COLLISION DETECTION AND AVOIDANCE IN
COMPUTER CONTROLLED MANIPULATORS
S h r i r a m M. Udupa
C a l i f o r n i a I n s t i t u t e o f Technology
Pasadena, C a l i f o r n i a
t h e 2 D and 3 D s o l u t i o n n o t e d , i t i s easy t o
v i s u a l i z e the s o l u t i o n f o r the 3D m a n i p u l a t o r .
ABSTRACT - The p r o b l e m o f p l a n n i n g s a f e t r a j e c t o r i e s f o r computer c o n t r o l l e d m a n i p u l a t o r s
w i t h two m o v a b l e l i n k s and m u l t i p l e d e g r e e s o f
f r e e d o m i s a n a l y z e d , and a s o l u t i o n t o t h e
problem proposed.
The k e y
1.
features of
S e c t i o n 2 o f t h i s p a p e r p r e s e n t s a n e x a m p l e , and
S e c t i o n 3 a s t a t e m e n t and a n a l y s i s o f t h e p r o b l e m .
S e c t i o n s 4 and 5 p r e s e n t t h e s o l u t i o n .
Section 6
s u m m a r i z e s t h e k e y i d e a s i n t h e s o l u t i o n and
i n d i c a t e s areas f o r f u t u r e work.
the s o l u t i o n are:
the i d e n t i f i c a t i o n of t r a j e c t o r y
p r i m i t i v e s and a h i e r a r c h y o f a b s t r a c t i o n spaces t h a t p e r m i t s i m p l e m a n i p u l a t o r models,
2.
t h e c h a r a c t e r i z a t i o n o f empty space b y
approximating it w i t h easily describable
e n t i t i e s c a l l e d charts - the approximat i o n i s d y n a m i c and can b e s e l e c t i v e ,
3.
a scheme f o r p l a n n i n g m o t i o n s c l o s e t o
obstacles that is computationally
v i a b l e , and t h a t s u g g e s t s how p r o x i m i t y
s e n s o r s m i g h t be used to do t h e
p l a n n i n g , and
2.
AN EXAMPLE
T h i s s e c t i o n d e s c r i b e s a n example ( F i g u r e 2 . 1 ) o f
t h e c o l l i s i o n d e t e c t i o n and a v o i d a n c e p r o b l e m f o r
a two-dimensional manipulator.
The e x a m p l e
h i g h l i g h t s f e a t u r e s o f t h e p r o b l e m and i t s
solution.
2.1
4.
t h e use o f h i e r a r c h i c a l d e c o m p o s i t i o n
to reduce the complexity of the
planning problem.
KEYWORDS - M a n i p u l a t o r , p l a n n i n g , r e p r e s e n t a tion, heuristics, abstraction.
1.
INTRODUCTION
The P r o b l e m
The m a n i p u l a t o r h a s two l i n k s and t h r e e d e g r e e s
o f f r e e d o m . The l a r g e r l i n k , c a l l e d t h e boom,
s l i d e s b a c k and f o r t h and c a n r o t a t e a b o u t t h e
origin.
The s m a l l e r l i n k , c a l l e d t h e f o r e a r m ,
has a r o t a t i o n a l d e g r e e o f f r e e d o m a b o u t t h e t i p
o f t h e boom.
The t i p o f t h e f o r e a r m i s c a l l e d
t h e h a n d . S a n d G a r e t h e i n i t i a l and f i n a l
configurations of the manipulator.
Any r e a l
m a n i p u l a t o r ' s l i n k s w i l l have p h y s i c a l d i m e n s i o n s .
The l i n e segment r e p r e s e n t a t i o n o f t h e l i n k i s a n
a b s t r a c t i o n ; t h e p h y s i c a l d i m e n s i o n s can b e
a c c o u n t e d f o r and how t h i s i s done i s d e s c r i b e d
later.
The p r o b l e m o f p l a n n i n g s a f e t r a j e c t o r i e s f o r
computer c o n t r o l l e d m a n i p u l a t o r s w i t h two movable
l i n k s and m u l t i p l e d e g r e e s o f f r e e d o m i s a n a l y z e d ,
and a s o l u t i o n t o t h e p r o b l e m i s p r e s e n t e d .
The c l o s e d p o l y g o n s i n t h e f i g u r e r e p r e s e n t
polygonal approximations to o b s t a c l e s ; these
p o l y g o n s may b e c o n c a v e o r c o n v e x , and t h e r e i s
n o l i m i t t o t h e number o f s i d e s .
The t r a j e c t o r y p l a n n i n g s y s t e m i s i n i t i a l i z e d
w i t h a d e s c r i p t i o n of the p a r t of the environment
t h a t t h e m a n i p u l a t o r i s t o maneuver i n .
When
g i v e n t h e g o a l p o s i t i o n and o r i e n t a t i o n o f t h e
hand, the system plans a complete t r a j e c t o r y
t h a t w i l l s a f e l y maneuver t h e m a n i p u l a t o r i n t o
the goal c o n f i g u r a t i o n .
The e x e c u t i v e s y s t e m i n
charge o f o p e r a t i n g t h e hardware uses t h i s
t r a j e c t o r y t o p h y s i c a l l y move t h e m a n i p u l a t o r .
The p r o b l e m i s t o p l a n a c o l l i s i o n f r e e t r a j e c t o r y that w i l l get the manipulator from S to G.
(A t r a j e c t o r y s p e c i f i e s the manipulator conf i g u r a t i o n as a f u n c t i o n of t i m e ) .
The s o l u t i o n i s a p p l i e d t o a s i m p l i f i e d t w o d i m e n s i o n a l m a n i p u l a t o r (2D s y s t e m ) and a f u l l fledged three-dimensional manipulator - the
S c h e i n m a n arm - (3D s y s t e m ) .
A l l t h e examples in
t h i s paper a r e f r o m t h e 2D system.
Once t h e 2 D
s o l u t i o n i s u n d e r s t o o d and s i m i l a r i t i e s b e t w e e n
*
Present Address
B e l l Telephone L a b o r a t o r i e s
Naperville, I l l i n o i s
60540
2.2
The S o l u t i o n
S i n c e t h e boom i s much l a r g e r t h a n t h e f o r e a r m ,
t h e boom i s t h e more c o n s t r a i n i n g o f t h e t w o
links.
T h e r e f o r e , a s a f e boom t r a j e c t o r y i s
f i r s t p l a n n e d , and t h e n t h e f o r e a r m i s m a n e u v e r e d
s a f e l y a l o n g t h e boom t i p l o c u s .
I f f o r some
reason, the forearm cannot be s a f e l y maneuvered,
t h e boom t r a j e c t o r y i s r e v i s e d and t h e p r o c e s s
repeated.
Boom P l a n n i n g :
The boom p l a n n i n g p r o b l e m
i s t o f i n d a t r a j e c t o r y f o r t h e two j o i n t s a s s o c i a t e d w i t h t h e boom.
W e can t r y t o g e t t h e
boom t i p f r o m S t o G a l o n g t h e s h o r t e s t p a t h
Robot ? c s - 2 :
737
Udupa
between the two boom t i p l o c a t i o n s - a s t r a i g h t
l i n e boom t i p l o c u s .
