which will mean all the more honor and glory for Helen
if she can get him to tell about his plans,” said Margaret.
“I’ll do my best,” promised Helen and her lips set in a
line that indicated the Blair fighting spirit was on the
job.
They were still more than two miles from Sandy Point
when a scarlet-hued plane shot into sight and climbed
dizzily toward the clouds. It spiralled up and up, the roar
of its motor audible even above the noise of the
speedboat’s engine.
“There’s ‘Speed’ Rand now!” cried Tom. “No one flies
like that but ‘Speed’.”
The graceful little plane reached the zenith of its climb,
turned over on its back and fell away in twisting series
of spirals that held the little group in the boat
breathless.
The plane fluttered toward the lake, seemingly without
life or power. Just before it appeared about to crash, the
propeller fanned the sunlight, the nose jerked up, and
the little ship skimmed over the waters of the lake.
It was coming toward the Liberty at 200 miles an hour.
On and on it came until the roar of its motor drowned
out every other sound. Helen, Tom and Margaret threw
themselves onto the floor of the boat and Jim Preston
crouched low behind his steering wheel.
There was a sharp crash and Helen held her breath. She
was sure the plane had struck the Liberty but the boat
moved steadily ahead and she turned quickly to look for
the plane.](https://image.slidesharecdn.com/2466081-250610042424-060046aa/85/Cabledriven-Parallel-Robots-Proceedings-Of-The-Second-International-Conference-On-Cabledriven-Parallel-Robots-1st-Edition-Andreas-Pott-85-320.jpg)
Cabledriven Parallel Robots Proceedings Of The Second International Conference On Cabledriven Parallel Robots 1st Edition Andreas Pott
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- 5. Andreas Pott Tobias Bruckmann Editors Cable-Driven Parallel Robots Proceedings of the Second International Conference on Cable-Driven Parallel Robots Mechanisms and Machine Science 32
- 6. Mechanisms and Machine Science Volume 32 Series editor Marco Ceccarelli, Cassino, Italy
- 7. More information about this series at http://www.springer.com/series/8779
- 8. Andreas Pott • Tobias Bruckmann Editors Cable-Driven Parallel Robots Proceedings of the Second International Conference on Cable-Driven Parallel Robots 123
- 9. Editors Andreas Pott Fraunhofer Institute for Manufacturing Engineering and Automation IPA Stuttgart Germany Tobias Bruckmann University of Duisburg-Essen Duisburg Germany ISSN 2211-0984 ISSN 2211-0992 (electronic) ISBN 978-3-319-09488-5 ISBN 978-3-319-09489-2 (eBook) DOI 10.1007/978-3-319-09489-2 Library of Congress Control Number: 2012943639 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
- 10. Committee General Chair Andreas Pott, Fraunhofer IPA, Stuttgart, Germany Tobias Bruckmann, University Duisburg-Essen, Germany Scientific Committee Sunil Agrawal, University of Delaware, United States Clément Gosselin, Laval University, Québec, Canada Marc Gouttefarde, LIRMM, Montpellier, France Jean-Pierre Merlet, INRIA, Sophia Antipolis, France Dieter Schramm, University Duisburg-Essen, Germany Alexander Verl, University Stuttgart, Germany v
- 11. Preface First application ideas and concepts for cable-driven parallel robots were presented in the late 1980s. Due to the unique properties of these robots, like huge size of the workspace, high payload, and outstanding dynamic capacities, the potential advantages became obvious and successful application projects seemed to be within grasp. During the following years it became clear that the mechanical simplicity is accompanied by practical issues and theoretical challenges. Accordingly, the real- ization of applications on a reliable and industrial level did not broadly succeed. Thanks to extensive research—also massively driven by many of the contribu- tors to this book—in the recent years numerous questions were answered and several prototypes were realized. Even more, projects in close cooperation with industry or directly funded by industrial companies are currently testing cable- driven parallel robots in productive environments and first products are expected soon. In 2012, leading experts from three continents gathered during the “First International Conference on Cable-Driven Parallel Robots” in Stuttgart, Germany. This conference initiated a forum for the cable robot community that is continued by the “Second International Conference on Cable-Driven Parallel Robots” at the University Duisburg-Essen in 2014. This book summarizes the contributions of the participants of this event. During the lectures it became obvious that practical investigations as well as the stable and reliable control of cable-driven parallel robots are attracting the focus of research teams around the world. We are sure that this pioneers future applications where cable-driven parallel robots enable outstanding solutions in the domains of logistics, handling, production, maintenance, and physical therapy. We are most grateful to the authors for their significant contributions, to the reviewers for their careful feedback, and for the support of the scientific committee that enabled this. We also thank the people at Springer for their efficient support and help. The conference was organized by the University of Duisburg-Essen and the Fraunhofer Institute for Manufacturing Engineering and Automation IPA under the vii
- 12. patronage of International Federation for the Promotion of Mechanism and Machine Science (IFToMM). It is supported by the Förderverein Ingenieurwissenschaften Universität Duisburg-Essen e.V. and the Duisburger Universitätsgesellschaft e.V. as well as by the Rectorate and the Faculty for Engineering of the University Duisburg-Essen. We would like to express our gratefulness to these institutions for their valuable sponsorship. June 2014 Andreas Pott Tobias Bruckmann viii Preface
- 13. Contents Part I Modeling The Forward Kinematics of Cable-Driven Parallel Robots with Sagging Cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Jean-Pierre Merlet and Julien Alexandre-dit-Sandretto An Elastic Cable Model for Cable-Driven Parallel Robots Including Hysteresis Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Philipp Miermeister, Werner Kraus, Tian Lan and Andreas Pott On the Improvement of Cable Collision Detection Algorithms . . . . . . . 29 Dinh Quan Nguyen and Marc Gouttefarde Workspace Analysis of Redundant Cable-Suspended Parallel Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Alessandro Berti, Jean-Pierre Merlet and Marco Carricato On the Static Stiffness of Incompletely Restrained Cable-Driven Robot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Hui Li Simulation and Control with XDE and Matlab/Simulink of a Cable-Driven Parallel Robot (CoGiRo) . . . . . . . . . . . . . . . . . . . . 71 Micaël Michelin, Cédric Baradat, Dinh Quan Nguyen and Marc Gouttefarde Part II Accuracy Presentation of Experimental Results on Stability of a 3 DOF 4-Cable-Driven Parallel Robot Without Constraints . . . . . . . . . . . . . . 87 Valentin Schmidt, Werner Kraus and Andreas Pott ix
- 14. Experimental Determination of the Accuracy of a Three-Dof Cable-Suspended Parallel Robot Performing Dynamic Trajectories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Clément Gosselin and Simon Foucault Efficient Calibration of Cable-Driven Parallel Robots with Variable Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Dragoljub Surdilovic, Jelena Radojicic and Nick Bremer Part III Control Robust Internal Force-Based Impedance Control for Cable-Driven Parallel Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Christopher Reichert, Katharina Müller and Tobias Bruckmann Adaptive Control of KNTU Planar Cable-Driven Parallel Robot with Uncertainties in Dynamic and Kinematic Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Reza Babaghasabha, Mohammad A. Khosravi and Hamid D. Taghirad Dynamic Analysis and Control of Fully-Constrained Cable Robots with Elastic Cables: Variable Stiffness Formulation . . . . . . . . . 161 Mohammad A. Khosravi and Hamid D. Taghirad Adaptive Terminal Sliding Mode Control of a Redundantly-Actuated Cable-Driven Parallel Manipulator: CoGiRo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Gamal El-Ghazaly, Marc Gouttefarde and Vincent Creuze Haptic Interaction with a Cable-Driven Parallel Robot Using Admittance Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Wei Yang Ho, Werner Kraus, Alexander Mangold and Andreas Pott A Kinematic Vision-Based Position Control of a 6-DoF Cable-Driven Parallel Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Ryad Chellal, Loïc Cuvillon and Edouard Laroche Analysis of a Real-Time Capable Cable Force Computation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Katharina Müller, Christopher Reichert and Tobias Bruckmann x Contents
- 15. First Experimental Testing of a Dynamic Minimum Tension Control (DMTC) for Cable Driven Parallel Robots . . . . . . . . . . . . . . 239 Saeed Abdolshah and Giulio Rosati Modeling and Control of a Large-Span Redundant Surface Constrained Cable Robot with a Vision Sensor on the Platform. . . . . . 249 Amber R. Emmens, Stefan A.J. Spanjer and Just L. Herder Part IV Application Cable Function Analysis for the Musculoskeletal Static Workspace of a Human Shoulder. . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Darwin Lau, Jonathan Eden, Saman K. Halgamuge and Denny Oetomo A Reconfigurable Cable-Driven Parallel Robot for Sandblasting and Painting of Large Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Lorenzo Gagliardini, Stéphane Caro, Marc Gouttefarde, Philippe Wenger and Alexis Girin ARACHNIS: Analysis of Robots Actuated by Cables with Handy and Neat Interface Software . . . . . . . . . . . . . . . . . . . . . . 293 Ana Lucia Cruz Ruiz, Stéphane Caro, Philippe Cardou and François Guay Upper Limb Rehabilitation Using a Planar Cable-Driven Parallel Robot with Various Rehabilitation Strategies . . . . . . . . . . . . . 307 XueJun Jin, Dae Ik Jun, Xuemei Jin, Jeongan Seon, Andreas Pott, Sukho Park, Jong-Oh Park and Seong Young Ko Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Contents xi
- 16. Part I Modeling
- 17. The Forward Kinematics of Cable-Driven Parallel Robots with Sagging Cables Jean-Pierre Merlet and Julien Alexandre-dit-Sandretto Abstract Solving the forward kinematics (FK) of parallel robots is known to be a difficult task and the problem is even more complex for cable driven parallel robot (CDPR): the system of equations that has to be solved is larger than with rigid legs as first the static equations have to be taken into account and, second because the deformation of the cables because of their elasticity and their mass may play a role, while being described by a relatively non algebraic complex model. We consider in this paper any arbitrary CDPR whose cables may present a significant deformation due to their elasticity and own mass and we present for the first time an interval analysis based generic algorithm that allows to calculate in a guaranteed manner all the FK solutions and illustrate its use for a CDPR with 8 cables. 1 Introduction Cable-driven parallel robot (CDPR) have the mechanical structure of the Gough plat- form with rigid legs except that the legs are cables whose length may be controlled. We will assume that the output of the coiling system for cable i is a single point Ai , while the cable is connected at point Bi on the platform (Fig.1). Classical kinematics problem are the inverse kinematics (find the lengths of the cables for a given pose of the platform) and the forward kinematics (FK) (find the pose(s) of the platform for given cable lengths). Solving the FK of parallel robots is one of the most challenging problem in modern kinematics. Two categories of FK may be distinguished: J.-P. Merlet (B) · J. Alexandre-dit-Sandretto INRIA, 2004, Route des Lucioles, Sophia-Antipolis, 06902 Cedex, France e-mail: Jean-Pierre.Merlet@inria.fr J. Alexandre-dit-Sandretto e-mail: julien.alexandre-dit-sandretto@ensta-paristech.fr © Springer International Publishing Switzerland 2015 A. Pott and T. Bruckmann (eds.), Cable-Driven Parallel Robots, Mechanisms and Machine Science 32, DOI 10.1007/978-3-319-09489-2_1 3
- 18. 4 J.-P. Merlet and J. Alexandre-dit-Sandretto Fig. 1 Cable driven parallel robots with sagging cables A1 A2 A3 A4 A5 A6 A7 B7 B3 B1 B2 B4 B5 B6 • real-time FK: used for control purposes with the objective of determining the current pose of the robot. It need to be fast (running within a sampling time of the robot controller) and uses the knowledge that the solution is “close” to a known pose. It may also be used for simulation purposes • full FK: the purpose is to determine all solutions of the FK. It is used off-line for determining the possible initial states of the robot when running a simulation We will see later on that the real time FK is not a problem and we will address only the full FK. This problem has been addressed for CDPRDs in a very preliminary stage only very recently and under restrictive assumptions on the behavior of the cables. Indeed we may consider: • non-deformable cables: they are aligned along the direction Ai Bi • elastic cables: they are also aligned along Ai Bi but their lengths depend upon the tension to which they are submitted • catenary cables: they exhibit elasticity and their own mass leads to a deformed shape Most of the kinematic works have assumed non-deformable cables. For robots having at least 6 cables the FK is equivalent to the one of classical parallel robots (for more than 6 cables at most 6 will be under tension simultaneously [1]). With less than 6 cables the problem is still open as the geometrical constraints relating the length of the cables to the pose leads to less equations than unknowns, which imposes to add the 6 additional statics equations and the cable tensions as additional unknowns. For a CDPR with m cables the minimal system has 6 + m equations in 6 + m unknowns, to be compared with the system of 6 equations for the Gough platform. Although there has been progress recently to solve these problems [2–4] there are still a lot of progress to be made in order to determine the maximal number of solutions according to m, solutions that should have only positive tensions and are stable.
