Solving Geodesic Equations with Composite Bernstein Polynomials for Trajectory Planning

TL;DR

This work introduces a geodesic-based trajectory planning framework that uses a continuous Gaussian cost surface to encode obstacles and solves for smooth, feasible paths with composite Bernstein polynomials in a symbolic optimization setting. By discretizing the direct optimal control problem via an integration-based, composite Bernstein approach and enforcing geodesic constraints on the cost surface, the method yields efficient, differentiable trajectories in 2D and 3D. Numerical results demonstrate reliable obstacle avoidance and the potential of geodesic-like trajectories to serve as fast warmstarts for more complex motion-planning problems, with substantial runtime gains at higher discretization orders. The approach is dimension-agnostic, supports real-time-inspired computation, and lays groundwork for extensions to dynamic environments and alternative cost surfaces, enabling scalable planning for aerial, ground, underwater, and space systems.

Abstract

This work presents a trajectory planning method based on composite Bernstein polynomials for autonomous systems navigating complex environments. The method is implemented in a symbolic optimization framework that enables continuous paths and precise control over trajectory shape. Trajectories are planned over a cost surface that encodes obstacles as continuous fields rather than discrete boundaries. Regions near obstacles are assigned higher costs, naturally encouraging the trajectory to maintain a safe distance while still allowing efficient routing through constrained spaces. The use of composite Bernstein polynomials preserves continuity while enabling fine control over local curvature to satisfy geodesic constraints. The symbolic representation supports exact derivatives, improving optimization efficiency. The method applies to both two- and three-dimensional environments and is suitable for ground, aerial, underwater, and space systems. In spacecraft trajectory planning, for example, it enables the generation of continuous, dynamically feasible trajectories with high numerical efficiency, making it well suited for orbital maneuvers, rendezvous and proximity operations, cluttered gravitational environments, and planetary exploration missions with limited onboard computational resources. Demonstrations show that the approach efficiently generates smooth, collision-free paths in scenarios with multiple obstacles, maintaining clearance without extensive sampling or post-processing. The optimization incorporates three constraint types: (1) a Gaussian surface inequality enforcing minimum obstacle clearance; (2) geodesic equations guiding the path along locally efficient directions on the cost surface; and (3) boundary constraints enforcing fixed start and end conditions. The method can serve as a standalone planner or as an initializer for more complex motion planning problems.

Figures (5)

• Figure 1: Geodesic vs. Non-Geodesic.
• Figure 2: Two Valid Geodesics.
• Figure 3: An example geodesic-constrained trajectory through a 12-obstacle environment.
• Figure 4: An example geodesic-constrained trajectory shown on top of the associated obstacles' cost surface.
• Figure 5: An example trajectory shown avoiding 15 objects.