Approximate joint trajectories for control of industrial robots along Cartesian paths

To approximate a preplanned robot path in Cartesian coordinates, such as the seam in arc welding, by functions of joint displacements, enough points on the path must be selected and transformed into joint coordinates. Instead of joining the adjacent, transformed joint points by a straight line, lower degree polynomials are first determined and then splined together. The approximation functions are required to have continuous second derivatives with specified boundary conditions so that the robot's position, orientation, velocity, and acceleration are all continuous along the path. To reduce the approximation error, cubic and quartic polynomials are used with least-squares fit. A comparison of the approximation errors of the methods is presented with an illustration by a numerical example.