Trajectory Planning for Point-to-Point Motion Using High-order Polynomials

In this paper, a trajectory planning problem based on high-order polynomials is formulated for a point-to-point motion. The problem aims to find suitable polynomial trajectories that connect an initial to a final configuration while satisfying other specified constraints. The constraints are considered as zero velocity at the endpoint as well as a limitation on acceleration for the whole motion time. However, this problem is very difficult to trace more particularly when the number of coefficients (decision-making variables) is large. As a new approach, a high-order polynomial equation containing only two-term is proposed to generate suitable trajectories between two configurations. The advantage of the proposed polynomial is that it can be traced analytically in order to get solutions for the two independent coefficients in a closed-form. The motion simulations show that the resulting high-degree trajectories with two-term polynomial satisfy the mentioned constraints as well as they are continuous and smooth. Additionally, comparing outputs of Genetic Algorithm with the closed-form solutions for the problem show that closed-form expressions generate coefficients that are near optimal.