In Figure 2 . 2 , the shaded
area represents the area t h a t the boom sweeps
when i t s t i p t r a c e s a s t r a i g h t l i n e from S to G.
Since the shaded area i n t e r s e c t s the L-shaped
o b j e c t , the boom w i l l c o l l i d e w i t h t h a t o b j e c t .
This c o l l i s i o n can be avoided if an i n t e r m e d i a t e
p o i n t P is chosen and the boom t i p is r e q u i r e d
to go through P. The above procedure is then
a p p l i e d r e c u r s i v e l y to the s e c t i o n s SP and PG
u n t i l a safe t r a j e c t o r y i s found.
Figure 2.3
shows the f i n a l boom t i p locus t h a t guarantees
boom s a f e t y .
Forearm P l a n n i n g : This r e f e r s to f i n d i n g a
t r a j e c t o r y f o r the forearm j o i n t .
The basic
idea is to j u g g l e the forearm back and f o r t h so
t h a t it avoids any c o l l i s i o n s as one end of it
t r a v e l s along the boom t i p l o c u s .
This is not
easy.
F u r t h e r c o m p l i c a t i o n s a r i s e because the
m a n e u v e r a b i l i t y of the forearm near the goal
c o n f i g u r a t i o n is very r e s t r i c t e d .
So as not to
c l u t t e r the diagram, the forearm t i p locus has
been supressed from Figure 2 . 3 .
Execution:
The above p l a n n i n g r e s u l t s in
a sequence of i n t e r m e d i a t e c o n f i g u r a t i o n s l e a d i n g
to the goal c o n f i g u r a t i o n .
The t r a j e c t o r y
c a l c u l a t i o n r o u t i n e s use t h i s sequence to generate
a trajectory.
The e x e c u t i v e system in charge of
o p e r a t i n g the hardware uses the t r a j e c t o r y to
move the m a n i p u l a t o r .
Embellishments:
The p l a n n i n g described is
called mid-section planning.
This mode of
p l a n n i n g does not make use of the n a t u r e of
o b s t a c l e c o n f i g u r a t i o n s , and is good f o r p l a n n i n g
maneuvers f a r from o b s t a c l e s .
Another mode of
p l a n n i n g , c a l l e d t e r m i n a l phase p l a n n i n g , makes
use of the n a t u r e of o b s t a c l e c o n f i g u r a t i o n
around the hand, and can be used f o r p l a n n i n g
maneuvers near the s t a r t and goal c o n f i g u r a t i o n s . The use of both the m i d - s e c t i o n and the
t e r m i n a l phase p l a n n i n g r e s u l t s in a simpler
t r a j e c t o r y (a s m a l l e r sequence of i n t e r m e d i a t e
configurations).
Figure 2.4 shows the boom t i p
locus f o r the s i m p l e r t r a j e c t o r y .
The r e p r e s e n t a t i o n of the m a n i p u l a t o r , o b s t a c l e s ,
empty space and t r a j e c t o r i e s , the two p l a n n i n g
modes and the associated h e u r i s t i c s , the use of
the nature o f o b s t a c l e c o n f i g u r a t i o n s f o r t e r m i n a l maneuvers e t c . are a l l discussed i n Sect i o n s 4 and 5.
2.3
H i s t o r i c a l Perspective
C o l l i s i o n avoidance problems became m a n i f e s t when
computer c o n t r o l l e d m a n i p u l a t o r s came i n t o
existence d u r i n g the s i x t i e s .
Pieper (1968) was
the f i r s t one to i n v e s t i g a t e the problem.
Paul
(1972) t a c k l e d t r a j e c t o r y c a l c u l a t i o n and
s e r v o i n g . Lewis (1974) a t t a c k e d a very r e s t r i c t e d v e r s i o n o f the c o l l i s i o n avoidance
problem, and Widdoes (1974) d i d the same. None
of the e a r l i e r attempts can handle c o m p l e x i t i e s
s i m i l a r t o the ones i l l u s t r a t e d i n the example
of F i g u r e 2 . 1 . The r e p r e s e n t a t i o n s used by
these e a r l i e r programs are i n c o n v e n i e n t , or the
p l a n n i n g s t r a t e g i e s inadequate f o r h a n d l i n g the
situation.
3.
THE PROBLEM AND ITS ANALYSIS
The problem is to p l a n a t r a j e c t o r y to get a
computer c o n t r o l l e d m a n i p u l a t o r w i t h two movable
l i n k s and m u l t i p l e degrees of freedom s a f e l y i n t o
a d e s i r e d goal c o n f i g u r a t i o n .
The p l a n n i n g system is i n i t i a l l y given a d e s c r i p t i o n o f the environment i n which the m a n i p u l a t o r
is to o p e r a t e .
This environment undergoes
minor changes when o b j e c t s i n i t are t r a n s p o r t e d
around by the m a n i p u l a t o r . The environment may
a l s o change d r a s t i c a l l y , as would happen if the
r o b o t ( t o which the m a n i p u l a t o r belonged) moved
to a new p l a c e .
It is assumed t h a t such d r a s t i c
changes are i n f r e q u e n t compared to the t o t a l
number of t r a j e c t o r i e s planned.
This assumption
i s r e f e r r e d t o a s the i n f r e q u e n t environment i n i t i a l i z a t i o n hypothesis.
The i n p u t to the p l a n n i n g system c o n s i s t s of the
p o s i t i o n and o r i e n t a t i o n of the hand in the goal
configuration.
The output i s a l i s t o f i n t e r mediate c o n f i g u r a t i o n s t h a t w i l l be used by the
t r a j e c t o r y c a l c u l a t i o n programs t o run the h a r d ware.
A s o l u t i o n t h a t w i l l perform w e l l i n simple and
commonly o c c u r i n g s i t u a t i o n s is d e s i r e d .
The
system should recognize when t h i n g s go awry and
should ask f o r human a s s i s t a n c e when t h a t happens. Optimal plans are not needed; at the same
t i m e , however, b l a t a n t l y s t u p i d plans are not
permitted.
Planning begins w i t h h y p o t h e s i z i n g a t r a j e c t o r y .
F o l l o w i n g t h i s i s a n i t e r a t i v e step t h a t i n v o l v e s
a check f o r c o l l i s i o n and a t r a j e c t o r y m o d i f i c a t i o n ( i f t h e r e i s danger).
Under normal c i r c u m stances, the loop terminates when a safe t r a j e c t o r y i s found.
Good h e u r i s t i c s f o r h y p o t h e s i z i n g t r a j e c t o r i e s
are e s s e n t i a l . I d e a l l y , the system should propose
a c o l l i s i o n - f r e e t r a j e c t o r y o n the f i r s t t r y .
Since a t r a j e c t o r y designed to pass through l a r g e
empty spaces is l i k e l y to be s a f e , a c h a r a c t e r i z a t i o n of l a r g e empty spaces is d e s i r a b l e .
A l s o , a c h a r a c t e r i z a t i o n of o b s t a c l e c o n f i g u r a t i o n s i n the immediate v i c i n i t y o f the hand i s
desirable; f o r , special h e u r i s t i c s , that i n crease the chances of proposing a c o l l i s i o n - f r e e
t r a j e c t o r y , can be associated w i t h these c o n figurations.
