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Robust Control Barrier Functions using Uncertainty Estimation with Application to Mobile Robots

Ersin Das, Joel W. Burdick

TL;DR

The paper addresses safety guarantees for nonlinear control-affine systems in the presence of both matched and unmatched uncertainties. It introduces an uncertainty estimator that bounds the estimation error and integrates it into control barrier function (CBF) and high-order CBF (HOCBF) conditions, yielding robust safety filters (UE-CBF-QP and UE-HOCBF-SOCP) that compensate matched uncertainty and account for unmatched disturbances. The approach extends to higher-order barrier functions to manage relative-degree differences, with second-order cone constraints arising when the disturbance relative degree is one less than the input relative degree. Experimental validation on a tracked mobile robot and simulations on an elastic actuator demonstrate robust safety with improved disturbance rejection and practical applicability beyond nominal models.

Abstract

This paper proposes a safety-critical control design approach for nonlinear control affine systems in the presence of matched and unmatched uncertainties. Our constructive framework couples control barrier function (CBF) theory with a new uncertainty estimator to ensure robust safety. We use the estimated uncertainty, along with a derived upper bound on the estimation error, for synthesizing CBFs and safety-critical controllers via a quadratic program-based feedback control law that rigorously ensures robust safety while improving disturbance rejection performance. We extend the method to higher-order CBFs (HOCBFs) to achieve safety under unmatched uncertainty, which may cause relative degree differences with respect to control input and disturbances. We assume the relative degree difference is at most one, resulting in a second-order cone constraint. We demonstrate the proposed robust HOCBF method through a simulation of an uncertain elastic actuator control problem and experimentally validate the efficacy of our robust CBF framework on a tracked robot with slope-induced matched and unmatched perturbations.

Robust Control Barrier Functions using Uncertainty Estimation with Application to Mobile Robots

TL;DR

The paper addresses safety guarantees for nonlinear control-affine systems in the presence of both matched and unmatched uncertainties. It introduces an uncertainty estimator that bounds the estimation error and integrates it into control barrier function (CBF) and high-order CBF (HOCBF) conditions, yielding robust safety filters (UE-CBF-QP and UE-HOCBF-SOCP) that compensate matched uncertainty and account for unmatched disturbances. The approach extends to higher-order barrier functions to manage relative-degree differences, with second-order cone constraints arising when the disturbance relative degree is one less than the input relative degree. Experimental validation on a tracked mobile robot and simulations on an elastic actuator demonstrate robust safety with improved disturbance rejection and practical applicability beyond nominal models.

Abstract

This paper proposes a safety-critical control design approach for nonlinear control affine systems in the presence of matched and unmatched uncertainties. Our constructive framework couples control barrier function (CBF) theory with a new uncertainty estimator to ensure robust safety. We use the estimated uncertainty, along with a derived upper bound on the estimation error, for synthesizing CBFs and safety-critical controllers via a quadratic program-based feedback control law that rigorously ensures robust safety while improving disturbance rejection performance. We extend the method to higher-order CBFs (HOCBFs) to achieve safety under unmatched uncertainty, which may cause relative degree differences with respect to control input and disturbances. We assume the relative degree difference is at most one, resulting in a second-order cone constraint. We demonstrate the proposed robust HOCBF method through a simulation of an uncertain elastic actuator control problem and experimentally validate the efficacy of our robust CBF framework on a tracked robot with slope-induced matched and unmatched perturbations.
Paper Structure (12 sections, 6 theorems, 58 equations, 4 figures)

This paper contains 12 sections, 6 theorems, 58 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Control Barrier Functions
  4. Analysis of the Matched and Unmatched Uncertainties
  5. Main Result
  6. Uncertainty Estimator
  7. Uncertainty Estimation-Based Safety with CBFs
  8. Uncertainty Estimation-Based Safety with HOCBFs
  9. Simulations And Hardware Experiments
  10. Elastic Actuator with Matched and Unmatched Uncertainties
  11. Experimental Validation on a Mobile Robot
  12. Conclusions and Future Work

Key Result

Theorem 1

If $h$ is a CBF for system on $\mathcal{C}$ with an ${\alpha \!\in\! \mathcal{K}_{\infty, e}}$, then any Lipschitz continuous controller ${\mathbf{k}\!:\! X \!\to\! U}$ satisfying renders eq:clsystem1 safe with respect to $\mathcal{C}$.

Figures (4)

  • Figure 1: Block diagram of the uncertainty estimator-based safe control framework. Augmenting a given, and potentially unsafe, controller with an error-bounded uncertainty estimator and a safety filter guarantees that uncertain system states remain in a subset of the safe set.
  • Figure 2: Simulations of the uncertain elastic actuator. (Left) (Top) State and reference tracking of the uncertain system operating under the UE-HOCBF-QP. (Bottom) Values of the safe control input, $u$ safe, and the nominal input, $u$ nominal, vs. time. (Center) (Top) The proposed estimator can effectively estimate actual model uncertainties within the quantified bounds. Because of ${\hat{\Delta}(0) \!=\! \bf 0}$ and ${\Delta_1 \!\equiv\! 0,~\Delta_3 \!\equiv\! 0}$, the estimator output is zero for these uncertainty components. (Bottom) The estimation error with the theoretical bound which is satisfied. (Right) (Top) The value of CBF $h$ vs. time and trajectory of the system with the boundary of the safe set. (Bottom) The proposed UE-HOCBF-QP safety filter maintains safety in the presence of unmodeled dynamics. While HOCBF-QP satisfies safety for the nominal system, it leaves the safe set with the uncertain system. Under the robustified HOCBF (R-HOCBF) approach tan2021high, the system leaves the safe set due to ignoring the unmatched uncertainty, which leads to safety violations.
  • Figure 3: Photo of a tracked mobile robot on an inclined surface during one experiment. The higher friction carpet patches on the slope result in slip uncertainty when only one track lies on a patch (Left). The geometry of full-state matched and unmatched uncertainties due to skidding and slipping, and description of the safe set ${\mathcal{C}}$ for Example \ref{['sec:exper']}(Right).
  • Figure 4: Experimental results for the mobile robot example of Fig. \ref{['fig:robot6d']}. (Left) (Top) State trajectories (${x^I~[m],~y^I~[m],~\theta~[rad]}$) and goal position of a robot operating under the UE-CBF-QP. (Middle) and (Bottom) The values of the safe linear velocity input, $v~[m/s]$ safe, the angular velocity input, $\omega~[rad/s]$ safe, and the nominal control inputs, $v$ nominal, $\omega$ nominal, vs. time. (Center) The uncertainty estimator tracksmodeing inaccuracies, as shown by the first and last components of the uncertainty vector,${\Delta_1}$ and ${\Delta_3}$. (Right) (Top) The uncertainty estimation error stays within the theoretical bounds. (Middle) The value of CBF $h$ vs. time. (Bottom) System trajectory is plotted with the safe set boundary. The proposed UE-CBF-QP-based safety filter maintains a safe distance from an edge on a slope in the presence of unmodeed dynamics. However, the CBF-QP controlled robot leaves the safe set due to the slope-induced uncertainty.

Theorems & Definitions (25)

  • Definition 1: IRD
  • Definition 2: CBF, ames2017control
  • Theorem 1
  • Definition 3: HOCBF, xiao2021high, tan2021high
  • Theorem 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 4: Full-State Matched/Unmatched Uncertainty
  • Definition 5: DRD
  • ...and 15 more