In advanced robot control problems, on-line computation of inverse Jacobian solution is frequently required. Parallel processing architecture is an effective way to reduce computation time. A parallel processing architecture is developed for the inverse Jacobian (inverse differential kinematic equation) of the PUMA arm. The proposed pipeline/parallel algorithm can be inplemented on an IC chip using systolic linear arrays. This implementation requires 27 processing cells and 25 time units. Computation time is thus significantly reduced

Han, W. H.

Hsia, T. C.

Lu, G. Z.

English

NASA Technical Reports Server

Parallel Processing Architecture for Computing Inverse 
Differential Kinematic Equations of the PUMA Arm r -  
T.C. Hsia cab y,Gf- 
University of California, Davis c<@ u-~ f  si 
\ Davis, CA 95616 
G.Z. Lu and W.H. Han 
~ankai University 
Tiangjin, China 
f . ~ p S  y /bl I 3 
I n  advanced robot con t ro l  problems, on- l ine canputat ion o f  inverse Jacobian s o l u t i o n  
canputat ion time. -, A.paralle1 processing a rch l tec tu re  i s  developed f o r  the  
inverse  Jacobian ( inverse d i f f e r e ~ ~ a l  kinematic equation) o f  PUMA ann,fQ). The proposed 
i s  f requent ly  required. P a r a l l e l  processing a rch l tec tu re  i s  an e f f e c t i v e  way t o  reduce 
p i p e l i n c / p a r a l l e l  a lgor i thm can be implemented on I C  ch ip  using sys to l i c  l i n e a r  arrays. 
Th is  implementation requires 27 processing c e l l s  and 25 t ime uni ts.  
thus s i g n i f i c a n t l y  reduced. 
2. I n t roduc t i on  
I n  many advanced robot con t ro l  problems, such as w i t h  sensor guided manipulations. i t  i s  essential t h a t  the 
end e f f e c t o r  be appropr iately cont ro l led  i n  Cartesian coordinates so tha t  the robot can adapt t o  a changing 
environnent. This means t h a t  we need t o  canpute the  inverse  Jacobian i n  real t ime t o  provide the required 
d i f f e r e n t i a l  change i n  j o i n t  var iables f o r  a deslred d i f f e r e n t i a l  change i n  p o s i t i o n  and orientation. The speed 
of t h i s  canputation d i r e c t l y  a f fec ts  the speed of robot operation. Thus e f f i c i e n t  algorithms for canputing the 
i nve rse  Jacobian are needed. 
Jacobian problem su i tab le  f o r  s e r i a l  computer implementation [l,Z]. 
a lgor i thm development f o r  implementation on p ipe l i ned  or p a r a l l e l  canputer [3]. These resu l ts  shlw t h a t  such 
p a r a l l e l  algorithms can reduce canputation time s i g n i f i c a n t l y .  
A more important requirement I n  robot manipulat lon 1s the canputing o f  the fnverse Jacobfan solutlon. 
i s  genera l l y  a troublesanc problem when we t ry t o  i n v e r t  the  Jacobian nuner ical ly.  
de r i ve  an e x p l i c i t  so lu t i on  o f  the inverse Jacobian f o r  a given robot. 
t h a t  such so lu t ions  can be obtained by d i f f e r e n t i a t i n g  the  kinematic equations. This approach has s h w n  t o  
r e s u l t  s impler inverse Jacobian solut ions w i th  regard t o  manipulator degeneracies and j o i n t  constraints. The 
inverse  Jacobian o f  the PUMA ann has been solved s p e c i f l c a l l y  i n  [Z]. 
Jacobian der ived i n  [ Z ] .  With rapid advances i n  VLSI technology, t h i s  type of a lgor l thm can be read i l y  
Implemented on I C  chlps. These special purpose chips can be connected t o  a host canputer system t o  achieve 
rea l - t ime Carteslan space con t ro l  a t  s u f f i c i e n t l y  h igh  sample rate. !t i s  noted t h a t  a study has b e w  made 