- 19. The Forward Kinematics of Cable-Driven Parallel Robots . . . 5 If we assume elasticity in the cables there has been some works for the IK [5, 6] but to the best of our knowledge the FK has never been addressed in the general case. This paper addresses the most general case of FK for CDPR having catenary type cables. Very few works have addressed the IK and FK of such a robot [7–9] and none have considered the general case. 2 Problem Statement 2.1 Cable Model We will assume cables with linear density μ, cross-section A0 and E will denote the Young modulus of the cable material. A reference frame O, x, y, z will be used and the coordinates of the Ai points are known in this frame. In the vertical plane of the cable we may assume that the cable is attached at point A with coordinates (0,0) while the other extremity is attached at point B with coordinates (xb, yb) (Fig.2). The vertical and horizontal forces Fz, Fx are exerted on the cable at point B and the cable length at rest is L0. With this notation the coordinates of B are related to the forces Fx , Fz [10] by: xb = Fx ( L0 E A0 + sinh−1(Fz) − sinh−1(Fz − μgL0 Fx ) μg ) (1) zb = F2 x + F2 z − F2 x + (Fz − μgL0)2 μg + Fz L0 E A0 − μgL2 0 2E A0 (2) 2.2 The FK Problem We consider a spatial robot with m cables whose lengths will be denoted L1 0, . . . , Lm 0 . Without loss of generality we will assume that the Young modulus, linear density and cross-section of the cables are identical. The problem we have to solve is to determine all the possible poses of the platform being given the L j 0 and the location of the Ai , together with the external forces/torques F that act on the platform. In terms of unknowns we will assume a minimal representation of the pose with 6 parameters
- 20. 6 J.-P. Merlet and J. Alexandre-dit-Sandretto A(0,0) B(xb,yb) Fx Fz Fig. 2 A deformed cable and we have also the 2m unknown forces Fi x , Fi z , for a total of 6 + 2m unknowns. Note that to express these forces in the reference frame we need to establish a rotation matrix Rot that rotates the planar frame around the z axis by an angle ui so that in the reference frame we get the force acting on the platform F as F = Rot(−Fx , 0, −Fz)T and hence for cable j Fj = (−F j x cos u j , F j x sin u j , F j z ). If we assume that the external force acting on the platform is the gravity and that the platform mass is M the mechanical equilibrium imposes that: j=m j=1 F j x cos u j = 0 j=m j=1 F j x sin u j = 0 j=m j=1 F j z = Mg (3) If C is the center of mass of the platform we get also j=m j=1 CBj × Fj = 0 (4) In terms of equations we have the 2m Eqs.(1), (2) and the 6 static equations that express the mechanical equilibrium of the platform. Hence we end up with 6 + 2m equations so that solving the FK requires to solve a square system, which will usually haveafinitenumberofsolutions.ItmayalsobeenseenthattheFKinthatcaseismuch more complex than the FK of the Gough platform (it has 2m additional equations) and that the classical methods used to determine an upper bound of the maximum number of solutions (Bezout number, elimination, Gröbner basis) cannot be applied here as Eqs.(1), (2) are not algebraic.
- 21. The Forward Kinematics of Cable-Driven Parallel Robots . . . 7 3 Solving the FK As a theoretical solving appears to be difficult to be used we will have to resort to a numerical solving method, that has to provide all the solutions. We will use an interval analysis (IA) approach, which guarantees to find all the solutions lying within some given ranges. The basis of IA is the interval evaluation: being given a function f (x1, . . . , xn) in n variables and assuming that each variable xi lies in the range [xi , xi ] the interval evaluation of f is a range [A, B] such that ∀xi ∈ [xi , xi ], i ∈ [1, n] we have A ≤ f (x1, . . . , xn) ≤ B. There are multiple ways to define an interval evaluation but the most simple is the natural evaluation: each mathematical operator has an interval equivalent (for example the addition interval operator + is defined as [a, b] + [c, d] = [a + c, b + d]) and transforming any function by using the interval operators allows to calculate the interval evaluation. One of the property of the interval evaluation [A, B] is that if A 0 of B 0, then f cannot cancel whatever is the value of the variables in their ranges. Note that an interval evaluation may be overestimated: there may not be value of the variables in their respective range such that f (x1, . . . , xn) = A or f (x1, . . . , xn) = B. Indeed overestimation may occur because of multiple occurrences of a given variable that are considered as independent: for example the evaluation of x − x when x ∈ [−1, 1] is [−2, 2]. But the size of the overestimation decreases with the width of the variable ranges. The second key ingredient of IA is the branch and bound algorithm. A box B is defined as a set of ranges for the variables. If for a given box we have f (B = [A, B] with A 0, B 0, then we select one of the variable, xi bisect its range in two and create two new boxes B1, B2 that are identical to B except for the range for xi which result from the bisection. These boxes are stored in a list and will be processed later on. We will see in the next section that if a box is tiny enough we may determine if it includes a single solution and compute this solution with an arbitrary accuracy. However we will not use the minimal set of equations for the FK. Indeed the pose of the platform will not be represented by the coordinates of C and three orientation angles. The motivation is that coordinates xb, zb in (1, 2) will be obtained after using the rotation matrix and will include several occurrences of the rotation angles, possibly leading to large overestimation. We prefer to represent the pose of the platform by the 12 coordinates in the reference frame of 4 of the points Bi , which are not coplanar (we assume here that B1, B2, B3, B4 are chosen). With this choice the coordinates of any point M on the platform may be obtained as OM = α1OB1 + α2OB2 + α3OB3 + α4OB4 where the α are known constants. Such a representation allows one to obtain the coordinates of the Bj , j 4 and of the center of mass C. As we have now 12 unknowns for representing the pose of the platform instead of 6 with the minimal representation we need 6 additional equations that are obtained by stating that the distance between a pair of points in the set B1, . . . B4 is a known constant. Note that these equations are not sufficient to fully describe the geometry of the platform (e.g. the equations does not allow to differentiate if a Bj point is over or under the plane that includes the three other points). Another test is needed and we will use the fact that for any point M of the platform there exist constants βj such that
- 22. 8 J.-P. Merlet and J. Alexandre-dit-Sandretto BjM = β1BjBk + β2BjBl + β3BjBk × BjBl, j, k,l ∈ [1, 4] (5) It remains to manage the angles ui : formally they can be obtained using the equation x j b sin(u j ) + y j b cos(u j ) = 0 (6) where x j b , y j b are the coordinates of Bj in the reference frame. But by so doing their interval evaluation will be relatively large even for small width for the x j b , y j b intervals, especially if j 4. Hence we prefer to add them as additional unknowns and to use (6) as additional equations. Hence we end up with 12 + 3m unknowns for 12 + 3m equations. Note that we have checked that solving this system in the context of a real-time FK is not difficult as soon as a certified strategy is used [11]: this strategy allows one to determine the current pose of the robot or eventually that the pose is too close to a singularity (in which case the FK has an infinite number of solutions). An IA approach impose to determine a domain in which are located all the solu- tions. This can be easily done for the x j b , y j b variable that are restricted to lie within the convex hull of the Ai points. This can also be done for z j b, the z coordinate of Bj that cannot be greater than the highest z coordinate of the Ai points and cannot be lower than the length obtained if we assume that the cable is vertical and bears the platform. As for the ui as the cable have to lie within the convex hull of the Ai we can also get bounds for these variables. It remains the variables Fx , Fz which have no natural bounds except that Fx cannot be negative. We will first define m new variables λj such that F j z = λj Fx (which allows one to have simpler expressions for (1, 2). We take then as upper bound for Fx 10 times the value of mg and for the λ a range of [−10, 10] (at the extremities of this range the cables are almost vertical). 3.1 Determining Exact Solutions The classical IA branch and bound algorithm assume that if the width of a box is smaller than a small value and the interval evaluations of all equations include 0, then we have found an approximate solution of the system. Here we proceed in another way: for each box of the algorithm we run a few iterations of the Newton-Raphson scheme with as estimate of the solution the center of the box H. Note that even if the NR algorithm converges there is no guarantee (1) that the result is indeed an approximate solution of the system, (2) that the solution lie within the box or even within the search space, (3) that the result satisfies the constraint (5). In order to check if the result is really a solution of the system we use Kantorovitch theorem [11] that allows one to verify that there is indeed a single solution of the system in a ball centered at H with a known radius. If this test succeed we have furthermore the property that the NR scheme, initialized with H as guess point, converges toward
- 23. The Forward Kinematics of Cable-Driven Parallel Robots . . . 9 the solution. We will see in the implementation section that this property will allow us to compute an approximation of the solution with an arbitrary accuracy. As soon as a solution H0 is found it is stored and in a first step we will assume that there is no other solution in a ball centered at H0 with a given radius, this being applied only on the 12 coordinates of the point B1, . . . , B4. Any box that is fully included in this ball will be eliminated and if a box has an intersection with the ball, then the intersection part will be removed from the box. Our purpose in this first step is to determine balls that include a solution and possibly others. In a second step we will run the algorithm on this ball and this check will be faster because the search domain will be drastically reduced. With this approach the IA algorithm is guaranteed to complete. 3.2 Heuristics A drawback of the usual IA branch and bound algorithm that eliminates boxes only according to the interval evaluation of the equations is that is not efficient as soon as we have complex equations with multiple occurrences of the variables. But several heuristics allows one to drastically improve the efficiency of the algorithm. A first set of heuristics is called the consistency approach, which is based on a rewriting of the equations. Consider for example the equation that described that the distance between the points of a pair of Bi point is constant. This equation is written as (xi b − x j b )2 + (yi b − y j b )2 + (zi b − z j b)2 = d2 i j (7) which may be rewritten as (xi b − x j b )2 = d2 i j − (yi b − y j b )2 − (zi b − z j b)2 Let [A, B] denote the interval evaluation of the right hand-side of this equation. We deduce that • if B 0 the equation has no solution • if A ≤ 0, B ≥ 0 then − √ B + x j b ≤ xi b ≤ −x j b + √ B • if A ≥ 0 then xi b belongs to [− √ B + x j b , − √ A + x j b ] ∪ [ √ A + x j b , √ B + x j b ] With this approach we may improve the range for any variable in the equation or even eliminate a box without having to use the bisection process. It is important to note that if the set of variable is denoted x and we are able to write an equation under the form g(xi ) = G(x) the consistency requires an inverse operator of g in order to be able to update xi . This also motivate our choice not to use the minimal representation of the pose but a more algebraic formulation whose inverse is trivial. In our implementation the consistency is applied on all equations of the system and for all variables. It is
- 24. 10 J.-P. Merlet and J. Alexandre-dit-Sandretto also used on Eq.(5) and on the equations BiBj.BiBk = di j dik cos(θ) where θ is the known angle between the lines going through (Bi , Bj ) and (Bi , Bk). Another efficient heuristic is the 3B method. Assume that we have a box and select one of the variable xi whose range is [xi , xi ]. We change this range to [xi , [xi + ε], where ε has a small value. Interval evaluation of the equations and the consistency are used to determine if this new box may include a solution. If the answer is negative we can safely modify the initial box by setting the range for xi to [xi + ε, xi ]. This process is applied for all variables but also on the right side of the interval for xi . Another approach is used for the equations that have multiple occurrences of the same variable. We calculate the gradient of these equations and it interval evaluation for the current box. If this evaluation has a constant sign we set the variable to the appropriate lower or upper bound of the range to improve the interval evaluation. This process has to be recursive: as soon as a variable is set to an extremity of its range, then the interval evaluation of the gradient for another variable may become of constant sign. Another important issue is the choice of the variable that will be selected for the bisection process. Our strategy is to bisect in priority the 12 coordinates of the B1, . . . B4 until the width of their interval is lower than a given threshold. Indeed these variables play an important role in the equations: if they are fixed the Eqs.(1), (2) admit a single solution that correspond to the minimal potential energy of the cables. 3.3 Implementation The algorithm is implemented using the interval arithmetics of BIAS/PROFIL1 while the higher level uses the functions of our ALIAS library2 that mixes a C++ library and a Maple interface. The Maple interface has allowed to generate most of the C++ code for the algorithm and includes an arbitrary accuracy Newton scheme which allow us to calculate an approximation of the solution with n digits, the n-th digit being guaranteed to be exact, n being a number given by the end-user. Another property of the Maple interface is that it allows one to implement the algorithm in a distributed manner, i.e. running the algorithm on several computers. Indeed it must be noticed that in the solving scheme the treatment of a given box is independent from the treatment of the other boxes to be processed. This allow to have a master program that manages the list of boxes to be processed and the list o solutions and an arbitrary number m of slave computers. The master computer process the initial box until it has a fixed number of boxes in its list. Then it sends the top boxes to the slave computer that a few iterations of the solving algorithm and send back to the master the eventual solution and the boxes that remain to be 1 http://www.ti3.tuhh.de/keil/profil/index_e.html. 2 http://www-sop.inria.fr/coprin/developpements/main.html.
