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Direct Shooting Method for Numerical Optimal Control: A Modified Transcription Approach

Jiawei Tang, Yuxing Zhong, Pengyu Wang, Xingzhou Chen, Shuang Wu, Ling Shi

TL;DR

This work tackles numerical optimal control for high-order systems using direct shooting, highlighting a contradictory dynamics issue when applying standard first-order transcription with augmented states. It introduces modified direct shooting formulations based on second-order Euler and RK4 schemes that preserve the intrinsic relationship between a state and its derivatives, and extends these ideas to general high-order dynamics with convergence guarantees. The authors prove global truncation error bounds for the proposed methods and demonstrate through benchmark problems that the modified approaches yield significantly more accurate solutions, while maintaining competitive runtimes. The practical impact lies in enabling more reliable and precise direct shooting for complex robotic and mechanical systems, with potential integration into DDP-based frameworks for further performance gains.

Abstract

Direct shooting is an efficient method to solve numerical optimal control. It utilizes the Runge-Kutta scheme to discretize a continuous-time optimal control problem making the problem solvable by nonlinear programming solvers. However, conventional direct shooting raises a contradictory dynamics issue when using an augmented state to handle {high-order} systems. This paper fills the research gap by considering the direct shooting method for {high-order} systems. We derive the modified Euler and Runge-Kutta-4 methods to transcribe the system dynamics constraint directly. Additionally, we provide the global error upper bounds of our proposed methods. A set of benchmark optimal control problems shows that our methods provide more accurate solutions than existing approaches.

Direct Shooting Method for Numerical Optimal Control: A Modified Transcription Approach

TL;DR

This work tackles numerical optimal control for high-order systems using direct shooting, highlighting a contradictory dynamics issue when applying standard first-order transcription with augmented states. It introduces modified direct shooting formulations based on second-order Euler and RK4 schemes that preserve the intrinsic relationship between a state and its derivatives, and extends these ideas to general high-order dynamics with convergence guarantees. The authors prove global truncation error bounds for the proposed methods and demonstrate through benchmark problems that the modified approaches yield significantly more accurate solutions, while maintaining competitive runtimes. The practical impact lies in enabling more reliable and precise direct shooting for complex robotic and mechanical systems, with potential integration into DDP-based frameworks for further performance gains.

Abstract

Direct shooting is an efficient method to solve numerical optimal control. It utilizes the Runge-Kutta scheme to discretize a continuous-time optimal control problem making the problem solvable by nonlinear programming solvers. However, conventional direct shooting raises a contradictory dynamics issue when using an augmented state to handle {high-order} systems. This paper fills the research gap by considering the direct shooting method for {high-order} systems. We derive the modified Euler and Runge-Kutta-4 methods to transcribe the system dynamics constraint directly. Additionally, we provide the global error upper bounds of our proposed methods. A set of benchmark optimal control problems shows that our methods provide more accurate solutions than existing approaches.
Paper Structure (17 sections, 4 theorems, 31 equations, 3 figures, 2 tables)

This paper contains 17 sections, 4 theorems, 31 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Numerical Optimal Control
  3. Nonlinear Optimal Control
  4. First-order Direct Shooting
  5. Downside of First-order Direct Shooting Method
  6. Modified Direct Shooting
  7. Second-order Euler Method
  8. Second-order RK4 Method
  9. Extension to the High-order System
  10. Convergence Analysis
  11. Numerical Experiments
  12. Problem Descriptions
  13. Performance Metrics
  14. Accuracy
  15. Timing
  16. ...and 2 more sections

Key Result

Proposition 1

Under the first-stage Runge-Kutta scheme, the second-order differential equation constraint (2ndDyn) is equivalent to the following equality constraints, i.e., $\forall k \in [0, N)$, Proof: For $t_k \leq t \leq t_{k+1}$, the Euler method assumes that $\dot{q}(t)$ is approximated by the first-order Taylor polynomial around knot point $k$. Hence, we have Through writing $q(t)$ in the integral for

Figures (3)

  • Figure 1: Schematics, configuration, and input vectors of the benchmark system dynamics.
  • Figure 2: The timing performance comparison of different methods in the cartpole swing-up problem in relation to the number of time intervals $N$.
  • Figure 3: The total transcription error comparison of different methods in the cartpole swing-up problem in relation to the number of time intervals $N$.

Theorems & Definitions (7)

  • Example 1
  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2