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Synthesizing Controller for Safe Navigation using Control Density Function

Joseph Moyalan, Sriram S. K. S Narayanan, Andrew Zheng, Umesh Vaidya

TL;DR

This paper extends density-based safe control to nonlinear systems with drift by introducing a Control Density Function (CDF). It shows that safety and convergence to a target can be imposed simultaneously via a QP-CDF, using an occupancy-based interpretation of the density to encode unsafe regions. A key contribution is almost-everywhere convergence, enabling the safe controller to coexist with a nominal controller through a deviation-minimizing objective. The approach is validated on nonlinear examples including a Duffing oscillator and a Dubin car, demonstrating safe navigation with obstacles and the ability to incorporate additional performance goals; the framework offers advantages over CBF-based methods by unifying convergence and safety constraints within a single, tractable optimization.

Abstract

We consider the problem of navigating a nonlinear dynamical system from some initial set to some target set while avoiding collision with an unsafe set. We extend the concept of density function to control density function (CDF) for solving navigation problems with safety constraints. The occupancy-based interpretation of the measure associated with the density function is instrumental in imposing the safety constraints. The navigation problem with safety constraints is formulated as a quadratic program (QP) using CDF. The existing approach using the control barrier function (CBF) also formulates the navigation problem with safety constraints as QP. One of the main advantages of the proposed QP using CDF compared to QP formulated using CBF is that both the convergence/stability and safety can be combined and imposed using the CDF. Simulation results involving the Duffing oscillator and safe navigation of Dubin car models are provided to verify the main findings of the paper.

Synthesizing Controller for Safe Navigation using Control Density Function

TL;DR

This paper extends density-based safe control to nonlinear systems with drift by introducing a Control Density Function (CDF). It shows that safety and convergence to a target can be imposed simultaneously via a QP-CDF, using an occupancy-based interpretation of the density to encode unsafe regions. A key contribution is almost-everywhere convergence, enabling the safe controller to coexist with a nominal controller through a deviation-minimizing objective. The approach is validated on nonlinear examples including a Duffing oscillator and a Dubin car, demonstrating safe navigation with obstacles and the ability to incorporate additional performance goals; the framework offers advantages over CBF-based methods by unifying convergence and safety constraints within a single, tractable optimization.

Abstract

We consider the problem of navigating a nonlinear dynamical system from some initial set to some target set while avoiding collision with an unsafe set. We extend the concept of density function to control density function (CDF) for solving navigation problems with safety constraints. The occupancy-based interpretation of the measure associated with the density function is instrumental in imposing the safety constraints. The navigation problem with safety constraints is formulated as a quadratic program (QP) using CDF. The existing approach using the control barrier function (CBF) also formulates the navigation problem with safety constraints as QP. One of the main advantages of the proposed QP using CDF compared to QP formulated using CBF is that both the convergence/stability and safety can be combined and imposed using the CDF. Simulation results involving the Duffing oscillator and safe navigation of Dubin car models are provided to verify the main findings of the paper.
Paper Structure (10 sections, 2 theorems, 39 equations, 3 figures, 1 algorithm)

This paper contains 10 sections, 2 theorems, 39 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Statement
  3. Density function for safe navigation
  4. Control Density Function (CDF)
  5. Computational framework
  6. Simulation Results
  7. Duffing oscillator
  8. Dubin car model
  9. Conclusion
  10. Appendix

Key Result

Theorem 1

Under Assumption assume:2, given system dynamics in eq:dyn_sys and density function given by eq:rho_S, the system trajectories can be driven from almost all initial conditions to a target set ${\mathbf X}_T$ while avoiding unsafe set $\mathcal{U}$ if there exists a control ${\mathbf u} \in U$ and $\ The proof is provided in the Appendix.

Figures (3)

  • Figure 1: (a) Environment setup with unsafe set $\mathcal{U}$ and target ${\mathbf X}_T$, (b) density function $\rho({\mathbf x})$ for this environment.
  • Figure 2: (a) Trajectories converging to the target (green) while avoiding the unsafe set (gray), (b) Control plots for the controlled Duffing oscillator dynamics obtained by solving QP-CDF.
  • Figure 3: (a) Trajectories in the $x_1 - x_2$ space converging to the target (green) while avoiding unsafe sets (gray), (b) state trajectory plot of $\theta$ for safe navigation.

Theorems & Definitions (4)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Lemma 1