- 25. The Forward Kinematics of Cable-Driven Parallel Robots . . . 11 Fig. 3 The robot developed for the ANR Cogiro project. Although the robot is real we present a CAD drawing that allows one to better figure out the CDPR Table 1 Coordinates of the attachment points on the base (in meters) x y z x y z −7.175 −5.244 5.462 −7.316 −5.1 5.47 −7.3 5.2 5.476 −7.161 5.3 5.485 7.182 5.3 5.488 7.323 5.2 5.499 7.3 −5.1 5.489 7.161 −5.27 5.497 treated. As the communication overhead is small compared to the computing time of the algorithm the distributed version allows to divide the processing time by m. 4 Example We consider the large scale robot developed by LIRMM and Tecnalia as part of the ANR project Cogiro [7] (Fig.3). This robot is a suspended CDPR (i.e. there is no cable pulling the platform downward) with 8 cables, whose Ai coordinates are given in Table1. The cables characteristics are E = 1009 N/m2, μ = 0.346kg/m and their diameter is 10mm. The mass of the platform of 10kg. The value of the L0s (Table2) are the non-deformed cable lengths for the pose (1, 0, 2) in meters and for an orientation such that the reference frame and the mobile frame are aligned. Table 2 Lengths of the cable at rest (in meters) 1 2 3 4 10.48215 9.838952 10.16035 8.96827 5 6 7 8 10.310003 8.421629 8.663245 8.655556
- 26. 12 J.-P. Merlet and J. Alexandre-dit-Sandretto Table 3 The coordinates of B1, B2, B3 for the 19 solutions (meter) Sol x1 y1 z1 x2 y2 z2 x3 y3 z3 1 1.55 0.43 4.26 0.04 0.21 4.89 1.05 −0.35 4.71 2 0.57 −0.08 3.53 1.47 −0.80 4.71 0.44 −0.89 4.15 3 1.45 −0.47 2.37 0.42 0.34 3.39 0.43 −0.27 2.38 4 1.27 0.65 4.41 −0.04 −0.31 4.11 1.10 −0.37 4.39 5 0.37 0.47 3.96 1.27 −0.33 5.10 1.34 0.11 4.01 6 0.91 0.98 3.45 0.94 0.04 4.81 1.64 0.64 4.09 7 1.53 −0.74 3.40 0.47 0.53 3.49 1.13 −0.18 4.16 8 1.46 −0.42 4.29 0.52 0.33 3.16 0.94 0.46 4.25 9 1.06 0.64 4.06 1.00 −0.60 2.97 0.93 −0.38 4.12 10 1.66 −0.07 3.45 0.46 0.77 4.20 0.94 −0.30 4.16 11 0.71 0.64 4.10 1.08 −0.64 3.11 0.58 −0.39 4.14 12 1.65 −0.32 4.99 0.73 −0.07 3.64 0.83 −0.78 4.57 13 0.29 0.69 3.64 1.11 −0.67 3.17 0.12 −0.32 3.71 14 1.39 0.45 3.81 0.55 −0.23 2.55 0.88 0.81 2.99 15 1.28 0.53 3.78 0.11 0.01 4.82 1.22 −0.20 4.50 16 1.04 0.90 3.63 0.54 −0.12 4.82 1.52 0.33 4.34 17 1.37 0.52 3.59 0.19 −0.64 3.63 0.44 0.40 3.15 18 0.71 0.92 3.69 0.53 −0.69 4.01 −0.08 0.31 3.94 19 1.86 −0.15 4.12 0.27 −0.60 4.19 1.18 −0.31 4.88 With 8 cables we have to solve a system of 36 equations and this is probably the most challenging FK task that has even been considered. The solving algorithm has been implemented using 10 computers and nineteen solutions were found in the search domain in a computation of about 24h. They are presented in Table3, while the cable tensions are given in Table4. The solutions are depicted in Fig.4. It is interesting to note that the solution poses are distributed all over the possible workspace: for example the x, y, z coordinates of B1 are in the ranges [0.29, 1.86], [−0.74, 0.98], [2.37, 4.99]. The Fx forces exhibit also a very large range. For example for cable 1 this force ranges from 20.08 to 417N. We observe the same variation for the Fz force: for the same cable its ranges from −30.25 to 15.91N. In 15 cases on 19 the FZ tension in cable 1 is positive, meaning that the cable exert a downward force on the platform. The number of cables that exert an upward force to support the load is either 2, 3 or 4, meaning that only a small subset of cables contributes to this support: this may be an useful information for dimensioning the cable.
- 27. The Forward Kinematics of Cable-Driven Parallel Robots . . . 13 Table 4 Cable tensions for the 19 solutions (Newton) 1 Fx 417.00 22.12 78.64 421.40 14.78 17.25 48.67 98.43 1 Fz −30.25 15.05 11.19 −43.41 14.86 12.21 4.61 12.48 2 Fx 20.08 138.03 48.65 160.06 292.98 26.46 361.66 11.28 2 Fz 12.79 5.90 10.45 3.60 −16.58 7.61 −38.66 11.63 3 Fx 53.41 51.34 48.02 57.21 55.95 56.22 53.79 61.07 3 Fz 0.49 5.01 1.03 −6.48 5.28 −0.82 −6.66 −1.12 4 Fx 63.89 17.85 429.43 41.10 14.32 73.80 129.88 305.89 4 Fz 11.08 13.29 −29.11 6.18 14.03 5.90 −13.06 −11.58 5 Fx 21.49 242.38 298.60 26.28 19.78 251.91 315.03 21.29 5 Fz 13.77 7.43 −26.40 10.61 16.49 −1.49 −36.82 13.14 6 Fx 76.90 68.82 245.02 15.36 17.30 197.57 129.34 101.20 6 Fz 2.37 11.95 −16.54 11.72 14.32 −16.60 −17.55 7.06 7 Fx 30.99 134.64 90.89 119.67 31.69 63.39 20.24 114.91 7 Fz 10.63 −11.15 5.19 −4.05 7.42 −7.84 8.26 −11.74 8 Fx 29.99 112.30 27.31 37.41 93.70 117.12 19.86 119.86 8 Fz 13.87 −10.68 13.34 9.01 −4.54 −17.91 10.88 −17.24 9 Fx 43.32 78.79 68.24 21.31 66.92 108.11 19.48 125.50 9 Fz 11.47 −4.42 7.80 10.23 −1.06 −18.03 9.69 −18.95 10 Fx 103.16 170.17 57.39 269.24 21.03 13.32 42.69 51.20 10 Fz −2.67 −5.56 9.40 −36.83 11.86 10.12 1.69 8.73 11 Fx 28.80 99.44 33.76 24.86 90.54 109.52 22.93 114.64 11 Fz 13.35 −8.23 12.24 10.64 −4.98 −18.65 10.11 −17.74 12 Fx 37.36 48.75 270.98 94.64 107.81 16.26 11.05 384.16 12 Fz 15.91 6.93 −7.06 −2.23 −0.10 11.65 12.15 −40.52 13 Fx 23.40 95.81 23.16 54.19 112.26 76.11 74.82 75.04 13 Fz 12.62 −6.67 12.17 6.41 −8.11 −11.21 1.71 −10.19 14 Fx 85.67 52.69 26.80 44.85 89.57 98.61 91.56 42.53 14 Fz 3.87 −0.39 9.28 2.15 −1.29 −12.44 −2.41 −2.02 15 Fx 70.67 21.20 442.79 124.43 14.12 26.91 145.75 307.32 15 Fz 5.97 14.92 −25.45 −2.25 14.31 9.76 −19.27 −1.26 16 Fx 96.61 27.71 316.92 20.56 15.64 175.25 131.51 167.57 16 Fz 0.35 14.57 −18.38 11.45 14.25 −12.87 −17.14 4.51 17 Fx 227.68 18.81 22.23 124.03 52.72 123.33 67.08 30.85 17 Fz −23.60 11.93 10.69 −9.49 10.80 −12.77 4.35 4.83 18 Fx 38.88 22.14 16.06 59.80 304.33 39.15 209.62 104.55 18 Fz 10.48 12.55 13.66 9.15 −28.04 3.34 −12.75 −11.67 19 Fx 142.07 19.07 208.49 104.84 101.23 68.68 11.11 333.35 19 Fz −0.72 13.37 4.85 6.58 −3.75 −2.84 11.52 −32.30
- 28. 14 J.-P. Merlet and J. Alexandre-dit-Sandretto Fig. 4 Solutions 1–19 5 Conclusions WehavepresentedforthefirsttimeagenericalgorithmtosolvetheFKforCDPRwith sagging cables. This a computer intensive algorithm (because of the complexity of the problem), that is however guaranteed to provide all solutions. A test case of a robot with 8 cables (probably one of the most complex that has been studied) has shown that we may obtain surprising poses. As prospective our objective is to determine a better balance between the various heuristics that are used in the solving. We will also study the stability of the solutions, possibly introducing stability condition as an additional solving heuristic in order to speed up the computation.
- 29. The Forward Kinematics of Cable-Driven Parallel Robots . . . 15 Acknowledgments This research has received partial funding from the European Community’s Seventh Framework Program under grant agreement NMP2-SL-2011-285404 (CABLEBOT). References 1. Merlet J-P (2012) The kinematics of the redundant N-1 wire driven parallel robot. In: IEEE international conference on robotics and automation, Saint Paul, 14–18 May 2012, pp 2313– 2318 2. Carricato M, Merlet J-P (2013) Stability analysis of underconstrained cable-driven parallel robots. IEEE Trans Robot 29(1):288–296 3. Carricato M, Abbasnejad G (2012) Direct geometrico-static analysis of under-constrained cable-driven parallel robots with 4 cables. In: 1st international conference on cable-driven parallel robots, Stuttgart, 3–4 Septembre 2012, pp 269–286 4. Jiang Q, Kumar V (2010) The inverse kinematics of 3-d towing. In: ARK, Piran, June 28– July 1, 2010, pp 321–328 5. Merlet J-P (2012) Managing the redundancy of N-1 wire-driven parallel robots. In: ARK, Innsbruck, 25–28 June 2012, pp 405–412 6. Such M et al (2009) An approach based on the catenary equation to deal with static analysis of three dimensional cable structures. Eng Struct 31(9):2162–2170 7. Gouttefarde M et al (2012) Simplified static analysis of large-dimension parallel cable-driven robots. In IEEE international conference on robotics and automation, Saint Paul, 14–18 May 2012, pp 2299–2305 8. Kozak K et al (2006) Static analysis of cable-driven manipulators with non-negligible cable mass. IEEE Trans Robot 22(3):425–433 9. Riehl N et al (2009) Effects of non-negligible cable mass on the static behavior of large workspace cable-driven parallel mechanisms. In: IEEE international conference on robotics and automation, Kobe, 14–16 May 2009, pp 2193–2198 10. Irvine HM (1981) Cable structures. MIT Press, Cambridge 11. Merlet J-P (2004) Solving the forward kinematics of a Gough-type parallel manipulator with interval analysis. Int J Robot Res 23(3):221–236
- 30. An Elastic Cable Model for Cable-Driven Parallel Robots Including Hysteresis Effects Philipp Miermeister, Werner Kraus, Tian Lan and Andreas Pott Abstract Experimental results indicate that time invariant linear elastic models for cable-driven parallel robots show a significant error in the force prediction during operation. This paper proposes the use of an extended model for polymer cables which allows to regard the hysteresis effects depending on the excitation amplitude, frequency, and initial tension level. The experimental design as well as the parameter identification are regarded. 1 Introduction A cable-driven parallel robot, in the following simply called cable robot, is a parallel kinematic machine mainly consisting of a platform, cables, and winches as shown in Fig.1. The cables connect the platform to the winches which in turn control the platform pose by changing the cable length. The control inputs for the winches usually are computed using a simplified kinematic robot model which regards the platform and frame geometry under the assumption that the attachment points at the platform and the contact points at the winch are time invariant. Methods for design and workspace computation of such systems can be found in [3, 4, 10, 12]. Extended kinematic models also include the pulley geometry at the winches and result in higher positioning accuracy [9]. The cable robot IPAnema at Fraunhofer IPA uses Dyneema cables instead of conventional steel cables which brings the advantage of the lower weight but at the same time introduces a more complex elastic behavior in the most relevant force transition element of the cable robot. It showed that the Dyneema polyethylene cables have a changing elastic behavior over time, are subject to settling effects, are sensitive to overload, and show hysteresis effects. Since it is very difficult tobuildthemodelsandidentifyingtherelatedparametersusingmodelsfromdifferent fields such as tribology, viscoelasticity, and multibody systems, here we propose a black box approach to model the drive chain. While white box modeling demands a P. Miermeister (B) · W. Kraus · T. Lan · A. Pott Fraunhofer IPA, Stuttgart, Germany e-mail: philipp.miermeister@ipa.fraunhofer.de © Springer International Publishing Switzerland 2015 A. Pott and T. Bruckmann (eds.), Cable-Driven Parallel Robots, Mechanisms and Machine Science 32, DOI 10.1007/978-3-319-09489-2_2 17
- 31. 18 P. Miermeister et al. Fig. 1 Overconstrained cable-driven parallel robot very good knowledge of the inner relations of a system, black box modeling tries to identify the system behavior by observing the input output behavior of a system. In the fist part of the paper the modeling of the cables is shown. The second part deals with the parameter identification, while the third part shows the evaluation of the model for different input signals. 2 Robot and Cable Elasticity Model The minimal robot model usually running in a numerical control to operate a robot is based on a solely geometrical model as shown in Fig.2 including the m platform and winch attachment points bi and ai with i = 1 . . . m, respectively. The cables can be modeled in different ways depending on the demanded degree of accuracy. The most simple model is just the geometrical model without elastic behavior, meaning that the inverse kinematic equation
- 32. An Elastic Cable Model for Cable-Driven Parallel Robots . . . 19 Fig. 2 Kinematic loop for one cable li = ai − (r + Rbi ) (1) can be used to compute the cable length li = li 2 from a given platform posi- tion r and rotation R. More complex models deal with pulleys by introducing time variant anchorage points and handle non negligible cable mass by utilizing catenary equations to model the cable sagging [5, 11]. Since the mass of the cables of the IPAnema robot is small in comparison to the applied tension level it is not necessary to regard the cable sagging in the model. For cable force control, admittance control, and parameter identification it is necessary to predict the cable force very accurately. The force and torque equilibrium of a cable robot is described by the structure equation u1 · · · um b1 × u1 · · · bm × um AT ⎡ ⎢ ⎣ f1 . . . fm ⎤ ⎥ ⎦ f + fp τp w = 0 (2) where AT is the well known structure matrix [12], ui = li li −1 2 is a unit vector in direction of the cables, f is the vector of cable forces and w is the external wrench. Going from the kinematic model to a linear elastic model of type
- 33. 20 P. Miermeister et al. fi = fi = ci Δli for Δli 0 0 for Δli 0 (3) with Δli = lSP,i − li where ci is the cable stiffness and lSP,i is the set point for the cable length while li is the actual cable length in the workspace one can achieve a more accurate prediction. Modeling the cable forces in this way gives acceptable results for positive cable tensions, but leads to numerical problems in case of Δli 0 due to the discontinuity and the zero area. Comparing the linear elastic model with experimental data showed that cables also come along with a hysteresis behavior which depends on the actuation frequency, amplitude, tension level and amplitude as well as the current angle of the cables with regards to the redirection pulleys of the robot. Different models such as the Bouc– Wen-Model, Bilinear Model, Bingham Model, or Polynomial model can be used to describe the hysteresis behavior [1, 2, 6]. A single tensioned cable of the robot can be seen as single degree of freedom system with an elastic–plastic behavior whose hysteresis by means of the cable elongation x can be described by two polynomial functions, one for the upper part and one for the lower part FU(x) = nP i=0 ai xi , ẋ 0 (4) and FL(x) = nP i=0 (−1)i+1 ai xi , ẋ 0, (5) respectively. Under the assumption of displacement anti-symmetry one can compute the hysteresis function by combining Eqs.(4) and (5) to F(x, ẋ) = g(x) + h(x)sgn(ẋ) (6) with the polynomials g(x) = a1 + a3x3 + · · · + am xng h(x) = a0 + a2x2 + · · · + anxnh (7) where ng is an odd number and nh = ng − 1. The degree of the polynomials can be chosen according to the expected accuracy. The sum of the polynomials g(x) and h(x) can be interpreted as the superposition of an anhysteretic nonlinear elastic part and hysteretic nonlinear damping part as shown in Fig.3. The expression in Eq.(6) allows to describe the hysteresis for a certain amplitude A independent of the current velocity state. Experiments showed that the hysteretic behavior of the cable is different comparing very small an very large amplitudes. Extending the model in order to deal with variable amplitudes and replacing the solely polynomial description by a velocity depended damping expression one can write
- 34. An Elastic Cable Model for Cable-Driven Parallel Robots . . . 21 Fig. 3 Decomposition of a hysteresis curve into its anhysteretic nonlinear elastic part and its nonlinear damping part F = K0 + K1(A)x + K2(A)x2 + K3(A)x3 + Sa|v|α(A) sgn(v) A(Aω)α(A) lπ (8) where the first amplitude dependent polynomial part, now describes the nonlinear non hysteretic spring behavior and the second part of the equation deals with the velocity depended hysteretic damping forces. The hysteretic part is parameterized by the area of the hysteresis Sa and the viscoelastic damping factor α. In case of α = 0 the second part of Eq.(8) gets independent of the velocity representing dry friction behavior. The energy dissipation in one cycle for a given friction behavior is l = T/2 0 sin (ωt)α(A)+1 dωt T/2 0 sin (ωt)2 dωt = 2π T/2 0 sin (ωt)α(A)+1 dωt (9) where T is the time of oscillation and ω is the angular frequency (Fig.3). 3 Design of Experiments and Parameter Identification Considering Eq.(8) for a given amplitude level, one has a five dimensional parameter space where the first parameter K0 determines the pretensioned cable state with A = 0 and v = 0 reducing the identification problem to four dimensions pM = K1 K2 K3 Sa T . (10)
