Control Barrier Corridors: From Safety Functions to Safe Sets

TL;DR

It is shown that individual state safety can be extended locally over control barrier corridors for convex barrier functions, provided the control convergence rate matches the barrier decay rate, highlighting a trade-off between safety and reactiveness.

Abstract

Safe autonomy is a critical requirement and a key enabler for robots to operate safely in unstructured complex environments. Control barrier functions and safe motion corridors are two widely used but technically distinct safety methods, functional and geometric, respectively, for safe motion planning and control. Control barrier functions are applied to the safety filtering of control inputs to limit the decay rate of system safety, whereas safe motion corridors are geometrically constructed to define a local safe zone around the system state for use in motion optimization and reference-governor design. This paper introduces a new notion of control barrier corridors, which unifies these two approaches by converting control barrier functions into local safe goal regions for reference goal selection in feedback control systems. We show, with examples on fully actuated systems, kinematic unicycles, and linear output regulation systems, that individual state safety can be extended locally over control barrier corridors for convex barrier functions, provided the control convergence rate matches the barrier decay rate, highlighting a trade-off between safety and reactiveness. Such safe control barrier corridors enable safely reachable persistent goal selection over continuously changing barrier corridors during system motion, which we demonstrate for verifiably safe and persistent path following in autonomous exploration of unknown environments.

Figures (2)

• Figure 1: Control barrier corridors turn CBF safety specification on control inputs into a geometric representation of safe goals for state feedback control. For instance, a control barrier corridor $\mathrm{BC}(\mathrm{x})$ around a safe state $\mathrm{x}$ of a fully actuated system $\dot{\mathrm{x}} = \mathrm{u}$ under proportional control $\mathrm{u} = -\kappa(\mathrm{x} - \mathrm{x}^{*})$ is the set of goal states $\mathrm{x}^{*}$ that ensure safe control based on the feasibility condition $\dot{h}_i(\mathrm{x})\geq -\alpha (h_i(\mathrm{x}))$ for a set of CBFs $h_1(\mathrm{x}), \ldots, h_{m}(\mathrm{x})$. (left) The control barrier corridor of a fully actuated circular robot, centered at $\mathrm{x}$ with radius $r > 0$, relative to a point obstacle at $\mathrm{q}$, constructed based on the power-distance barrier function $h(\mathrm{x}) = \|\mathrm{x} - \mathrm{q}\|^ p - r^p$. The control barrier corridor enlarges with an increasing ratio of the barrier decay rate to the proportional control gain, $\frac{\alpha}{\kappa}$, and shrinks with the increasing order $p$ of the power distance. (right) The control barrier corridor (yellow) of a fully actuated circular robot (cyan+red arrow), constructed relative to multiple sensed obstacle points (red), captures a local collision-free space around the robot for convex barrier functions (e.g., the power distance with $p \geq 1$) under identical linear barrier decay rate and proportional control gain, $\alpha = \kappa$. This allows for selecting a safely reachable goal (cyan) for following a reference path (blue).
• Figure 2: The influence of the proportional control gain $\kappa$, barrier decay rate $\alpha$, and the order $p$ of the power distance on control barrier corridors (yellow) of a fully actuated robot (cyan) relative to sensed obstacle points (red). A control barrier corridor corresponds to a local safe zone around the robot (cyan), free of obstacles (red), for a pair of matching barrier rate and control gain ($\alpha=\kappa$), and convex barrier functions ($p \geq 1$). The conservativeness of the barrier corridor increases with higher control gain ($\alpha < \kappa$) and higher barrier convexity ($p > 1$), and it becomes less accurate with increasing barrier rate ($\alpha > \kappa$) and increasing barrier concavity ($p < 1$).

Theorems & Definitions (8)

• Definition 1
• Definition 2
• Example 1
• Proposition 1
• proof
• Proposition 2
• proof
• Remark 1