MOTION PLANNING FOR TWO-DIMENSIONAL ARM MANIPULATORS in VS .NET

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Figure 5.3 Obstacles A and B form shadows ; the arm endpoint cannot reach points inside a shadow. For example, point P1 is in the shadow of the circular obstacle A and thus cannot be reached. The shadow of the circular obstacle B forms two disconnected subshadows.
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the M-line, the arm endpoint can continue through points b9 , b10 , b11 , meeting the M-line again at point b12 as under option 2. In fact, when starting at point b2 under any of the two options, if one continues rotating the arm around obstacle A while keeping in contact with it, the arm endpoint will make a complete closed curve, passing through the points b2 , b3 , b4 . . . , b8 , b9 . . . , b13 , b14 and eventually arriving at the same point b2 . This indicates that the paths produced under both options are complementary to each other: When added together, they form a closed curve. Regarding this curve, consider the area whose curvilinear boundary passes through points b , b2 , b3 , b4 , then the segment b4 , b10 of the workspace boundary, then points b10 , b9 , b of our curve, and nally the smaller part of the obstacle A boundary between points b and b . This area is called the shadow of obstacle A: Though this is a part of free space, no point (such as P1 ) inside this area can be reached by the arm endpoint. This suggests that an obstacle shadow will be perceived by the arm as an obstacle, as real as an actual physical obstacle. The arm cannot penetrate either
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Figure 5.4 An example of interaction between obstacles. The shadow (shaded area) behind obstacle A is the result of interaction between obstacles A and B. If the arm moves through the positions (a1 , b1 ), (a2 , b2 ), . . . , (a17 , b17 ), at any moment it is in contact with either obstacle A or B. This means that the arm will perceive these two obstacles as one obstacle. Because of obstacle C, link l1 cannot realize any angle values 1 in the range 1 < 1 < 1 .
of them. The shape of a shadow depends on the shape, size, and position in W -space of the corresponding actual obstacle that creates the shadow, as well as on the arm links shapes and dimensions. An obstacle can form disconnected shadows, as in the case of obstacle B (Figure 5.3). Or, obstacles can interact in forming shadows; this happens, for example, when two or more points of the arm body touch two or more actual obstacles simultaneously, as at position (a8 , b8 ) in Figure 5.4. De nition 5.2.1. A virtual obstacle X is an area (or areas) in W -space, no points of which can be reached by the arm endpoint because of the arm s possible interference with the actual obstacle X. Thus a virtual obstacle consists of the corresponding actual obstacles and their shadows. In W -space a virtual obstacle forms one or more compact areas (see Figure 5.4). Whereas topologically this combination presents little of interest in W -space, we will see below that it possesses interesting properties in the arm s C-space.
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De nition 5.2.2. Passing around an obstacle presents a continuous motion of the arm, during which the arm is constantly in contact with the corresponding physical obstacle(s). It is clear from Figure 5.4 that two or more actual obstacles may be interpreted by the arm as a single virtual obstacle. In Figure 5.4, at any position from the set (a1 , b1 ), (a2 , b2 ), . . . , (a17 , b17 ) the arm is in contact with at least one of the actual obstacles A and B. Hence the two obstacles will be interpreted as one. De nition 5.2.3. A virtual line is a curve in W -space that the arm endpoint follows when passing around an obstacle. The virtual line forms the boundary of a virtual obstacle in W -space. A virtual line is not necessarily a smooth curve. For example, if the arm endpoint follows a sharp corner on an obstacle, or if the arm contacts some obstacle while passing around another obstacle [as in the link position (a8 , b8 ), Figure 5.4], the virtual line may form sharp turns. Nor is a virtual line necessarily a non-self-intersecting curve (see virtual boundary of obstacle B, Figure 5.3), differing in this respect from the boundaries of physical two-dimensional objects. We will discuss this issue later, when analyzing the arm C-space properties. Points of contact on the arm may undergo a discontinuous jump when passing around obstacles. This can happen because of the shapes of obstacles and arm links involved, or because of the arm obstacle interaction. In Figure 5.4, for example, during link l2 motion through positions (a1 , b1 ), (a2 , b2 ), and so on, an instant before position (a8 , b8 ) link l2 is in contact with obstacle A; an instant after position (a8 , b8 ) the link is in contact with obstacle B. Accordingly, in this short period the contact point on the arm jumps from a point of contact on one side of link l2 to a completely different point on the link s other side. Note, however, that even in such cases there will be no discontinuity in the virtual curve.2 For example, in the area of point b8 , which corresponds to the jump of the contact point mentioned above (Figure 5.4), the virtual line remains continuous. There will be more on the virtual line continuity in our analysis of the arm C-space. Observe also that some distinct pieces of the virtual line may be associated with the same physical curve. Such is, for example, a part of the virtual line (b14 , b8 ) (Figure 5.3), which is a part of obstacle A boundary. When trying to do a complete rotation by the arm around A, the arm endpoint will follow the curve segment (b14 , b8 ) twice, once in each of the two directions. The requirement of continuous contact while passing around the obstacle is equivalent to adding a constraint on the arm motion. In general, the arm s position relative to obstacles is described by one of these three situations: 1. No contact with obstacles takes place; the motion is unconstrained, and all points in the vicinity of the arm endpoint are available for its next position.
Given the physics of the underlying phenomenon, this is not surprising: Physical motion is continuous, so the arm endpoint must be moving through a continuous curve.