- 35. 22 P. Miermeister et al. Table 1 Parameter sensitivity K1 K2 K3 Sa Sensitivity 1.1 0.1 0.03 0.3 The amplitude model is considered separately. While the parameter identification problem for pM has to deal with the problem of finding the inverse mapping function from four to one dimensions pM = f −1 M (rM), the identification of pM,i = fA,i(A) is just a one dimensional problem. For an efficient measurement and identification process it is important to have a good selection of model parameters and measure- ment samples. A good selection of model parameters means a high sensitivity with regards to the objective function and a good selection of samples means maximized information gain with a minimal number of samples. To obtain the optimal number of parameters necessary for the model identification, the sensitivity of the parameters is computed by S = O2 − O1 Ō12 I2 − I1 ¯ I12 −1 . (11) This sensitivity metric gives the relative normalized change of the output O with regards to the input I using the averages Ō and ¯ I. Parameters with a low sensitivity do not have to be regarded in the model. As can be seen in Table1, elements with an order higher than three can be neglected in the model. The influence of the parameters K1 and Sa are visualized in Figs.4 and 5. The selection of measurement samples was done according to a D-optimal design of experiments giving a set of tuples (pM,i , rM,i ), i = 1 . . . nD where nD is the num- ber of measurements and rM,i is the ith error function for the parameter identification problem given by ri (p) = F(p, xi ) − FM,i (12) Fig. 4 Parameter sensitivity of K1
- 36. An Elastic Cable Model for Cable-Driven Parallel Robots . . . 23 Fig. 5 Parameter sensitivity of Sa Assuming only small deviations Δp it is possible to use linearization around p0 which gives the Jacobian Jrp. From that the optimum parameter set can be found by minimizing the least squared error using the objective function popt = min 1 2 nD i=0 F(p, xi ) − FM,i 2 . (13) Using the linearization one can compute the minimal solution for Eq.(13) by solving the well known normal equation JT rpJrpΔp = JT rpΔr. (14) In case that the error of the model is small, a local optimization scheme such as the Levenberg Marquardt algorithm can be used to find the optimal parameter set. Having a good initial guess p0 is essential for a successful parameter identification. It showed that local optimization is a good choice for long term parameter tracking or repeated adjustment after certain time periods, but did not work well for the very first identification process. Nonlinearities and the huge initial errors demanded for more global optimization techniques such as simulated annealing or genetic optimization procedures which can be used to identify the global optimum. 4 Experimental Results For measurement, the platform is fixed at the origin such that no interaction between the cables can occur. The excitation function for the cable elongation and it’s asso- ciated velocity is chosen as
- 37. 24 P. Miermeister et al. Table 2 Parameter sensitivity K1 K2 K3 Sa Sensitivity −16.8863 2.411 1.0472 1.1698 Δl = −A cos(ωt) (15) Δ˙ l = v = Aω sin(ωt) For this excitation function the damping force can be calculated as Fh = c|Aω sin(ωt)|α sgn(sin(ωt)) (16) where c is damping function c = Sa ⎛ ⎝2A(Aω)α T/2 0 sin (ωt)α+1 dωt ⎞ ⎠ . (17) with Sa describing the area of the hysteresis function. The cables were actuated with an amplitude ranging from 0.1mm up to 1.5mm. The pretension level was chosen at 60N. Running the test was done by generating the cable length set points according to the experimental design using Matlab. The length for each cable in the joint space was commanded by a TwinCat3 controller connected to industrial synchronous servo motors. Force sensors between the cables and the platform were used to measure the cable forces. The force data together with the actual cable lengths were stored in a csv file by TwinCAT and used for model identification in Matlab. Using the actual cable lengths instead of the commanded cable lengths is important to reduce errors introduced by system dead time and controller delay. The results of the parameter identification process can be found in Table2. The comparison of measurements and simulation results after the identification process gave an average model error of 0.4N for a static pose. Checking the model prediction after a few days of operation, the model and the real system already started to diverge as can be seen in Fig.6 where the cable force already shows an offset of 0.4N to the previously created model. This may be caused by temperature changes, cable settlement and high tension states resulting in lasting changes in the cable’s elastic behavior. Using the model for a sinusoidal excitation, a randomly shaped signal can be approximated by a Fourier decomposition. A trapezoidal and a triangular signal where used as test signals to verify the model for different shaped inputs. The Fourier decomposition of the triangular function for example is given by f (t) = 8A π2 (sin(ωt) − 1 32 sin(3ωt) + 1 52 (sin(5ωt)) − · · · ). (18) The comparison of the curve progression of the Fourier based force model and the actual measurements are shown in Fig.7 for an approximation with the first three
- 38. An Elastic Cable Model for Cable-Driven Parallel Robots . . . 25 Fig.6 Comparisonofhysteresismodel andmeasurementsafterapproximatelyfordaysofoperation Fig. 7 Triangular test signal with Fourier approximation Fourier coefficients. The related error shown in Fig.8 has a maximum magnitude of ±1.5N at the peak points. The model prediction is based on the whole set of measurement samples which provides knowledge about the past and future of the signal at a certain time stamp. Using the model in a real scenario one has to use
- 39. 26 P. Miermeister et al. Fig. 8 Hysteresis function and error for a triangular test signal Fig. 9 Pulley assembly of the cable-robot demonstrator. a Pulley for cable redirection. b Cable and bearing forces a continuous-time Foruier transformation to deal with the unknown future of the signal, or the numerical control has to feed forward the positioning set points of the cables. While the prediction for a small area around the initial pose was accurate, experimental results showed that the model error increased depending on the platform pose in the workspace. This is is caused by the elasticity of the pulley mounting and the damping behavior of the pulley bearings. To get better model prediction for the whole workspace, these influences have to be regarded in the elasticity model of the power trains. Introducing the angle α to measure the wrapping length of the cable as shown in Fig.9b, its influence on the stiffness of a single powertrain is experimentally determined as can be seen in Fig.10. The influence of angle α on the hysteresis behavior is shown in Fig.11.
- 40. An Elastic Cable Model for Cable-Driven Parallel Robots . . . 27 Fig. 10 Hysteresis and stiffness behavior in relation to the pulley angle α Fig. 11 Damping behavior in relation to the pulley α The effect may be caused by the increased wrapping angle and the direction of the changing force vector as shown in Fig.9b, increasing the reaction and friction force at the pulley bearing according to Fi = f 2 cos(αi + 1). (19) The angle of attack also influences the torque applied to the bearing on the frame and therefore influences the observed elasticity in the cables.
- 41. 28 P. Miermeister et al. 5 Conclusion and Outlook In this paper an improved cable model was presented which allows to regard the hysteresis effect during force computation. The approach can be used to improve force algorithms and the identification of geometrical robot parameters in an auto calibration procedure which relies on the force sensors for data acquisition. The sim- plicity of the model allows to compute the force values in a deterministic time slot, meeting the demand of real-time algorithms. While the model gives significantly better results than the linear elastic model it showed that the cable behavior not only depends on local parameters such as the amplitude, but also on more global and time variant parameters such as the platform pose. Beside that, the actual robot and the model tend to diverge over time depending on the operating load and environ- mental conditions. Further experiments will be executed to evaluate the long term parameter stability of the robot parameters depending on operational time. It also would be interesting to investigate the influence of overload on the cables alone and in interaction with the surrounding support structure. References 1. Badrakhan F (1987) Rational study of hysteretic systems under stationary random excitation. Non-Linear Mech 22(4):315–325 2. Bouc R (1971) Modle mathmatique dhystrsis. Acustica 24:16–25 3. Gouttefarde M, Merlet JP, Daney D (2007) Wrench-feasible workspace of parallel cable-driven mechanisms. In: IEEE international conference on robotics and automation, Roma, Italy, pp 1492–1497 4. HillerM,FangS,MielczarekS,VerhoevenR,FranitzaD(2005)Design,analysisandrealization of tendon-based parallel manipulators. Mech Mach Theory 40(4):429–445 5. Irvine M (1981) Cable structures. MIT Press, Cambridge 6. Ismail M, Ikhouane F, Rodellar J (2009) The hysteresis Bouc–Wen model, a survey. Arch Comput Methods Eng 16(2):161–188 7. Kossowski C, Notash L (2002) CAT4 (Cable actuated truss—4 Degrees of Freedom): a novel 4 DOF cable actuated parallel manipulator. J Rob Syst 19:605–615 8. Merlet J-P, Daney D (2007) A new design for wire-driven parallel robot. In: 2nd international congress, design and modelling of mechanical systems, 2007 9. Pott A (2012) Influence of pulley kinematics on cable-driven parallel robots. In: Advances in robot kinematics, Austria, pp 197–204 10. Pott A, Bruckmann T, Mikelsons L (2009) Closed-form force distribution for parallel wire robots. In: Computational kinematics. Springer-Verlag, Duisburg, Germany, pp. 25–34 11. Thai H, Kim S (2011) Nonlinear static and dynamic analysis of cable structures. Finite Elem Anal Des 47(3):237–246 12. Verhoeven R (2004) Analysis of the workspace of tendon-based Stewart platforms. PhD thesis, University of Duisburg-Essen, Duisburg
- 42. On the Improvement of Cable Collision Detection Algorithms Dinh Quan Nguyen and Marc Gouttefarde Abstract This paper presents several algorithms to detect the cable interferences for a general spatial Cable-Driven Parallel Robot (CDPR). Two types of cable interfer- ences are considered. The first type is the collisions between cables and cables. The second type is the interferences between cables and the CDPR mobile platform. In each case, an algorithm is proposed to efficiently verify the cable interferences. The useoftheproposedalgorithmsisthenillustratedbyaverificationprocedureofthecol- lision free condition over a given Cartesian workspace and orientation workspace of a CDPR. These tools can be used in the design or planning phase of a general CDPR. 1 Introduction For cable-driven parallel robots, collision detections happen in several cases: • Interferences between cables and cables • Interferences between cables and mobile platform • Interferences between mobile platform and surrounding environment • Interferences between cables and surrounding environment. Efficientmethodstodetectsuchcollisionsbecomenecessary,especiallyforspatial CDPR having a large number of cables (e.g. m ≥ 6) such as the NIST robot crane [1], the Marionet CDPR [2, 3] and the CoGiRo prototype [4]. These methods could be used in two main situations: • Design/planning: required to check the capability of CDPR (e.g. compute the bounds on the orientation and Cartesian spaces within which there is no cable interference). • Control: required to guarantee safety issues in operating CDPR in real-time. D.Q. Nguyen (B) · M. Gouttefarde Laboratoire d’Informatique, de Robotique et de Micro-électronique de Montpellier (LIRMM-CNRS-UM2),161 rue Ada, 34392 Montpellier Cedex 5, France e-mail: dinhquan.nguyen@lirmm.fr M. Gouttefarde e-mail: marc.gouttefarde@lirmm.fr © Springer International Publishing Switzerland 2015 A. Pott and T. Bruckmann (eds.), Cable-Driven Parallel Robots, Mechanisms and Machine Science 32, DOI 10.1007/978-3-319-09489-2_3 29
- 43. 30 D.Q. Nguyen and M. Gouttefarde In most situations, the latter case can be avoided if all the safety constraints are dealt with from checking the capability of CDPR over a desired workspace. In this paper, we mainly discuss the verification of collision free conditions for a CDPR in the design or offline planning phases. In fact, CDPR cable interference problem has not extensively been addressed. Studies on this aspect can be listed as [5–10]. In term of the collisions between cables or CDPR mobile platform with surround- ing environment, one can use AABB or OOBB tree methods [11]. These methods are fast and effective for large and complex shape objects (triangulations of the mobile platform and obstacles may consist of a lot of vertices). For CDPR with light-weight cables, the cables can be considered as straight lines. For CDPR using hefty cables, the mobile platform weight is expectedly large, the cable sagging effects may not really affect the algorithms, thus it may also be sufficient enough to consider the cables as straight lines. In either cases, the inter- ferences between cables and cables can be treated as interferences between straight line segments. The interferences between cables and CDPR mobile platform can be considered as collisions between straight line segments and triangles (the latter triangulating the surface of the CDPR mobile platform). Usual methods [11] can be applied to detect cable interferences. However, these methods are only suited for real-time situations and are not satisfactory enough to our application of interest which is verifying the cable interference free conditions for a given desired Cartesian workspace and set of orientation ranges. In [5], Merlet discussed algorithms to detect interferences between cables and cables as well as between cables and mobile plat- form. However, the proposed methods were only applied to fixed orientation cases and did not applied for a range of orientations. In [9], Perreault presented an analysis of the cable interference-free workspace of CDPR. The analysis was also mainly applied to the cases of CDPR with constant orientation. In this paper, we aim to develop algorithms that could improve the efficiency of the verification of collision free operation with respect to given CDPR Cartesian workspace and orientation workspace. These algorithms concern only the interfer- ences between cables and cables, and cables and CDPR mobile platform. The paper is organized as follows. Section 2 presents the algorithm to detect interferences between cables and cables. The collision detection algorithm for the interferences between cables and CDPR mobile platform is discussed in Sect.3. An illustrating example of using the proposed algorithms to verify the collision free operation of a CDPR over a given Cartesian workspace and orientation workspace is given in Sect.4. Finally, Sect.5 gives an illustration of the performance of the presented methods. 2 Interferences Between Cables and Cables Let us consider a m-cable CDPR as shown in Fig.1. We will consider the interference between two cables AiBi and AjBj (i = j).
