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Solving Geodesic Equations With Composite Bernstein Polynomials for Trajectory Planning
Conference proceeding

Solving Geodesic Equations With Composite Bernstein Polynomials for Trajectory Planning

Nick Gorman, Gage MacLin, Maxwell Hammond and Venanzio Cichella
Proceedings - IEEE Aerospace Conference, pp.1-11
03/07/2026
DOI: 10.1109/AERO66936.2026.11519773

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Abstract

This work presents a trajectory planning method based on composite Bernstein polynomials, designed 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. Areas near obstacles are assigned higher costs, naturally encouraging the trajectory to maintain a safe distance while still allowing for efficient routing through constrained spaces. The use of composite Bernstein polynomials allows the trajectory to maintain continuity while enabling fine control over local curvature to meet geodesic constraints. The symbolic representation supports exact derivatives, improving the efficiency of the optimization process. The method is applicable to both two- and three-dimensional environments and is suitable for ground, aerial, underwater, and space system applications. In the context of 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, operations in cluttered gravitational environments, and planetary exploration missions where on-board computational resources are limited. Demonstrations show that the approach can efficiently generate smooth, collision-free paths in scenarios with multiple obstacles, maintaining clearance without requiring extensive sampling or post-processing. The optimization is subject to several constraints: (1) a Gaussian surface implemented as an inequality constraint that ensures minimum clearance from obstacles; (2) geodesic equations are used such that the path follows the most efficient direction relative to the cost surface; (3) boundary constraints to enforce fixed start and end conditions. This approach can serve as a standalone planner or as an initializer for more complex motion planning problems.
 Optimization Polynomials Costing Costs Equations Printing Surfaces Tagging Timing Trajectory

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