Functions in Polar Coordinates to Determine the Kinematic Manipulability in Serial Robots

Let nDoF denote a mechanism of n degrees of freedom based on point o_1. The problem considered is: assume that in a neighborhood of point p_0 the vectorial description of the movement of the End-Effector(Ef) is given by the function u. Once the system arrives at u(p_0 ), in which direction is the magnitude(the norm) of the velocity vector maximal (or minimal)?. For 2DoF the answer is given by the velocity ellipses at every point. The real problem turns out to be practical calculation of the ellipses. Until now only partial answer has been given via the eigenvalues of A, the matrix of the system and a function of the Jacobian of uat p_0, because the lengths of the semiaxes are functions of them. The problem of finding the sought direction in the workspace has not been practically considered. In this paper a complete procedure is given to calculate the ellipses at every point in that workspace. First it is shown how the introduction of polar coordinates for the eigenvalues calculations greatly simplifies them. The main contribution is a practical procedure, based on geometrical considerations, to calculate the angle of the major semiaxis with the coordinate axis and thus the orientation (or equivalently, the eigenvectors). It is shown that there is no need to recalculated eigenvalues and eigenvectors at each point. It is enough to calculate them at a given set of points and then interpolate to know the corresponding values. The use of the condition number of A gives important insights in the situation. A complete map of ellipses is given for 2DoF.