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Robotics Research
lechnical Report






Motion-Planning with Inertial Constraints


Colm O'Diinlaing

Technical Report No. 230

Robotirs Report No. 73

July, 1986



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Clj -H a (v + Mt) - 2M'r. (3.6)

Case (ii)

.V = - 1 , so 2x-vt > 0. Reasoning in the same way, the left-hand inequality becomes
redundant and the right-hand inequality becomes

4/V/.V < 2M't - (v-Mt)^. (3.7)

Summarizing, the region F/?q,(0.0) may be expressed as the union of two regions,
each region being bounded by a parabolic segment and a straight-line segment (the straight
line satisfying the equation 2x — vi = 0). A straightforward calculation reveals that both par-
abolas intersect the straight Ime 2x = vt at the same two points in phase space, namely (as is
hardly surprising, since they represent the points reached by maintaining acceleration ±.V/)
±(V2Aft~,Mt). Therefore the region FR^ ,(0,0) may be expressed more succinctly as the
region bounded by rwo parabolic segments in the ( t ,v)-plane:

f/?o,/(0,0) = [(x.v):iv + Mtr-2M^-r < 4A/.v < 2.1/-/ - ( v-,V/r)-}. (3.8)

It is trivial to show that FR^ ,(0.0) = F/?q ,_^(0,0) , which trivially generalizes equa-
tion (3.8) to nonzero values of a. If x{s) is an acceleration-bounded trajectory (i.e., one


C^Mt^ Mt)

Illustrating F/?o,,(0,0)

I > . n: ■'■iisupa.'ii .-iri' ,-j ,:

satisfying the constraint (2.2)) passing through (a,Xa,Va')' ^^^" xis)—x^ — (s—a)v^ is an
acceleration-bounded trajectory passing through (a, 0,0). This, combined with the obvious
inverse transformation, establishes a bijection between the set of acceleration-bounded trajec-
tories through {a,x^.,Vij) and (i;,0,0) respectively, and therefore we conclude

FR,/x^,v^) = ((.v, + (f-a)v^,r,)+(.r,v):(,r,v)^f-/?o,,-a(0,0) }. (3.9)

We can express this more succinctly using the notation of Minkowski sum of sets of vectors
(given two sets X and Y of vectors, the Minkowski sum X+Y is defined as
(.r +y: .r €X,y ^ K}). Let 7"^ , be the linear transformation which takes a point (x^,v^) to
{Xa + {t — a)v^,v^). Then, for any phase region / at time a,

FRaAD = r,.,(/)+F/?o,,-.(0,0).


4. Phase region formed from a reachable pair of pursuit functions.

The phase region at time a formed from a pair {f ,g) of pursuit functions is the set of ail
points ((3,.r,v') in phase space at time a such that there exists an admissible trajectory begin-
ning at {a,x,v). Similarly the phase region at time h is the set of all endpoints of admissible

In this section we shall construct e.\plicitly the phase region (at time h) formed from a
pair of pursuit functions assuming they are 'reachable.'

Definition. A pomt ( T,.V-,i-j in parametrized phase space at time T (where a

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