In this paper an approach for the application of parallel processing to the dynamic analysis of robots based on the multibody system method is presented. The inherent structure of the symbolic equations of motion is used for partitioning those into independent modules for concurrent evaluation. The applied strategies for parallelization include the parallel evaluation of subsystem equations and the parallel computation of the inertia matrix along with its factorization, and of the force vectors and control inputs. The implementation of the parallel structures is discussed with respect to optimal utilization of transputer resources. The presented approach yields a strong reduction of computing time and supports real-time simulations of controlled robots, which is also shown in an example

Schiehlen, Werner

Schäfer, Peter

Online Publikationen der Universität Stuttgart

I'ruc.'t.>(l inI(S ,,{ the I~I "CS SIC E lmernll t iuna l Symposium on Robot ics. ;" Iechatronics and 
~l;mu fi1c t urinl( SyMf'fIls'!f2 ""be, Japan, Sep. 1 6 ~2/J. 1!/92 
Application of Parallel Computing to Robot Dynamics 
P. Schafer and W. Schiehlen 
Insti tute B of Mechanics, University of Stuttgart, 
Pfaffcnwaldring 9, W-7000 Stuttgart 80, Germany 
Abstract 
599 
In this paper an approach for the application of parallel processing to the dynamic 
analysis of robots based on the multibody system method is presented. The inherent 
structure of the symbolic equations of motion is used for partitioning those into 
independent modules for concurrent evaluation. The applied strategies for parallelization 
include the parallel evaluation of subsystem equations and the parallel computation of the 
inertia matrix along with its factorization, and of the force vectors and control inputs. The 
implementation of the parallel structures is discussed with respect to optimal utilization of 
transputer resources. The presented approach yields a strong reduction of computing time 
and supports real-time simulations of controlled robots, which is also shown in an example. 
1. INTRODUCTION 
For the modeling of robots and walking machines the method of multibody systems is 
well qualified and it provides reliably the system's state equations, see e.g Schweitzer; 
Mansour [1]. However, the resulting equations of motion and the equations of reaction are 
highly nonlinear and real-time solutions cannot be obtained by standard computing 
technology. Therefore, special real-time hardware such as, for instance, a transputer 
network has to be used for this type of application, e.g. Schiehlen; Schafer [2] . Transputers 
are specifically designed for distributed memory parallel processing. In order to exploit the 
features of this computer hardware the equations of motion have to be investigated with 
respect to the potential of parallel computation. The approach to be presented is based on 
the automatic generation of symbolic equations of motion for robots, Kreuzer; Leister [3]. 
An efficient parallel implementation has to take into account that communication between 
processors is costly and coarse grain parallelism has to be achieved. Combining different 
strategies for paratlelizing the robot dynamics equations leads to a considerable reduction 
of computation time and supports real-time simulation. 
2. MODELING OF ROBOT DYNAMICS 
For the modeling of the dynamical behavior of a robot the links arc assumed to be rigid 
bodies interconnected by joints, The links are subject to applied forces and torques, e.g. tool 
forces, or forces and torques due to actively controlled drives. For a system consisting of p 
bodies subject to q holonomic and scleronomic constraints the position vector Ti of the 
center of mass Ci and the rotation tensor Si of body i, j; ] (J )p, can be expressed in terms of 
[=p-q generalized coordinates represented by the f x 1 position vector y as 
ri ; r;iy,t), Si; S,(y,t), (I) 
sec Figure 1. Differentiation with respect to time yields the linear and angular velocities v I 
and "'i and accelerations aj and ail respectively, 
Vj - JnY+ii"j. 
Qi -}nY+i1j • 
"' . ;J~y. +", . 
I " , . 