- 44. On the Improvement of Cable Collision Detection Algorithms 31 Ai Aj Am Bi Bj Bm Fig. 1 A general m-cable CDPR i B i A i B i A j A j B ij d ε i B j A i A j B ij d ε j B j A ij d ε (a) (b) (c) Fig. 2 Usual method of checking interference between two cables. a Not colliding, b (Going to) collide, c Colliding Figure2 illustrates a general method to detect the collision between two cables. In the first case (Fig.2a), the two cables i and j are not colliding since the distance between the two cables dij is greater than a given small value ε (this value can be chosen as the cable diameter). In the second case (Fig.2b) when the cable i moves toward the cable j, according to the collision condition dij ε, the two cables collide. Note that, in this state, the cable j is “behind” the plane (AiBiBj). In this case, one can say that the two cables are going to collide but a real collision has not yet happened. In the third case (Fig.2c), the same collision condition is valid. However, the cable j is “in front of” the plane (AiBiBj). In this case, a real collision between the two cables i and j has occurred. This algorithm can be formalized as follows [6, 7, 9]:
- 45. 32 D.Q. Nguyen and M. Gouttefarde A real collision between the two cables i and j will occur if the two following condi- tions are met (i) the distance between the two cables is very small: dij ε (ii) the position of the cable j (or the cable exit point Aj) with respect to the plane (AiBiBj) changes sign (e.g. switch from “behind” position to “in front of” posi- tion) It is enough to use this algorithm for real-time collision detection where the positions of cable exit point Ai and cable anchor point Bi are updated online in each sample time (while the mobile platform is following a trajectory). The computation of dij can be found in [11]. However, in the design phase, where the collision free conditions need to be verified with respect to a range of orientations and a volume of Cartesian space, this usual method may not be really effective. In fact, to check the cable interferences, it could be enough to consider the second condition (ii) (in the usual algorithm) while neglecting the first condition (i) (dij ε). The two cables i and j can be far away (the distance dij can be large) but their relative positions will tell us whether or not there was a collision when the mobile platform “moved” from an arbitrary pose Xp to another pose Xq in the CDPR workspace. Figure3 illustrates the method proposed in this paper to detect the interference between the two cables i and j. Suppose that the CDPR mobile platform moves from an initial pose Xp to an arbitrary pose Xq where a rotation and/or a translation occur. In the first case (Fig.3a), when the mobile platform “moves” from pose Xp to pose Xq, the cable j is always “behind” the plane (AiBiBj). There should be no collision between the two cables i and j. In the second case (Fig.3b), the position of cable j with respect to the plane (AiBiBj) changed sign so that a collision probably occurred between the two cables. i B i A i B i A j A j B i B j A i A j B ij d j B j A ij d ij d i B i A j B j A ij d (a) (b) Fig. 3 Checking interference between two cables. a Not colliding, b Colliding
- 46. On the Improvement of Cable Collision Detection Algorithms 33 The collision detection algorithm between the two cables used in this paper consists of the following steps: • Step 1. At pose Xp, compute the position of AjBj with respect to plane (AiBiBj) and store it in the variable sij(Xp): ⎧ ⎪ ⎨ ⎪ ⎩ sij(Xp) = 1, if AjBj is “in front of” the plane (AiBiBj) sij(Xp) = 0, if AjBj lies on the plane (AiBiBj) sij(Xp) = −1, if AjBj is “behind” the plane (AiBiBj) (1) with sij = sign − − → AiBi × − − → AiBj T · − − → AiAj (2) Here, the two cable exit points Ai and Aj must not be coincident: Ai = Aj. • Step 2. At pose Xq, compute the projection image A jBj of cable AjBj onto the plane (AiBiBj): A j = Aj − t ∗ n (3) with n = − − → AiBi × − − → AiBj t = nT · − − → AiAj nT n • Step 3. At pose Xq, compute the position of AjBj with respect to plane (AiBiBj) and store it in the variable sij(Xq) by using (2). If sij(Xq) == sij(Xp), then no collision should have occurred. If sij(Xq) = sij(Xp) and A jBj is not intersecting AiBi, then no collision should have occurred. If sij(Xq) = sij(Xp) and A jBj is intersecting AiBi, then we consider that a colli- sion between the two cables i and j occurred. In this algorithm, for a m-cable CDPR, step 1 requires to compute sij in (2) Ncc times: Ncc = m · (m − 1) 2 (4) The algorithm stops if there exists any i, j for which a collision occurs (thus, the times of performing steps 2 and 3 is N2,3 ≤ Ncc). The presented algorithm considers that given the two arbitrary poses Xp and Xq, if a collision is detected, then there exists no collision free trajectory that allows the CDPR mobile platform to move from pose Xp to pose Xq (regardless of any trajectory planning method).
- 47. 34 D.Q. Nguyen and M. Gouttefarde 3 Interferences Between Cables and CDPR Mobile Platform 3.1 Method 1 Suppose that the CDPR mobile platform is triangulated into N triangles. The first approach to detect the interferences between cables and CDPR mobile platform is quite straightforward using the method to detect collision between line segments and triangles [11]. At pose Xp, for each cable i, we check for the interferences between cable i and all the triangles that do not belong to the planes which contain the cable end point Bi. If there is a collision then we stop the checking process and give out a warning. Figure4 shows an example of the collision between cables i and the mobile plat- form. The computational time in this case depends on the number of vertices of the mobile platform as well as the number of cables. This method is quite “heavy” and not really effective if the mobile platform has a complex shape (triangulated with a large number of triangles N 1). To avoid excessive computational time, we can approximate the mobile platform shape by a more simple convex shape whose number of triangles is reduced considerably, e.g. in Fig.5 (the simplified shape should enclose the CDPR mobile platform). Fig. 4 Interferences between cables and CDPR mobile platform. a Not colliding, b Colliding i B i A i B i A (a) (b) Fig. 5 Simplification of the mobile platform shape. a Real shape, b Simplified shape (a) (b)
- 48. On the Improvement of Cable Collision Detection Algorithms 35 Fig. 6 Detecting collision between cable i and the mobile platform. a Not colliding, b Colliding i B i A i B i A ( 1) i k D + Bi iN D ik D M ( 1) i k D + M Bi iN D ik D (a) (b) 3.2 Method 2 Although the first method (Sect.3.1) to detect the interferences between cables and mobile platform is simple, the issue of heavy computational time may remain if the mobile platform has a complex shape and the simplification procedure cannot significantly reduce the number of its vertices. We propose a second heuristic method which consists in checking whether or not the cable AiBi belongs to the subspace (convex cone) spanned by its nearest edges. Fig.6 shows an illustrating example of this approach. The algorithm is given in the following steps: • Step 1. Perform a simplification of the mobile platform to transform it into a simpler convex shape while keeping important vertices. This simplified convex shape should enclose the CDPR mobile platform. • Step 2. Determine the nearest neighbor vertices Dik (k = 1, NBi) of anchor points Bi in such a way that the convex cone spanned by the vectors ( − − → BiDi1, − − → BiDi2, ..., − − → BiDiNBi ) includes the CDPR mobile platform. NBi should be the minimum number of such nearest neighbor vertices of Bi. • Step 3. Compute the positions (or the signs) of an arbitrary point M lying within the mobile platform shape with respect to the planes (BiDikDi(k+1)) and store them into vector SBi of size [NBi × 1]: SBi(k) = sign − − → BiDik × − − − − − → BiDi(k+1) T · − − − → DikM (5) For instance, the point M can be chosen as the origin of the local frame attached to the mobile platform or as its center of mass. Note that all the cases where SBi(k) = 0 in (5) are considered invalid (the point M must lie strictly inside the mobile platform shape). • Step 4. At an arbitrary pose Xp, compute the signs SAi of the cable exit point Ai with respect to the NBi planes (BiDikDi(k+1)) (vector SAi is of size [NBi × 1]): SAi(k) = sign − − → BiDik × − − − − − → BiDi(k+1) T · − − → DikAi (6)
- 49. 36 D.Q. Nguyen and M. Gouttefarde If ∃ i and ∃ k (i = 1, m, k = 1, NBi) such that SAi(k) = 0, then the cable AiBi is considered to be colliding with the mobile platform. If SAi ≡ SBi then the cable AiBi is considered to be colliding with the mobile platform. If SAi = SBi and SAi(k) = 0 (∀ k) then there is no collision. Note that the steps 1–3 should only be done at the initial step of an optimization or operation process. Then, step 4 will be used to check the interferences between the cables and the mobile platform for each considered robot configuration. This approach utilizes the fact that the positions of a point lying within the mobile platform with respect to the planes (BiDikDi(k+1)) never change. One only need to evaluate (6) Ncp times to check the collision, where: Ncp = m i=1 NBi (7) This method is fast and reliable. However, there are still a few limitations to this approach. The algorithm only works under the condition that the mobile platform has a convex shape. In case the mobile platform shape is concave, a pre-process (Step 1) is needed to convert it into a convex object (with a number of vertices as small as possible) in order to apply the algorithm. Currently, we are not aware of an efficient (fast) method of selecting the right number of the nearest neighbor vertices Dik of anchor point Bi. One still has to manually select the vertices Dik. The process of simplifying the mobile platform shape to reduce its complexity can be done with available CAD softwares e.g. [12]. 4 Verification of Collision Free Condition for a Given Workspace Let us consider an application where one want to verify the cable interferences of a CDPR with respect to a given Cartesian workspace and orientation range. The CDPR workspace is given as follows: xmin ≤ x ≤ xmax ymin ≤ y ≤ ymax zmin ≤ z ≤ zmax θx min ≤ θx ≤ θx max θy min ≤ θy ≤ θy max θz min ≤ θz ≤ θz max where X = (x y z, θx θy θz) denotes the mobile platform pose. Assume that the Cartesian workspace is discretized into a finite set of Np points and the orientation workspace is discretized into a finite set of Nq points (these points can be chosen as extreme points which lie on the workspace boundaries). Let us take an arbitrary pose
- 50. On the Improvement of Cable Collision Detection Algorithms 37 Xc of the given workspace where we assume that there is no cable interference: Xc = (xc yc zc, θxc θyc θzc) (8) The verification of collision free condition of the CDPR with respect to the as- signed workspace is illustrated in the following pseudocode: Compute sij(Xc) in (2); Simplify the CDPR mobile platform shape (if it is necessary); Determine the NBi nearest neighbor vertices Dik of Bi; Compute SBi (Xc)in(5); OK = 1; (there is no collision) for k = 1 : Np for l = 1 : Nq X = (xk yk zk, θxl θyl θzl); OK = Check the interferences between cables and cables; if (OK == 0) break; OK = Check the interferences between cables and mobile platform; if (OK == 0) break; end; end; In short, we perform the verification process at each discrete points in Cartesian space and orientation space. The reference (initial) state of the mobile platform is computed at the pose Xc. In the step checking the interferences between cables and cables, the initial pose is always Xc, and the destination pose is X. This means that when the mobile platform “moves” from pose Xc to pose X, the checking process stops if there is any interference detected. In the step checking the interferences between the cables and the mobile platform, the second approach is used. There is no collision if the returned value of the checking variable is OK = 1. In this way, the collision free condition of the CDPR with respect to a given workspace is ensured in the sense that, when OK = 1 is returned, there should always exist one collision free path starting from the home pose Xc to any pose (among the considered discrete set of poses) in the workspace. When OK = 0 is returned, there very probably exists no collision free trajectory that allows the mobile platform to move freely within the given workspace (regardless of any path planning method). Currently, this approach has only been validated on examples. One can select just one home pose Xc to check the collision free conditions with respect to the given workspace (Xc can be chosen as the center pose of the given workspace). To increase the reliability, we can apply the algorithm to a set of Nc home poses Xc and with large numbers of discrete poses (Np and Nq are large). The computational time is proportional to Nc × Np × Nq.
