It is often proposed that the redundancy in choosing a force distribution for multiple arms grasping a single object should be handled by minimizing a quadratic performance index. The performance index may be formulated in terms of joint torques or in terms of the Cartesian space force/torque applied to the body by the grippers. The former seeks to minimize power consumption while the latter minimizes body stresses. Because the cost functions are related to each other by a joint angle dependent transformation on the weight matrix, it might be argued that either method tends to reduce power consumption, but clearly the joint space minimization is optimal. A comparison of these two options is presented with consideration given to computational cost and power consumption. Simulation results using a two arm robot system are presented to show the savings realized by employing the joint space optimization. These savings are offset by additional complexity, computation time and in some cases processor power consumption

Alberts, Thomas E.

Soloway, Donald I.

English

NASA Technical Reports Server

N90-29373
CONPARISON OF JOINT SPACE VERSUS TASK FORCE LOAD DISTRIBUTION
OPTINIZATION FOR A NIJLTIAJU4NANIPUI_ATOR SYSTEN
Donald I. Soloway
NASA Langley Research Center
Hampton, Virginia 23665-5225
Thomas E. Alberts
Old Dominion University
Norfolk, Virginia
Abstract
It is often proposed that the redundancy in choosing a force distribution
for multiple arms grasping a single object should be handled by minimizing a
quadratic performance index. The performance index may be formulated in terms
of joint torques or in terms of the Cartesian space force/torque applied to the
body by the grippers. The former seeks to minimize power consumption while the
latter minimizes body stresses. Because the cost functions are related to each
other by a joint angle dependent transformation on the weight matrix, it might
be argued that either method tends to reduce power consumption, but clearly the
joint space minimization is optimal. In this paper, a comparison of these two
options is presented with consideration given to computational cost and power
consumption. Simulation results using a two arm robot system are presented to
show the savings realized by employing the joint space optimization. These
savings are offset by additional complexity, computation time and in some cases
processor power consumption.
1. Introduction
Some of the recent developments in multiple arm manipulation of a commonly
grasped body include the work of Hayati [1], Alberts [2], and Carnignan [3].
The common thread between these papers is that each employs the minimization of
some form of quadratic performance index to choose an appropriate load
distribution. Hayati proposed an extension of Mason's [4] hybrid position/force
control in which the inertia of each arm is artificially extended to include a
portion of the payload inertia. From a practical point of view, the method may
be difficult to implement effectively, because it requires precise knowledge of
the inertial properties of the arms and of the jointly manipulated objects as
well as the solution of inverse dynamics. Alberts closes a force feedback loop
around a kinematic resolved rate controller for multiple arms, thus realizing
the Damping Control Method. As in Hayati's work the problem of redundancy in
determining the distribution of load among the manipulators is handled by
minimizing a quadratic cost function in task-space force and torque. This tends
to minimize internal forces in the body while maintaining control over a
prescribed force and torque interaction with the external environment. Alberts
formulation does not consider the closed chain dynamics of the manipulators and
i::,./._. ........... ; .... :. L_..., ;;_ _:,_'FILMED 431
https://ntrs.nasa.gov/search.jsp?R=19900020557 2020-03-19T21:38:27+00:00Z
payload, but rather each manipulator is viewed as an actuator with an
independent control system. Carnignan used a quadratic cost function to
minimize joint space torques including those due to manipulator kinetics, but
due to the inclusion of the manipulator kinetic effects, undesirable internal
forces may be produced in the body. This method is computationally more
expensive than task space optimization.
It can be argued that minimizing a quadratic cost function in joint space
has greater power efficiency than a minimization in task space, but at what
computational cost? This paper compares the task space versus joint space cost
functions with respect to power efficiency and computational cost. The
development is to be based on Alberts' task space cost function and a joint
space cost function developed along similar lines. The computational cost of
using the joint space minimization scheme is similar to Carnignan's method. The
torques required to compensate for manipulator kinetics are not included in the
minimization so as to avoid imposing unnecessary internal forces and torques
within the manipulated body.
