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5.8.2 Interaction Between the Robot and Obstacles Below we will need the property of space uniform local connectedness (ULC). To derive it, we need to properly de ne the notion of a contact between the robot and an obstacle. To this end, four conditions will be stated (Conditions 5.8.1 to 5.8.4) that together de ne a contact. Mathematically, at position (joint vector) j , robot L is in contact with obstacle O if L(j ) O = and L(j ) O = (5.7)
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The rst relation in (5.7) states that j OC (De nition 5.8.4), while the second relation states that j OC , where OC refers to the boundary of OC . However, there may be situations where both relations of (5.7) hold but no obstacle exists in CS . Consider a robot manipulator with a xed base, one link, and one revolute joint, along with a circular obstacle centered at the robot base O, as shown in Figure 5.34. Here relation (5.7) is satis ed at every robot con guration. Note, though, that the link can rotate freely in WS; this means that there are no obstacles, and hence no obstacle boundaries, in CS . Therefore, robot con gurations that satisfy Eq. (5.7) do not necessarily correspond to points on CSO boundaries. We modify the notion of contact by imposing additional conditions on the admissible robot and obstacle spatial relationships. As with any physical system, the term contact implies an existence of a force at the point of contact between the robot and the obstacle. In other words, for an object to present an obstacle for the robot, it must be possible for the robot to move in the direction of the force if the object were removed. With this de nition of a contact, the robot shown in Figure 5.34 is not in contact with the obstacle at any position because at a point of contact it cannot exert a force upon
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Figure 5.34 Shown is a single-link robot with a revolute joint at point O, along with a circular obstacle (shaded) also centered at O. With no obstacles in CS , the link can freely rotate about point O.
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the obstacle. Mathematically, the removal of such false contacts translates into the following condition, which guarantees that each CSO component has at least one interior point: Condition 5.8.1. Let j C satisfy (5.7); that is, there exists u L such that w = u(j ) O. For given > 0 and > 0, de ne O = O U (w, ), L = L U (u, ), and OC = {j U (j , ) : L(j ) O = }. For any given > 0, there must exist (0, ) and (0, ) such that OC = . Theorem 5.8.2. An obstacle in WS can map into any large but nite number of CSO components in CS. Proof: We rst design a simpli ed example showing that a simple obstacle in WS can map into two CSO components in CS. In Figure 5.35, the WS obstacle O produces two separate CSO components, each resulting from the interaction between O and each of the two vertical walls on the robot. Clearly, one can add additional vertical walls to the robot (and reduce the size of the obstacle if necessary) so that the number of CSO components will increase as well. This way one can create as many CSO components as one wishes. On the other hand, by Condition 5.8.1, a CSO component must have an interior point. Also, by Theorem 5.8.1, CSO is an open set, and so its any interior point must have a neighborhood of positive radius r that is entirely enclosed in a CSO component. Thus the CSO component must occupy in CS a nite volume (area). By Lemma 5.8.1, C has a nite volume or area; hence the number of CSO components in CS must be nite. Q.E.D. Figure 5.36a demonstrates another case of a false contact, more complicated than the previous one. The corresponding CSO indeed has interior points, Figure 5.36b. By our de nition of contact, at the con guration shown the robot is not in contact with the obstacle because it cannot exert any force upon the
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