MOTION PLANNING FOR THREE-DIMENSIONAL ARM MANIPULATORS

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Figure 6.17 Paths g1 , g2 , and g3 constitute a complete set of generic paths. A hexagon is obtained by (a) cutting the square along g1 , g2 , and g3 , (b) pasting along b, and (c) pasting along a. (d) The resulting hexagon.

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Now consider the effects of the above operation on obstacles (see Figure 6.18a). Obstacle boundaries and the generic paths partition our hexagon into occupied areas (shaded areas in Figure 6.18b) and free areas (numbered I , II, III, IV and V in Figure 6.18b). Each free area is not necessarily simple, but it must be homeomorphic to a disc, possibly with one or more smaller discs removed (e.g., area I of Figure 6.18b has the disc that corresponds to obstacle O2 removed). The free area s inner boundaries are formed by obstacle boundaries; its outer boundary consists of segments of obstacle boundaries and segments of the generic paths. Any arbitrary obstacle-free path p that connects points S and T consists of one or more segments, p1 , p2 , . . . , pn , in the hexagon. Let xi , yi be the end points of segment pi , where x1 = S, xi+1 = yi for i = 1, 2, . . . , n 1, and yn = T . Since p is obstacle-free, xi and yi must be on the outer boundary of the free area that contains pi . Therefore, xi and yi can be connected by a path segment pi of the outer boundary of the free area. The path p = p1 p2 . . . pn that connects S and T and consists of segments of the generic paths and segments of obstacle boundaries is therefore entirely on the connectivity graph G.

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THREE-LINK XXP ARM MANIPULATORS

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Figure 6.18 Illustration for Theorem 6.3.13. Shown are two obstacles O1 , O2 (shaded areas) and path p (thicker line). The torus is represented, respectively, as (a) a unit square with its opposite sides a and b identi ed in pairs and (b) as a hexagon, with generic paths as its sides. Segments p1 , p2 and p3 in (b) are connected; they together correspond to the path p in (a).

Figure 6.18a presents a torus shown as a unit square, with its opposite sides a and b identi ed in pairs. O1 and O2 are two obstacles. Note that the three pieces of obstacle O1 in the gure are actually connected. Segments g1 , g2 and g3 are (any) three of the four generic paths. For an XXP arm, we now de ne generic paths and the connectivity graph in B, which is homeomorphic to a torus. De nition 6.3.14. For any two points a, b J, let ab be the straight line segment connecting a and b. A vertical plane is de ned as

1 Vab = Pm (Pc (ab))

MOTION PLANNING FOR THREE-DIMENSIONAL ARM MANIPULATORS

where Pc and Pm are respectively the conventional and minimal projections as in De nition 6.3.7 and De nition 6.3.8. In other words, Vab contains both a and b and is parallel to the l3 axis. The degenerate case where ab is parallel to the l3 axis is simple to handle and is not considered. De nition 6.3.15. Let Ls and Lt be the given start and target con gurations of the arm, and let S = {j J|L(j ) = Ls } J and T = {j J|L(j ) = Lt } J, respectively, be the sets of points corresponding to Ls and Lt . Let f : J C be the projection as in De nition 6.3.12. Then the vertical surface V C is de ned as V = {f (j ) C|j Vst for all s S and t T}

For the RRP arm, which is the most general case among XXP arms, V consists of four components Vi , i = 1, 2, 3, 4. Each Vi represents a class of vertical planes in J and can be determined by the rst two coordinates of a pair of points drawn s s s t t t respectively from S and T. If js = ( 1 , 2 , l3 ) and jt = ( 1 , 2 , l3 ) are the robot s start and target con gurations, the four components of the vertical surface V can be represented as follows:

t t s s V1 : ( 1 , 2 ) ( 1 , 2 ) t t t s s s V2 : ( 1 , 2 ) ( 1 , 2 2 sign( 2 2 ))

(6.7)

t s 2 sign( 2 2 ))

V3 : V4 :

s s ( 1 , 2 )) s s ( 1 , 2 )

t ( 1 t ( 1

2 2

t sign( 1 t sign( 1

s t 1 ), 2 ) s t 1 ), 2

where sign() takes the values +1 or 1 depending on its argument. Each of the components of V -surface determines a generic path, as follows: gi = Vi B, i = 1, 2, 3, 4

Since B is homeomorphic to a torus, any three of the four generic paths can be used to form a connectivity graph. Without loss of generality, let g = 3 gi i=1 and denote g = Bf g. A connectivity graph can be de ned as G = g Bf . If there exists a path in Cf , then at least one such path can be found in the connectivity graph G. Now we give a physical interpretation of the connectivity graph G; G consists of the following curves:

Cp the boundary curve of the oor, identi ed by the fact that the third link of the robot reaches its lower joint limit (l3 = 0) and, simultaneously, one or both of the other two links reach their joint limits.