- 51. 38 D.Q. Nguyen and M. Gouttefarde 5 Simulation Let us consider the 8-cable CDPR shown in Fig.7. The mobile platform is a cube with 8 vertices. In this test, we show the computation time for each function call to check the collisions between cables and cables, and between cables and the CDPR mobile platform (using method 1 and method 2) while assuming that the mobile platform moves from the home pose Xc to the destination pose Xd. The home pose is Xc = (0 0 0, 0 0 0) (m, rad). The destination pose is varied. The results are given in Table1. We use MATLAB to run the simulation on a PC with CPU core i7-2620M 2.7GHz. In the case of checking the collision between cables and cables, we have the number of cables is m = 8. The computation time is quite expensive. The maximum computation time for checking the collision is around tcc ≈ 4.65 ms. InthecaseofcheckingthecollisionbetweenthecablesandCDPRmobileplatform using the first method, the mobile platform surface is triangulated into 12 triangles. For each cable, one need to verify potential collision with a maximum of 6 triangles. The maximum number of calls of the primitive test used to detect the collision Fig. 7 Example of 8-cable CDPR Table 1 Collision detection computation time Destination pose Cables-cables (ms) Cables-platform Cables-platform Xd (m, rad) (method 1) (ms) (method 2) (ms) (0 0 0, 0 0 − π/4) 4.63 (no collision) 4.61 (collision) 0.74 (collision) (2 0 1, π/3 − π/3 0) 4.57 (collision) 4.79 (collision) 0.92 (collision) (0 0 0, 0 − π/3 − π/4) 4.54 (collision) 4.50 (collision) 0.39 (collision) (0 0 1, 0 0 π/4) 4.65 (no collision) 5.28 (no collision) 1.47 (no collision) (0 2 0, 0 0 0) 4.61 (no collision) 5.30 (no collision) 1.46 (no collision) (0 0 2, π/4 0 0) 4.63 (no collision) 5.27 (no collision) 1.44 (no collision)
- 52. On the Improvement of Cable Collision Detection Algorithms 39 between a line segment and a triangle is 6 × m = 48. The maximum computation time is around tcp1 ≈ 5.3 ms. On the other hand, in the second method, the mobile platform has a convex shape. For each vertex Bi, there are a minimum of 3 neighboring vertices. The computation time in this case is significantly reduced compared to the first method. The maximum computation time is around tcp2 ≈ 1.47 ms Assume that one want to verify the CDPR capability over a given workspace where the Cartesian workspace is discretized into Np = 20 points and the orientation workspace is discretized into the minimum number Nq = 8 points (taking only the extreme values of each angle into account). The number of considered home poses is Nc = 10 points. If we choose to use the second method to check the collision between the cables and the CDPR mobile platform then the maximum computation time to verify the collision free collision condition for the given workspace is around: tmax = Nc × Np × Nq × (tcc + tcp2) = 10 × 20 × 8 × (4.65 + 1.47) ms = 9.792 s Currently, it is up to the user to choose appropriate values of Nc, Np and Nq, consid- ering the trade off between reliability of the result and computation time. It is worth noting that, by using parallel computing (taking the advantages of both powerful CPU and GPU), one can also greatly reduce the computation times of the presented methods. 6 Conclusion Several algorithms to detect the cable interferences of a CDPR have been discussed in this paper. The presented heuristic approaches improve the usual methods of detecting cable collisions in term of efficiency. Two types of cable interferences have been considered: collisions between cables and cables as well as collisions between cables and the CDPR mobile platform. The application of these tools was illustrated by an example of checking the collision free condition of a CDPR with respect to given Cartesian workspace and orientation workspace. The proposed approach offers the user a fast and reliable method to quantify the CDPR capability in term of position and orientation in the design or path planning phases. In our future work, we will aim to explicitly “prove” the presented algorithms (to verify the collision free conditions of a CDPR with respect to a given range of orientations and given Cartesian workspace) as well as improve the method of simplifying the CDPR platform shape. Acknowledgments The research leading to these results has received funding from the European Community’s Seventh Framework Programme under grant agreement No. NMP2-SL-2011-285404 (CABLEBOT).
- 53. 40 D.Q. Nguyen and M. Gouttefarde References 1. Albus J, Bostelman R, Dagalakis N (1993) The NIST robocrane. J Robot Syst 10(5):709–724 2. Merlet J-P (2008) Kinematics of the wire-driven parallel robot marionet using linear actuators. Proceedings of IEEE international conference on robotics and automation. pp 3857–3862 3. Merlet J-P, Daney D (2010) A portable, modular parallel wire crane for rescue operations. In: Proceedings of IEEE international conference on robotics and automation (ICRA) , May 2010. pp 2834–2839 4. Lamaury J, Gouttefarde M (2013) A tension distribution method with improved computational efficiency. Cable-driven parallel robots, mechanisms and machine science, vol 12. Springer, Heidelberg, pp 71–85 5. Merlet J-P (2004) Analysis of the influence of wires interference on the workspace of wire robots. In: Advances in Robot Kinematics. Kluwer Academic Publishers, Dordrecht, pp 211– 218 6. Maeda K, Tadokoro S, Takamori T, Hiller M, Verhoeven R (1999) On design of a redundant wire-driven parallel robot WARP manipulator. In: Proceedings of IEEE international confer- ence on robotics and automation. pp 895–900 7. Otis M, Perreault S, Nguyen-Dang T-L, Lambert P, Gouttefarde M, Laurendeau D, Gosselin C (2009) Determination and management of cable interferences between two 6-DOF foot platforms in a cable-driven locomotion interface. IEEE Trans Syst Man Cybern Part A Syst Hum 39(3):528–544 8. Perreault S, Gosselin C (2008) Cable-driven parallel mechanisms: application to a locomotion interface. J Mech Des 130:10 9. Perreault S, Cardou P, Gosselin C, Otis MJ-D (2010) Geometric determination of the interference-free constant-orientation workspace of parallel cable-driven mechanisms. J Mech Robot 2:011012 10. Wischnitzer Y, Shvalb N, Shoham M (2008) Wire-driven parallel robot: permitting collisions between wires. Int J Robot Res 27(9):1007–1026 11. Ericson C (2005) Real-time collision detection. Morgan Kaufmann, San Fransisco 12. Meshlab. http://meshlab.sourceforge.net/
- 54. Workspace Analysis of Redundant Cable-Suspended Parallel Robots Alessandro Berti, Jean-Pierre Merlet and Marco Carricato Abstract Thispaperfocusesoncable-suspendedparallelrobots(CSPRs),asubclass of cable-driven parallel robots, and particularly on the analysis of their workspace. CSPRs present, among other interesting characteristics, large workspaces and high reconfigurability, which make them attractive for a large variety of applications, especially for pick and place operations over wide spaces. This paper is based on the assumption that the safest (and cheapest) control scheme for a redundant CSPR consists, at the current state of development, in actuating only 6 cables at a time. This paper shows how, under this assumption, it is still possible to take advantage of redundancy to enhance the workspace and eventually reduce the maximal ten- sion among cables. A simple interval-analysis routine is presented as a tool for the workspace and trajectory analysis of a redundant CSPR, and the results of a case study on an existing prototype are discussed. 1 Introduction Cable-drivenparallelrobots(CDPRs)employcablesinplaceofrigid-bodyextensible legs in order to control the end-effector pose. CDPRs strengthen classic advantages characterizing closed-chain architectures versus serial ones, like reduced mass and inertia, a larger payload to robot weight ratio, high dynamic performances, etc., while providing peculiar advantages, such as a larger workspace, reduced manufacturing A. Berti · M. Carricato (B) Department of Industrial Engineering and Interdepartmental Center for Health Sciences and Technologies, University of Bologna, Bologna, Italy e-mail: marco.carricato@unibo.it A. Berti e-mail: alessandro.berti10@unibo.it J.-P. Merlet HEPHAISTOS Project, French National Institute for Research in Computer Science and Control (INRIA), Sophia-Antipolis, France e-mail: jean-pierre.merlet@inria.fr © Springer International Publishing Switzerland 2015 A. Pott and T. Bruckmann (eds.), Cable-Driven Parallel Robots, Mechanisms and Machine Science 32, DOI 10.1007/978-3-319-09489-2_4 41
- 55. Another Random Document on Scribd Without Any Related Topics
- 56. CHAPTER XI New Plans With the end of the school year Tom and Helen were able to give their complete time and energies to the Herald. When Monday, the first of June arrived, they were working on their fourth issue of the Herald and Helen had written a number of stories on the last week’s activities at school, the graduation exercises, the junior- senior dinner and the senior class play. She praised Miss Weeks highly for her work with the class play and lauded the seniors for their fine acting. Although urged that she say something about her own part, Helen steadfastly refused and her brother finally gave up in disgust and delved in to the ledger for on his shoulders fell the task of making out the monthly bills and handling all of the business details of the paper. When Tom had completed his bookkeeping he turned to his sister. “Helen,” he began, “we’re not making enough.” “But, Tom,” she protested, “the paper is carrying more advertising than when Dad ran it.”
- 57. [161] [162] “Yes, but our expenses are high,” said Tom. “We’ve got to look ahead all the time. Dad will have used all of the money he took with him in a little less than six months. After that it will be up to us to have the cash in the bank. Right now we’ve just a little under a hundred dollars in the bank. Current bills will take more than that, and our own living expenses, that is for mother and we two, will run at least $100 a month. With our total income from the paper only slightly more than $200 a month on the basis of the present amount of advertising, you see we’re not going to be able to save much toward helping Dad.” “Then we’ll have to find ways of increasing our volume of business,” said Helen. “That won’t be easy to do in a town this size,” replied Tom, “and I won’t go out and beg for advertising.” “No one is going to ask you to,” said Helen. “We’ll make the Herald such a bright, outstanding paper that all of the business men will want to advertise.” “We’ll do the best we can,” agreed Tom. “Then let’s start right now by putting in a farm page,” suggested Helen. “But there won’t be many farm sales from now on,” argued Tom. “No,” conceded his sister, “but there is haying, threshing and then corn picking and all of the stores have supplies to sell to the farmers.” “I believe you’re right. If you’ll do the collecting this afternoon, I’ll go down to Gladbrook and see if we can
- 58. [163] get the cooperation of the county agent. Lots of the townships near here have farm bureaus and I’ll get the names of all of their leaders and we’ll write and tell them what we plan to do.” After lunch Tom teased the family flivver into motion and set out for Gladbrook while Helen took the sheaf of bills and started the rounds of the business houses. She had no trouble getting her money from all of the regular advertisers and in every store in which she stopped she took care to ask the owner about news of the store and of his family. She noticed that it flattered each one and she resolved to call on them at least once a week. Tom returned from Gladbrook late in the afternoon. He was enthusiastic over the success of his talk with the county agent. “He’s a fine chap,” Tom explained. “Had a course in agricultural journalism in college and knows news and how to write it. The Gladbrook papers, the News and the Times, don’t come up in this section of the county and he’ll be only too glad to send us a column each week.” “When will he start?” “Next week will be the first one. He’ll mail his column every Tuesday evening and we’ll have it on the Wednesday morning mail. Now, here’s even better news. I went to several of the department stores at Gladbrook and told them we were going to put out a real farm page. They’re actually anxious to buy space and by driving down there once a week I can get two or three good ads.” “How will the local merchants feel?” asked Helen.
- 59. [164] “They won’t object,” replied Tom, “for I was careful to stress that I would only accept copy which would not conflict with that used by our local stores.” “That was a wise thing to do,” Helen said. “We can’t afford to antagonize our local advertisers. I made the rounds and collected all of the regular accounts. There’s only about eighteen dollars outstanding on this month’s bills and I’ll get all but about five dollars of that before the week is over.” “Want to go to Cranston Friday or Saturday?” asked Tom. “I surely do,” Helen replied. “But what for, Tom, and can we afford it?” “One of us will have to make the trip,” her brother said. “Putting on this farm page means we’ll have to print two more pages at home, six altogether, and will need only two pages of ready-print a week from the World Printing Company. We’ll go down and talk with their manager at Cranston and select the features we want for the two pages they will continue to print for us.” “Our most important features in the ready-print now are the comics, the serial story and the fashion news for women,” said Helen. “Then we’ll have one page of comics,” said Tom, “and fill the other page with features of special interest to our women readers.” The next three days found the young Blairs so busy getting out the current edition of the paper that they had little time to talk about their plans.
- 60. [165] They had decided to go to Cranston Friday but when Helen found that there were special rates for Saturday, they postponed the trip one day. When the Friday morning mail arrived, Helen was glad they had changed their plans. While sorting the handful of letters, most of them circulars destined for the wastepaper basket, she came upon the letter she had been looking forward to for days. The words in the upper left hand corner thrilled her. It was from the Cranston bureau of the Associated Press. With fingers that trembled slightly, she tore it open. Would she get the job as Rolfe correspondent? A green slip dropped out of the envelope and Tom, who had come in from the composing room, reached down and picked it up. “Ten dollars!” he whistled. “What’s that?” demanded Helen, incredulously. “It’s your check from the Associated Press for covering the tornado,” explained Tom. “Look!” Helen took the slip of crisp, green paper. She wasn’t dreaming. It was a check, made out in her name and for $10. “But there must be some mistake,” she protested. “They didn’t mean to pay me that much.” “If you think there’s a mistake,” grinned Tom, “you can go and see them when we reach Cranston tomorrow. However, if I were you, I’d tuck it in my pocket, invite my brother across the street to the drug store, and buy him a big ice cream soda.”