recent ly  t o  implement d i r e c t  kinematic so lu t i on  on VLSI chips t o  speed up canputat ion time [4]. The goal here 
i s  t o  f u r t h e r  exp lo i t  the advantages of VLSI technology f o r  the oeslgn of customized chips dedicJted t o  the  
canput ing o f  the inverse Jocobian of PUMA ann. 
3. D i f f e r e n t i a l  Kinematic So lu t i on  of PUMA A n n  
~,I 
l F -  - 
Canputation time i s  
There have been e f f o r t s  made recent ly i n  developlng computationally e f f i c i e n t  algorthfm: t o  solve the  
I n  add i t i on  sane work  has been repor ted  i n  
This 
A more d i rec t  approach i s  t o  
Paul, Shimano. and Mayer [2] have shown 
I n  t h i s  paper. we present a p ipe l i ne /pa ra l l e l  a lgor i thm and arch i tec tu re  f o r  canputing the PUMA ann inverse  
D l f f e r e n t i a l  changes i n  j o i n t  variables dq i  can be re la ted  t o  the d i f f e r e n t  changes I n  t rans la t ion  and 
r o t a t i o n  dx. dy, dr. 6 x .  6y. and 62 of the end e f f e c t o r  by the re la t ionsh ip  
(1) T [dx. dy. dZ. 6 ~ .  6, 6z.IT = J Cdqi, dqz. ..., dqn l  
i n  whlch n i s  the nunber o f  j o i n t s .  and J i s  the Jacobian matr ix.  
need the so lu t i on  of dqi g i ven  the desired d i f f e r e n t i a l  change dx, dy. d, 6*, 6,. 62 . 
canpute the inverse problem 
But i n  advanced robot  control problems, we 
That i s  we need t o  
(2) 
T 
Cdql. dq2,...,dqnIT 9 J-' Idx .  dy. dZ. 6X. 0. 623 
Th is  represents the inverse d i f f e r e n t i a l  kinematic s o l l i t i o n  ( inverse Jacobian) o f  the robot ann. 
t h e  inverse Jacobian problem can be frequent ly obtained. and such a so lu t i on  f o r  the PUMA ann i s  given i n  [2]. 
For the  PUMA ann, the j o i n t  var iables are the s i x  r o t a t i o n a l  j o i n t  angles 9. e2, ..., e6. Furthennore, t he  
Instead o f  re l y ing  on t h e  d i r e c t  canputing o f  t h e  inverse  Jacobian matr ix J-', an ana ly t i ca l  s o l u t i o n  o f  
317 
https://ntrs.nasa.gov/search.jsp?R=19890017201 2020-03-20T02:05:36+00:00Z
d i f f e r e n t l a 1  cha e t  I n  t r a n s l a t i o n  and ro ta t i on  can be n l a t c d  t o  the d i f f e r e n t i a l  change of the erd e ? f c c t w  
haageon u t r t x ?  ~23: 
d T - T - 9  (31 
where 
O -62 6y dx 
Therefore d i f f e r e n t i a l  chan es i n  t rans la t i cn  and r o t a t i o n  can a lso  be speci f ied’  i n  t e n s  o f  the x ,  ?. z 
elements of dp, do. and da 9dn vector i s  redundant). The desired solut ions of do l  i n  te rns  o f  dp. do, d d  da 
f o r  t he  PUMA ann obtained i n  [Z] are given i n  the appendix. 
c a p u t i q  these equations i s  ncw developed below. 
4. S y s t o l i c  Array Processing 
VLSI technology has created a new archi tecture ho r i zon  i n  implementing p a r a l l e l  a lgor i thms d i rec t l y  on 
hardware. Central t o  t h i s  a rch i tec tu re  i s  the we o f  s y s t o l i c  l i n e a r  arrays which consist of In te r -  
connected simple and most ly i d e n t i c a l  processing ce l l s .  Algorithms that can be executed using loent ical  
operat ions simultaneously can take advantage of the s y s t o l i c  array arch i tec tu re  t o  reduce canputation time. 
A p t p e l i n d p a r a l l e l  processing archi tecture ftr 
The processing c e l l  ? t r u c t u r e  we w f i l  employ i s  the “inner product step processor” which p e r f o m  
mat r ix -vec tor  m u l t i p l i c a t i o n  using one-way p ipe l ine  algorithms. For example, ccnput ing 
Ab = p 
where A i s  mm and b i s  m x l .  can be car r ied  out i n  the  f o l l o w i n g  recurrence manner: 
k = 1.m. 1 = 1.n 