2. Power Cost Calculation
In a multiarm robotic system, minimization of a joint torque based
quadratic cost function would tend to minimize the dissipated power used by the
motors under static conditions. 1
A minimization scheme to establish an "optimal" force distribution could be
formulated in either joint space or task space. From the standpoint of power
used to drive the joints, it would be optimal to formulate the minimization in
joint space. In this paper the question considered is that of how much power
really can be saved by optimizing in joint space as opposed to task space.
Based upon a task space quadratic performance index
T
Qt = -FtWt_t (I)
the task space optimal load distribution force equation [2] is
Zt = At£ (2)
where
At = wtlHT[HwtlHT] -1 (3)
and _t is a vector of wrench vectors (as in Equation 15, Appendix I) such
that each components F(_I of_t is a wrench vector applied to the body at
the grasp point of the-i manlpulator and subscript t denotes task space. H
is a matrix of Jacobian transformations that depends upon the locations of the
grasp points of each arm relative to the applied external force F. The weight
1To truly minimize power consumption under dynamic conditions the cost
function should include a term representing the product of joint torque and
joint velocity. This is discussed further later in reference to Equation (9).
432
matrix is given by Wt = diag (v1, v2,...,v N) where N is the number
of arms and vi is a diagonal weight matrix reflecting the relative cartesian
end effector force and torque capabilities of the ith arm.
Using a similar development (Appendix I) but, based upon a joint space
quadratic performance index
Qj = TTwjz (4)
the joint space optimal load distribution force equation is
_rj= AjF (5)
where
Aj = j-Twj-Ij-1H[Hj-Twjj'IH-T] -I (6)
with W" = diag (w1, w2, ...,wN) and wi is a diagonal matrix whose
element_ are the squares of the reciprocals of the relative joint torque
capabilities of the ith arm. Subscript j denotes joint space.
The cost function is an indication of power used since torque (3) is
proportional to current and current squared is proportional to power
dissipated. To compare power consumption on an equal basis the cost of task
space optimized load distribution was evaluated in terms of the joint space cost
function. Thus, for evaluative purposes, the torque-squared cost of executing
the task space optimization scheme can be expressed as
Qtj = TTwj Tt (7)
where Tt = JTAt_FF
and the cost associated with joint space optimization is given by Equation (4),
that is Qil = Qi- The cost resulting from the joint space formulation will
always be _i_ass n that of the task space formulation according to the above
criterion.
The power used by a motor is
2 KE KE
Power = laR + l_T mTm + l_TmT l
(8)
where Ia is the armature current
R is the armature resistance
KE is the back emf constant
KT is the torque constant
is the angular speed of the rotor
Tl is the torque of the load
433
Tm = Tf + Dm where Tf is constant friction
and D is the viscous damping coefficient.
The current Ia is related to the dynamcs of the motor by
Ia : 1 [(Jm + Jl)dm + Tm + TI] (9)
where Jm is the motor inertia and Jl is the load inertia. Observe that
in the jointspace minimization presented here (7) minimizes the power due to
the term laZR in (8) but does not account for the power associated with
mTl in the last term of (8). From a practical point of view it appears that
the contribution of this term will normally be small. In simulations conducted,
in which the load velocity was 0.1 m/s, the mT l term resulted in power
difference of less than I%.
The percent difference in power is defined as follows
PN = Pt - Pj * 100
Pj
(10)
where Pt and Pj represent the power used by task and joint space
respectively.
The power difference is
PD = Pt - Pj where PD is in watts (11)
A simulation 2 was used with the system parameters of the NASA LTM 3 (given
in Appendix Ill) to compute the power used by the two-arm LTM system in moving a
40 lb. payload on Earth. The center of gravity of the object was at the grasp
point of one of the arms. The two arms moved horizontally without rotating the
object. For this trajectory PD and PN were plotted in figures i and 2
respectively. The initial Pj = 1093 watts.