Good techniques f o r making t r a j e c t o r y m o d i f i c a t i o n s are r e q u i r e d ; the m o d i f i c a t i o n s should
ensure t h a t the same problem does not r e c u r and
t h a t new problems do not a r i s e .
A few terms t h a t w i l l enable us to t a l k about the
c o l l i s i o n d e t e c t i o n problem are now i n t r o d u c e d .
Robotics-2 :
738
Udna
The m a n i p u l a t o r ' s s t a t e can be described e i t h e r
as a v e c t o r s p e c i f y i n g i t s j o i n t angles or as a
p o s i t i o n and o r i e n t a t i o n of the hand. The former
i s a r e p r e s e n t a t i o n i n j o i n t v a r i a b l e space o r
j o i n t space, and the l a t t e r i s a r e p r e s e n t a t i o n
in c a r t e s i a n space.
The subspace of j o i n t space
generated by the boom j o i n t s is c a l l e d boom space.
When the m a n i p u l a t o r moves i t s l i n k s t r a c e a
volume ( s u r f a c e , in two dimensions) c a l l e d the
t r a j e c t o r y envelope (see F i g u r e 2 . 2 ) .
C o l l i s i o n d e c t e c t i o n i n v o l v e s checking i n t e r s e c t i o n s of the t r a j e c t o r y envelope and o b s t a c l e s .
T r a j e c t o r y envelopes are most c o n v e n i e n t l y
described i n j o i n t space and o b s t a c l e s i n c a r t e s i a n space.
Consequently, i n t e r s e c t i o n checking
r e q u i r e s constant t r a n s f o r m a t i o n s between the two
spaces.
This is expensive and we need to d e t e r mine whether obstacles should be represented in
j o i n t space o r t r a j e c t o r i e s i n c a r t e s i a n space,
or whether it is p o s s i b l e to use the two spaces
e f f e c t i v e l y and avoid the t r a n s f o r m a t i o n problem.
We need to see whether t r a j e c t o r y envelopes and
o b s t a c l e d e s c r i p t i o n s can b e s i m p l i f i e d since i t
would lead to inexpensive i n t e r s e c t i o n checks.
A g a i n , safe t r a j e c t o r y p l a n n i n g can be viewed as
maneuvering in f r e e space or as a v o i d i n g o b s t a c l e s .
Can these complementary views be used to advantage?
The s o l u t i o n presented in the next two s e c t i o n s
provides answers to a l l the p o i n t s and questions
r a i s e d i n the above a n a l y s i s , making f a s t c o l l i s i o n avoiders a d i s t i n c t p o s s i b i l i t y .
4.
REPRESENTATION
T r a j e c t o r y p l a n n i n g gets e a s i e r w i t h simple
t r a j e c t o r y envelopes.
Simple envelopes are
p o s s i b l e only w i t h w e l l - c h o s e n t r a j e c t o r y p r i m i t i v e s and simple m a n i p u l a t o r models. Simple
m a n i p u l a t o r models make the task of h y p o t h e s i z i n g
and m o d i f y i n g t r a j e c t o r i e s easy.
And simple
t r a j e c t o r y envelopes and n u m e r i c a l l y manageable
o b s t a c l e r e p r e s e n t a t i o n s make c o l l i s i o n d e t e c t i o n
inexpensive.
With t h i s i n mind, polyhedra models
of o b s t a c l e s , a h i e r a r c h y of a b s t r a c t i o n spaces
p e r m i t t i n g s i m p l i f i e d m a n i p u l a t o r models and
u s e f u l t r a j e c t o r y p r i m i t i v e s are i n t r o d u c e d i n
this section.
T h e i r use i n p l a n n i n g i s presented
in section 5.
S t a r t i n g w i t h a simple and d i r e c t model of two
connected c y l i n d e r s , the a b s t r a c t i o n spaces
p e r m i t the m a n i p u l a t o r to be modelled as two
connected l i n e segments, as a s i n g l e l i n e segment
and, i n c r e d i b l y , as a p o i n t ! Furthermore, the
t r a n s f o r m a t i o n s g e n e r a t i n g the a b s t r a c t i o n
spaces are i n e x p e n s i v e .
This is i m p o r t a n t ;
o t h e r w i s e , the advantages gained by o p e r a t i n g in
these a l t e r n a t e spaces would be l o s t in the
process of g e n e r a t i n g them.
The m a n i p u l a t o r and i t s environment are described
f i r s t , the t h r e e problem spaces next and f i n a l l y
the t r a j e c t o r y p r i m i t i v e s are p r e s e n t e d .
Table 1
summarizes the r e l a t i o n s h i p between the d i f f e r e n t
problem spaces.
4.1
The M a n i p u l a t o r And I t s Environment
The m a n i p u l a t o r of Section 2 is an a b s t r a c t i o n of
the class of computer c o n t r o l l e d manipulators
w i t h two movable l i n k s and m u l t i p l e degrees of
freedom. The Scheinman Arm shown in Figure 4 . 1
is another example.
D e t a i l s on the hardware and
a l g o r i t h m s f o r t r a j e c t o r y c a l c u l a t i o n and servoing
o f t h i s m a n i p u l a t o r are described i n Paul (1972),
D o b r o t i n and Scheinman (1973) and Lewis (1974).
The m a n i p u l a t o r is a s i x degree freedom device
a l l o w i n g the hand to be p o s i t i o n e d anywhere
( w i t h i n the maneuverable space) and w i t h any
orientation.
Figure 4 . 1 shows the s i x j o i n t s and the l i n k s
between the p o i n t s .
L i n k l i s c a l l e d the p o s t ,
l i n k 2 the s h o u l d e r , and l i n k 3 the boom. Link4
and l i n k 5 are n o n - e x i s t e n t because the manipulat o r design has the l a s t t h r e e j o i n t s a t the t i p
of the boom.
Link6 is c a l l e d the forearm.
Except f o r j o i n t 3 , which i s a s l i d i n g j o i n t , a l l
the j o i n t s are r e v o l u t e .
J o i n t l is called phi,
j o i n t 2 t h e t a , j o i n t 3 r, j o i n t 4 f t h e t a , j o i n t 5 f_
p h i , and j o i n t 6 f _ p s i . The p r e f i x " f " i n d i c a t e s
t h a t the angles r e f e r to the forearm.
The
forearm t i p i s c a l l e d the hand.
When l o o k i n g along the boom at the hand, the boom
i s e i t h e r o n the r i g h t o r the l e f t side o f the
shoulder.
This gives r i s e to the n o t i o n of a
r i g h t - h a n d e d and l e f t - h a n d e d manipulator respect i v e l y , and i s c a l l e d the l a t e r a l p r o p e r t y . T o
s i m p l i f y the p r e s e n t a t i o n , the l a t e r a l p r o p e r t y
o f the m a n i p u l a t o r w i l l h e n c e f o r t h b e i g n o r e d .
This concludes the d e s c r i p t i o n of the m a n i p u l a t o r .
The term environment w i l l be used to denote the
set of o b s t a c l e s in the workspace of the manipulator.
Since the hand can touch the manipulator
p o s t , the p o s t , t o o , i s considered a n o b s t a c l e .
Obstacles can be b o t h r e g u l a r and i r r e g u l a r in
shape. Objects on a robot such as the wheels,
the TV r a c k , the p l a t f o r m e t c . are r e g u l a r and
can be described as c y l i n d e r s , p a r a l l e l o p i p e d s ,
or unions of these shapes. Boulders and rocks in
the maneuverable space would be examples of
i r r e g u l a r shaped o b j e c t s .
