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Harmonic Control Lyapunov Barrier Functions for Constrained Optimal Control with Reach-Avoid Specifications

Amartya Mukherjee, Ruikun Zhou, Haocheng Chang, Jun Liu

TL;DR

This work addresses safety-critical constrained control by unifying reach-avoid objectives with harmonic analysis through harmonic CLBFs. By solving a boundary-value problem with $\nabla^2 V = 0$ and appropriate boundary data, the authors obtain a unique harmonic CLBF whose gradient guides a gradient-based controller, optionally augmented with small noise to escape saddle points. They also discuss a Poisson variant with $\nabla^2 V = -6$ and provide a sufficient condition involving a positive $\lambda$ to guarantee avoidance of unsafe sets. Numerical experiments across Roomba, DiffDrive, CarRobot, and a 2D quadrotor demonstrate high safety and reliable goal-reaching, often outperforming baselines and avoiding the need for trajectory-based training.

Abstract

This paper introduces harmonic control Lyapunov barrier functions (harmonic CLBF) that aid in constrained control problems such as reach-avoid problems. Harmonic CLBFs exploit the maximum principle that harmonic functions satisfy to encode the properties of control Lyapunov barrier functions (CLBFs). As a result, they can be initiated at the start of an experiment rather than trained based on sample trajectories. The control inputs are selected to maximize the inner product of the system dynamics with the steepest descent direction of the harmonic CLBF. Numerical results are presented with four different systems under different reach-avoid environments. Harmonic CLBFs show a significantly low risk of entering unsafe regions and a high probability of entering the goal region.

Harmonic Control Lyapunov Barrier Functions for Constrained Optimal Control with Reach-Avoid Specifications

TL;DR

This work addresses safety-critical constrained control by unifying reach-avoid objectives with harmonic analysis through harmonic CLBFs. By solving a boundary-value problem with and appropriate boundary data, the authors obtain a unique harmonic CLBF whose gradient guides a gradient-based controller, optionally augmented with small noise to escape saddle points. They also discuss a Poisson variant with and provide a sufficient condition involving a positive to guarantee avoidance of unsafe sets. Numerical experiments across Roomba, DiffDrive, CarRobot, and a 2D quadrotor demonstrate high safety and reliable goal-reaching, often outperforming baselines and avoiding the need for trajectory-based training.

Abstract

This paper introduces harmonic control Lyapunov barrier functions (harmonic CLBF) that aid in constrained control problems such as reach-avoid problems. Harmonic CLBFs exploit the maximum principle that harmonic functions satisfy to encode the properties of control Lyapunov barrier functions (CLBFs). As a result, they can be initiated at the start of an experiment rather than trained based on sample trajectories. The control inputs are selected to maximize the inner product of the system dynamics with the steepest descent direction of the harmonic CLBF. Numerical results are presented with four different systems under different reach-avoid environments. Harmonic CLBFs show a significantly low risk of entering unsafe regions and a high probability of entering the goal region.
Paper Structure (16 sections, 6 theorems, 35 equations, 3 figures, 2 tables)

This paper contains 16 sections, 6 theorems, 35 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Control Lyapunov Barrier Functions
  4. Harmonic Functions
  5. Harmonic CLBF
  6. Avoiding the saddle points
  7. Conditions on the noise
  8. Numerial Results
  9. Problem I
  10. Problem II
  11. 2D Quadrotor
  12. Conclusions
  13. Roomba
  14. DiffDrive
  15. CarRobot
  16. ...and 1 more sections

Key Result

Theorem 1

If $u$ is harmonic on a domain $\mathcal{S}$, then $u$ satisfies the mean value property in $\mathcal{S}$. Furthermore, if $x\in\mathcal{S}$ and $r>0$ are such that $\overline{B_r(x)}\subset\mathcal{S}$, where $B_r(x)$ is a sphere of radius $r$ centered at $x$, then

Figures (3)

  • Figure 1: Contour plots of the CLBFs for problems I and II. (a) & (b) The plots for problem I with the CLBF obtained by solving Laplace equation (left) and Poisson's equation (right) respectively; (c) & (d) The plots for problem II with the CLBF obtained by solving Laplace equation (left) and Poisson's equation (right) respectively.
  • Figure 2: Countour plot of the harmonic CLBF for the 2D Quadrotor environment
  • Figure 3: Plot of trajectories on the 2D Quadrotor environment with 12 different initial conditions.

Theorems & Definitions (13)

  • Definition 1: Control Lyapunov Barrier Function dawson2021safedu2023reinforcement
  • Definition 2: Harmonic Function gilbarg1983elliptic
  • Theorem 1: Mean Value Theorem protter1983maximum
  • Theorem 2: Strong Maximum Principle) gilbarg1983elliptic
  • Theorem 3: Weak Maximum Principle gilbarg1983elliptic
  • Definition 3: Harmonic CLBF
  • Theorem 4
  • proof
  • Proposition 1
  • proof
  • ...and 3 more