On robotic trajectory planning using polynomial interpolations

Trajectory planning is a classic topic in robotics. Although cubic splines are widely used to interpolate the trajectory in the literature, they have drawbacks in satisfying more boundary conditions for high dynamic performance, say, the control of acceleration at the end points. On the other hand, if quartic splines are employed, more conditions are needed to determine the coefficients of the fourth-order polynomials for the trajectory. In this paper, we present a method to interpolate the trajectory by combining third-order and fourth-order polynomials. We use two quartic polynomials for the first and last segments of the trajectory, and cubic ones for other segments. The trajectory can be uniquely determined by a set of path points for boundary conditions including accelerations at end points. We also investigate the planning of optimal trajectories in the case of variant intervals between adjacent path points. In the optimization, we use multiple objectives including accelerations and jerks. How to determine succinctly the polynomial coefficients is described. Examples are provided for illustration, and comparisons with previous planning method are reported. Good results of application in humanoid robot motion planning for obstacle stepping-over are obtained