4.2
Real Problem Space
The r e a l problem space is a simple-minded and
d i r e c t computer r e p r e s e n t a t i o n of the m a n i p u l a t o r
and i t s environment.
The m a n i p u l a t o r is modelled
as a sequence of connected c y l i n d e r s , one each
f o r the p o s t , boom and forearm. The boom and
forearm r e p r e s e n t a t i o n s correspond to the minimum
bounding c y l i n d e r s t h a t enclose them.
The t r a j e c t o r y envelope i s , consequently, a two-element
three-dimensional s o l i d .
Obstacles are approximated by polyhedra - p l a n e faced o b j e c t s . There is no r e s t r i c t i o n on the
number of f a c e s , and both concave and convex
polyhedra are a l l o w e d .
B e t t e r approximation
i m p l i e s more f a c e s , and, consequently, more
storage r e q u i r e d to save the d e s c r i p t i o n and
R o b o t i c s - 2 : tMupa
739
more time r e q u i r e d to analyze c o l l i s i o n s .
Thus
t h e r e is a t r a d e - o f f i n v o l v e d .
However, the
r e p r e s e n t a t i o n is compact and i n t e r s e c t i o n
checks are inexpensive because o b s t a c l e surfaces
are l i n e a r .
The set of p o l y h e d r a , each a p p r o x i mating a r e a l o b s t a c l e , is c a l l e d a map.
The maneuverable space is the complement of the
volume occupied by elements of the map, w i t h
respect to the m a n i p u l a t o r ' s work space.
4.3
Primary Problem Space
The primary problem space p e r m i t s the m a n i p u l a t o r
to be viewed as c o n s i s t i n g of a s i n g l e l i n e
segment and having no l a t e r a l p r o p e r t y .
F i r s t consider a t w o - l i n e segment model of the
m a n i p u l a t o r . The f i n i t e axes of the c y l i n d e r s
bounding the boom and forearm are used f o r t h i s
model. The equivalence between the primary
problem space and r e a l problem space is preserved
by e n l a r g i n g the polyhedra by the r a d i u s of the
manipulator l i n k s .
The enlarged polyhedra are a
c a l l e d primary o b s t a c l e s .
The set of primary
o b s t a c l e s is c a l l e d the primary map.
With
line-segment models of the m a n i p u l a t o r l i n k s ,
the t r a j e c t o r y envelope is two connected s u r f a c e s ,
one c a l l e d the boom surface and the o t h e r the
forearm s u r f a c e .
The maneuverable space is
c a l l e d primary f r e e space and is the complement
of the volume occupied by primary o b s t a c l e s
w i t h respect to the m a n i p u l a t o r ' s workspace.
Note t h a t the enlargement of o b s t a c l e s needs to
be done j u s t once f o r a given environment.
F i n a l l y , the s i n g l e l i n e segment d e s c r i p t i o n o f
the m a n i p u l a t o r is made p o s s i b l e by a t r a n s f o r m a t i o n c a l l e d survey.
Survey p e r m i t s the boom
to be viewed as a s i n g l e p o i n t , and the t r a j e c t o r y
envelope then reduces to the forearm surface generated by the motion of the forearm. The
t r a n s f o r m a t i o n survey i s a p p l i e d t o f r e e space
and r e s u l t s in a c h a r t .
A c h a r t f o r primary f r e e
space in c a l l e d a primary c h a r t .
Consider the s e t o f a l l p o i n t s i n f r e e space such
t h a t the e n t i r e boom i s safe from c o l l i s i o n i f
the boom t i p were p o s i t i o n e d t h e r e .
This subset
of f r e e space is c a l l e d navspace ( n a v i g a t i o n a l
space).
The survey t r a n s f o r m a t i o n approximates
navspace by boxes in r - t h e t a - p h i space c a l l e d
regions and the set of regions is c a l l e d a c h a r t .
Regions are s t r u c t u r e d e n t i t i e s .
They are made
up of s e c t o r o i d s and ( i n 3D) s e c t o r o l d s are
composed of pases. The pasc ( p a r a l l e l o p i p e d in
s p h e r i c a l c o o r d i n a t e s ) i s the s m a l l e s t u n i t .
Figure 5 . 1 shows s i x 2D regions bounded by the
r a d i a l arrows; i t a l s o shows one o f the regions
(R) and i t s f o u r component s e c t o r o i d s - S1,S2,S3
and S4. Pases, s e c t o r o i d s and regions are bound
by constant p h i and constant t h e t a s u r f a c e s .
All
pases in a s e c t o r o i d have the same p h i l i m i t s ,
and a l l s e c t o r o i d s in a r e g i o n have the same
theta l i m i t s .
Pases, s e c t o r o i d s and r e g i o n s a l l
have associated w i t h them a maximum and minimum r
v a l u e , c a l l e d rmax and rmin r e s p e c t i v e l y , i n d i c a t i n g the safe l i m i t s of the boom e x t e n s i o n .
The
d i f f e r e n c e between the maximum and minimum r
values i s c a l l e d the safe l i m i t i n t e r v a l . A
r e g i o n , s e c t o r o i d or pasc is considered impassable
i f the safe l i m i t i n t e r v a l i s l e s s than some
prespecified value.
These items are i l l u s t r a t e d
i n Figure 5 . 1 .
Regions are an approximation to the p o i n t s in
navspace. This approximation is dynamic and can
be changed by h i g h e r - l e v e l programs. The approxi m a t i n g procedure i s c a l l e d r e f i n e m e n t , and the
refinement l e v e l i s c a l l e d r e s o l u t i o n . The i n i t i a l approximation i s done t o a d e f a u l t r e s o l u tion.
I f the r e s o l u t i o n o f a p a r t i c u l a r p a r t o f
the environment is not adequate, the system
f u r t h e r r e f i n e s t h a t p o r t i o n o f the c h a r t .
This is termed the s e l e c t i v e r e f i n e m e n t capability.
This c a p a b i l i t y makes i n c r e m e n t a l
m o d i f i c a t i o n s ( n e c e s s i t a t e d by minor changes to
the environment) to the c h a r t s i n e x p e n s i v e .
Since refinement is dynamic, survey is not a
one-time t r a n s f o r m a t i o n .
This i s the p r i c e t h a t
has to be p a i d f o r the f l e x i b i l i t y .
Since t h e r e
is a l i m i t to the p r e c i s i o n of placement of the
hardware, the process o f refinement w i l l not
continue i n d e f i n i t e l y .
The concept of navspace permits c o n s i d e r i n g the
boom as a s i n g l e p o i n t . Navspace and i t s app r o x i m a t i o n b y c h a r t s i s thus c r u c i a l t o safe
trajectory planning.
The reason f o r imposing a
s t r u c t u r e on c h a r t s ( i n terms of r e g i o n s , sect o r o i d s and pases) is to have, some s e l e c t i v i t y in
terms of what p a r t s of navspace should be r e f i n e d
and t o what l e v e l .
