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Mixed Bernstein-Fourier Approximants for Optimal Trajectory Generation with Periodic Behavior

Liraz Mudrik, Sean Kragelund, Isaac Kaminer

TL;DR

This work develops a mixed Bernstein-Fourier framework for optimal trajectory generation with periodic dynamics, aligning the optimization grid to uniform onboard sampling while capturing both nonperiodic and periodic components. It establishes uniform convergence, error bounds, and a regulated LS for coefficient determination, then proves feasibility and consistency of the approximated NLP and extends the covector mapping theorem to the mixed basis. The approach is validated through numerical examples including disturbance rejection, observer-trajectory optimization, and autonomous mine countermeasure search, showing substantial gains in accuracy and computational efficiency over Bernstein-only and pseudospectral methods. The results demonstrate robust dual-variable approximation and practical applicability to periodic robotic tasks with strict sampling constraints. This methodology offers a theoretically grounded, computationally efficient pathway for advanced autonomous trajectory planning under periodic dynamics.

Abstract

Efficient trajectory generation is crucial for autonomous systems; however, current numerical methods often struggle to handle periodic behaviors effectively, particularly when the onboard sensors require equidistant temporal sampling. This paper introduces a novel mixed Bernstein-Fourier approximation framework tailored explicitly for optimal motion planning. Our proposed methodology leverages the uniform convergence properties of Bernstein polynomials for nonperiodic behaviors while effectively capturing periodic dynamics through the Fourier series. Theoretical results are established, including uniform convergence proofs for approximations of functions, derivatives, and integrals, as well as detailed error bound analyses. We further introduce a regulated least squares approach for determining approximation coefficients, enhancing numerical stability and practical applicability. Within an optimal control context, we establish the feasibility and consistency of approximated solutions to their continuous counterparts. We also extend the covector mapping theorem, providing theoretical guarantees for approximating dual variables crucial in verifying the necessary optimality conditions from Pontryagin's Maximum Principle. Numerical examples illustrate the method's superior performance, demonstrating substantial improvements in computational efficiency and precision in scenarios with complex periodic constraints and dynamics. Our mixed Bernstein-Fourier methodology thus presents a robust, theoretically grounded, and computationally efficient approach for advanced optimal trajectory planning in autonomous systems.

Mixed Bernstein-Fourier Approximants for Optimal Trajectory Generation with Periodic Behavior

TL;DR

This work develops a mixed Bernstein-Fourier framework for optimal trajectory generation with periodic dynamics, aligning the optimization grid to uniform onboard sampling while capturing both nonperiodic and periodic components. It establishes uniform convergence, error bounds, and a regulated LS for coefficient determination, then proves feasibility and consistency of the approximated NLP and extends the covector mapping theorem to the mixed basis. The approach is validated through numerical examples including disturbance rejection, observer-trajectory optimization, and autonomous mine countermeasure search, showing substantial gains in accuracy and computational efficiency over Bernstein-only and pseudospectral methods. The results demonstrate robust dual-variable approximation and practical applicability to periodic robotic tasks with strict sampling constraints. This methodology offers a theoretically grounded, computationally efficient pathway for advanced autonomous trajectory planning under periodic dynamics.

Abstract

Efficient trajectory generation is crucial for autonomous systems; however, current numerical methods often struggle to handle periodic behaviors effectively, particularly when the onboard sensors require equidistant temporal sampling. This paper introduces a novel mixed Bernstein-Fourier approximation framework tailored explicitly for optimal motion planning. Our proposed methodology leverages the uniform convergence properties of Bernstein polynomials for nonperiodic behaviors while effectively capturing periodic dynamics through the Fourier series. Theoretical results are established, including uniform convergence proofs for approximations of functions, derivatives, and integrals, as well as detailed error bound analyses. We further introduce a regulated least squares approach for determining approximation coefficients, enhancing numerical stability and practical applicability. Within an optimal control context, we establish the feasibility and consistency of approximated solutions to their continuous counterparts. We also extend the covector mapping theorem, providing theoretical guarantees for approximating dual variables crucial in verifying the necessary optimality conditions from Pontryagin's Maximum Principle. Numerical examples illustrate the method's superior performance, demonstrating substantial improvements in computational efficiency and precision in scenarios with complex periodic constraints and dynamics. Our mixed Bernstein-Fourier methodology thus presents a robust, theoretically grounded, and computationally efficient approach for advanced optimal trajectory planning in autonomous systems.

Paper Structure

This paper contains 31 sections, 11 theorems, 115 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical Background
  3. Bernstein Approximation
  4. Fourier Series
  5. Mixed Bernstein-Fourier Basis for Approximation Theory
  6. Foundations of the Mixed Bernstein-Fourier Basis
  7. Mathematical Properties and Convergence
  8. Approximation Using Regulated Least Squares
  9. Optimal Motion Planning Problem
  10. Problem Formulation
  11. Analysis
  12. Necessary Conditions for Optimal Motion Planning
  13. Problem Formulation
  14. Applying PMP to Problem P
  15. KKT Conditions of Problem Pn
  16. ...and 16 more sections

Key Result

Lemma II.1

Let $\textup{f} \in \mathcal{C}^{0}_{m_{\textup{f}}}$ on $[0,t_{f}]$. Its Bernstein approximation, defined in Eq. eq:br_app, satisfies for any $n \in \mathbb{N}$and all $t \in [0, t_f]$: where $C_{0}$ is a positive constant independent of $n$ and $W_{\textup{f}}(\cdot)$ is the modulus of continuity of $\textup{f}$ in $[0,t_{f}]$. If $\textup{f} \in \mathcal{C}^{1}_{m_{\textup{f}}}$ on $[0,t_{f}]$,

Figures (10)

  • Figure 1: Approximation of a periodic function using mixed Bernstein-Fourier (B-F) versus Bernstein-only approximants.
  • Figure 2: The data to be approximated.
  • Figure 3: Mean RMS error comparison for different approximation bases.
  • Figure 4: Schematic of the Covector Mapping Theorem for mixed Bernstein-Fourier approximants.
  • Figure 5: Primal solution comparison. The Mixed basis ($N=4$) achieves smooth control reconstruction with minimal coefficients.
  • ...and 5 more figures

Theorems & Definitions (25)

  • Lemma II.1: Uniform Convergence, see floater_convergence_2005
  • Lemma II.2: Uniform Convergence of the Integral, see cichella_optimal_2021
  • Lemma II.3
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem III.1
  • proof
  • Example 1
  • Corollary III.1.1
  • ...and 15 more