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A High-Speed Time-Optimal Trajectory Generation Strategy via a Two-layer Planning Model

Haotian Tan, Yuan-Hua Ni

TL;DR

This work tackles time-optimal trajectory generation for autonomous systems in environments with obstacles by introducing a two-layer planning framework that operates on short, fixed-horizon planning cycles solved via convex subproblems. It decomposes the original nonconvex problem into a relaxed convex problem $\mathcal{P}_r$ and a strict variant $\mathcal{P}_s$, enabling incremental construction of a global solution through trajectory concatenation, with dynamics linearized around the current state. The authors integrate two trajectory-search methods, CVAPF and CDWA, to generate nominal segments and convex feasible regions, supported by theoretical guarantees that ensure contraction of the auxiliary cost and preservation of local optimality upon concatenation, plus conditions for near-global optimality in affine cases. Numerical experiments in static and dynamic maps demonstrate higher robustness and significantly faster performance compared with standard SCP approaches, validating the practical impact for real-time motion planning in uncertain and time-varying environments. The framework thus offers a scalable, reliable path to near time-optimal control under nonconvex constraints, with potential extensions to broader nonlinear vehicle models and richer obstacle representations.

Abstract

MPC (Model predictive control)-based motion planning and trajectory generation are essential in applications such as unmanned aerial vehicles, robotic manipulators, and rocket control. However, the real-time implementation of such optimization-based planning faces significant challenges arising from non-convex problem structures and inherent limitations of nonlinear programming -- notably the difficulty in guaranteeing solution quality and the unpredictability of computation time. To improve robustness and computational efficiency, this paper introduces a two-layer motion planning algorithm for intelligent ground vehicles based on convex optimization. The proposed algorithm iteratively constructs discrete optimal control subproblems with small, fixed terminal times, referred to as planning cycles. Each planning cycle is further solved within progressively constructed convex sets generated by utilizing customized search algorithms. The entire solution to the original problem is obtained by incrementally composing the solutions of these subproblems. The proposed algorithm demonstrates enhanced reliability and significantly reduced computation time. We support our approach with theoretical analysis under practical assumptions and numerical experiments. Comparative results with standard sequential convex programming (SCP) methods demonstrate the superiority of our method -- include a significant improved computational speed under dynamic environments while maintain a near optimal final time.

A High-Speed Time-Optimal Trajectory Generation Strategy via a Two-layer Planning Model

TL;DR

This work tackles time-optimal trajectory generation for autonomous systems in environments with obstacles by introducing a two-layer planning framework that operates on short, fixed-horizon planning cycles solved via convex subproblems. It decomposes the original nonconvex problem into a relaxed convex problem and a strict variant , enabling incremental construction of a global solution through trajectory concatenation, with dynamics linearized around the current state. The authors integrate two trajectory-search methods, CVAPF and CDWA, to generate nominal segments and convex feasible regions, supported by theoretical guarantees that ensure contraction of the auxiliary cost and preservation of local optimality upon concatenation, plus conditions for near-global optimality in affine cases. Numerical experiments in static and dynamic maps demonstrate higher robustness and significantly faster performance compared with standard SCP approaches, validating the practical impact for real-time motion planning in uncertain and time-varying environments. The framework thus offers a scalable, reliable path to near time-optimal control under nonconvex constraints, with potential extensions to broader nonlinear vehicle models and richer obstacle representations.

Abstract

MPC (Model predictive control)-based motion planning and trajectory generation are essential in applications such as unmanned aerial vehicles, robotic manipulators, and rocket control. However, the real-time implementation of such optimization-based planning faces significant challenges arising from non-convex problem structures and inherent limitations of nonlinear programming -- notably the difficulty in guaranteeing solution quality and the unpredictability of computation time. To improve robustness and computational efficiency, this paper introduces a two-layer motion planning algorithm for intelligent ground vehicles based on convex optimization. The proposed algorithm iteratively constructs discrete optimal control subproblems with small, fixed terminal times, referred to as planning cycles. Each planning cycle is further solved within progressively constructed convex sets generated by utilizing customized search algorithms. The entire solution to the original problem is obtained by incrementally composing the solutions of these subproblems. The proposed algorithm demonstrates enhanced reliability and significantly reduced computation time. We support our approach with theoretical analysis under practical assumptions and numerical experiments. Comparative results with standard sequential convex programming (SCP) methods demonstrate the superiority of our method -- include a significant improved computational speed under dynamic environments while maintain a near optimal final time.

Paper Structure

This paper contains 19 sections, 7 theorems, 88 equations, 9 figures, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Classic Framework
  4. Our Proposed Framework
  5. Algorithm and Analysis
  6. Reformulation of Planning Cycles
  7. Entire Algorithm
  8. Customized Vortex Artificial Potential Field
  9. Customized Dynamic Window Approach
  10. Theoretical Guarantees
  11. Contribution Summary
  12. Numerical Experiments
  13. Experiments with Static Maps
  14. Experiments with Dynamic Maps
  15. Conclusion
  16. ...and 4 more sections

Key Result

Theorem 1

For given $\gamma$, $z^{(K+1)}_{N+1}$ can be obtained with $\| z^{(K+1)}_{N+1} - z_f \|_{Q_0} \leq \gamma$ after executing finite planning cycles.

Figures (9)

  • Figure 1: A single irregular obstacle (top left) can be approximated by a closed ball (top right), while multiple densely clustered small obstacles (bottom left) may be merged into a larger enclosing ball (pink circle, bottom right) for simplified representation.
  • Figure 2: A Visualization of obtaining $\mathcal{Z}_i$
  • Figure 3: Processes solving $\mathcal{P}_0$ in the proposed framework
  • Figure 4: The solid red line and the dashed blue line represent two possible paths on a map. However, sharp turns in the red path typically signal that the vehicle is slowing down at those points, leading to longer travel times and lower solution quality.
  • Figure 5: Trajectories generated by 2 distinct algorithm
  • ...and 4 more figures

Theorems & Definitions (13)

  • Remark 1
  • Definition 1
  • Theorem 1
  • Remark 2
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • Remark 3
  • Definition 2
  • Theorem 4
  • ...and 3 more