A Closed-Form Control for Safety Under Input Constraints Using a Composition of Control Barrier Functions

TL;DR

This paper introduces a novel closed-form optimal control method that ensures safety and input constraints are satisfied simultaneously by composing multiple control barrier functions using a soft-minimum approach.

Contribution

It develops a new method to construct a relaxed control barrier function from multiple CBFs with different degrees, enabling a closed-form optimal control respecting safety and input limits.

Findings

Successfully guarantees safety constraints in simulation

Respects actuator input limits in control design

Applicable to nonholonomic ground robot scenarios

Abstract

We present a closed-form optimal control that satisfies both safety constraints (i.e., state constraints) and input constraints (e.g., actuator limits) using a composition of multiple control barrier functions (CBFs). This main contribution is obtained through the combination of several ideas. First, we present a method for constructing a single relaxed control barrier function (R-CBF) from multiple CBFs, which can have different relative degrees. The construction relies on a log-sum-exponential soft-minimum function and yields an R-CBF whose zero-superlevel set is a subset of the intersection of the zero-superlevel sets of all CBFs used in the composition. Next, we use the soft-minimum R-CBF to construct a closed-form control that is optimal with respect to a quadratic cost subject to the safety constraints. Finally, we use the soft-minimum R-CBF to develop a closed-form optimal control that not only guarantees safety but also respects input constraints. The key elements in developing this novel control include: the introduction of the control dynamics, which allow the input constraints to be transformed into controller-state constraints; the use of the soft-minimum R-CBF to compose multiple safety and input CBFs, which have different relative degrees; and the development of a desired surrogate control (i.e., a desired input to the control dynamics). We demonstrate these new control approaches in simulation on a nonholonomic ground robot.