Provably Safe Control of Lagrangian Systems in Obstacle-Scattered Environments

TL;DR

This paper introduces a hybrid control method for Lagrangian systems that ensures safety and stability in environments with obstacles by combining geometric barrier functions and Lyapunov functions within a QP framework.

Contribution

It presents a novel control approach that avoids trajectory planning by using obstacle-free ellipsoids and a switching mechanism for safe, stable navigation.

Findings

Successfully guarantees safety and stability in simulations

Uses geometric constraints to create feasible control barrier functions

Employs a switching mechanism for QP transitions

Abstract

We propose a hybrid feedback control law that guarantees both safety and asymptotic stability for a class of Lagrangian systems in environments with obstacles. Rather than performing trajectory planning and implementing a trajectory-tracking feedback control law, our approach requires a sequence of locations in the environment (a path plan) and an abstraction of the obstacle-free space. The problem of following a path plan is then interpreted as a sequence of reach-avoid problems: the system is required to consecutively reach each location of the path plan while staying within safe regions. Obstacle-free ellipsoids are used as a way of defining such safe regions, each of which encloses two consecutive locations. Feasible Control Barrier Functions (CBFs) are created directly from geometric constraints, the ellipsoids, ensuring forward-invariance, and therefore safety. Reachability to each location is guaranteed by asymptotically stabilizing Control Lyapunov Functions (CLFs). Both CBFs and CLFs are then encoded into quadratic programs (QPs) without the need of relaxation variables. Furthermore, we also propose a switching mechanism that guarantees the control law is correct and well-defined even when transitioning between QPs. Simulations show the effectiveness of the proposed approach in two complex scenarios.