Safety-Critical Control for Autonomous Systems: Control Barrier Functions via Reduced-Order Models

TL;DR

This work addresses the challenge of ensuring safety in high‑dimensional autonomous systems by leveraging reduced‑order models (ROMs) to construct control barrier functions (CBFs). It unifies construction methods through backstepping, Lyapunov‑certified tracking, and ISSf notions, enabling safe control of complex robots via simple ROM CBFs lifted to full systems. The tutorial synthesizes theory, numerical examples, and hardware case studies across aircraft, quadrotors, legged/wheeled robots, manipulators, and trucks, illustrating robust, real‑time safety guarantees and, in some cases, model‑free safety concepts. It also discusses practical limitations (actuation bounds, underactuation, ROM selection) and outlines open research directions, including backup CBFs and integration with planning to enhance safety in realistic operating conditions.

Abstract

Modern autonomous systems, such as flying, legged, and wheeled robots, are generally characterized by high-dimensional nonlinear dynamics, which presents challenges for model-based safety-critical control design. Motivated by the success of reduced-order models in robotics, this paper presents a tutorial on constructive safety-critical control via reduced-order models and control barrier functions (CBFs). To this end, we provide a unified formulation of techniques in the literature that share a common foundation of constructing CBFs for complex systems from CBFs for much simpler systems. Such ideas are illustrated through formal results, simple numerical examples, and case studies of real-world systems to which these techniques have been experimentally applied.

Figures (16)

• Figure 1: Vector field of the inverted pendulum in Example \ref{['example:inverted-pendulum']} without any controller (left) and with the safety filter from \ref{['eq:safety-filter-qp-closed-form']} (right). In each plot, the red ellipse denotes the boundary of $\mathcal{C}$, the black vertical lines denote $|\theta|=\bar{\theta}=\frac{\pi}{4}$, and the arrows of varying color illustrate the system vector field. The varying colors of the arrows characterize the magnitude of each vector, with lighter colors corresponding to larger magnitudes.
• Figure 2: Smooth universal formulas for safety-critical control compared to the $\operatorname{ReLU}$ function associated with quadratic programs. The left plot illustrates the variation of $\lambda(a,b)$ with respect to $a$ for a fixed $b>0$ while the right plot illustrates the variation of $\lambda(a,b)$ with respect to $b$ for a fixed $a>0$ for each of the formulas in \ref{['eq:smooth-formulas']}.
• Figure 3: Safe set constructed for the one-dimensional double integrator via backstepping. Here, the colored curves represent the zero level set of $h$ as defined in \ref{['eq:h-backstepping']} for various $\mu$, where $k_0$ is constructed using the Softplus universal formula from \ref{['eq:smooth-formulas']} with $\sigma=0.1$. Note that as $\mu$ is increased the resulting safe set approaches the original constraint set $\mathcal{C}_0$ from \ref{['eq:C0']}.
• Figure 4: Results of the double integrator obstacle avoidance scenario from Example \ref{['example:double-int-obstacle-avoidance']}. (a) The trajectories of the double integrator's position, (b) its velocities, (c) the values of the safety constraint $h_0$ along the system's trajectory, and (d) the norm of the control input over time. This figure has been adapted from AndrewCDC22.
• Figure 5: Safe set constructed for the one-dimensional double integrator using the extended CBF approach. Here, the colored curves represent the boundary of $\hat{\mathcal{C}}_0\cap\mathcal{C}$ for different choices of $\alpha_0$, the black lines denote the boundary of $\hat{\mathcal{C}}_0$, and the transparent curves of corresponding color denote the boundary of $\mathcal{C}$ for different choices of $\alpha_0$.

Theorems & Definitions (40)

• definition 1: Safety AmesECC19
• Theorem 1: Nagumo's Theorem nagumo1942lage
• definition 2: Regular value AbrahamMarsdenRatiu
• Lemma 1: AbrahamMarsdenRatiu
• Corollary 1
• definition 3: Barrier function AmesADHS15
• Theorem 2: AmesADHS15
• Theorem 3: AmesADHS15
• definition 4: Controlled invariance Blanchini
• definition 5: Control barrier function AmesTAC17