- 61. [166] [167] “Wait until I see what the letter says,” replied Helen. She pulled it out of the envelope and Tom leaned over to read it with her. “Dear Miss Blair,” it started, “enclosed you will find check for your fine work in reporting the tornado near Rolfe. Please consider this letter as your appointment as Rolfe correspondent for the Associated Press. Serious accidents, fires of more than $5,000 damage and deaths of prominent people should be sent as soon as possible. Telegraph or telephone, sending all your messages collect. In using the telegraph, send messages by press rate collect when the story is filed in the daytime. If at night, send them night press collect. And remember, speed counts but accuracy must come first. Stories of a feature or time nature should be mailed. We are counting on you to protect us on all news that breaks in and near Rolfe. Very truly yours, Alva McClintock, Correspondent in charge of the Cranston Bureau.” “He certainly said a lot in a few words,” was Tom’s comment. “Now you’re one up on me. You’re editor of the Herald and Associated Press correspondent and I’m only business manager.” “Don’t get discouraged,” laughed Helen, “I’ll let you write some of the Associated Press stories.” “Thanks of the compliment,” grinned Tom. “I’m still waiting for that ice cream soda, Miss Plutocrat.” “You’ll grumble until I buy it, I suppose, so I might as well give in right now,” said Helen. “Come on. I’m hungry for one myself.” Tom and Helen boarded the nine forty-five Saturday morning and arrived at the state capital shortly after
- 62. [168] noon. It was Helen’s first trip to Cranston and she enjoyed every minute of it, the noise and confusion of the great railroad terminal, the endless bobbing about of the red caps, the cries of news boys heralding noonday editions and the ceaseless roar of the city. They went into the large restaurant at the station for lunch and after that Tom inquired at the information desk for directions on how to reach the plant of the World Printing Company. He copied the information on a slip of paper and the two young newspaper people boarded a street car. Half an hour later they were on the outskirts of the industrial district and even before the conductor called their stop, Tom heard the steady roar of great presses. “Here we are,” he told Helen as they stepped down from the car and looked up at a hulking ten story building that towered above them. “The Cranston plant of the Rolfe Herald,” chuckled Helen. “Lead on.” They walked up the steps into the office, gave their names and indicated their business to the office girl. After waiting a few minutes they were ushered into an adjoining office where an energetic, middle aged man who introduced himself as Henry Walker, service manager, greeted them. “Let’s see, you’re from the Rolfe Herald?” he asked. “My sister and I are running the paper while Dad is in the southwest regaining his health,” explained Tom. “We’ve got to expand the paper to increase our
- 63. [169] advertising space and the only thing we can see to do is cut down our ready-print to two pages.” “Explain just what you mean,” suggested the service manager. Tom outlined their advertising field and how they hoped to increase business by adding two more pages of home print, one of which would be devoted to farm advertising and news and the other to be available for whatever additional advertising they could produce. “We’ll be sorry to have you drop two pages of ready- print,” said Mr. Walker, “but I believe you’re doing the right thing. Now let’s see what you want on the two pages you’ll retain.” “Helen is editor,” Tom explained, “and it’s up to her to pick out what she wants.” “You’re doing a splendid job on the Herald,” the service manager told Helen. “I get copies of every paper we serve and I’ve been noticing the changes in make-up and the lively stories. However, I am sorry to hear about your father but with you two youngsters to give him pep and courage he ought to be back on the job in a few months.” “We’re sure he will,” smiled Helen as she unfolded a copy of their last edition of the Herald. “I’ve pasted up two pages of the features I want to retain,” she explained as she placed them in front of the service manager. “I see,” he said. “You’re going to be quite metropolitan with a full page of comics and a page devoted to women. I’m glad of that. Too many editors of weeklies
- 64. [170] fail to realize that the women and not the men are the real readers of their papers. If you run a paper which appeals to women and children you’ll have a winner. Comics for the youngsters and a serial story with a strong love element and fashions and style news for the women.” “How about cost?” asked Tom. “Dropping the two pages won’t quite cut your bill with us in half,” explained Mr. Walker, “for you’re retaining all of our most expensive features. However, this new plan of yours will reduce your weekly bill about 40 per cent.” “That’s satisfactory,” agreed Tom, “and we’d like to have it effective at once. Helen has written the headings she wants for each page.” “We’ll send the pages, made up in the new way, down at the usual time next week,” promised the service manager, “and when there is anything else we can do, don’t hesitate to let us know.” When they were out of the building, they paused to decide what to do next. “I liked Mr. Walker,” said Helen. “He didn’t attempt to keep us from making the change. It means less money for his company yet he didn’t object.” “It was good business on his part,” replied Tom. “Now we feel kindly toward him and although he has lost temporarily he will gain in the end for we’ll give him every bit of business we can in the way of ordering supplies for job printing and extra stock for the paper.”
- 65. [171] [172] “If we have time,” suggested Helen, “I’d like to go down to the Associated Press office.” “Good idea,” agreed Tom. “I’d like to see how they handle all of the news.” They boarded the first down town street car and got off fifteen minutes later in the heart of Cranston’s loop district. Across the street was the building which housed the Cranston Chronicle, the largest daily newspaper in the state. They consulted the directory in the lobby of the building and took the elevator to the fifth floor where the Associated Press offices were located. They stepped out of the elevator and into a large room, filled with the clatter of many machines. A boy, his face smeared with blue smudges off carbon paper, rushed up to them and inquired their business. “I’m Helen Blair, a new correspondent at Rolfe,” explained the editor of the Herald, “and I’d like to see Mr. McClintock, the chief correspondent.” “Okay,” grinned the boy. “I’ll tell him. You wait here.” The youngster hurried across the room to a large table, shaped like a half moon and behind which sat a touseled haired chap of indeterminate age. He might be 30 and he might be 40, decided Helen. “Glad to know you, Miss Blair,” he said. “You did a nice piece of work on the storm.” “Thank you, Mr. McClintock,” replied Helen. “But my brother, Tom, deserves all of the credit. He suggested calling the story to you.”
- 66. [173] “Then I’ll thank Tom, too,” laughed the head of the Cranston bureau of the Associated Press. “We’re here today on business for our paper,” explained Helen, “and with a few minutes to spare before train time hoped you wouldn’t mind if we came in and saw how the ‘wheels go round’ here.” “I’ll be happy to show you the ‘works’,” replied Mr. McClintock, and he took them over to a battery of electric printers. “These,” he explained, “bring us news from every part of the country, east, south and far west. In reality, they are electric typewriters controlled from the sending station in some other city. We take the news which comes in here, sift it out and decide what will interest people in our own state, and send it on to daily papers in our territory.” “Do these electric printers run all day?” asked Tom. “Some of them go day and night,” continued Mr. McClintock, “for the A.P. never sleeps. Whenever news breaks, we’ve got to be ready to cover it. That’s why we appreciated your calling us on the storm. We knew there was trouble in your part of the state but we didn’t have a correspondent at Rolfe. It was a mighty pleasant surprise when you phoned.” They visited with the Associated Press man for another fifteen minutes and would have continued longer if Tom had not realized that they had less than twenty minutes to make their train. The last two blocks to the terminal were covered at a run and they raced through the train gates just before they clanged shut.
- 67. [174] “Close call,” panted Tom as they swung onto the steps of the local and it slid out of the train shed. “Too close,” agreed Helen, who was breathless from their dash. “Had to make it, though,” added Tom, “or we’d have been stranded here flat broke with the next train for home Monday night.” “Don’t worry about something that didn’t happen,” Helen said. “I’ve enjoyed every minute of our trip and we’re all ready now to start our expansion program for the Herald in earnest.” Adding two more pages of home print to the paper meant more work than either Tom or Helen had realized. There was more news to be written and more ads to be set and another run to be made on the press. With early June at hand the summer season at the resorts on the lower end of Lake Dubar got under way and Helen resolved to make a trip at least once a week and run a column or two of personals about people coming and going. She also gave liberal space to the good roads election in July, stressing the value the paved scenic highway would be to Rolfe. The two pages of ready-print arrived on Tuesday and Tom and Helen were delighted with the appearance of the comic page and the feature page for women readers. “We’ll have the snappiest looking paper in the county,” chuckled Tom. “Dad won’t know the old paper when he sees this week’s issue.”
- 68. [175] The county agent kept his promise to send them at least a column of farm news and Helen made it a point to gather all she could while Tom went to the county seat Tuesday morning and solicited ads for the page. The result was a well-balanced page, half ads and half news. Careful solicitation of home town merchants also brought additional ads and when they made up the last two pages Thursday noon they felt the extra work which increasing the size of the paper meant was more than repaid in extra advertising. “I’m printing a number of extra copies this week,” explained Tom. “There are lots of people around here who ought to take the Herald. With our expansion program we may pick up some extra subscriptions and we might get a chance at the county printing.” “Tom!” exclaimed Helen. “Do you really think we might get to be an official county paper.” “I don’t see why not,” said Tom. “Of course the two Gladbrook papers will always be on the county list but there are always three who print the legal news and the third one is the Auburn Advocate. Auburn isn’t any larger than Rolfe and I know darned well we have almost as many subscriptions as they do.” “How do they decide the official papers?” Helen wanted to know. “The county board of supervisors meets once a year to select the three official papers,” Tom explained, “and the three showing the largest circulation are selected. It would mean at least $2,000 extra revenue to us, most of which would be profit.” “Then why didn’t Dad try for it?” Helen asked.
- 69. [176] [177] “I’m not sure,” said Tom slowly. “There are probably several reasons, the principal one being that he wasn’t strong enough to make the additional effort to build up the circulation list. The other is probably Burr Atwell, owner and publisher of the Auburn Advocate. I’ve heard Dad often remark that Atwell is the crookedest newspaperman in the state.” “How much circulation do you think the Advocate has now?” Helen asked. “Their last postoffice statement showed only 108 more than ours,” replied Tom. “And when do the supervisors have their annual meeting?” “About the 15th of December,” said Tom. “Now what’s up?” “Nothing much,” smiled Helen. “Only, when the supervisors meet next the Rolfe Herald is going to have enough circulation to be named an official county paper. “Why Tom,” she went on enthusiastically, “think what it would mean to Dad?” “I’m thinking of that,” nodded her brother, “but I’m also thinking of what Burr Atwell might do to the Herald.”
- 70. CHAPTER XII Special Assignment The enlarged edition of the Herald attracted so much comment and praise from the readers that Tom and Helen felt well repaid for their additional efforts. Tom sat down and figured out the profit, deducted all expenses, and announced that they had made $78 on the edition, which, they agreed, was a figure they should strive to reach each week. “If we can keep that up,” commented Tom, “we’ll be sitting on top of the world.” “But if we were only an official county paper we’d have the moon, too,” Helen said. They discussed the pros and cons of getting enough additional circulation to beat the Auburn Advocate and the danger of arousing the anger of Burr Atwell, its publisher. “We don’t need to make a big campaign for subscriptions,” argued Helen. “We’ve taken the biggest step right now—improving and expanding the amount of local and country reading matter. Whenever I have an
- 71. [178] [179] extra afternoon this summer I’ll drive out in the country and see if I can’t get some people who haven’t been subscribers to take our paper.” Tom agreed with Helen’s suggestion and that very afternoon they took the old family touring car, filled it with gas and oil, and ambled through the countryside. Tom had a list of farmers who were non-subscribers and before the afternoon was over they had added half a dozen new names to the Herald’s circulation list. In addition, they had obtained at least one item of farm news at every place they stopped. “I call that a good afternoon’s work,” Helen commented when they drove the ancient flivver into the garage at home. “Not bad at all,” Tom agreed. “Only, we’ll keep quiet about our circulation activities. No use to stir up Burr Atwell until he finds it out for himself, which will be soon enough.” The remaining weeks of June passed uneventfully. The days were bright and warm with the softness of early summer and the countryside was green with a richness that only the middle west knows. Helen devoted the first part of each week to getting news in Rolfe and on Fridays and Saturdays took the old car and rambled through the countryside, stopping at farmhouses to make new friends for the Herald and gather news for the farm page. The revenue of the paper was increasing rapidly and they rejoiced at the encouraging news which was coming from their father. The Fourth of July that year came on Saturday, which meant a two day celebration for Rolfe and the summer
- 72. [180] resorts on Lake Dubar. Special trains would be routed in over the railroad and the boats on the lake would do a rushing business. The managers of Crescent Beach and Sandy Point planned big programs for their resorts and ordered full page bills to be distributed throughout that section of the state. The county seat papers had usually obtained these large job printing orders but by carefully figuring, Tom put in the lowest bids. Kirk Foster, the manager of Crescent Beach, ordered five thousand posters while Art Provost, the owner of Sandy Point, ordered twenty thousand. Crescent Beach catered to a smaller and more exclusive type of summer visitors while Sandy Point welcomed everyone to its large and hospitable beach. There was not much composition for the posters but the printing required hours and it seemed to Helen that the old press rattled continuously for the better part of three days as Tom fed sheet after sheet of paper into the ancient machine. The wonder of it was that they had no breakdowns and the bills were printed and delivered on time. “All of which means,” said Tom when he had finished, “that we’ve added a clear profit of $65 to our bank account.” “If we keep on at this rate,” Helen added, “we’ll have ample to take care of Dad when he needs more money.” “And he’ll be needing it sometime this fall,” Tom said slowly. “Gee whizz, but it sure does cost to be in one of those sanitariums. Lucky we could step in and take hold here for Dad.”