I t i s i m p o r t a n t t o note t h a t
the exact n a t u r e of a r e g i o n and i t s components
is i r r e l e v a n t , and the choice of a box in rt h e t a - p h i space as the u n i t was d i c t a t e d by the
choice of a p a r t i c u l a r p l a n n i n g s t r a t e g y described
in Section 5.
4.4
Secondary Problem Space
The secondary problem space permits the manipul a t o r to be viewed as a s i n g l e p o i n t .
To get the s i n g l e p o i n t d e s c r i p t i o n of the
m a n i p u l a t o r , l e t u s s t a r t w i t h the t w o - l i n e
segment model of the m a n i p u l a t o r . We ignore the
f o r e a r m , and account f o r it by e n l a r g i n g the
primary o b s t a c l e s by the l e n g t h of the f o r e a r m ;
t h i s enlargement r e s u l t s i n secondary o b s t a c l e s
and a secondary map, and the maneuverable space
i s c a l l e d secondary f r e e space.
With the forearm
accounted f o r , the m a n i p u l a t o r c o n s i s t s o f only
the boom and the t r a j e c t o r y envelope is the boom
surface.
I t i s now p o s s i b l e t o a r r i v e a t the s i n g l e p o i n t
d e s c r i p t i o n of the m a n i p u l a t o r . We apply the
survey t r a n s f o r m a t i o n to secondary f r e e space.
This r e s u l t s in a secondary c h a r t , composed of
secondary r e g i o n s .
Whenever the boom t i p is in a
secondary r e g i o n the f o l l o w i n g h o l d s :
Robotics-2:
740
Udupa
1.
b y d e f i n i t i o n o f the r e g i o n , the
e n t i r e boom is f r e e of c o l l i s i o n , and
2.
s i n c e secondary regions are generated
u s i n g secondary o b s t a c l e s , the forearm
i s f r e e from c o l l i s i o n i r r e s p e c t i v e o f
its orientation.
2.
These c o n s t r a i n t s on the boom and forearm t r a j e c t o r i e s r e s u l t i n the decomposition o f the t r a j e c t o r y s u r f a c e i n t o a sequence of p a r a l l e l o g r a m s
and s e c t o r s of a c i r c l e , c o n s i d e r a b l y s i m p l i f y i n g
the c o l l i s i o n d e t e c t i o n t a s k .
The t r a j e c t o r y envelope a t t h i s l e v e l then i s
the l i n e generated by the motion of the boom
tip.
A complex t r a j e c t o r y s o l i d t h a t c o n s i s t e d
of two s o l i d s has thus been reduced to a l i n e .
The refinement process f o r secondary c h a r t s is
s i m i l a r t o primary c h a r t s and s o are a l l the
a t t r i b u t e s and t r a n s f o r m a t i o n s discussed i n the
c o n t e x t o f primary c h a r t s .
5.
H i e r a r c h y , s e p a r a b i l i t y and r e v e r s i b i l i t y are the
key concepts in p l a n n i n g .
The p r i n c i p l e of
r e v e r s i b i l i t y s t a t e s t h a t i f a t r a j e c t o r y from S
to G is c o l l i s i o n f r e e then the same t r a j e c t o r y
backwards from G to S is a l s o c o l l i s i o n f r e e .
S e p a r a b i l i t y means the decomposition of the goal
i n t o independent subgoals.
Hierarchy i s used i n
the usual sense. For each g o a l , the most import a n t aspects are t a c k l e d f i r s t and the l e s s e r
ones n e x t .
This is a p p l i e d to every stage of the
process.
If some d e c i s i o n s made at a h i g h e r
l e v e l do not pan o u t , l o c a l c o r r e c t i o n s are made.
I f the l o c a l f i x u p s d o not s o l v e the problems,
the system r e t u r n s t o the next h i g h e r l e v e l f o r
replanning.
A t each stage i t i s ensured t h a t the
system w i l l t e r m i n a t e i t s a c t i v i t i e s i n a f i n i t e
amount o f t i m e .
I f the system i s unsuccessful i n
s o l v i n g the problem, it g i v e s up and asks f o r
human h e l p .
Looked at from a d i f f e r e n t a n g l e , the ideas of
secondary problem space r e p r e s e n t a t i o n s ( t h e
secondary c h a r t s i n p a r t i c u l a r ) , are a f o r m a l
c h a r a c t e r i z a t i o n o f the i n t u i t i v e ideas o f ease
of maneuvering in l a r g e chunks of empty space.
The s i m p l i f i c a t i o n of the t r a j e c t o r y envelope
from two s o l i d s to a l i n e makes the e x p e c t a t i o n
come t r u e .
Trajectory Primitives
Since o b s t a c l e faces are planes i n c a r t e s i a n
"space, if the t r a j e c t o r y envelope were a plane
(primary problem space) or a l i n e (secondary
problem space) i n c a r t e s i a n space, c o l l i s i o n
checking would be s i m p l e .
5.1
Overview Of Planning
The n a t u r a l approach to h i e r a r c h i c a l p l a n n i n g
i s t o p l a n i n secondary problem space f i r s t
( s i n c e the t r a j e c t o r y envelope i s the s i m p l e s t
t h e r e ) and then r e f i n e the t r a j e c t o r y i n p r i mary problem space.
The d i f f i c u l t y of i n t e r f a c i n g the two problem spaces makes t h i s
approach u n a t t r a c t i v e . So i n s t e a d , p l a n n i n g
is "done" in p r i m a r y problem space and secondary problem space is used f o r s i m p l i f i c a tions.
The d e t a i l s o f t h i s approach are
presented n e x t .
Since the m a n i p u l a t o r j o i n t s can be operated
i n d e p e n d e n t l y , the boom t i p can be made to t r a c e
c a r t e s i a n space s t r a i g h t l i n e s .
Planning the
c a r t e s i a n space s t r a i g h t l i n e locus f o r the boom
t i p i s easy i n the 2 D case, w h i l e i t i s beset
w i t h problems in the 3D case. Hence in the 3D
case, we s e t t l e f o r boom space - subspace of
j o i n t space generated by the boom j o i n t s s t r a i g h t l i n e locus f o r the boom t i p .
The t r a j e c t o r y p l a n n i n g problem i s separated i n t o
t h r e e phases.
The f i r s t i s a goal f e a s i b i l i t y
a n a l y s i s phase, the second is the m i d - s e c t i o n
p l a n n i n g phase and the l a s t is the t e r m i n a l p l a n n i n g phase.
A t the f e a s i b i l i t y a n a l y s i s s t a g e ,
g o a l f e a s i b i l i t y is checked and any necessary
r e f i n e m e n t s o f the c h a r t s i s c a r r i e d o u t .
The
t e r m i n a l phase a c t i v i t i e s use the r e v e r s i b i l i t y
p r i n c i p l e and p l a n t r a j e c t o r i e s near the i n i t i a l
and f i n a l c o n f i g u r a t i o n s .
The m i d - s e c t i o n phase
deals w i t h midway t r a j e c t o r y p l a n n i n g .
For the
t e r m i n a l phase, forearm and boom p l a n n i n g i t e r a t e
u n t i l a s a t i s f a c t o r y boom t i p l o c a t i o n f o r
s t a r t i n g the m i d - s e c t i o n t r a j e c t o r y i s f o u n d .
For the m i d s e c t i o n , p l a n n i n g proceeds h i e r a r c h i cally.