dj=Jr,Y + Cij (2) 
with the Jacobian matrices of translation In and rotation JRj . The vectors Vj' W i' ii; and aj 
contain the terms that arc not linearly dependent on j and j. respectively. The dynamic 
equations can be derived by applying Newton's and Euler's law for each frcc body with mass 
11\ and inertia tensor 1;. They read as 
m · Q. =r.a+r.c 
I I Jj Jj • / ." . + (J) . X / ·OJ · = I~ + I~ I I I " , , , (3) 
where f;Q and Ii represent the applied forces and torques and !;C and If comprise the 
constraint forces and torques, respectively. After assembling the masses and inertia tensors 
as well as the Jacobians to the global matrices 
T 
1:1 = diag!m,E, ... ,mpE''/" ... ,/pj, J; [4 ..... .Jrp.J~,,, ... .J~1 ' (4) 
and assembling the applied forces qa in an analogous manner the Newton- Euler equations 
of the overall system can be written as a 6p x 1 vector equation 
g J j + f ; iI' + fl g, (5) 
where f contains Coriolis and centrifugal forces. The constraint forces arc represented by 
the q x 1 vector gof generalized constraint forces and a corresponding 6p x q distribution 
matrix Q. According to d'Alcmbert's principle the constraint forces can now be eliminated 
by pre multiplying (5) with the transpose J T of the global Jacobian matrix to obtain the 
equations of motion 
M(y)j(t) + k(y,j ,t) ; q(y,j,u,t) (6) 
with the f x f symmetric and positive definite inertia matrix M, the f x 1 vector of 
generalized Coriolis and centrifugal forces k and the f x I vector of generalized applied 
forces q. The vector u summarizes the control inputs. The equations of reaction, on the 
..,.- , 
other hand, can be found by premultiplying (5) with fl T M which yields 
N(y)g(t) - k(y,j,t) ; - q(y,j ,u,t) . (7) 
For numerical integration (6) has to be solved for the generalized accelerations j which 
can be done by Cholesky decomposition M ; LLT with a lower triangular f x f matrix L to 
be determined . The required number of operations is of order 0(f3) and this procedure 
x 
bOdY).._-~ 
/" ... Ci / I / 
~- '" 
y 
601 
becomes computationally expensive if the 
number of degrees of freedom increases. 
Therefore, the recursive dynamics 
approach is widely being applied in robot 
dynamics analysis, see e.g. Brandl, 
Johanny, Olter [4 J. In this approach the 
kinematic quantities are determined 
recursively starting from the base body. 
Likewise, the forces and torques are 
determined in a backward recursion 
starting from the endeffector. The inertia 
matrices 3rc inverted locally in the 
Newton-Euler equations (3) for each 
body. A final recursion solves a set of 
linear equations with triangular structure 
to obtain explicit equations of motion in 
the generalized accelerations, 
j = fCy ,y,u,I). (8) 
Figure 1. Robot structure and tree body 
Thus. the inversion of total inertia matrix is avoided. However, the evaluation of the 
generalized accelerations requires some overhead not necessary in the nonrecursive 
formulation. 
3. PARALLEL EVALUATION OF ROBOT DYNAMICS EQUATIONS 
The aim of the parallelization of robot dynamics equations is to implement the 
computations on transputers, which arc distributed memory parallel computers. Therefore, 
fine grain parallelization is not considered. Instead, it is necessary to partition the equations 
to be evaluated into blocks that are independent of each other as far as possible. Since 
function evaluations are by far the most costly part of robot simulations, the discussion of 
parallelism is restricted to the evaluation of the equations of motion. Furthermore, the 
investigations are based on symbolic equations of motion of the form (6) and the serial 
recursive dynamics are not considered. 
There arc several possibilities for obtaining partially independent computations in the 
symbolic equations of motion (6). The first strategy is motivated by the partitioning of the 
overall system into subsystems as shown in Figure 2. The local coordinates are represented 
by the vectors Yl and Y2J respectively, and the local equations of motion read as 
(9) 
with matrices Mi, and vectors ki and qi ' i= I (1)2, corresponding to (6). Here, g represents 
the constraint forces and torques fi2 and li2 introduced by cutting the system and Q l' Q2 
are the corresponding distribution matrices. The coupling of the subsystems can be 
performed by relating the local coordinates y 1 and Y2 to the global generalized coordinates, 
(10) 
with the subsystem Jacobian matrices 11 
and 12. The global equations of motion (6) 
follow by substituting the accelerations j I 
and y, derived from (10) into (9), 
premultiplying the resulting equations by 
i[and If, respectively, and adding them 
up in order to eliminate the constraint 
forces g. The coupling of subsystems can 
also be achieved for local equations of the 
form (8), Schiehlen [5J. This method 
avoids the inversion of the global inertia 
matrix M. but the constraint forces have to 
be computed instead using (7). This 
approach may be beneficial if systems with 
kinematic loops are treated. 
o 
o 
subsystem 2 
subsystem I 
Figure 2. Subsytem partitioning 
Equations (9) can be implemented for concurrent evaluation and the coupling is done 
subsequently, as shown in Figure 3. However, the inherent dependency between the 
subsystem kinematics exploited by the recursive dynamics approach leads to a strong data 
dependency. Consequently, the parallel computation is serialized by the communication of 
interface variables and only minor benefits of a concurrent evaluation can be achieved. For 
tree structured systems, on the other hand, subsytem equations of independent branches 
can be evaluated fully in parallel. In vehicle dynamics, e.g., this strategy proves to be most 
appropriate, Knaupp [6J. 
