Lyapunov based dynamic controller designs for reach-and-avoid problems

TL;DR

This paper addresses reach-and-avoid for nonlinear systems by pairing a virtual, fully actuated dynamics that generates a safety-oriented reference path with a Lyapunov-based controller for the plant. A hybrid systems formulation enables discrete decisions to switch between stabilization and obstacle avoidance, yielding obstacle avoidance for the virtual path and asymptotic convergence of the plant output to the origin. The main contributions are the construction of local and global avoidance laws for the virtual dynamics, the integration with a Lyapunov-based plant controller, and demonstration on extended unicycle dynamics. The approach enables online safe navigation in the presence of static spherical obstacles and scales to nonlinear systems with local information.

Abstract

Safe obstacle avoidance and target set stabilization for nonlinear systems using reactive feedback control is under consideration. Based only on local information and by considering virtual dynamics, a safe path is generated online. The control law for the virtual dynamics is combined with a feedback controller for the dynamics of interest, where Lyapunov arguments and forward invariance are used to ensure that the state of the system remains in a vicinity of the path. To allow for discrete decisions in the avoidance controller design, the closed-loop dynamics are formulated using the hybrid systems framework. The results are illustrated by a numerical example for unicycle dynamics.

Figures (2)

• Figure 1: Illustration of $\xi$ safely bypassing an obstacle $q$ and corresponding functions $\sigma^i(\xi), \alpha^i_s(\xi)$, $\alpha^i_a(\xi)$. The cyan circle represents the set $Q^=(q)$.
• Figure 2: Top: Simulation of system \ref{['eq:UniCycleDynamics']}, feedback law \ref{['eq:ExampleInput']} and paths given by \ref{['eq:HybridPath']}. Magenta areas refer to the sets $M_0^i, M_1^i$, cyan lines represent level sets of Lyapunov functions \ref{['eq:ExampleLyapunovFunction']} centered at the small red circles, and solid and dashed lines result from different choices of $\bar{q}_i$ in \ref{['eq:HybridFeedbackLaw']}. Bottom: Time evolution of the input $u_{\zeta(t,j)}(x(t,j))$, the virtual input $\bar{\mu}(\zeta(t,j),\rho(t,j))$, and the hybrid state $-\rho(t,j)$.

Theorems & Definitions (17)

• Remark 2.1
• Lemma 3.1
• Remark 3.1
• Lemma 3.2
• Definition 1: Activation radius
• Definition 2: Activation and switching functions
• Lemma 3.3
• Theorem 3.1
• Remark 3.2
• Lemma 3.4