Mathematical formalism for the fast evaluation of the configuration space

In this paper a mathematical formalism for the configuration space evaluation of robots is presented, that tries to optimise the computational time. The use of the proposed method and the choice of a suitable coordinate system, in the workspace, as well as in the configuration space, lead to the fact that the obstacle representation in the configuration space could be seen as a convolution of two functions that describe the robot and the obstacles respectively. The method has been applied to mobile and articulated robots in the two dimensional plane, but its application to the most popular robots can be easily done. Additionally, the computational load is independent of the shape and number of obstacles and of the robot shape. The mathematical tool that has been used is the FFT (fast Fourier transform) since it eases the parallel implementation of the resulting algorithms reducing significantly the computational load