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Figure 5.32 A 9-DOF robot with two arms attached to a common base.
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motion of the robot in WS. By introducing FCS, the robot motion planning problem can be studied under a uni ed mathematical framework [108]. For sensor-based motion planning algorithms to work, it is essential that the CSO boundary presents manifolds. This topological property is not trivial and cannot be simply assumed. It has been shown in Ref. 109 that in general CSO boundaries are not manifolds. Consider, for example, the example shown in Figure 5.33a. The setting is such that the mobile robot R can barely squeeze into the opening in the obstacle O, while touching both opposite walls of O simultaneously. As a consequence, the CSO boundary, which consists of two rectangles and a straight-line segment that connects them (Figure 5.33b), is not a manifold.
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Figure 5.33 Interaction between a square-shaped mobile robot R and an obstacle O. (a) WS: The robot can barely squeeze into the opening of obstacle O. (b) CS : The corresponding CSO boundary consists of the inner and outer rectangles plus a straight-line segment that connects them.
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Under what conditions will CSO boundaries present manifolds We will show below that if a certain unrestrictive spatial relationship between the robot and obstacle is satis ed for example, under no condition is the robot immobilized, nor does it need to squeeze between two obstacles as in Figure 5.33 then FCS is uniformly locally connected (ULC). Although the ULC property is not suf cient to ensure manifold boundaries for the general case, we will show that for the two-dimensional (2D) case the ULC property guarantees that FCS is bounded by manifolds in this case, simple closed curves. We proceed as follows. First, a general robot with d translational and/or revolute degrees of freedom will be de ned. CS is de ned as a Euclidean space formed by the robot parameter variables. A physical obstacle is the interior of a connected compact point set in WS. We will show that CSO is a closed subset of CS (Corollary 5.8.1). Then we will study the interaction between the robot and obstacles, and will de ne a set of conditions that correspond to certain undesirable degenerate situations (Conditions 5.8.1 to 5.8.4), such as when a part of or the whole robot is immobilized or when the robot can move between two obstacles only by simultaneously touching them both (Figure 5.33). We will show in Section 5.8.3 that after these situations are removed, CSO presents a uniformly locally connected subset of CS . We will then show in Section 5.8.4 that ULC is a necessary condition for an open subset of a compact space to have manifold boundaries. This is also a suf cient condition for a 2D open subset of a compact space to have manifold boundaries that is, simple closed curves. We will thus conclude that FCS of a 2-DOF robot is bounded by simple closed curves. More details pertinent to the material in this section can be found in [107]. 5.8.1 Workspace; Con guration Space As said above, kinematically a robot arm manipulator is an assembly of rigid links connected to each other by joints that permit the links motion relative to each other [8]. Joints and links form kinematic pairs. As in prior sections, without loss of generality we limit the types of kinematic pairs to either translational (prismatic) or rotational (revolute). The degrees of freedom (DOF) of a robot are often referred to as its mobility, which is the number of independent variables that must be speci ed in order to locate all the links relative to each other. The 9-DOF robot shown in Figure 5.32 has nine joints and nine links. The robot base is a link in which two translational joints l1 and l2 are implemented. The number of degrees of freedom of a robot is not necessarily equal to the number of links or the number of joints. Closed kinematic chains often have fewer DOF than the number of their links (joints). For example, a triangle-shaped planar closed kinematic chain with three links and three revolute joints has mobility zero. We choose arbitrarily d independent joints, J1 , . . . , Jd , to form a d-DOF robot, and we parameterize the robot con guration using the corresponding joint variables. With a reference system de ned at its connecting joint, each kinematic pair can be speci ed by four scalar parameters. So, for the joint i, ai is the link
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