Solving Geodesic Equations with Composite Bernstein Polynomials for Trajectory Planning

2026-02-10

AI Summary

This work addresses the challenge of continuous trajectory planning for autonomous systems in complex environments, where dynamic feasibility, collision avoidance, and computational efficiency must be simultaneously satisfied. The authors propose a symbolic optimization framework based on composite Bernstein polynomials that naturally incorporates obstacle information by solving geodesic equations in a continuous cost field, thereby eliminating the need for discrete representations or post-processing. By integrating Gaussian inequality constraints with geodesic conditions, the method enables fine-grained control over local curvature while ensuring global smoothness. Validated in both two- and three-dimensional cluttered scenarios, the approach efficiently generates dynamically feasible, collision-free trajectories that satisfy prescribed boundary conditions for diverse platforms—including spacecraft—without requiring extensive sampling.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• trajectory planning
• geodesic equations
• obstacle avoidance
• autonomous systems
• cost surface

Innovation

• Composite Bernstein Polynomials
• Geodesic Constraints
• Continuous Cost Field
• Symbolic Optimization
• Trajectory Planning