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Feasible Space Monitoring for Multiple Control Barrier Functions with application to Large Scale Indoor Navigation

Hardik Parwana, Mitchell Black, Bardh Hoxha, Hideki Okamoto, Georgios Fainekos, Danil Prokhorov, Dimitra Panagou

TL;DR

The paper tackles the challenge of guaranteeing the existence of a feasible solution for QP-based controllers when enforcing multiple CBF constraints by introducing a feasible-space CBF (FS-CBF) that regulates the volume $\mathcal{V}(t,x)=\mathrm{vol}(\mathcal{U}_c)(t,x)$. It establishes persistent compatibility as a sufficiency condition via a forward-invariant domain $D(t)=\{x:\mathcal{V}(t,x)\ge\epsilon\}$ and proposes practical volume-estimation surrogates (Monte Carlo, Chebyshev ball, inscribed ellipsoid) plus a smooth approximation to enable real-time control synthesis. The approach is validated through simulations and a large-scale indoor navigation scenario in AWS Hospital Gazebo, showing improved feasibility and reduced sensitivity to nominal control parameters compared to standard CBF-QP. The work offers a pragmatic pathway to safer, more reliable multi-CBF control in constrained environments and suggests extensions to non-smooth analysis and MPC-like optimization frameworks.

Abstract

Quadratic programs (QP) subject to multiple time-dependent control barrier function (CBF) based constraints have been used to design safety-critical controllers. However, ensuring the existence of a solution at all times to the QP subject to multiple CBF constraints (hereby called compatibility) is non-trivial. We quantify the feasible control input space defined by multiple CBFs at a state in terms of its volume. We then introduce a novel feasible space (FS) CBF that prevents this volume from going to zero. FS-CBF is shown to be a sufficient condition for the compatibility of multiple CBFs. For high-dimensional systems though, finding a valid FS-CBF may be difficult due to the limitations of existing computational hardware or theoretical approaches. In such cases, we show empirically that imposing the feasible space volume as a candidate FS-CBF not only enhances feasibility but also exhibits reduced sensitivity to changes in the user-chosen parameters such as gains of the nominal controller. Finally, paired with a global planner, we evaluate our controller for navigation among other dynamically moving agents in the AWS Hospital gazebo environment. The proposed controller is demonstrated to outperform the standard CBF-QP controller in maintaining feasibility.

Feasible Space Monitoring for Multiple Control Barrier Functions with application to Large Scale Indoor Navigation

TL;DR

The paper tackles the challenge of guaranteeing the existence of a feasible solution for QP-based controllers when enforcing multiple CBF constraints by introducing a feasible-space CBF (FS-CBF) that regulates the volume . It establishes persistent compatibility as a sufficiency condition via a forward-invariant domain and proposes practical volume-estimation surrogates (Monte Carlo, Chebyshev ball, inscribed ellipsoid) plus a smooth approximation to enable real-time control synthesis. The approach is validated through simulations and a large-scale indoor navigation scenario in AWS Hospital Gazebo, showing improved feasibility and reduced sensitivity to nominal control parameters compared to standard CBF-QP. The work offers a pragmatic pathway to safer, more reliable multi-CBF control in constrained environments and suggests extensions to non-smooth analysis and MPC-like optimization frameworks.

Abstract

Quadratic programs (QP) subject to multiple time-dependent control barrier function (CBF) based constraints have been used to design safety-critical controllers. However, ensuring the existence of a solution at all times to the QP subject to multiple CBF constraints (hereby called compatibility) is non-trivial. We quantify the feasible control input space defined by multiple CBFs at a state in terms of its volume. We then introduce a novel feasible space (FS) CBF that prevents this volume from going to zero. FS-CBF is shown to be a sufficient condition for the compatibility of multiple CBFs. For high-dimensional systems though, finding a valid FS-CBF may be difficult due to the limitations of existing computational hardware or theoretical approaches. In such cases, we show empirically that imposing the feasible space volume as a candidate FS-CBF not only enhances feasibility but also exhibits reduced sensitivity to changes in the user-chosen parameters such as gains of the nominal controller. Finally, paired with a global planner, we evaluate our controller for navigation among other dynamically moving agents in the AWS Hospital gazebo environment. The proposed controller is demonstrated to outperform the standard CBF-QP controller in maintaining feasibility.
Paper Structure (17 sections, 2 theorems, 17 equations, 6 figures, 1 table)

This paper contains 17 sections, 2 theorems, 17 equations, 6 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. Notation
  3. System Description
  4. Problem Formulation
  5. Feasible Volume Control Barrier Function
  6. Polytope Volume Barrier Function
  7. Polytope Volume as Candidate CBF
  8. Polytope Volume Estimation
  9. Monte Carlo
  10. Chebyshev Ball of a Polytope
  11. Inscribing Ellipsoid
  12. Smooth Approximation of Polytope Volume
  13. Simulation Results
  14. Case Study 1: Valid FS-CBF computation
  15. Case Study 2: Improving Feasibility and Sensitivity
  16. ...and 2 more sections

Key Result

Theorem 1

Let $h_i, i\in \{1,..,N\}$ be $N$ barrier functions corresponding to $N$ safe sets defined as in eq::safeset. Let $\mathcal{V}(t,x)$ be the volume of the feasible CBF space $\mathcal{U}_c(t,x)$ in the control domain defined in eq::cbf_control_intersection. For $0<\epsilon<<1$, let $D(t)=\{x \in \mat

Figures (6)

  • Figure 1: Indoor navigation scenario (a) Robot (green circle) navigating in AWS Hospital Gazebo environment using proposed CBF controller. (b) Rviz visualization of the global map, global planner's path (green), humans (blue) and the nearest static obstacles (red, found in the way described in Section \ref{['section::case_study_3']}) used by the robot for collision avoidance.
  • Figure 2: Feasible space (green) of a single integrator agent with X, Y velocity inputs (in m/s) as it navigates to its goal location with N=3 in \ref{['eq::qp_controller']} while avoiding obstacles (black). (a) The path in configuration space, (b), (c), (d) feasible control space visualization at different times. The dotted lines represent hyperplane equations that constrain the control input. The $2\times 2$ square is formed by control input bound. Other hyperplanes are formed as a result of CBF equations.
  • Figure 3: (a) Workspace of black robot with illustrative orange obstacles, (b) The state space. In the violet region, CBFs are compatible, i.e., the volume of the polytope $\mathcal{U}_c$ is $>0$. The yellow region is the largest control invariant set contained in the violet region. (c) The feasible control space $\mathcal{U}_c$ defined by multiple CBFs at the current state and time. It's volume is given by the area of the green region in this 2D example.
  • Figure 4: Feasible space (green) defined by hyperplanes H1-H5. H3, H4 are overlapping. Suppose H4, H5 are free to move horizontally. It can be shown that the gradient of the area of the green region exists w.r.t position of H5 but not w.r.t H4.
  • Figure 5: Compatible space vs refined FS-CBF values at different $X,Y$ and heading $\theta = 45^0$. The color bar represents the numerical values of FS volume and FS-CBF clipped to the interval [0,1]. (a),(b) Case 1 - initiating with three valid CBFs. (c)-(f) initiating with three candidate CBFs with aggressive (Case 2) and conservative (Case 3) class-$\mathcal{K}$ function parameters for HOCBF. The compatible space (purple) is a subset of the forward invariant set (yellow) as also illustrated in Fig. \ref{['fig::mainfig']}
  • ...and 1 more figures

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Theorem 2
  • proof