A 3-D collision-free trajectory planning method for a mobile robot is proposed in the paper. The geometric shapes of the objects in the workspace are modeled as ellipsoids of 3-D quadric model for simple mathematical representation and easy geometric approximation. By a series of coordinate transformations between the mobile robot and obstacles, the collision detection problem in trajectory planning is reduced to test a point falling outside or inside the transformed ellipsoids, which models obstacles geometrically. Finally, the collision probability, which is defined by projecting the quadric ellipsoid onto a 3-D Gaussian distribution contour, plays a very significant role in search the optimal path through the defined objective function.

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