2The simulation is a combination of ROBSIM (a full dynamics robot simulator
developed by NASA LaRC) and user coded. Robsim simulated the robotics dynamics
and the user code calculated the load distributed joint torques and costs.
3LTM is an acronym for the Laboratory Telerobotic Manipulator developed for
NASA LaRC by Oak Ridge National Laboratory. The LTM is a seven-degree-of-
freedom arm employing differential friction drive joints. For simplicity the
differential joint drive are treated as conventional gear driven joints.
434
24. g
p 24.8
D
W
24.6
G
t
t
s 24.5
24.4
O .5 1 1.5 2 2.5 3 3.5
Time in Seconds
Figure 1. The difference between task space and joint space power comsumption.
4
2.4
2.2
1.8
1.5
1.4
1.2
0
Ftgure 2.
.5 1 1.5 2 2.5 3 3.5 4
Time in Seconds
The percentage of power saved wtth respect to the total power
used In jotnt space,
435
3. Computational Cost Calculation
In a real-time computer program it is desirable to keep computationally
expensive operations to a minimum. The number of operations it takes to
calculate task space and joint space load distribution are determined below.
For this analysis the following assumptions are made:
1. The manipulators grasp points are fixed while the body is in motion,
thus H remains a constant matrix.
2. W does not change over the motion of the body.
3. All manipulators have the same number of degrees of freedom.
In task space A is constant thus requiring only a matrix vector multiply to
calculate rt. A is a 6N x 6 matrix where N is the number of manipulators
grasping the body, and F is a 6 x 1 vector needing:
36N multiplies;
30N additions
to calculate rt.
Joint space will have the same operations as above plus the operations to
calculate A. The total number of operation needed to calculated _j as
derived in Appendix II is:
N(48n + 72) + 195
71
N(48n + 30) + 206
multiplies
divides
additions
With the need to calculated the jacobian inverse the total number of operations
to calculate rj as derived in Appendix II is:
N(60n + 267) + 195
71N + 71
N(58n + 236) + 206
multiplies
divides
additions
Currently the most powerful space qualified procesor is the Harris 80C86. Based
on using this processor with a 5 mhz clock and the number of arms and degrees of
freedom (DOF), the results Table I, II, and Ill are obtained. The entries in
the tables represent the number of times the computation can be executed in 1
second.
Number of Arms
D F
e r 6
g o e
r f e 7
e d
e o 8
s m
17.5
16.4
15.3
3
12.4
11.5
10.7
4
9.54
8.86
8.27
Joint Space with Jacobian Inverse Calculated
Table I
436
D F
e r
g o e
r f e
e d
e o
s m
2
29.8
27.2
25.0
3
21.9
19.8
18.1
4
17.3
15.6
14.2
Joint Space without Jacobian Inverse Calculated
Table II
D F
e r
g o e
r f e
e d
e o
s m
6
7
8
2
413
413
413
3
275
275
275
4
206
206
206
Task Space
Table Ill
The computational cost of task space optimization is small enough to execute the
code on an existing processor whereas joint space optimization would likely
require a separate processor. In this case the power required to operate
additional processing equipment must be considered. The power requirement of an
64K 80C86 board based on Harris radiation hardened components is 30 watts.
4. Summary
It has been shown that an apparently significant amount of power can be
saved by employing the joint space optimized load distribution. In the example
presented the largest savings over task space optimization realized was about 25
watts. This can be viewed as an extreme case. It is important to note,
however, that the joint space optimization represents a substantial increase in
computational burden. If this results in a need for additional processors, the
power required to operate them might offset the savings realized by the
optimization scheme.
In the future, space qualified processors will be available that are much
more powerful and possibly more power efficient than those now used. In the
case where a surplus of efficient computing power is available, the joint space
optimization may prove to be the method of choice.
