Simplified Robotics Joint-Space Trajectory Generation with a via Point Using a Single Polynomial

This paper presents novel fourth- and sixth-order polynomials to solve the problem of joint-space trajectory generation with a via point. These new polynomials use a single-polynomial function rather than two-polynomial functions matched at the via point as in previous methods. The problem of infinite spikes in jerk is also addressed. 1. Introduction Joint-space trajectory generation is in common usage in robotics to provide smooth, continuous motion from one set of joint angles to another, for instance, for moving between two distinct Cartesian poses for which the inverse pose solution has yielded two distinct sets of joint angles. The joint-space trajectory generation occurs at runtime for all joints independently but simultaneously. There is an entire body of literature devoted to trajectory generation (aka motion planning and path planning) at the joint level. Paul and Zhong [1] were among the first to suggest the use of polynomials for robot trajectory generation. A common joint-space trajectory generation method (linearly changing joint velocity using starting and ending parabolic blends) is identically presented by many authors [2–7]. Despite the widespread popularity of this method, it suffers from infinite spikes in jerk (the derivative of acceleration) and requires three separate functions instead of one. An identical third-order polynomial joint-space trajectory generation approach is also presented by many authors [2, 4–7]. Further, an identical fifth-order polynomial joint-space trajectory generation approach is presented by many authors [2, 4, 5, 7]. From these authors’ lists, it may appear that Koivo was the first to present these methods, when in fact they were already presented in Craig’s first edition in 1986. Fu et al. [8] depart from these standard methods, suggesting initial, intermediate, and final polynomials of order 4-3-4, 3-5-3, or 5 third-order polynomials, for a single joint motion. These are by far the most (unnecessarily) complicated methods and are presented without justification or comparison with simpler methods. For dealing with a via point in which the robot need not stop at the via point (such as for obstacle avoidance), Craig [5] suggests matching two third-order polynomials. Apparently alone amongst all of the major robotics textbook authors, Angeles [9, Section？？6.5] derives a 4-5-6-7 seventh-order polynomial to fit two via points and ensure finite joint jerk at the start and end of motion. Angeles [9, Section？？6.6] also presents approximation of this seventh-order polynomial with a cubic spline and discusses the