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These attractive points of sensor-based planning stands out when comparing it with the paradigm of motion planning with complete information (the Piano Mover s model). The latter requires the complete information about the scene, and it requires it up front. Except in very simple cases, it also requires formidable calculations; this rules out a real-time operation and, of course, handling moving or shape-changing obstacles. From the standpoint of theory, the main attraction of sensor-based planning is the surprising fact that in spite of the local character of robot sensing and the high level of uncertainly after all, practically nothing may be known about the environment at any given moment SIM algorithms can guarantee reaching a global goal, even in the most complex environment. As mentioned before, those positive sides of the SIM paradigm come at a price. Because of the dynamic character of incoming sensor information namely, at any given moment of the planning process the future is not known, and every new step brings in new information the path cannot be preplanned, and so its global optimality is ruled out. In contrast, the Piano Mover s approach can in principle produce an optimal solution, simply because it knows everything there is to know.1 In sensor-based planning, one looks for a reasonable path, a path that looks acceptable compared to what a human or other algorithms would produce under similar conditions. For a more formal assessment of performance of sensorbased algorithms, we will develop some bounds on the length of paths generated by the algorithms. In 7 we will try to assess human performance in motion planning. Given our continuous model, we will not be able to use the discrete criteria typically used for evaluating algorithms of computational geometry for example, assessing a task complexity as a function of the number of vertices of (polygonal or otherwise algebraically de ned) obstacles. Instead, a new pathlength performance criterion based on the length of generated paths as a function of obstacle perimeters will be developed. To generalize performance assessment of our path planning algorithms, we will develop the lower bound on paths generated by any sensor-based planning algorithm, expressed as the length of path that the best algorithm would produce in the worst case. As known in complexity theory, the dif culty of this task lies in ghting an unknown enemy we do not know how that best algorithm may look like. This lower bound will give us a yardstick for assessing individual path planning algorithms. For each of those we will be interested in the upper bound on the algorithm performance the worst-case scenario for a speci c algorithm. Such results will allow us to compare different algorithms and to see how far are they from an ideal algorithm. All sensor-based planning algorithms can be divided into these two nonoverlapping intuitively transparent classes:
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In practice, while obtaining the optimal solution is often too computationally expensive, the everincreasing computer speeds make this feasible for more and more problems.
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Class 1. Algorithms in which the robot explores each obstacle that it encounters completely before it goes to the next obstacle or to the target. Class 2. Algorithms where the robot can leave an obstacle that it encounters without exploring it completely. The distinction is important. Algorithms of Class 1 are quite thorough one may say, quite conservative. Often this irritating thoroughness carries the price: From the human standpoint, paths generated by a Class 1 algorithm may seem unnecessarily long and perhaps a bit silly. We will see, however, that this same thoroughness brings big bene ts in more dif cult cases. Class 2 algorithms, on the other hand, are more adventurous they are more human , they take risks. When meeting an obstacle, the robot operating under a Class 2 algorithm will have no way of knowing if it has met it before. More often than not, a Class 2 algorithm will win in real-life scenes, though it may lose badly in an unlucky scene. As we will see, the sensor-based motion planning paradigm exploits two essential topological properties of space and objects in it the orientability and continuity of manifolds. These are expressed in topology by the Jordan Curve Theorem [57], which states:
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Any closed curve homeomorphic to a circle drawn around and in the vicinity of a given point on an orientable surface divides the surface into two separate domains, for which the curve is their common boundary.
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The threateningly sounding orientable surface clause is not a real constraint. For our two-dimensional case, the Moebius strip and Klein bottle are the only examples of nonorientable surfaces. Sensor-based planning algorithms would not work on these surfaces. Luckily, the world of real-life robotics never deals with such objects. In physical terms, the Jordan Curve Theorem means the following: (a) If our mobile robot starts walking around an obstacle, it can safely assume that at some moment it will come back to the point where it started. (b) There is no way for the robot, while walking around an obstacle, to nd itself inside the obstacle. (c) If a straight line for example, the robot s intended path from start to target crosses an obstacle, there is a point where the straight line enters the obstacle and a point where it comes out of it. If, because of the obstacle s complex shape, the line crosses it a number of times, there will be an equal number of entering and leaving points. (The special case where the straight line touches the obstacle without crossing it is easy to handle separately the robot can simply ignore the obstacle.) These are corollaries of the Jordan Curve Theorem. They will be very explicitly used in the sensor-based algorithms, and they are the basis of the algorithms convergence. One positive side effect of our reliance on topology is that geometry of space is of little importance. An obstacle can be polygonal or circular, or of a shape that for all practical purposes is impossible to de ne in mathematical terms; for our algorithm it is only a closed curve, and so handling one is as easy as the other. In practice, reliance on space topology helps us tremendously in
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