437
Appendix I
5. Joint Space Optimal Load Distribution
Assume that a desired net cartesian space wrench vector F acting on the
body with known point of application p is specified. The wrench vector is
made up of cartesian space force f and moment m vectors such that
b
(12)
This wrench could be to counteract gravitational loading or inertial reactions
due to body acceleration, or to apply a force to the external environment
through the jointly manipulated object. Now consider a system of N
manipulators, where F(i) denotes a cartesian space wrench applied to the
jointly manipulated body by the end effector of arm i. The following
conditions must be satisfied in order to establish equilibrium:
N N
f = Z (13) and m = Z (m(i) + r_p(i) X f(i) )
- i=1 - i=1 - -
(14)
where f(i) and m(i) are the force and moment vectors, respectively, that
make up- _(i)
(15)
and _[p(i) is a vector drawn from the point of application p of the force
f to the grasp point for manipulator i. The minimization procedure operates on
quadratic function Q in the vector T, with positive definite symmetric
weighting matrix W
Q = T W _ (16)
where
= JTr,
Z--
F(
jT is the jacobian transpose of the manipulator
i] and W = diag (Wl,
W 2 , , W N )D 0 e
Conditions (13) and (14) are expressed in matrix form as
(17)
438
I o Eri+ ;13!_3 ,---. -i_ix ""rp(1)X]| 13 ][:]13 I 0 (1)[rp(N)X] I 13
where, 13 is a 3 by 3 identity matrix and [_(i)X] is a matrix that
operates on f(i) to form the cross product --_[p(i)X _f(i). A more compact
expression fo-r (18) is
(18)
=F
HJT T = F
Using a Lagrange multiplier
cost function is obtained•
(19)
to append equation (19) to (16) the augmented
= _TwT - xT(Hj'TT -F_) (20)
The optimal solution must satisfy
@--_= O, that is 2TTw - xTHj-1 = 0 (21)
Rearranging and applying equation (19) to equation (21)
T = 1 HW-1j-1HTx which yields X = 2(Hj'Tw-1j-IHT)-I[ (22)
Now upon eliminating X between equations (21) and (22) the final expression
for joint torques is obtained•
T = W-Ij-1HT(Hj-Tw-Ij-1HT)-IF = AF (23)
Appendix II
To count the operations in A it is necessary to first simplify
two matrices A and B, such that
where A = AB-1, A = j-Twj-Ij-1HT and B = HA
Calculating A:
J;10J21 0 1 w-l=
• j_l I ' j ]0 andw wil w_I HT=
A into
HT
:I
439 ,
where j_l is the jacobian inverse for manipulator i,
wT1 is the weight matrix inverse for manipulator
and HT
i
SO
is the transform grasp point to controlled force F
manipulator i.
for
and
A = j-TwTIj-1HT- =
J
o ]"-1 -1 -T T
L 0 JN WN JN HN
B = HA =
0 -1 -1 -T T
HIJN WN JN HN
.-T.-I .-1.T
The cost to calculate ui Hi ui _i is:
-I T
Ji Hi =
at a cost of
[I[ioozn x6 1 -z ofull 0 I I y -xmatrix J 1
0 1
Io
n*(6 multiplies + 6 additions)
X
0
0
i
WTIFJTIHTIFnx6 1: lf.ll I
I-I I- Lmatrixj
at a cost of n*6 multiplies
-T -1 -1 T
Ji [Wi Ji Hi] =
6x61
ful I J
matrix]
at a cost of n36 multiplies + (n-1)36 additions•
N[48n multiplies + (48n-36) additions].
