Control Density Function for Robust Safety and Convergence

TL;DR

This work presents the Control Density Function (CDF) as a density-based, occupancy-inspired alternative to control barrier functions for safe navigation of nonlinear systems. By analytically constructing a density $\rho(\mathbf{x})$ and formulating a convex quadratic program that enforces both safety and convergence via a single constraint, the authors unify safety with goal attainment in a principled control framework. Robustness is addressed through bounded dynamics perturbations and a scenario-based approach to state uncertainty, with simulations spanning a single integrator, double-gyre flow, bicycle model, and autonomous lane-keeping to validate safety, convergence, and robustness. The results demonstrate a scalable, differentiable feedback law with continuity guarantees and practical applicability to complex robotic systems.

Abstract

We introduce a novel approach for safe control design based on the density function. A control density function (CDF) is introduced to synthesize a safe controller for a nonlinear dynamic system. The CDF can be viewed as a dual to the control barrier function (CBF), a popular approach used for safe control design. While the safety certificate using the barrier function is based on the notion of invariance, the dual certificate involving the density function has a physical interpretation of occupancy. This occupancy-based physical interpretation is instrumental in providing an analytical construction of density function used for safe control synthesis. The safe control design problem is formulated using the density function as a quadratic programming (QP) problem. In contrast to the QP proposed for control synthesis using CBF, the proposed CDF-based QP can combine both the safety and convergence conditions to target state into single constraints. Further, we consider robustness against uncertainty in system dynamics and the initial condition and provide theoretical results for robust navigation using the CDF. Finally, we present simulation results for safe navigation with single integrator and double-gyre fluid flow-field examples, followed by robust navigation using the bicycle model and autonomous lane-keeping examples.

Figures (10)

• Figure 1: (a) Environment setup with unsafe set ${\mathbf X}_u$ and target ${\mathbf X}_T$, (b) density function $\rho({\mathbf x})$ for this environment.
• Figure 2: Comparison with CBFs: Performance of QP-CBF (dashed lines) with varying $e_1$ compared with QP-CDF (solid lines) with varying $b_j$ for single integrator dynamics.
• Figure 3: Comparison with CBFs: Control plots for QP-CBF (dashed lines) with varying $e_1$ compared with QP-CDF (solid lines) with varying $b_j$ for single integrator dynamics.
• Figure 4: Double-Gyre flow-field: Safe navigation trajectory in the position states.
• Figure 5: Bicycle model: (a) Trajectory plots with zero uncertainty in the state estimate and (b) trajectory plots with uncertainty in the state estimate where uncertainty bound $\beta=0.5$.

Theorems & Definitions (14)

• Definition 1
• Theorem 1: zheng2023safe
• Theorem 2
• Theorem 3
• Lemma 1
• Theorem 4
• proof
• Definition 2
• Theorem 5
• Lemma 2