- 73. [181] “We owe him more than we’ll ever repay,” said Helen, “and the experience we’re getting now will be invaluable. We’re working hard but we find time to do the things we like.” Helen planned special stories for the edition just before the Fourth and visited the managers of both resorts to get their complete programs for the day. Kirk Foster at Crescent Beach explained that there would be nothing unusual there except the special display of night fireworks but Art Provost over at Sandy Point had engaged a line of free attractions that would rival any small circus. Besides the usual boating and bathing, there would be free acts by aerialists, a high dive by a girl into a small tank of water, half a dozen clowns to entertain the children, a free band concert both afternoon and evening, two ball games and in addition to the merry-go-round on the grounds there would be a ferris wheel and several other “thrill” rides brought in for the Fourth. “You ought to have a great crowd,” said Helen. “Goin’ to be mighty disappointed if I don’t,” said the old resort manager. “Plannin’ a regular rip-snorter of a day. No admission to the grounds, but Boy! it’ll cost by the time they leave.” “Going to double the prices of everything?” asked Helen. “Nope. Goin’ to have so many things for folks to do they’ll spend everything they got before they leave.” “In that case,” replied Helen, “I see where I stay at home. I’m a notorious spendthrift when it comes to
- 74. [182] celebrating the Fourth.” “I should say you’re not goin’ to stay home,” said Mr. Provost. “You and your mother and Tom are goin’ to be my guests. I’ve got your passes all filled out. Swim, ride in the boats, dance, roller skate, see the ball games, enjoy any of the ‘thrill rides’ you want to. Won’t cost you a cent.” “But I can’t accept them,” protested Helen. “We’ll pay if we come down. Besides, we didn’t give you all of those bills for nothing.” “Seemed mighty near nothin’ compared with the prices all the other printers in the county wanted,” smiled Mr. Provost. “You’ve been down every week writin’ items about the folks who come here and, believe me, I appreciate it. These passes are just a little return of the courtesy you’ve shown me this summer.” “When you put it that way, I can scarcely refuse them,” laughed Helen. “As a matter of fact,” she added, “I wanted them terribly for we honestly couldn’t afford to come otherwise.” When Helen returned to the office she told Tom about the passes and he agreed that acceptance of them would not place the Herald under obligation to the resort owner. “I always thought old man Provost a pretty good scout,” he said, “but I hardly expected him to do this. And say, these passes are good for both Saturday and Sunday. What a break!”
- 75. [183] [184] “If we see everything Saturday we’ll be so tired we won’t want to go back Sunday,” Helen said. “Besides, Mother has some pretty strong ideas on Sunday celebrations.” The telephone rang and Helen hastened into the editorial office to answer. She talked rapidly for several minutes, jotting down notes on a pad of scratch paper. When she had finished, she hurried back into the composing room. “Tom,” she cried, “that was Mr. Provost calling.” “Did he cancel the passes?” “I should say not. He called to say he had just received a telegram from the Ace Flying Circus saying it would be at Sandy Point to do stunt flying and carry passengers for the Fourth of July celebration.” “Why so excited about that? We’ve had flying circuses here before.” “Yes, I know, Tom, but ‘Speed’ Rand is in charge of the Ace outfit this year.” “‘Speed’ Rand!” whistled Tom. “Well, I should say that was different. That’s news. Why Rand’s the man who flew from Tokyo to Seattle all alone. Other fellows had done it in teams but Rand is the only one to go solo. He’s big news in all of the dailies right now. Everyone is wondering what daredevil stunt he’ll do next.” “He’s very good looking and awfully rich,” smiled Helen.
- 76. [185] “Flies just for fun,” added Tom. “With all of the oil land he’s got he doesn’t have to worry about work. Tell you what, I’ll write to the Cranston Chronicle and see if they’ll send us a cut of Rand. It would look fine on the front page of this week’s issue.” “Oh,” exclaimed Helen “I almost forgot the most important part of Mr. Provost’s call. He wants you to get out 10,000 half page bills on the Ace Flying Circus. Here are the notes. He said for you to write the bill and run them off as soon as you can.” The order for the bills put Tom behind on his work with the paper and it was late Thursday afternoon before Helen started folding that week’s issue. But they didn’t mind being late. The bill order from Sandy Point had meant another piece of profitable job work and Mr. Provost had also taken a half page in the Herald to advertise the coming of his main attraction for the Fourth. Mrs. Blair came down to help with the folding and Margaret Stevens, just back from a vacation in the north woods with her father, arrived in time to lend a hand. “Nice trip?” Helen asked as she deftly folded the printed sheets. “Wonderful,” smiled Margaret, “but I’m glad to get back. I missed helping you and Tom. Honestly, I get a terrific thrill out of reporting.” “We’re glad to have you back,” replied Helen, “and I think Mr. Provost down at Sandy Point will be glad to give me an extra pass for the Fourth. I’ll tell him you’re our star reporter.”
- 77. [186] “I’d rather go to Crescent Beach for the Fourth,” said Margaret. “It’s newer and much more ritzy than Sandy Point.” “You’d better stop and look at the front page carefully,” warned Tom, who had shut off the press just in time to hear Margaret’s words. She stopped folding papers long enough to read the type under the two column picture on the front page. “What!” she exclaimed, “‘Speed’ Rand coming here?” “None other and none such,” laughed Tom. “Guaranteed to be the one and only ‘Speed’ Rand. Step right this way folks for your airplane tickets. Five dollars for five minutes. See the beauty of Lake Dubar from the air. Don’t crowd, please.” “Do you still want me to get a pass?” Helen asked. “It will be honored any place at Sandy Point during the celebration and Mr. Provost says we can all have rides with the air circus ‘Speed’ Rand is running.” “I should say I do want a pass,” said Margaret. “At least it’s some advantage to being a newspaper woman besides just the fun of it.” The famous Ace air circus of half a dozen planes roared over Rolfe just before sunset Friday night and the whole town turned out to see them and try to identify the plane which “Speed” Rand was flying. The air circus was flying in two sections, three fast, trim little biplanes that led the way, followed by three large cabin planes used for passenger carrying. Every ship was painted a brilliant scarlet and they looked like
- 78. [187] tongues of flames darting through the sky, the afternoon sun glinting on their wings. The air circus swung over Rolfe in a wide circle and the leading plane dropped down out of the sky, its motor roaring so loud the windows in the houses rattled in their frames. “He’s going to crash!” cried Margaret. “Nothing of the kind,” shouted Tom, who had read widely of planes and pilots and flying maneuvers. “That’s just a power dive—fancy flying.” Tom was right. When the scarlet biplane seemed headed for certain destruction the pilot pulled its nose up, levelled off, shot over Rolfe at dizzying speed and then climbed his craft back toward the fleecy, lazy white clouds. “That’s Rand,” announced Tom with a certainty that left no room for argument. “He’s always up to stunts like that.” “It must be awfully dangerous,” said Helen as she watched the plane, now a mere speck in the sky. “It is,” agreed Tom. “Everything depends on the motor in a dive like that. If it started to miss some editor would have to write that particular flyer’s obituary.” The morning of Saturday, the Fourth, dawned clear and bright. Small boys whose idea of fun was to arise at four o’clock and spend the next two hours throwing cannon crackers under windows had their usual good time and Tom and Helen, unable to sleep, were up at six o’clock. Half an hour later Margaret Stevens, also awakened by
- 79. [188] [189] the almost continuous cannonading of firecrackers, came across the street. “Jim Preston is going to take us down the lake on his seven-thirty trip before the special trains and the big crowds start coming in,” said Tom. “But I’d like to see the trains come in,” protested Helen. “If we wait until then,” explained Tom, “we’ll be caught in the thick of the rush for the boats and we may never get to Sandy Point. We’d better take the seven-thirty boat.” From the hill on which the Blair home stood they looked down on the shore of Lake Dubar with its half dozen boat landings, each with two or three motorboats awaiting the arrival of the first special excursion train. Mrs. Blair called them to breakfast and they were getting up to go inside when Margaret’s exclamation drew their attention back to the lake. “Am I seeing things or is that the old Queen?” she asked, pointing down the lake. Tom and Helen looked in the direction she pointed. An old, double decked boat, smoke rolling from its lofty, twin funnels, was churning its way up the lake. “We may all be seeing things,” cried Tom, “but it looks like the Queen. I thought she had been condemned by the steamboat inspectors as unfit for further service.” “The news that ‘Speed’ Rand is going to be at Sandy Point is bringing hundreds more than the railroad expected,” said Helen. “I talked with the station agent
- 80. [190] last night and they have four specials scheduled in this morning and they usually only have two.” “If they vote the paved roads at the special election next week,” commented Tom, “the railroad will lose a lot of summer travel. As it is now, folks almost have to come by train for the slightest rain turns the roads around here into swamps and they can’t run the risk of being marooned here for several days.” The Queen puffed sedately toward shore. They heard the clang of bells in the engine room and the steady chouf-chouf of the exhaust cease. The smoke drifted lazily from the funnels. Bells clanged again and the paddle wheel at the stern went into the back motion, churning the water into white froth. The forward speed of the Queen was checked and the big double-decker nosed into its pier. “There’s old Capt. Billy Tucker sticking his white head out of the pilot house,” said Tom. “He’s probably put a few new planks in the Queen’s rotten old hull and gotten another O. K. from the boat inspectors. But if that old tub ever hits anything, the whole bottom will cave in and she’ll sink in five minutes.” “That’s not a very cheerful Fourth of July idea,” said Margaret. “Come on, let’s eat. Your mother called us hours ago.” They had finished breakfast and were leaving the table when Mrs. Blair spoke. “I’ve decided not to go down to Sandy Point with you,” she said. “The crowd will be so large I’m afraid I wouldn’t enjoy it very much.”
- 81. [191] “But we’ve planned on your going, Mother,” said Helen. “I’m sorry to disappoint you,” smiled her mother, “but Margaret’s mother and I will spend the day on the hill here. We’ll be able to see the aerial circus perform and really we’ll enjoy a quiet day here at home more than being in the crowd.” “It won’t be very quiet if those kids keep on shooting giant crackers,” said Tom. “They’ll be going to the celebration in another hour or two and then things will quiet down,” said Mrs. Blair. “How about a plane ride if the circus has time to take us?” asked Tom. Helen saw her mother tremble at Tom’s question, but she replied quickly. “That’s up to you, Tom. You know more about planes than I do and if you’re convinced the flying circus is safe, I have no objection.” But Helen made a mental reservation that the planes would have to look mighty safe before any of them went aloft. They hurried down the hill to the pier which Jim Preston used. The boatman and his helpers had just finished polishing the three speed boats Preston owned, the Argosy, the Liberty and the Flyer, which had been raised from the bottom of the lake and partially rebuilt. “All ready for the big day?” asked the genial boatman. “We’re shy a few hours sleep,” grinned Tom. “Those cannon crackers started about four o’clock but outside of that we’re all pepped up and ready to go.”
- 82. [192] “About three or four years ago,” reminded the boatman, “you used to be gallivantin’ around town with a pocketful of those big, red crackers at sun-up. Guess you can’t complain a whole lot now.” Tom admitted that he really couldn’t complain and they climbed into the Liberty. “I’m takin’ some last minute supplies down to the hotel at Sandy Point,” said the boatman, “so we won’t wait for anyone else.” He switched on the starter and the boat quivered as the powerful motor took hold. They were backing away from the pier when the pilot of one of the other boats shouted for them to stop. A boy was running down Main Street, waving a yellow envelope in his hand. Jim Preston nosed the Liberty back to the pier and the boy ran onto the dock. “Telegram for you,” he told Helen. “It’s a rush message and I just had to get it to you.” “Thanks a lot,” replied Helen. “Are there any charges?” “Nope. Message is prepaid.” Helen ripped open the envelope with nervous fingers. Who could be sending her a telegram? Was there anything wrong with her father? No, that couldn’t be it for her mother would have received the message. She unfolded the single sheet of yellow paper and read the telegraph operator’s bold scrawl.
- 83. [193] [194] “To: Helen Blair, The Herald, Rolfe. Understand ‘Speed’ Rand is at Rolfe for two days. Have rumor his next flight will be an attempted non-stop refueling flight around the world. See Rand at once and try for confirmation of rumor. Telephone as soon as possible. McClintock, The AP.” Helen turned to Tom and Margaret. “I’m to interview ‘Speed’ Rand for the Associated Press,” she exclaimed. “Let’s go!”
- 84. CHAPTER XIII Helen’s Exclusive Story While the Liberty whisked them through the glistening waters of Lake Dubar toward Sandy Point, Margaret and Tom plied Helen with questions. “Do you think Rand will give you an interview?” demanded Tom. “I’ve got to get one,” said Helen, her face flushed and eyes glowing with the excitement of her first big assignment for the Associated Press. “What will you ask him? How will you act?” Margaret wanted to know. “Now don’t try to get me flustered before I see Rand,” laughed Helen. “I think I’ll just explain that I am the local correspondent for the Associated Press, show him the telegram from Mr. McClintock and ask him to confirm or deny the story.” “I’ll bet Rand’s been interviewed by every famous reporter in the country,” said Tom.
- 85. [195] [196] “Which will mean all the more honor and glory for Helen if she can get him to tell about his plans,” said Margaret. “I’ll do my best,” promised Helen and her lips set in a line that indicated the Blair fighting spirit was on the job. They were still more than two miles from Sandy Point when a scarlet-hued plane shot into sight and climbed dizzily toward the clouds. It spiralled up and up, the roar of its motor audible even above the noise of the speedboat’s engine. “There’s ‘Speed’ Rand now!” cried Tom. “No one flies like that but ‘Speed’.” The graceful little plane reached the zenith of its climb, turned over on its back and fell away in twisting series of spirals that held the little group in the boat breathless. The plane fluttered toward the lake, seemingly without life or power. Just before it appeared about to crash, the propeller fanned the sunlight, the nose jerked up, and the little ship skimmed over the waters of the lake. It was coming toward the Liberty at 200 miles an hour. On and on it came until the roar of its motor drowned out every other sound. Helen, Tom and Margaret threw themselves onto the floor of the boat and Jim Preston crouched low behind his steering wheel. There was a sharp crash and Helen held her breath. She was sure the plane had struck the Liberty but the boat moved steadily ahead and she turned quickly to look for the plane.
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