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Constructive Safety-Critical Control: Synthesizing Control Barrier Functions for Partially Feedback Linearizable Systems

Max H. Cohen, Ryan K. Cosner, Aaron D. Ames

TL;DR

The paper addresses safety for nonlinear control‑affine systems by linking partial feedback linearization with control barrier function (CBF) synthesis. It develops a backstepping‑style construction that converts smooth inequality constraints on outputs into CBFs for the full‑order dynamics, relaxing uniform relative degree requirements and enabling application to underactuated robots. The approach is instantiated for robotic systems, deriving tractable CBFs with relative degree 2 and providing explicit formulas for the CBFs and corresponding safety filters. The framework is validated through pendulum‑on‑cart, planar quadrotor, and hardware quadrotor experiments, demonstrating guaranteed forward invariance of prescribed safe sets and practical safety guarantees in real hardware.

Abstract

Certifying the safety of nonlinear systems, through the lens of set invariance and control barrier functions (CBFs), offers a powerful method for controller synthesis, provided a CBF can be constructed. This paper draws connections between partial feedback linearization and CBF synthesis. We illustrate that when a control affine system is input-output linearizable with respect to a smooth output function, then, under mild regularity conditions, one may extend any safety constraint defined on the output to a CBF for the full-order dynamics. These more general results are specialized to robotic systems where the conditions required to synthesize CBFs simplify. The CBFs constructed from our approach are applied and verified in simulation and hardware experiments on a quadrotor.

Constructive Safety-Critical Control: Synthesizing Control Barrier Functions for Partially Feedback Linearizable Systems

TL;DR

The paper addresses safety for nonlinear control‑affine systems by linking partial feedback linearization with control barrier function (CBF) synthesis. It develops a backstepping‑style construction that converts smooth inequality constraints on outputs into CBFs for the full‑order dynamics, relaxing uniform relative degree requirements and enabling application to underactuated robots. The approach is instantiated for robotic systems, deriving tractable CBFs with relative degree 2 and providing explicit formulas for the CBFs and corresponding safety filters. The framework is validated through pendulum‑on‑cart, planar quadrotor, and hardware quadrotor experiments, demonstrating guaranteed forward invariance of prescribed safe sets and practical safety guarantees in real hardware.

Abstract

Certifying the safety of nonlinear systems, through the lens of set invariance and control barrier functions (CBFs), offers a powerful method for controller synthesis, provided a CBF can be constructed. This paper draws connections between partial feedback linearization and CBF synthesis. We illustrate that when a control affine system is input-output linearizable with respect to a smooth output function, then, under mild regularity conditions, one may extend any safety constraint defined on the output to a CBF for the full-order dynamics. These more general results are specialized to robotic systems where the conditions required to synthesize CBFs simplify. The CBFs constructed from our approach are applied and verified in simulation and hardware experiments on a quadrotor.
Paper Structure (9 sections, 3 theorems, 25 equations, 5 figures)

This paper contains 9 sections, 3 theorems, 25 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. CBFs for Feedback Linearizable Systems
  4. CBFs for Underactuated Robotic Systems
  5. Simulations and Hardware Experiments
  6. Pendulum on a Cart
  7. Planar Quadrotor
  8. Quadrotor Hardware Experiments
  9. Conclusions

Key Result

Lemma 1

Let $\psi\,:\,\mathbb{R}^p\rightarrow\mathbb{R}$ be a smooth function defining a set $\mathcal{C}_1\subset\mathbb{R}^p$ as: Let $\mathcal{D}_1\supset\mathcal{C}_1$ be an open set and suppose that: Then, for any smooth $\alpha\in\mathcal{K}_{\infty}^e$ there exists a smooth $\mathbf{k}_1\,:\,\mathcal{D}_1\rightarrow\mathbb{R}^p$ such that for all $\mathbf{y}\in\mathcal{D}_1$: For any $\sigma>0$,

Figures (5)

  • Figure 1: We present a methodology to systematically generate control barrier functions for high-dimensional underactuated systems from inequality constraints on the system's output. A video of an experimental demonstration of our approach can be found at https://youtu.be/GYvQjcojLIQ.
  • Figure 2: Safe sets and simulations results of the pendulum on a cart with a CBF placed on the cart's position (left) and the pendulum's angle (right). In each plot the dashed black lines denote the boundary of the constraint set and the gray regions denote the states where the constraint is violated. In the first and third plots, the green regions correspond to $\mathcal{S}$ and the purple region to $\mathcal{C}\setminus\mathcal{S}$.
  • Figure 3: Simulations of the planar quadrotor whose CBF ensures that $z\geq z_{\min}$ where $\mathbf{q}_0$ and $\mathbf{q}_f$ denote the initial and final position of the quadrotor.
  • Figure 4: Simulation results illustrating the evolution of the planar quadrotor's height (top left), orientation (top right), the safety constraint and CBF (bottom left), and control inputs (bottom right). In the top plots, the dashed lines denote the limits imposed on $z$ and $\theta$ by the safety constraint.
  • Figure 5: Experimental results (cf. Fig. \ref{['fig:master_figure']}) illustrating the evolution of the quadrotor's height (blue) and CBF (red).

Theorems & Definitions (8)

  • Definition 1: AmesTAC17
  • Definition 2: Isidori
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • proof