Soft-Minimum and Soft-Maximum Barrier Functions for Safety with Actuation Constraints

TL;DR

This work tackles safety under actuator constraints for nonlinear dynamics by developing two barrier-function–based safety controllers that operate via real-time convex optimization. The first approach uses a smooth soft-minimum barrier $h(x)$ obtained from a finite-horizon trajectory under a known backup control, yielding a feasible QP formulation and forward-invariance of a subset of the safe set. The second approach generalizes to multiple backups with a soft-maximum barrier, enabling larger safe regions while preserving feasibility and continuity through a blended control and boundary-switching strategy. Demonstrations on an inverted pendulum and a nonholonomic ground robot illustrate safe operation under actuation limits and improved safety margins when using multiple backups. Overall, the methods provide scalable, continuous safety filters that respect actuator constraints and guarantee forward invariance under practical conditions.

Abstract

This paper presents two new control approaches for guaranteed safety (remaining in a safe set) subject to actuator constraints (the control is in a convex polytope). The control signals are computed using real-time optimization, including linear and quadratic programs subject to affine constraints, which are shown to be feasible. The first control method relies on a soft-minimum barrier function that is constructed using a finite-time-horizon prediction of the system trajectories under a known backup control. The main result shows that the control is continuous and satisfies the actuator constraints, and a subset of the safe set is forward invariant under the control. Next, we extend this method to allow from multiple backup controls. This second approach relies on a combined soft-maximum/soft-minimum barrier function, and it has properties similar to the first. We demonstrate these controls on numerical simulations of an inverted pendulum and a nonholonomic ground robot.

Figures (9)

• Figure 1: $\SSS_\rms$, $\SSS_\rmb$, $\SSS$, $\bar{\SSS}_*$, and closed-loop trajectories for 8 initial conditions.
• Figure 2: $\theta$, $\dot \theta$, $u$, $u_\rmd$, $u_\rmb$, and $u_*$ for $x_0=[0.5\,\,0]^\rmT$.
• Figure 3: $h$, $h_\rms$, $\bar{h}_*$, $\frac{h-\epsilon}{\kappa_h}$, $\frac{\beta}{\kappa_\beta}$, and $\sigma$ for $x_0=[0.5\,\,0]^\rmT$.
• Figure 4: $\SSS_\rms$, $\SSS_\rmb$, and 3 closed-loop trajectories.
• Figure 5: $q_\rmx$, $q_\rmy$, $v$, $\theta$, $u$, $u_\rmd$, $u_\rmb$, and $u_*$ for $r_{\rmd}=[\,2 \quad 4.5\,]^\rmT$.