THE CASE OF THE PPP (CARTESIAN) ARM in .NET

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THE CASE OF THE PPP (CARTESIAN) ARM
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Figure 6.2 The work space of a 3D Cartesian arm: l1 , l2 , and l3 are links; J1 , J2 , and J3 are prismatic joints; P is the arm endpoint. Each link has the front and rear end; for example, J3 is the front end of link l2 . O1 , O2 , and O3 are three physical obstacles. Also shown in the plane (l1 , l2 ) are obstacles projections. The cube abcodefg indicates the volume whose any point can be reached by the arm endpoint.
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of the xed reference system. Value li also denotes the joint variable for link li ; it changes in the range li = [li min , li max ]. Assume for simplicity zero minimum values for all li , li = [0, li max ]; all li max are in general different. Each link presents a generalized cylinder (brie y, a cylinder) that is, a rigid body characterized by a straight-line axis coinciding with the corresponding joint axis, such that the link s cross section in the plane perpendicular to the axis does not change along the axis. A cross section of link li presents a simple closed curve; it may be, for example, a circle (then, the link is a common cylinder), a rectangle (as in Figure 6.2), an oval, or even a nonconvex curve. The link cross section may differ from link to link.2 The front ends of links l1 and l2 coincide with joints J2 and J3 , respectively; the front end of link l3 coincides with the arm endpoint P (Figure 6.2). The opposite end of link li , i = 1, 2, 3, is its rear end. Similarly, the front (rear) part of link li is the part of variable length between joint Ji and the front (rear) end of the link. When joint Ji is in contact with an obstacle, the contact is considered to be with link li 1 .
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More precisely, we will see that only link l3 has to be a generalized cylinder to satisfy the motion planning algorithm; links l1 and l2 can be of arbitrary shape.
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MOTION PLANNING FOR THREE-DIMENSIONAL ARM MANIPULATORS
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For the sensing mechanism, we assume that the robot arm is equipped with a kind of sensitive skin that covers the surfaces of arm links and allows any point of the arm surface to detect a contact with an approaching obstacle. Other sensing mechanisms are equally acceptable as long as they provide information about potential obstacles at every point of the robot body. Depending on the nature of the sensor system, the contact can be either physical as is the case with tactile sensors or proximal. As said above, solely for presentation purposes we assume that the arm sensory system is based on tactile sensing.3
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The Task. Given the start and target positions, S and T , with coordinates S = (l1 S , l2 S , l3 S ) and T = (l1 T , l2 T , l3 T ), respectively, the robot is required to generate a continuous collision-free path from S to T if one exists. This may require the arm to maneuver around obstacles. The act of maneuvering around an obstacle refers to a motion during which the arm is in constant contact with the obstacle. Position T may or may not be reachable from S; in the latter case the arm is expected to make this conclusion in nite time. We assume that the arm knows its own position in space and those of positions S and T at all times. Environment and Obstacles. The 3D volume in which the arm operates is the robot environment. The environment may include a nite number of obstacles. Obstacle positions are xed. Each obstacle is a 3D rigid body whose volume and outer surface are nite, such that any straight line may have only a nite number of intersections with obstacles in the workspace. Otherwise obstacles can be of arbitrary shape. At any position of the arm, at least some motion is possible. To avoid degeneracies, the special case where a link can barely squeeze between two obstacles is treated as follows: We assume that the clearance between the obstacles is either too small for the link to squeeze in between, or wide enough so that the link can cling to one obstacle, thus forming a clearance with the other obstacle. The number, locations, and geometry of obstacles in the robot environment are not known. W-Space and W-Obstacles. The robot workspace (W-space or W) presents a subset of Cartesian space in which the robot arm operates. It includes the effective workspace, any point of which can be reached by the arm end effector (Figure 6.3a), and the outside volumes in which the rear ends of the links may also encounter obstacles and hence also need to be protected by the planning algorithm (Figure 6.3b). Therefore, W is the volume occupied by the robot arm when its joints take all possible values l = (l1 , l2 , l3 ), li = [0, li max ], i = 1, 2, 3. Denote the following:
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vi is the set of points reachable by point Ji , i = 1, 2, 3; Vi is the set of points (the volume) reachable by any point of link li . Hence,
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On adaptation of tactile motion planning algorithms to more complex sensing, see Sections 3.6 and 5.2.5.
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