M, k, q 
Figure 3. Parallel evaluation of subsystem equations, strategy 1 
The second strategy for parallelization follows from the independence of ], f and 1[' in 
(5) which can be computed completely concurrently. The basic structure of the 
computational evaluations is shown in Figure 4. This approach is particularly interesting for 
extensive control law calculations which again can be performed on separate processors, as 
indicated with dashed lines in Figure 4. In some cases it may be of advantage either to 
transfer selected clements of the Jacobian matrix to the neighboring processors or to 
compute them locally, thus enabling these processors to compute k =] Tf or q =] T 1[, 
respectively, instead of evaluating k or if only. 
"i,! 
I,M,M 
q 
I r----' 
-"I u 1-1 L ___ J 
k q 
6<)3 
IrT L-l(q-k)~ 
Y 
Figure 4. Parallel evaluation of inertia matrix, force vectors and control law, strategy 2 
A drawback of lhe first two strategies presented is that the cost of the Cholesky 
decomposition fully adds to the execution time required for the preceding computations. In 
order to avoid this, a separate processor can be assigned to this task running in parallel to 
the computation of the force vectors k and q, FigureS. This third strategy pays off in any case 
and contributes strongly to overcome the disadvantage of the non recursive approach due to 
matrix factorization . The recursive formulation, on the other hand, leads to a coupling of all 
computations and has tess potential for paraIlciization. 
-y,y-., 1--· [I ~1~'M~'~M:::Jl~~;M~~;L;L~~T [~q=}1 I'.-.-IL-....,--T:L--~l-(q~~k-)-:~.--;:· 
Figure S. Pa rallel evaluation of force vectors and matrix factorization , strategy 3 
The number of operations in each of the different strategies is strongly problem 
dependent. In general,optimal processor utilization and speed-up can only be achieved by 
a combination of the different parallelization strategies. For this, the timing of the separate 
threads has to be monitored and processor assignment has to be done accordingly. 
In the above, the equations of reaction (7)were not considered, since they are completely 
independent of the solution of the equations of motion (6). However, the computation of 
the constraint forces can be evaluated in parallel to the equations of motion and the 
parallelization strategies may be applied accordingly. 
4. IMPLEMENTATION ON TRANSPUTER NElWORK 
The transputer is a powerful microprocessor designed for real- time parallel processing. 
It has local memo!), and four high-speed serial links for interconnection with neighboring 
transputers. A software controlled link switch allows to establish problem specific processor 
topologies. For real- time applications, two on-chip timers with 1 flS and 64 ~s resolution, 
respectively, can be accessed. Also, a wide variety of peripheral devices such as, for 
.," 
instance, analog interfaces arc available for control applications. The root-transputer 
communicates with the host-computer, as shown for a problem specific processor topology 
in Figure 6. Software development can be done with standard programming languages. The 
parallel processing language OCCAM, however, allows the most efficient utilization of the 
transputer resources. For details about transputers and OCCAM, see e.g. INMOS [71. 
I,M, 
,-----t LL T .. 1----, 
k q u 
host-workstation root- transputer transputcr- network 
Figure 6. Transputer network for robot simulation 
The parallelization strategies shown above lead to coarse grain parallelism and arc 
therefore well suited for the implementation on transputer networks. The implementation 
of parallel robot dynamics simulation starts with modeling by the program NEWEUL, 
Kreuzer, Leister [31. The formalism NEWEUL generates the symbolic equations (5) and 
(6) in FORTRAN-compatible form. A post-processor analyzes these equations and 
performs the partitioning into separate modules for concurrent evaluation according to a 
configuration description provided by the user, Rlikgauer [81. The composition of 
subsystems can be specified by using symbolic names for the rigid body reference frames. 
Also, processor numbers can be assigned to the various parallel threads according to 
Figures 3-5. Thus, arbitrary combinations of the different parallclization strategies can be 
achieved. A subsequent analysis of data dependencies provides information for setting up 
communication vectors that contain a minimum number of data to be exchanged between 
processors. It also allows to separate computation blocks that are independent of the 
interface variables and, therefore, can be computed in advance of the data exchange, thus 
minimizing idling times of the processors. At this point the advantages of symbolic 
equations with respect to code optimization become particularly obvious. Since 
communication times may be crucial for the resulting efficiency of the implementation, it is 
important to use the ability of the transputer to communicate through the links and compute 
in parallel due to its internal hardware structure. The post-processor finally generates 
OCCAM source code ready to be compiled and linked with a real-time simulation 
environment also written in OCCAM. 
