Inclusion of ellipsoids

Abstract

Nowadays, path planning for mobile robots has taken a new dimension. Due to many failures when experimenting the following of geometrical paths, determined by the first generation of planners (Latombe, 1991), with real robots, searchers concluded that those too simple paths were no longer enough. Planning method must now guarantee that the paths proposed are safe, i.e. that the robot will be able to follow a path without any risk of failure, or at least in warranting a high success rate. To achieve this goal, some parameters must be considered : uncertainties of the model used (not-so-perfect mapping, inaccuracy of the sensors, slipping of the robot on the floor, etc.). Collision detection is very important in mobile robotic and furthermore when trying to find a safe path in an uncertain-configuration space. Thus, searchers tend to integrate evolved collision detection in their planners. Thus, after having used circular disk to approximate the shape taken by the mobile robot, more and more searchers use elliptic disks as they offer a better accuracy. Used in the context of safepath planning (Pierre and Lambert, 2006; Lozano-Pérez and Wesley, 1979; Gonzalez and Stentz, 2005; Pepy and Lambert, 2006), ellipsoids allow to approximate the shape of the set of positions where the mobile could be(ellipsoids thus take the mobile robot’s geometry and the uncertainties on its position into account). The the Safe A* with Towers of Uncertainty (SATU*) planner (Pierre and Lambert, 2006) is one of those safe path planner that use ellipsoid to approximate the shape of the set of positions where the mobile could be. Ellipsoids are used in the SATU* to perform collision detection between the mobile robot and its environment. However, the authors of the SATU* have also proposed a new mean of organising the performing of the planner so that the ellipsoids can be used to detect very early beginning of useless paths. In order to achieve this goal, inclusion detection must be performed between two ellipsoids (that correspond to two different ways to come to the same position). In this paper, we are going to present an algebraic method using the resultant of Sylvester (Lang, 1984) to solve this problem. The SATU* algorithm (Alg. 1) has already been presented in (Pierre and Lambert, 2006) and three tests of inclusion of uncertainties (lines 7, 26 and 36) are used. However the authors did not explain how they implemented those tests nor give the algorithms used. As the model of uncertainties used in the SATU* corresponds to an ellipsoid in 3 dimensions, this test of inclusion of uncertainties can be seen as a test of inclusion of ellipsoids. In the present paper, we are going to propose an algorithm of test of inclusion of ellipsoids. In a first part, the uncertain configuration space will be described. Then, Sylvester’s resultant will be used to defined a ready for use inclusion detection test.