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MOTION PLANNING FOR THREE-DIMENSIONAL ARM MANIPULATORS
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below). By treating links interaction with obstacles individually and by making use of the information on what speci c part front or rear of a given link is currently in contact with obstacles, the path planning algorithm takes advantage of the obstacle monotonicity property. Because this information is not available in C-space, the following holds:
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Information Loss due to Space Transition. Information is lost in the space transition W C. Since some of this information namely, the location of contact points between the robot arm and obstacles is essential for the sensor-based planning algorithm, from time to time the algorithm may need to utilize some information speci c to W -space only.
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We will now consider some elemental planar interactions of arm links with obstacles, and we will show that if a path from start to target does exist, then a combination of elemental motions can produce such a path. De ne the following:
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Type I obstacle corresponds to a W - or C-obstacle that results from the interaction of link l1 with a physical obstacle. Type II obstacle corresponds to a W - or C-obstacle that results from the interaction of link l2 with a physical obstacle. Type III obstacle corresponds to a W - or C-obstacle that results from the interaction of link l3 with a physical obstacle.
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We will use subscripts + and to further distinguish between obstacles that interact with the front and rear part of a link, respectively. For example, a Type III+ obstacle refers to a C-obstacle produced by interaction of the front part of link l3 with some physical obstacle. In the next section we will analyze separately the interaction of each link with obstacles. Each time, three cases are considered: when an obstacle interacts with the front part, the rear part, or simultaneously with both parts of the link in question. We will also consider the interaction of a combination of links with obstacles, setting the foundation for the algorithm design.
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Interaction of Link l1 with Obstacles Type I Obstacles. Since, according to our model, sliding along an obstacle does not constitute an interference with the link l1 motion, we need to consider only those cases where the link meets an obstacle head-on. When only the front end of link l1 is in contact with an obstacle say, at the joint value l1 a Type I+ obstacle is produced, which extends from C-space oor to ceiling and side to side (see Figure 6.6) which effectively reduces the C-space cubicle by the volume (l1 max l1 ) l2 max l3 max . A similar effect appears when only the rear end of link l1 interacts with an obstacle say, at a joint value l1 . Then the C-space is effectively decreased by the volume l1 l2 max l3 max . Finally, a simultaneous contact of both front and rear ends with obstacles at a value l1 corresponds to a degenerate case where no motion of link l1 is possible; that is, the C-obstacle occupies the whole C-space. Formally the property of Type I obstacle monotonicity is expressed as follows:
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THE CASE OF THE PPP (CARTESIAN) ARM
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Figure 6.6 C-space with a Type I obstacle.
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Type I Monotonicity. For any obstacle interacting with link l1 , there are three directions corresponding to the joint axes li , i = 1, 2, 3, respectively, along which the C-obstacle behaves monotonically, as follows: If a position (l1 , l2 , l3 ) cannot be reached by the arm due to an obstacle interference, then no position (l1 , l2 , l3 ), such that l1 > l1 in case of the (obstacle s) contact with the link s front part, or l1 < l1 in case of the contact with the link s rear part, and l2 [0, l2 max ], l3 [0, l3 max ], can be reached either. Interaction of Link l2 with Obstacles Type II Obstacles Front Part of Link l2 Type II+ Obstacles. Consider the case when only the front part of link l2 interferes with an obstacle (Figure 6.2). Because link l2 effectively has two degrees of freedom, the corresponding Type II+ obstacle will look in C-space as shown in Figure 6.7. The monotonicity property in this case is as follows: Type II+ Monotonicity. For any obstacle interacting with the front part of link l2 , there are two axes (directions), namely l2 and l3 , along which the C-obstacle behaves monotonically, as follows: If a position (l1 , l2 , l3 ) cannot be reached by the arm due to an obstacle interference, then no position (l1 , l2 , l3 ), such that l2 > l2 and l3 [0, l3 max ], can be reached either.
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As a result, a Type II+ collision, as at point H in Figure 6.7, indicates that any motion directly upward or downward from H along the obstacle will necessarily bring the C-point to one of the side walls of the C-space cubicle. This suggests
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