605 
5. EXAMPLE: STANFORD MANIPULATOR 
The application of the parallelization methodology presented above is shown for the 
so-called Stanford manipulator, Kane, Levinson [9]. This robot represents a kinematic 
chain with five revolute and one prismatic joints and six degrees of freedom. The nonlinear 
equations of motion of this system arc quite large and real- time evaluation is a nontrivial 
task. The timing of the different subtasks for the implementation on one processor is shown 
in Figure 7. The shaded areas indicate pure communication without any computations. The 
times arc given in microseconds and refer to single precision (32 bit real) computations. The 
minimum cycle time turns out to be 1596 )JS. Figure 8 shows the corresponding timing 
diagram for the parallel implementation on three processors. This implementation is based 
on a combination of the strategies 2 and 3. As already indicated, the subsystem approach is 
not beneficial due to the chain topology. The result shows that the cycle time could be 
reduced to 1141 )JS and a speed-up of 1.4 could be achieved. This allows the real-time 
simulation by a second order integration scheme. for instance. the Adams- Moulton 
method which requires two function evaluations per time-step. It is important to note that 
certain communications run in paranel to computations, thus reducing the communication 
overhead considerably. Figure 8. 
i I 
o 
M 
t 
500 
k I u,q 
I j 
I y ~ I j. 
1000 1500 I [us] 
Figure 7. Task timing for Stanford manipulator implemented on one transputer 
M Processor 1 
k idle Processor 2 
idle idle Processor 3 
o 500 1000 I [us] 
Figure 8. Task timing for Stanford manipulator implemented on three transputers 
The transputer implementation in Figure 8 becomes particularly interesting if models 
for robot drives and sophisticated controllers arc to be coupled with the robot simulation. 
In this case one or more transputers can perform these tasks in parallel to the transputers 
evaluating the pure multibody model. In the implementation shown in Figure 8, for 
instance, processor 3 has about 500 ~s left for additional control computations that would 
not affect the total cycle time at all, which raises the achievable speed-up to 1.8. 
6. CONCLUSIONS 
The symbolic equations of motion of robots were investigated with respect to 
parallelization for transputer implementation. The evaluation of different matrices and 
vectors of the equations of motion on separate processors results in a considerable 
reduction of computing time. This approach is particularly advantageous for the simulation 
of robots with sophisticated control algorithms. The achievable cycle times allow real-time 
simulations with simple integration schemes. The next transputer generation promises a 
further reduction of cycle times for the presented implementation. thus allowing 
high-speed real-time simulations with more sophisticated integration codes. too. 
7. REFERENCES 
1 Schweitzer, G.; Mansour, M. (cds): Dynamics of Controlled Mechanical Systems. 
Berlinf ... :Springer, 1989. 
2 Sehiehlen, W , Schafer, P.: Modeling of Vehicles with Controlled Components. In: 
Anderson, R. (cd.): The Dynamics of Vehicles on Roads and Tracks. Proceedings of the 
11th IAVSD Symposium, Kingston, Canada, 1989. Lisse: Swets & ZeitJinger, 1990. 
3 Kreuzer, E.; Leister, G.: Programmsystem NEWEUI:90. Stuttgart: University of 
Stuttgart. Institute B of Mechanics, 1991. 
4 Brandl, H.; Johanni, R.; Otter, M.: A Very Efficient Algorithm for the Simulation of 
Robots and Similar Multibody Systems Without Inversion of the Mass Matrix. In: Proc. 
of the IFAC, IFIP, IMACS International Symposium on Theory of Robots. Vienna, 
(1986). pp.95-100. 
5 Schiehlen. w.: Computational Aspects in Multibody System Dynamics. In: Computer 
Methods in Applied Mechanics and Engineering. Vol. 90, Nos. 1-3, pp. 562-582. 
Amsterdam: North-Holland. 1991. 
6 Knaupp, E.: Simulation cines caumlichen Fahrzcugmodclls mit einem 
Transputernetzwerk. Stuttgart: University of Stuttgart, Institute B of Mechanics, Student 
Thesis STUD - 82, 1992. 
7 !NMOS Ltd.: The Transputer Applications Notebook - Applications and Software. 
Bristol: !NMOS Ltd., 1989. 
8 Rukgauer, A.: Parallele Implementierung von Bewegungsgleichungen aufTransputern. 
Stuttgart: University of Stuttgart, Institute B of Mechanics, Student Thesis STUD-83, 
1992. 
9 Kane, T.; Levinson, D.: The Use of Kane's Dynamical Equations in Robotics. 
Massachusetts: International Journal of Robotics Research Vol.2, No.3. 1983. pp. 3-21. 


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