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Robotics Research

lechnical Report

vner.

â– â– 3fo/

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Motion-Planning with Inertial Constraints

by

Colm O'Diinlaing

Technical Report No. 230

Robotirs Report No. 73

July, 1986

\.

(0

J5 -P

-P C

O -H -H

CO 5 Â«J

CM E M

I rH CP4J

Â« O C M

E-t C-Â» -H C

CJ en 10

CO C rH M

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 â€” 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 Â±.V/)

Â±(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

f-^Mt^-Mt;

C^Mt^ Mt)

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

I > . n: â– 'â– iisupa.'ii .-iri' ,-j ,:

satisfying the constraint (2.2)) passing through (a,Xa,Va')' ^^^" xis)â€”x^ â€” (sâ€”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 â‚¬X,y ^ K}). Let 7"^ , be the linear transformation which takes a point (x^,v^) to

{Xa + {t â€” a)v^,v^). Then, for any phase region / at time a,

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

(3.10)

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

trajectories.

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

lechnical Report

vner.

â– â– 3fo/

"'umc

â€¢:SM&B

jyajy?

Motion-Planning with Inertial Constraints

by

Colm O'Diinlaing

Technical Report No. 230

Robotirs Report No. 73

July, 1986

\.

(0

J5 -P

-P C

O -H -H

CO 5 Â«J

CM E M

I rH CP4J

Â« O C M

E-t C-Â» -H C

CJ en 10

CO C rH M

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 â€” 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 Â±.V/)

Â±(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

f-^Mt^-Mt;

C^Mt^ Mt)

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

I > . n: â– 'â– iisupa.'ii .-iri' ,-j ,:

satisfying the constraint (2.2)) passing through (a,Xa,Va')' ^^^" xis)â€”x^ â€” (sâ€”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 â‚¬X,y ^ K}). Let 7"^ , be the linear transformation which takes a point (x^,v^) to

{Xa + {t â€” a)v^,v^). Then, for any phase region / at time a,

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

(3.10)

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

trajectories.

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

Online Library → C O'Dunlaing → Motion-planning with inertial constraints → online text (page 1 of 4)