Boom t r a j e c t o r y i s f i r s t planned u s i n g
Safety of the boom t i p locus i m p l i e s the s a f e t y
of the e n t i r e m a n i p u l a t o r only when the locus
passes through a secondary c h a r t ; elsewhere the
s a f e t y of the forearm needs to be ensured.
To
make forearm s a f e t y checks t r a c t a b l e , the f o l l o w i n g are chosen as p r i m i t i v e s f o r the forearm
trajectory.
1.
PLANNING
The process of p l a n n i n g a t r a j e c t o r y was d i s cussed in S e c t i o n 3; the aim is to p l a n a safe
t r a j e c t o r y , and p l a n i t f a s t .
I f the m a n i p u l a t o r needs t o maneuver c l o s e t o
o b s t a c l e s , secondary problem space is of no use
s i n c e the ' g r o s s ' r e p r e s e n t a t i o n of the forearm
e n g u l f s f r e e space near o b s t a c l e s .
Of course
t h i s does not imply t h a t a t r a j e c t o r y does not
exist.
The f i n e r model of the forearm as a l i n e
segment (as in primary problem space) should be
used.
4.5
When the boom is s t a t i o n a r y , the
forearm s h a l l move in a s i n g l e p l a n e ;
t h i s c a l l e d c i r c l e motion (2D) o r
sphere motion (3D).
When the boom is moving, the hand s h a l l
t r a c e a l i n e p a r a l l e l t o the c a r t e s i a n
space s t r a i g h t l i n e a p p r o x i m a t i o n o f
the boom t i p l o c u s ; t h i s i s c a l l e d
pgram motion ( f o r p a r a l l e l o g r a m m o t i o n ) .
Robotics-2 :
741
Udupa
the p r i m a r y c h a r t s a l o n e .
For p o r t i o n s of the
boom t i p t h a t d o n o t l i e i n the secondary c h a r t ,
forearm p l a n n i n g is done. The s e p a r a b i l i t y
p r i n c i p l e i s used i n boom p l a n n i n g ; the t r a j e c t o r y f o r the t h e t a - p h i j o i n t s i s planned f i r s t
and the r - j o i n t i s f i x e d n e x t .
If a safe forearm t r a j e c t o r y cannot be found, the
boom t r a j e c t o r y is m o d i f i e d and another attempt
a t forearm p l a n n i n g made.
I f the system i s
unable to come up w i t h a safe t r a j e c t o r y even
a f t e r a p r e s p e c i f i e d number o f a t t e m p t s , i t
r e s o r t s to a c o n f i g u r a t i o n s w i t c h .
The same
techniques are used to p l a n a t r a j e c t o r y to get
the m a n i p u l a t o r t o the g o a l , t h i s time however,
in a different lateral configuration.
If this
a l s o f a i l s , the system gives up.
Note t h a t p l a n n i n g i n c o r p o r a t e s simple s t r a t e gies.
It may so happen t h a t the system f a i l s to
f i n d a s o l u t i o n when t h e r e e x i s t s one in the r e a l
world.
I t i s u n l i k e l y t h a t such s i t u a t i o n s w i l l
be encountered except in some p a t h o l o g i c a l o b s t a cle configurations.
5.2
Initialization
The system is i n i t i a l i z e d w i t h a d e s c r i p t i o n of
the environment. The system uses the i n p u t
polyhedra and generates primary and secondary
o b s t a c l e s f o r the l e f t and r i g h t , secondary and
primary maps.
A l l the c h a r t s are generated f o r a
d e f a u l t r e s o u l t i o n (see F i g u r e 5 . 1 , f o r example).
The regions o f the c h a r t s w i l l b e f u r t h e r r e f i n e d
as and when necessary. The i n i t i a l i z a t i o n needs
to be done once f o r every new environment.
5.3
Goal F e a s i b i l i t y and Impossible S i t u a t i o n s
Goal f e a s i b i l i t y i s done b e f o r e p l a n n i n g b e g i n s .
It i n c l u d e s boom placement and forearm placement
s a f e t y checks.
It determines whether the boom
t i p l i e s w i t h i n a pasc o f a primary r e g i o n .
If
n o t , the a p p r o p r i a t e r e g i o n i s r e p e a t e d l y r e f i n e d
u n t i l e i t h e r the goal boom t i p p o s i t i o n i s w i t h i n
a pasc or the r e s o l u t i o n l i m i t is reached and the
system r e t u r n s complaining t h a t the g o a l i s not
feasible.
The forearm f e a s i b i l i t y study i n v o l v e s
checking whether i n the f i n a l c o n f i g u r a t i o n , the
forearm i s safe from c o l l i s i o n ; i f n o t , the g o a l
i s not f e a s i b l e .
Figures 5.2 and 5.3 are r e f i n e ments of F i g u r e 5 . 1 to get p o i n t s S and G i n s i d e
the c h a r t s .
The environment is the one used in
the example of S e c t i o n 2.
During m i d - s e c t i o n phase boom p l a n n i n g , the
system keeps a watch f o r s i t u a t i o n s which would
get the boom s t u c k (see F i g u r e 5 . 4 , f o r example).
If the boom cannot maneuver out of an a r e a , the
system complains. A g a i n , d u r i n g forearm p l a n n i n g
along a proposed boom t i p l o c u s , the system looks
out f o r s i t u a t i o n s which would get the forearm
stuck (see F i g u r e 5 . 5 , f o r example).
Such
s i t u a t i o n s are c a l l e d i m p o s s i b l e s i t u a t i o n s .
5.4
M i d - S e c t i o n Planning
Boom Planning
Boom p l a n n i n g is e q u i v a l e n t to f i n d i n g the path
of a p o i n t through c h a r t s .
C a r t e s i a n space
s t r a i g h t l i n e locus f o r the boom t i p are used i n
the 2D system. The f e a t u r e s of the 2D system and
the r e q u i r e d extensions f o r the 3D system are
presented.
P o i n t p a t h p l a n n i n g is based on an a d a p t a t i o n of
a well-known a l g o r i t h m used f o r approximating a
curve by a sequence of s t r a i g h t l i n e s ; the
approximation is such t h a t every p o i n t on the
curve i s w i t h i n a s p e c i f i e d d i s t a n c e from the
l i n e segment (approximating the p o r t i o n of the
curve the p o i n t i s o n ) .
The r e c u r s i v e a l g o r i t h m
is i l l u s t r a t e d in Figure 5.6.
P o i n t C i s the
f a r t h e s t p o i n t on the curve from the approximat i n g s t r a i g h t l i n e AB.
I f the d i s t a n c e o f C
from AB exceeds the t o l e r a n c e l i m i t , the curve
i s s p l i t a t C and the a l g o r i t h m a p p l i e d r e c u r s i v e l y to s e c t i o n s AC and CB.
The above a l g o r i t h m works even if d i f f e r e n t
t h r e s h o l d s are used f o r d i f f e r e n t p a r t s o f the
curve.
Each component ( r e g i o n , s e c t o r o i d , or
pasc ( i n the 3D case)) has a s s o c i a t e d w i t h it
an rmax and rmin t h a t s p e c i f y the safe i n t e r v a l l i m i t , w i t h i n which the boom t i p must l i e
when it goes through t h a t component.
F i g u r e 5.7 shows the working of the m o d i f i e d
l i n e a r approximation a l g o r i t h m .
