MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS in Visual Studio .NET

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MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS
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3. A segment that is a part of one of the base circles (e.g., line 11-12, Figure 5.24b); the inside points of this segment cannot be reached by the arm. In order to apply the basic path planning procedure to this arm, the algorithm has to re ect speci cs of moving along the C-space cylinder. Similar to the RR arm, one concern in our case is whether obstacle boundaries may be formed by more than one simple curve. Recall that if a virtual boundary is formed by one simple (closed) curve, it is called a Type I obstacle, and if the virtual boundary has more than one simple (closed) curve, it is a Type II obstacle (see Section 5.2.2). Starting with one speci c case, observe that if a ring-like actual obstacle appears in W -space, positioned so that it separates the arm from the W -space outer boundary, the result will be a band-like virtual obstacle in Cspace formally, a Type II obstacle. One simple closed curve of the band can be reached by the arm, whereas the other, formed by one of the base circles, is inaccessible to the arm. Because of this, and in spite of the fact that the virtual boundary has two closed curves, from the standpoint of path planning we will treat it as a Type I obstacle. As another case, observe the arm shown in Figure 5.1c, where l1 = 0. If an obstacle extends from W -space into the dead zone, it is easy to see that in C-space a swath-like virtual obstacle will appear, whose virtual boundary in C-space includes two separate simple curves, plus two vertical lines each connecting the opposite base circles of the C-space cylinder. This is a real Type II obstacle. Similar to the RR arm, if during the arm motion one such curve of a Type II obstacle has been completely explored by the arm without ever meeting the M-line, it is clear that the second curve has to be explored as well. To do that, the complementary M2 -line will be used. As with the Cartesian arm studied above, the choice of the local direction for following the virtual obstacle by our RP arm happens to be unique. Once the arm encounters an obstacle, one of two possible cases arises. If the contact is a front contact that it, it corresponds to the front part of the arm contacting the obstacle then only such a local direction is meaningful that corresponds to decreasing values of l2 . As one can see in Figure 5.24a, the opposite local direction would never bring the arm any closer to the target. If, on the other hand, the contact is a rear contact, then only such local direction should be chosen that corresponds to increasing values of l2 . The reachability test is built in a manner similar to this test for the RR arm (Section 5.2.2), taking into account the simpler structure of the RP arm s C-space (see the algorithm below). How will the arm tell a front contact from a rear contact By our model, the arm s sensing lets it know which point of its body contacts the obstacle. The arm also knows at all times which point of link l2 is at the joint point of the link. This information allows the arm to always distinguish a front contact from a rear contact.
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REVOLUTE PRISMATIC (RP) ARM WITH PARALLEL LINKS
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Because of the unique choice of the local direction, there is no need to investigate the whole curve of the virtual boundary. If, while passing around the obstacle in the chosen local direction, the arm reaches one of the limits of l2 , it can safely conclude that it is dealing with a Type II obstacle, so the arm should start looking for the second curve of the virtual boundary using the complementary M-line. The procedure is further simpli ed through the use of the following statement similar to the one in Section 5.2.2: Lemma 5.5.1. For the two-link revolute prismatic (RP) arm, if position T is reachable from the starting position S, then there exists a path from S to T such that it corresponds to a monotonic change of the joint value 1 . In the motion planning procedure, a ag is used to indicate processing of each of the two curves of a Type II virtual boundary. When the complementary M-line is introduced, the numbering of hit and leave points starts over; Lo = S. The distance used is a Euclidean distance in W -space. Assume the M1 -line is the shorter of the two complementary M-lines. The procedure RP-Arm Algorithm includes the following steps. 1. Establish an M1 -line as the M-line. Set the ag down. Set j = 1. Go to Step 2. 2. From point Lj 1 , the arm moves along the M-line until one of the following occurs: (a) Target T is reached. The procedure stops. (b) An obstacle is encountered and a hit point, Hj , is de ned. In case of a front contact, choose the local direction such that it corresponds to decreasing values of l2 . In the case of a rear contact, choose the local direction such that it corresponds to increasing values of l2 . Go to Step 3. 3. The arm follows the virtual boundary until one of the following occurs: (a) The target is reached. The procedure stops. S T (b) Current joint value 1 is outside the interval ( 1 , 1 ). The target cannot be reached. The procedure stops. (c) The M-line is met at a distance d from T such that d < d(Hj , T ). Point Lj is de ned. Increment j . Go to Step 2. (d) The value l2 approaches one of its limits, and the ag is down (i.e., the rst curve of the virtual boundary of a Type II obstacle has been processed). Set the ag up. Set j = 1. Establish an M2 -line as the M-line. Move the arm back to S. Go to Step 2. (e) The value l2 approaches one of its limits, and the ag is up (i.e., the second curve of the virtual boundary of a Type II obstacle has been processed). The target cannot be reached. The procedure stops.
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