Motion Planning for Three-Dimensional Arm Manipulators in VS .NET

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Motion Planning for Three-Dimensional Arm Manipulators
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The robot is going to lose. Not by much. But when the nal score is tallied, esh and blood is going to beat the damn monster.
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Adam Smith, philosopher and economist, 1723 1790
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6.1 INTRODUCTION We are continuing developing SIM (Sensing Intelligence Motion) algorithms for robot arm manipulators. The cases considered in 5 all deal with arm manipulators whose end effectors (hands) move along a two-dimensional (2D) surface. Although applications do exist that can make use of those algorithms for example, assembly of microelectronics on a printed circuit board is largely limited to a 2D operation most robot arm manipulators live and work in three-dimensional (3D) space. From this standpoint, our primary objective in 5 should be seen as preparing the necessary theoretical background and elucidating the relevant issues, before proceeding to the 3D case. Sensorbased motion planning algorithms should be able to handle 3D space and 3D arm manipulators. Developing such strategies is the objective of this chapter. As before, the arm manipulators that we consider are simple open kinematic chains. Is there a fundamental difference between motion planning for two-dimensional (2D) and 3D arm manipulators The short answer is yes, but the question is not that simple. Recall a similar discussion about mobile robots in 3. From the standpoint of motion planning, mobile robots differ from arm manipulators: They have more or less compact bodies, kinematics plays no decisive role in their motion planning, and their workspace is much larger compared to their dimensions. For mobile robots the difference between the 2D and 3D cases is absolute and dramatic: Unequivocally, if the 2D case has a de nite and nite solution to the planning problem, the 3D case has no nite solution in general. The argument goes as follows. Imagine a bug moving in the two-dimensional plane, and imagine that on its way the bug encounters an object (an obstacle).
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Sensing, Intelligence, Motion, by Vladimir J. Lumelsky Copyright 2006 John Wiley & Sons, Inc.
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MOTION PLANNING FOR THREE-DIMENSIONAL ARM MANIPULATORS
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Assume the bug s target location is somewhere on the other side of the obstacle. One way for it to continue its motion is to rst pass around the obstacle. The bug has only two options: It can pass the obstacle from the left or it can pass it from the right, clockwise or counterclockwise. If neither option leads to success let us assume it is a smart bug, with a reasonably good motion planning skills the goal is not achievable. In this case, by slightly exaggerating the bug s stubbornness, we will note that eventually the bug will come back to the point on the obstacle where it started. It took only one simple going around, all in one direction, to explore the whole obstacle. That is the essence of the 2D case. Imagine now a y that is ying around a room that is, in 3D space. Imagine that on its way the y encounters a (3D) obstacle say, a child s balloon hanging on a string. Now there is an in nite number of routes the y can take to pass around the obstacle. The y would need to make a great many loops around the obstacle in order to explore it completely. That s the fundamental dif culty of the 3D case; in theory it takes an in nitely long path to explore the whole obstacle, even if its dimensions and volume are nite and modest.1 The point is that while in the 2D case a mobile robot has a theoretically guaranteed nite solution, no such solution can be guaranteed for a 3D mobile robot. The 3D sensor-based motion planning problem is in general intractable. The situation is more complex, but also not as hopeless, for 3D arm manipulators. Try this little experiment. Fix your shoulder and try to move your hand around a long vertical pole. Unlike a y that can make as many circles around the pole as it wishes, your hand will make about one circle around the pole and stop. What holds it from continuing moving in the same direction is the arm s kinematics and also the fact that the arm s base is nailed down. The length of your arm links is nite, the links themselves are rigid, and the joints that connect the links allow only so much motion. These are natural constraints on your arm movement. The same is so for robot arm manipulators. In other words, the kinematic constraints of an arm manipulator impose strong limitations on its motion. This fact makes the problem of sensor-based motion planning for 3D arm manipulators manageable. The hope is that the arm kinematics can be effectively exploited to make the problem tractable. Furthermore, those same constraints promise a constructive test of target reachability, similar to those we designed above for mobile robots and 2D arm manipulators. As noted by Brooks [102], the motion planning problem for a manipulator with revolute joints is inherently dif cult because (a) the problem is nondecomposable, (b) there may be dif culties associated with rotations, (c) the space representation and hence the time execution of the algorithm are exponential in the number of robot s degrees of freedom of the objects involved, and (d) humans are especially poor at the task when much reorientation is needed, which makes it dif cult to
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One may argue that the y can use its vision to space its loops far enough from each other, making the whole exercise quite doable. This may be true, but not so in general: The room may be dark, or the obstacle may be terribly wrinkled, with caves and overhangs and other hooks and crannies so that the y s vision will be of little help.
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