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Accelerated Spline-Based Time-Optimal Motion Planning with Continuous Safety Guarantees for Non-Differentially Flat Systems

Dries Dirckx, Jan Swevers, Wilm Decré

Abstract

Generating time-optimal, collision-free trajectories for autonomous mobile robots involves a fundamental trade-off between guaranteeing safety and managing computational complexity. State-of-the-art approaches formulate spline-based motion planning as a single Optimal Control Problem (OCP) but often suffer from high computational cost because they include separating hyperplane parameters as decision variables to enforce continuous collision avoidance. This paper presents a novel method that alleviates this bottleneck by decoupling the determination of separating hyperplanes from the OCP. By treating the separation theorem as an independent classification problem solvable via a linear system or quadratic program, the proposed method eliminates hyperplane parameters from the optimisation variables, effectively transforming non-convex constraints into linear ones. Experimental validation demonstrates that this decoupled approach reduces trajectory computation times up to almost 60% compared to fully coupled methods in obstacle-rich environments, while maintaining rigorous continuous safety guarantees.

Accelerated Spline-Based Time-Optimal Motion Planning with Continuous Safety Guarantees for Non-Differentially Flat Systems

Abstract

Generating time-optimal, collision-free trajectories for autonomous mobile robots involves a fundamental trade-off between guaranteeing safety and managing computational complexity. State-of-the-art approaches formulate spline-based motion planning as a single Optimal Control Problem (OCP) but often suffer from high computational cost because they include separating hyperplane parameters as decision variables to enforce continuous collision avoidance. This paper presents a novel method that alleviates this bottleneck by decoupling the determination of separating hyperplanes from the OCP. By treating the separation theorem as an independent classification problem solvable via a linear system or quadratic program, the proposed method eliminates hyperplane parameters from the optimisation variables, effectively transforming non-convex constraints into linear ones. Experimental validation demonstrates that this decoupled approach reduces trajectory computation times up to almost 60% compared to fully coupled methods in obstacle-rich environments, while maintaining rigorous continuous safety guarantees.
Paper Structure (13 sections, 9 equations, 2 figures, 3 tables, 2 algorithms)

This paper contains 13 sections, 9 equations, 2 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction: Autonomous Motion Planning
  2. Related Work
  3. Spline Parameterisations
  4. Contributions
  5. Spline-based Motion Planning
  6. Collision Avoidance between Polytopes
  7. Optimal Control for Time-Optimal Motion Planning
  8. Bernstein Parameterisation
  9. Accelerated Continuous Collision Avoidance
  10. Least-Squares Support Vector Machines
  11. Two-step Filter Procedure
  12. Experimental Validation
  13. Conclusion and Future Work

Figures (2)

  • Figure 1: Illustration of a validation environment with ten obstacles, a collision-free trajectory through it and the variation in start and end position to generate the full experimental data set.
  • Figure 2: Results of the comparison between the coupled and decoupled approach in terms of (top) computation time per iteration and (bottom) total computation time to solve Eq. \ref{['eq:ocp_motplan']} per $N_{\text{o}}$.