The d o t t e d
l i n e s show adjacent regions of a c h a r t and
t h e i r safe l i m i t i n t e r v a l s .
Every p a i r o f
r e g i o n s has a n o n t r i v i a l i n t e r s e c t i o n o f t h e i r
( r m i n , rmax) i n t e r v a l . S and G are the s t a r t
and goal boom t i p l o c a t i o n s .
The arrow in the
F i g u r e shows the p o i n t on SG t h a t is f a r t h e s t
from being i n s i d e the r e g i o n s . A subgoal
P i s i n t r o d u c e d i n the r e g i o n c o n t r i b u t i n g t o
the v i o l a t i o n and the a l g o r i t h m i s r e c u r s i v e l y
a p p l i e d to SP and PG.
F i g u r e 5.8 shows how adjacent r e g i o n s Rl and
R2, which have t r i v i a l ( r m i n , rmax) i n t e r v a l
i n t e r s e c t i o n s are handled by the i n t r o d u c t i o n
of a d d i t i o n a l subgoals PO and P I . Note t h a t
the l i n e along P0P1 is safe from the rmin of
R2 to rmax of R l .
A f u r t h e r g e n e r a l i z a t i o n of the simple r e c u r s i v e
c u r v e - a p p r o x i m a t i o n a l g o r i t h m i s t o make the
approximating l i n e segments be any d e s i r a b l e
curve.
I n f a c t , f o r the 3 D system t h i s
g e n e r a l i z a t i o n is used to p l a n a boom t i p
l o c u s t h a t i s l i n e a r i n the boom j o i n t a n g l e s .
The above p l a n n i n g procedure i s f i r s t a p p l i e d
a t the r e g i o n l e v e l , then a t the s e c t o r o i d
l e v e l and f i n a l l y a t the pasc l e v e l .
In the 3D m a n i p u l a t o r , the system f i r s t plans
a t r a j e c t o r y in the t h e t a - p h i space making o n l y
c e r t a i n minimal checks on the safe l i m i t
P o b o t i c s - 2 : Udupa
74 2
i n t e r v a l s of the regions through which the
t r a j e c t o r y passes.
Once t h i s is done, the rj o i n t p l a n n i n g i s done using the g e n e r a l i z e d
l i n e a r approximation a l g o r i t h m .
I n s t e a d o f using s o f t w a r e f o r g a t h e r i n g i n f o r m a t i o n f o r p l a n n i n g motions c l o s e t o o b s t a c l e s ,
p r o x i m i t y sensors can be mounted on the forearm
to provide t h i s information.
The l o g i c f o r
a n a l y z i n g r e a l t i m e data from an a r r a y of such
sensors i s s i m p l e ; the l o g i c i n c o r p o r a t e s the
simple a d j u s t motion h e u r i s t i c s described
above.
Forearm Planning
Forearm p l a n n i n g is j u s t a sequence of a p p l i c a t i o n s of the two p r i m i t i v e s f o r the forearm
trajectory.
F i g u r e 5.9 i l l u s t r a t e s c i r c l e
m o t i o n ; i t shows the angular i n t e r v a l the
forearm can move i n , the most favored o r i e n t a t i o n o f the forearm f o r the given d i r e c t i o n o f
t r a v e l of the boom, the forearm o r i e n t a t i o n
chosen and the boom t i p locus SG. F i g u r e 5.10
i l l u s t r a t e s pgram m o t i o n ; i t shows the d i r e c t i o n
of t r a v e l (SG being the boom t i p locus) the
i n i t i a l forearm o r i e n t a t i o n and the p a r a l l e l o gram generated by the forearm m o t i o n .
5.5
6.
CONCLUSIONS
A s o l u t i o n to the safe t r a j e c t o r y p l a n n i n g
problem f o r computer c o n t r o l l e d manipulators
w i t h two movable l i n k s and m u l t i p l e degrees of
freedom was p r e s e n t e d .
The s o l u t i o n t r e a t s
m a n i p u l a t o r s w i t h a s l i d i n g j o i n t , and p e r m i t s
t r a n s p o r t i n g of o b j e c t s which can be enclosed
w i t h i n the minimum bounding c y l i n d e r of the
manipulator l i n k .
For m o d i f i c a t i o n s o f the
s o l u t i o n that permit handling larger objects,
extensions t o the s o l u t i o n f o r t r e a t i n g manipu l a t o r s w i t h only r o t a r y j o i n t s , and d e t a i l s o n
how to account f o r the l a t e r a l p r o p e r t y of the
m a n i p u l a t o r , the reader is r e f e r r e d to Udupa
(1976).
A s i g n i f i c a n t p o r t i o n of the ideas
described here have been implemented in SAIL on
a DEC PDP-10 computer. The o u t p u t of the
c o l l i s i o n d e t e c t i o n and avoidance system has
y e t to be i n t e r f a c e d w i t h a r e a l m a n i p u l a t o r .
Terminal Phase Planning
Terminal phase p l a n n i n g plans t r a j e c t o r i e s c l o s e
t o o b s t a c l e s , p r i m a r i l y near the s t a r t and the
goal c o n f i g u r a t i o n s .
The s t r a t e g y c o n s i s t s of
p l a n n i n g p a i r s of a d j u s t and move m o t i o n s . A
sequence of such p a i r s of motions puts the boom
t i p a t a safe p o i n t , from which the m i d - s e c t i o n
s t r a t e g i e s take over.
A safe p o i n t , is a p o i n t
in a secondary pasc, or if t h e r e is no secondary
pasc w i t h a reasonable safe i n t e r v a l then i t i s
a p o i n t in a primary pasc whose safe l i m i t
i n t e r v a l exceeds a p r e s p e c i f i e d v a l u e .
6.1
Key Ideas
1.
S i m p l i f i e d M a n i p u l a t o r D e s c r i p t i o n s And
Trajectory Primitives:
Alternate
problem spaces of i n c r e a s i n g a b s t r a c t i o n t h a t
permit s i m p l i f i e d m a n i p u l a t o r models and
p r i m i t i v e t r a j e c t o r y types are i d e n t i f i e d ;
these s i m p l i f y c o l l i s i o n d e t e c t i o n and t r a j e c t o r y hypothesis and m o d i f i c a t i o n .
During the move motion the boom t i p moves along
a l i n e c o l l i n e a r w i t h the forearm and away from
the hand, and the forearm m a i n t a i n s i t s o r i e n t a t i o n i n c a r t e s i a n space.
This motion continues
u n t i l e i t h e r the boom t i p reaches a safe p o i n t
or a p o t e n t i a l c o l l i s i o n is recognized.
2. Navspace and C h a r t s : The concept of
navspace t h a t p e r m i t s the r e d u c t i o n of the boom
to a single point is i d e n t i f i e d .
Odd-shaped
navspace is approximated by e a s i l y d e s c r i b a b l e
e n t i t i e s c a l l e d c h a r t s ; the approximation i s
dynamic and can be s e l e c t i v e , thus p e r m i t t i n g
easy i n c r e m e n t a l m o d i f i c a t i o n s to the c h a r t s .
The a d j u s t motion o r i e n t s the forearm to reduce
chances of c o l l i s i o n d u r i n g the subsequent move
motion.
