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

TL;DR

This paper addresses safety of nonlinear systems under both state and input constraints by composing multiple control barrier functions into a single relax-constraint RCBC using a log-sum-exponential soft minimum $softmin_\rho$. It then derives a closed-form optimal controller for safety (and later for safety with input constraints) by solving a relaxed quadratic program, yielding explicit expressions for $u(x)$, a slack variable $\mu(x)$, and a Lagrange multiplier $\lambda(x)$ with a feasible set defined by $\Omega$ and $\omega(x)$. To handle actuator limits, the authors introduce control dynamics that transform input constraints into controller-state constraints, forming an augmented cascade and a composite RCBC $\hat h$; they obtain a closed-form surrogate controller $\hat u(\hat x)$ that ensures both safety and input-constraint satisfaction. Demonstrations on a nonholonomic ground robot show that the soft-minimum RCBF approach guarantees constraint satisfaction and outperforms multiple-HOCBF formulations in feasibility and computation, illustrating practical impact for safe, real-time constrained control. The framework provides a principled, scalable path to enforce multiple, potentially different-degree safety constraints together with actuator limits in nonlinear systems.

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.

Figures (13)

• Figure 1: Illustration of the relationships between $\SSS_\rms$, $\SC$, $\SH$, and $\SSS$ for $\ell = 2$ with $d_1 = 1$ and $d_2 = 3$. (a) shows $\SSS_\rms = \SC_{1,0} \cap \SC_{2,0}$ and $\SC = \SC_1 \cap \SC_2 = \SC_{1,0}\cap \SC_{2,0} \cap \SC_{2,1}$. (b) shows $\SC_{1,0} \cap \SC_{2,2}$ and $\SH\subset \SC_{1,0} \cap \SC_{2,2}$. (c) shows $\SSS_\rms$, $\SC$, $\SH$, and $\SSS \triangleq \SC \cap \SH$. (d) shows $\SH$, $\SSS$ and $\SB \triangleq \{ x \in \hbox{bd} \SH \colon L_fh(x) \le 0 \}$.
• Figure 2: Closed-form optimal and safe control using the composite soft-minimum R-CBF \ref{['eq:h_def']}. Control minimizes cost subject to safety constraint.
• Figure 3: Safe set $\SSS_\rms$, and 2,500 closed-loop trajectories using the control \ref{['eq:b_def', 'eq:h_def', 'eq:u_sol', 'eq:omega', 'eq:Omega', 'eq:mu_sol', 'eq:lambda_sol', 'eq:den']}.
• Figure 4: $q_\rmx$, $q_\rmy$, $v$, $\theta$, $u$ and $u_\rmd$ for $q_{\rmd}=[\,3 \quad 4.5\,]^\rmT$.
• Figure 5: $h$, $\min b_{j,i}$, and $\min h_j$ for $q_{\rmd}=[\,3 \quad 4.5\,]^\rmT$.

Theorems & Definitions (13)

• proof
• proof
• proof
• proof
• proof
• proof : Proof of Proposition \ref{['prop:softmin_bound']}
• proof : Proof of \ref{['prop:cascade_constraints']}
• proof : Proof of \ref{['prop:cascade']}
• proof : Proof of \ref{['proposition:error']}
• proof