Bernstein Polynomial-Based Method for Solving Optimal Trajectory Generation Problems

Abstract

This paper presents a method for the generation of trajectories for autonomous system operations. The proposed method is based on the use of Bernstein polynomial approximations to transcribe infinite dimensional optimization problems into nonlinear programming problems. These, in turn, can be solved using off-the-shelf optimization solvers. The main motivation for this approach is that Bernstein polynomials possess favorable geometric properties and yield computationally efficient algorithms that enable a trajectory planner to efficiently evaluate and enforce constraints along the vehicles’ trajectories, including maximum speed and angular rates as well as minimum distance between trajectories and between the vehicles and obstacles. By virtue of these properties and algorithms, feasibility and safety constraints typically imposed on autonomous vehicle operations can be enforced and guaranteed independently of the order of the polynomials. To support the use of the proposed method we introduce BeBOT (Bernstein/Bézier Optimal Trajectories), an open-source toolbox that implements the operations and algorithms for Bernstein polynomials. We show that BeBOT can be used to efficiently generate feasible and collision-free trajectories for single and multiple vehicles, and can be deployed for real-time safety critical applications in complex environments.

Minimum-Time Trajectory Generation for Wheeled Mobile Systems Using Bézier Curves with Constraints on Velocity, Acceleration and Jerk

“…Often, path planning techniques must also comply with geometric constraints [ 3 , 24 ]. A significant part of path planning methods is the choice of geometric curves, which can be polynomials [ 25 ], Bézier curves [ 6 , 26 , 27 , 28 ], cubic splines [ 29 ], B-splines [ 30 ], Dubins curves, clothoids [ 31 ], hypocycloids [ 32 ], and others, as presented in Ref. [ 23 ].…”

Orthonormal Bernstein Galerkin technique for computations of higher order eigenvalue problems

“…The optimal stable, nonnegative Bernstein polynomial basis gained popularity among several researchers and useful properties have been applied to solve regular as well as singular second to higher order BVPs and eigenvalue problems [ [9] , [10] , [11] , 22 , 23 , [29] , [30] , [31] , [32] , [33] , [34] , [35] , [36] , [37] , [38] ]. As stated in [22] , the polynomials can be used to form a basis over [0, 1] which is complete.…”

Fitness Dependent Optimizer Based Computational Technique for Solving Optimal Control Problems of Nonlinear Dynamical Systems

“…The BPs provide a resourceful approximation technique that is endowed with several essential properties, making it an indispensable method for refining approximated solutions [17], [50], [51].…”