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# Motion-planning with inertial constraints online

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 Font size Robotics Research lechnical Report vner. ■■3fo/ "'umc •: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 «J CM E M I rH CP4J « O C M E-t C-» -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 — 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 f-^Mt^-Mt; 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). (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 1 2 3 4