When the safe path for a robot manipulator is planned in the presence of obstacles, it is necessary to detect the collisions among the manipulator links and the obstacles. In a three-dimensional workspace, it is not easy to detect the interference and to compute the distance between the polyhedral objects, which also needs a large calculation effort. Besides, to plan the safe path in a short time, computations have to be rapid. To settle such problems, several methods based on mathematical programming are proposed. They are applications of linear programming, multiple goal programming, and quadratic programming and are adapted to collision detection, distance calculation, and safe path planning. As all the equations to be solved are linear, the requirements are accomplished with a small calculation effort with the proposed method.
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