The total cost of A is
440
Calculating B:
Bi = HiAi = full full
matrix matri x
at a cost of 36 multiplies 30 additions. Using triangular factorization to
count the operation for B-1, the cost is 195 multiplies 71 divides 206
additions. Total cost for Aj is
N(48n + 36) + 195 multiplies
71 divides
N(48n) + 206 additions
To calculate the cost for a jacobian inverse a generalized inverse formulation
is used. Ji-1 = JiT[jiJiT] -I where Ji is the 6 x n
jacobian for the ith manipulator and n is the degree of freedom of the
manipulator. The cost is 12n + 6 multiplies 71 divides and 1On + 206
additions giving a total cost for Aj with calculation of the jacobian
inverse as
N(60n + 231) + 195
71N + 71
N(58n + 206) + 206
multiplies
divides
additions
Appendix Ill
6. LTM System Parameters
Motor constants (for all 7 joints)
Torque Constant KT 8.5
Back emf constant KE 6.3
Armature resistance R 2.5
Armature inertia Jm 0.0015
Viscous Damping D 0.3
Static Friction Tf 0.8
oz-in/A
V/KRPM
OHM
oz-in-sec 2
oz-in-KRPM
oz-in
Gear ratio from motor shaft to joint (assuming conventional gear driven joints)
Joint 1 522
Joint 2 522
Joint 3 522
Joint 4 522
Joint 5 121
Joint 6 121
Joint 7 25
Denavit and Hartenberg parameters
441
Joint
where
joint
I
2
3
4
5
6
7
Inertia Matrix
Denavi t-Hartenberg Parameters
d a a o
0
0
0
0
0
0
5.9"
_90 °
90°
_90 °
90°
_90 °
0
23"
0
20"
0
eI + 90 °
o2
e3 - 90 °
o4
e5
e6 + 90 °
e7 - 90 °
90°
0
0
0
Link
Number
1
4
6
7
Inertia Matrix
KG-M
0.029
0.0
0.0
0.0
0.0145
0.0
0.0
0.0
0.0145
0.029
0.0
0.0
0.0
O.2989
0.0
0.0
0.0
O.2989
0.029
0.0
0.0
0.0
0.0145
0.0
0.0
0.0
0.0145
0.029
0.0
0.0
0.0
0.2296
0.0
0.0
0.0
0.2296
0.0163
0.0
0.0
0.0
O.0082
0.0
0.0
0.0
O.0082
0.0163
0.0
0.0
0.0
O.0269
0.0
0.0
0.0
O.0269
0.0182
0.0
0.0
0.0
O.0099
0.0
0.0
0.0
O.0099
the orlentation matrix is referenced to
Orientation
Matrix
1.0 0.0 0.0
0.0 0.0 -I.0
0.0 1.0 0.0
0.0 1.0 0.0
0.0 0.0 1.0
1.0 0.0 0.0
1.0 0.0 0.0
0.0 0.0 -1.0
0.0 1.0 0.0
0.0 -i.0 0.0
0.0 0.0 1.0
-I.0 0.0 0.0
1.0 0.0 0.0
0.0 0.0 -1.0
0.0 1.0 0.0
1.0 0.0 0.0
0.0 0.0 1.0
0.0 -1.0 0.0
0.0 -1.0 0.0
0.0 0.0 1.0
-1.0 0.0 0.0
the base of LTM.
442
.1.
.
1
1
.
Bibliography
Hayati, S.: "Hybrid Position/Force Control of Multiarm Cooperating Robots."
IEEE Conference on Robotics and Automation, April 1986.
Alberts, T., and Sol,way, D.: "Force Control of Multiarm Robot System."
IEEE Conference on Robotics and Automation, August 1988.
Carnignan, Craig R., and Akin, David L.: "Cooperative Control of two arms
in The Transport of an Inertial Load in Zero G." IEEE Conference on
Robotics and Automation, August 1988.
Mason, M. T.:
Manipulators."
pp. 418-432.
"Compliance and Force Control for Computer Controlled
IEEE Trans. on Systems, Man Cybernetics, v. 11, June 1981,
Denavit, J., and Hartenberg, R. S.: "A Kinematic Notation for Lower Pair
Mechanisms Based on Matrices." ASME Journal of Applied Mechanics, June
1955, pp. 215-221.
443



Comparison of joint space versus task force load distribution optimization for a multiarm manipulator system

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