Figure 5.11 shows a d j u s t motion
h e u r i s t i c s - the " b i n a r y " choice of f a v o r a b l e
o r i e n t a t i o n s - f o r the 2D case.
The numbers
i n d i c a t e the sequence i n which these o r i e n t a tions w i l l be t r i e d .
For a p a r t i c u l a r o r i e n t a t i o n , i f i t t u r n s out t h a t the subsequent move
motion makes no p r o g r e s s , the next o r i e n t a t i o n
i n the sequence i s t r i e d .
I f the m a n i p u l a t o r
j o i n t angles remain unchanged, even a f t e r a
p r e s p e c i f i e d number of t r i e s , the system r e t u r n s
a failure'.
3. Transformations f o r g e n e r a t i n g the
primary and secondary maps and c h a r t s need to be
computed o n l y once or a few t i m e s ; otherwise the
advantages of u s i n g the a l t e r n a t i v e problem
spaces would have been o f f s e t by the expensive
computations r e q u i r e d to generate them.
In 3D a 3 x 3 square of s o l i d angles is d e t e r mined about the c u r r e n t forearm o r i e n t a t i o n .
Each square is 5 degress in s i z e , and has a 0 or
1 a s s o c i a t e d w i t h i t according a s i t i s safe o r
not f o r the forearm t o maneuver w i t h i n the s o l i d
angle represented by the square. The 2D a d j u s t
motion s t r a t e g y i s a p p l i e d i n the plane t a k i n g
the forearm through the center of the safe s o l i d
angle i n t e r v a l .
I f more than one s o l i d angle
square i s s a f e , obvious g e n e r a l i z a t i o n s o f the
b i n a r y choice searching s t r a t e g y can be t r i e d .
4.
T r a j e c t o r y Planning In Empty Space
v s . C o l l i s i o n Avoidance: These two
complementary views can be used to advantage in
the safe t r a j e c t o r y p l a n n i n g problem.
Boom
planning is treated as planning t r a j e c t o r i e s in
empty space ( t h e c h a r t s ) , and forearm p l a n n i n g
is t r e a t e d as a c o l l i s i o n avoidance problem.
5. C a r t e s i a n Space v s . J o i n t Space: By
decomposing p l a n n i n g i n t o boom and forearm
p l a n n i n g and maneuverable space i n t o navspace
R o b o t i c s - 2 : Udupn
743
and s p a c e o c c u p i e d b y o b s t a c l e s , t h e a d v a n t a g e s
o f t h e r e p r e s e n t a t i o n s i n these a l t e r n a t e spaces
can b e c a p i t a l i z e d o n .
6. Planning:
Hierarchical decomposition,
d i f f e r e n t s t r a t e g i e s f o r maneuvering f a r from
o b s t a c l e s and f o r m a n e u v e r i n g c l o s e t o o b s t a c l e s ,
and a f o r m a l c h a r a c t e r i z a t i o n o f l a r g e c h u n k s o f
empty space a l l s i m p l i f y t h e p l a n n i n g t a s k .
to express my s i n c e r e a p p r e c i a t i o n to Sankaran
S r i n v a s , S c o t t R o t h and M e i r W e i n s t e i n o f C a l i f o r n i a I n s t i t u t e o f T e c h n o l o g y , and L e n F r i e d m a n
and B i l l W h i t n e y o f JPL f o r t h e many t h o u g h t f u l
d i s c u s s i o n s , and t h e i r c o n c e r n and e n c o u r a g e m e n t
at various stages of the research reported here.
1 w o u l d l i k e t o t h a n k E d B a l l a r d o f B e l l Labs
f o r reviewing a d r a f t of t h i s paper.
REFERENCES
7.
Planning At Execution Time:
Guidelines
have been suggested f o r i n c o r p o r a t i n g p r o x i m i t y
sensors i n t o the m a n i p u l a t o r system f o r doing
t e r m i n a l phase p l a n n i n g .
6.2
D o b r o t i n , B . M . and S c h e i n m a n , V . D . ,
" D e s i g n o f a Computer C o n t r o l l e d M a n i p u l a t o r f o r Robot R e s e a r c h " , 3 I J C A I , S t a n f o r d
U n i v e r s i t y , August 1973.
2.
L e w i s , R . A . , "Autonomous M a n i p u l a t i o n i n
a R o b o t : Summary o f M a n i p u l a t o r S o f t w a r e
F u n c t i o n s " , JPL T e c h n i c a l R e p o r t 3 3 - 6 7 9 ,
March 1974.
3.
P a u l , R., " M o d e l l i n g , T r a j e c t o r y C a l c u l a t i o n and S e r v o i n g o f a Computer C o n t r o l l e d
A r m " , PhD t h e s i s , S t a n f o r d U n i v e r s i t y ,
November 1 9 7 2 .
4.
P i e p e r , D . L . , "The K i n e m a t i c s o f M a n i p u l a t o r s Under Computer C o n t r o l " , PhD t h e s i s ,
S t a n f o r d U n i v e r s i t y , October 1968.
5.
U d u p a , S . M . , " C o l l i s i o n D e t e c t i o n and
A v o i d a n c e i n Computer C o n t r o l l e d M a n i p u l a t o r s " , PhD t h e s i s , C a l i f o r n i a I n s t i t u t e o f
T e c h n o l o g y , September 1976.
6.
Widdoes, C , " A H e u r i s t i c C o l l i s i o n Avoider
f o r t h e S t a n f o r d Robot A r m " , S t a n f o r d C .
S. Memo 2 2 7 , J u n e 1 9 7 4 .
S u g g e s t i o n s F o r F u t u r e Work
T r a n s p o r t i n g o b j e c t s comparable i n s i z e t o the
m a n i p u l a t o r , c o l l i s i o n avoidance f o r m u l t i p l e
manipulators, handling of a r i c h e r class of
c o n s t r a i n t s ( k e e p t h e hand v e r t i c a l d u r i n g
m o t i o n , f o r e x a m p l e ) , and o t h e r m a n i p u l a t o r
hardware designs ( t e l e s c o p i n g m a n i p u l a t o r s , f o r
e x a m p l e ) a r e some t o p i c s t h a t need i n v e s t i g a t i o n .
C o l l i s i o n a v o i d a n c e i n humanoid m a n i p u l a t o r s
( a l l r o t a r y j o i n t s ) can b e h a n d l e d b y a s l i g h t
extension of the s o l u t i o n presented in t h i s
paper.
L i t t l e i s known a b o u t m o d i f y i n g t r a j e c t o r i e s
d y n a m i c a l l y b a s e d o n any s e n s o r y d a t a t h e
s y s t e m may a c q u i r e d u r i n g e x e c u t i o n .
Further
i n v e s t i g a t i o n s o n t h e use o f p r o x i m i t y s e n s o r s ,
f o r c e and t a c t i l e s e n s o r s , and v i s u a l f e e d b a c k
to simplify planning is required.
7.
1.
ACKNOWLEDGEMENTS
The r e s e a r c h r e p o r t e d h e r e was done i n c o n n e c t i o n w i t h the Jet Propulsion L a b o r a t o r y ' s
(JPL) R o b o t i c s Research Program.
I would l i k e
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R o b o t i c s - 2 : Udupa
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Robotics-2: Udupa
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747
R o b o t i c s - 2 : Udupa
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