Justify using parallelogram determined by link vector

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Unformatted text preview: ddition) so, assume w.l.o.g. first link is longest n 0 i Theorem 8.6.3: Reachability region for n-link arm is origin-centered annulus with outer radius r = l r = l and inner radius 0 if longest link is at most half the total length of links, and r = l - l otherwise. i =1 i M i M iM i L4 L3 L1 L2 ri = l1 - (l2 + l3 + l4 ) Robot Arm Motion: Finding Configurations Find a single solution Can arm tip reach p? 2-Link Case C1 l1 C2 l2 Intersect circle C1 of radius l1 (centered on origin J0) with circle C2 of radius l2 (centered on origin p). In general there are 2 solutions (depends on how circles intersect) p = point to be reached Source: O'Rourke Robot Arm Motion: Finding Configurations Align L1 with L2 Case 1 : R C 0 "Anti-Align" L1 with L2 / O I Theorem: Every 3-link problem can be solved by one of these 2-link problems: (l1 + l2 , l3) [Fig 8.22(a)] (l1, l2 + l3) [Fig 8.22(b),(c), Fig 8.23] j0 = 1st joint angle C is centered at p and has radius l3. C does not enclose J0 C encloses J0 Case 2 : R C = 0 / Align L2 with L3 Solution exists for every j0! j0= 0 and (l2 , l3 ) [Fig 8.22(d)] Alternative to "Anti-Aligning" L1 with L2 (align L2 with L3) Boundary of annulus represents extreme 2-link configurations for "single link" of length l1+l2 or |l1-l2| Source: O'Rourke Robot Arm Motion: Finding Configurations Recursive, linear algorithm for n-link reachability: annulus R represents n-1 links of n-link arm with circle C of radius ln centered on p cases of Figure 8.22 apply Case 1 : R C 0 / Case 2 : R C [Fig 8.22(a),(b)] Choose one of (in general) 2 points of intersection [Fig 8.22(c),(d)] Choose any point on C (e.g. furthest from J0) Recursively find configuration for An-1=(l1,...,ln-1) Append last link Ln to this solution to connect to p Given point p to reach, first determine if p is reachable (via Theorem 8.6.3); if so, find configuration recursively. Source: O'Rourke Robot Arm Motion: n-Link Reachability Two Kinks Theorem: If an n-link arm A can reach a point, it can re...
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