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happens to be coming in real time from robot sensors, and thus there is always uncertainty about the robot s surroundings, one is forced to turn to the second approach, SIM. In other words, as a rule, only one approach applies to a given task. Consider, for example, a maze-searching task (called also a mouse-in-the-labyrinth problem). One starts at some starting point S inside the labyrinth and attempts to reach some target point T , also in the labyrinth.2 Imagine we have in our possession complete information about the labyrinth. We can feed these data into the computer, produce the bird s-eye view of the maze, and study the problem in great detail using this map. We can investigate different paths between points S and T , gure out the optimal (shortest) path, and so on. This is planning with complete information, and the Piano Mover s model should be the preferred approach. On the other hand, if all of a sudden you nd yourself in a maze, at any given moment you would see only the surrounding walls of the maze and perhaps remember a few corridors that you have just passed. You do not know what is ahead; input information is scant; what you learn comes from your sensors. Any movement, including the unfortunate deviations into dead end corridor appendices, becomes a part of the path. Doing anything approaching an optimal path is of course out of the question. Here you deal with incomplete information and produce the path as you go. This is planning with incomplete information, and so you need to turn to the SIM techniques. Since robot motion planning is the topic of this book, in Sections 2.8 and 2.9 we will further explore differences between these two paradigms for motion planning, the Piano Mover s model and the Sensing Intelligence Motion model.
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Provable Versus Heuristic Algorithms. Another important distinction between algorithms is between provable (other terms: nonheuristic, exact, algorithmic) and heuristic approaches. A provable motion planning algorithm is one for which there is a guarantee that if a path between the starting and target points exist, the algorithm will nd one in nite time and without an exhaustive search or else will conclude in nite time that there is no path if such is the case. We then say that the algorithm converges. To obtain such a guarantee, people go through the trouble of proving the algorithm convergence. An algorithm itself should allow such a proof; for example, the so-called common sense strategies we call them heuristic algorithms do not allow a proof of convergence and are not likely to be convergent. Whereas for some applications, having a guarantee of convergence may be a moot point as, for example, when the user s knowledge or intuition pretty much replaces it for more complex cases, seeking convergence re ects more than a love for academic purity. As we will see in 7, in complex problems most motion planning problems with robot arm manipulators t this
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In other variations of this problem, one starts inside the maze and tries to nd an exit from it; or, one starts outside the maze and tries to reach the location with a hidden treasure somewhere inside the labyrinth.
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category human intuition is not a good advisor. If, while operating under some reasonably sounding algorithm with unproven convergence, the robot fails to nd a path, the failure may simply mean that feasible paths do exist but the algorithm has missed them. A guarantee of convergence then becomes a very practical issue.
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2.8 MOTION PLANNING WITH COMPLETE INFORMATION In this type of motion planning, input information is processed before the actual motion starts. This means that the input information must exist beforehand. The model with complete information is formulated as follows.3 Given a solid object (robot), or a combination of solid objects, in two- or three-dimensional space, whose size, shape, and initial and target position and orientation are fully described, and given a set of obstacles whose shapes, positions, and orientations in space are likewise known, the task is to nd a continuous path for the object from the initial to the target position while avoiding collisions with obstacles along the way. An important assumption used in the model is that the surfaces of the moving object and the obstacles are algebraic or semialgebraic. This guarantees a nal description of the input data. In some works a stricter requirement of planar surfaces is imposed. Because complete information about the problem is assumed, the whole operation of path planning is a one-time, off-line operation. The main dif culty is not in proving existence of algorithms that would guarantee a solution (they obviously exist), but in assessing the problem complexity and obtaining a computationally ef cient procedure. Reaching a solution means either nding a path or concluding in nite time that no path exists. Since a solution is always feasible, cases of arbitrary complexity can in principle be considered. Another apparent advantage of dealing with complete information is that various optimization criteria nding the shortest path, or the minimum-time path, or the safest path, and so on can be introduced easily. Historically, Piano Mover s approach strategies were the rst to come, starting in late 1960s. Most of the people who formulated the problem of robot collision avoidance were computer scientists. For them, collision avoidance was a purely computational problem, and the question of handling input information boiled down to a search in the database that contained that information. They often perceived sensing, partial information, uncertainty, control, and all such issues as small conceptual bumps that only interfered with the beautiful computational problem in hand. By the late 1980s, the area became one of the richest and popular areas in computational geometry. Hundreds of planning algorithms with complete information have been published; the problem s computational complexity has been studied in depth, and ingenuous ways of dealing with it were reported [11].
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3 A good survey of the work on provable algorithms for the Piano Mover s problem can be found in Ref. 11; specialized maze search algorithms are considered in Ref. 12.
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