Th is  operat ion can be implemented by a l i nea r  array of  m i nne r  product step processors shcun i n  Figure 
I n  the f o l l o r i n g  sect ion,  we w i l l  reformulate the  inverse  d i f f e r e n t i a l  kinematic equat ion given i r  
appendix i n  terns of a s e t  o f  matr ix-vector m u l t i p l i c a t i o n s  which can be canputed i n  p a r a l l e l  and pipel 
f as h i on. 
5. A1 g o r i  thm Jevel opment 
(4 )  
t he  
n i  ng 
I n  t h i s  section, we present the  matrix-vector m u l t i p l i c a t i o n  processing schemes f o r  canputing the 
d i f f e r e n t i a l s  dot. i = 1.2.....6. 
T y p i c a l l y  these funct ions can’be generated by employing OH look-up echniques c5.63. The a lgor i thn  1s broken  
dcwn !n to  15 steps as descr ibed below. The notat ion Si 1 S i n q  Ci 5 Cosoi are used. 
Here we assime t h a t  t h e  t r igonanet r fc  funct ions requ i red  are avaflable. 
tipY dp, pX pY ay ax dax -ax day oy OX dox ~ O Y  
dpy py -px -ax ay day -ay -day -OX oy doy -dox 
318 
9: 
h i  [ h  l..h15] 
l P l Z  p17 p14 p112 p1131 
(3)  AZbZ P2 
'h21 h22 'h24 
h22 h21 h25 
1 J 
p5 ips1  ~ 5 2 1  
319 
A7 I 1" "1 b7 
p32 'h13 p7 
The data f l o w  t iming tab le  f o r  these computations are given i n  Tables 1 and 2. 
The resu l t s  or 25 time un i t s  i s  a s i g n i f i c a n t  reduct ion of canputat ion time i n  canparison w i t h  tha t  when a 
The t o t a l  nunber o f  m u l t i p l i c a t i o m  o f  t h a t  solut ion 
This i s  equivalent t o  150 time un i t s  i n  the  sys to ic  array processing system as opposed t o  the 25 
I t  i s  shcwn t h a t  the 
s o l u t i o n  requires 25 t ime un i t s  and 27 processing ce l l s .  
s e r i a l  cmpu te r  i s  used t o  canpute the o r ig ina l  solut ion.  
i s  about 150. 
-time un i t s  we have achieved by exp lo i t i ng  paral le l ism. 
3 20 
2 
-dPx 
dPx 
Table 1. 
3 4  
Data flaw timing table for steps 
on top rm indicate  time units.  
5 6 7 8 9 1 0  
1 through 8 which canpute d e l  dQ3 
11 12 13 14 15 16 
and 
17 
p1s 
P18 
Nunbcn 
C1 
51 
dol* p1z 
p19 
PllZ 
p113 
c3 
53 
p17 
dPZ 
h l  1 
h2 1 
h22 
dP Z 
hll. dc3 
c23 
523 
321 
Table 2. Data f l o w  t iming tab le  fo r  steps 9 through 15 whlch canpute deq. des, d e .  
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 
p64 
p63 
p68 
'p6 7 
p15 p63 
h 3 l  -h13 
p15 'p63 
h 3 l  h13 
p67 Pl lO 
P l l O  'p67 
h33 h15 
'p67 'pl10 
h15 4 3 3  
p8 1 
p9 2 
4 4 3  h51 
p6 5 
966 
p69 
p610 
p15 
p63 
c4 
54 
6. Conclusion 
I t  has been demonstrated i n  th i s  paper tha t  p a r a l l e l  canputlng arch i tec tu re  can be developed f o r  t h e  
inverse d i f f e r e n t f a 1  kinematic equation of the PUMA am. By using s y s t o l i c  l i nea r  arrays employed i n  VLSI chlo 
design, the canputat ion can be canpleted w i th  27 processing c e l l s  i n  25 time units. 
one m u l t i p l i c a t i o n  i s  counted as one time un i t .  the p a r a l l e l  a rch i tec tu re  d e f i n i t e l y  provides a subs tan t i a l  
reduc t ion  i n  canputat icn time. A custan!zed I C  ch ip  dedicated t o  t h i s  algor i thm can be fabricated. 
The d i f f e r e n t i a l  kinematic equation i n  i t s  o r i g ina l  fo rn  requires about 150 m u l t i p l i c a t i o n s  t o  canpute. I f  
References 
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Kinematics." proceedings. 1986 IEEE 
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322 
32 3 
Figure  1. Inner product step processor 
324 


Parallel processing architecture for computing inverse differential kinematic equations of the PUMA arm

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