An Analytic Closed-form Solution for Trajectory Generation on a Path along an Arc of a Circle

A polynomial trajectory is a time-traveled distance function used to describe trajectory of the robot. Optimal high-degree polynomial trajectories considering initial and the final velocity conditions besides the acceleration constraints are desired. In this paper, a trajectory optimization problem aiming travel maximum distance for a robot that follows an arc based path is formulated. Along the path, the robot requires observing initial and final zero velocity conditions as well as certain acceleration limits. A high-degree polynomial equation along the trajectory is proposed inside of the optimization problem. The closed-form solution of the problem had been obtained analytically. The solution includes the coefficients of the any high-degree trajectory polynomial equation where the coefficients are obtained in closed-form. Simulations several experiments show that the resulting high-degree trajectories satisfy the initial and final zero velocity conditions as well as acceleration constraint.