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Safe Navigation under Uncertain Obstacle Dynamics using Control Barrier Functions and Constrained Convex Generators

Hugo Matias, Daniel Silvestre

TL;DR

This work tackles safe navigation for an agent amid obstacles whose dynamics are uncertain but linear by marrying time-varying Control Barrier Functions (CBFs) with guaranteed state estimation based on Constrained Convex Generators (CCGs). A finite-horizon CCG estimator yields guaranteed obstacle state enclosures, which are then evolved into CBFs through a convex-constrained conversion procedure proven via the Implicit Function Theorem; a smooth LogSumExp-based merging yields a single global CBF used in a Quadratic Program safety filter. The method naturally handles rigid-body geometries via Minkowski sums and extends from first-order control-affine dynamics to second-order strict-feedback dynamics using CBF backstepping with a Gaussian-weighted centroid controller for smoothness. Simulations demonstrate collision-free trajectories around static and dynamic obstacles with mixed geometries, validating the approach’s applicability to complex, uncertain environments. The framework provides a principled, computationally tractable path toward robust, real-time safe navigation in engineering settings with uncertain obstacle behavior and arbitrary agent/obstacle shapes.

Abstract

This paper presents a sampled-data framework for the safe navigation of controlled agents in environments cluttered with obstacles governed by uncertain linear dynamics. Collision-free motion is achieved by combining Control Barrier Function (CBF)-based safety filtering with set-valued state estimation using Constrained Convex Generators (CCGs). At each sampling time, a CCG estimate of each obstacle is obtained using a finite-horizon guaranteed estimation scheme and propagated over the sampling interval to obtain a CCG-valued flow that describes the estimated obstacle evolution. However, since CCGs are defined indirectly - as an affine transformation of a generator set subject to equality constraints, rather than as a sublevel set of a scalar function - converting the estimated obstacle flows into CBFs is a nontrivial task. One of the main contributions of this paper is a procedure to perform this conversion, ultimately yielding a CBF via a convex optimization problem whose validity is established by the Implicit Function Theorem. The resulting obstacle-specific CBFs are then merged into a single CBF that is used to design a safe controller through the standard Quadratic Program (QP)-based approach. Since CCGs support Minkowski sums, the proposed framework also naturally handles rigid-body agents and generalizes existing CBF-based rigid-body navigation designs to arbitrary agent and obstacle geometries. While the main contribution is general, the paper primarily focuses on agents with first-order control-affine dynamics and second-order strict-feedback dynamics. Simulation examples demonstrate the effectiveness of the proposed method.

Safe Navigation under Uncertain Obstacle Dynamics using Control Barrier Functions and Constrained Convex Generators

TL;DR

This work tackles safe navigation for an agent amid obstacles whose dynamics are uncertain but linear by marrying time-varying Control Barrier Functions (CBFs) with guaranteed state estimation based on Constrained Convex Generators (CCGs). A finite-horizon CCG estimator yields guaranteed obstacle state enclosures, which are then evolved into CBFs through a convex-constrained conversion procedure proven via the Implicit Function Theorem; a smooth LogSumExp-based merging yields a single global CBF used in a Quadratic Program safety filter. The method naturally handles rigid-body geometries via Minkowski sums and extends from first-order control-affine dynamics to second-order strict-feedback dynamics using CBF backstepping with a Gaussian-weighted centroid controller for smoothness. Simulations demonstrate collision-free trajectories around static and dynamic obstacles with mixed geometries, validating the approach’s applicability to complex, uncertain environments. The framework provides a principled, computationally tractable path toward robust, real-time safe navigation in engineering settings with uncertain obstacle behavior and arbitrary agent/obstacle shapes.

Abstract

This paper presents a sampled-data framework for the safe navigation of controlled agents in environments cluttered with obstacles governed by uncertain linear dynamics. Collision-free motion is achieved by combining Control Barrier Function (CBF)-based safety filtering with set-valued state estimation using Constrained Convex Generators (CCGs). At each sampling time, a CCG estimate of each obstacle is obtained using a finite-horizon guaranteed estimation scheme and propagated over the sampling interval to obtain a CCG-valued flow that describes the estimated obstacle evolution. However, since CCGs are defined indirectly - as an affine transformation of a generator set subject to equality constraints, rather than as a sublevel set of a scalar function - converting the estimated obstacle flows into CBFs is a nontrivial task. One of the main contributions of this paper is a procedure to perform this conversion, ultimately yielding a CBF via a convex optimization problem whose validity is established by the Implicit Function Theorem. The resulting obstacle-specific CBFs are then merged into a single CBF that is used to design a safe controller through the standard Quadratic Program (QP)-based approach. Since CCGs support Minkowski sums, the proposed framework also naturally handles rigid-body agents and generalizes existing CBF-based rigid-body navigation designs to arbitrary agent and obstacle geometries. While the main contribution is general, the paper primarily focuses on agents with first-order control-affine dynamics and second-order strict-feedback dynamics. Simulation examples demonstrate the effectiveness of the proposed method.
Paper Structure (15 sections, 6 theorems, 119 equations, 3 figures, 3 algorithms)

This paper contains 15 sections, 6 theorems, 119 equations, 3 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Main Contributions and Organization
  3. Notation and General Definitions
  4. Mathematical Background
  5. Control Barrier Functions
  6. Safety Filters
  7. Problem Formulation
  8. Proposed Solution
  9. CBF Backstepping
  10. Guaranteed Obstacle-State Estimation using Constrained Convex Generators
  11. Constrained Convex Generators
  12. Explicit Finite-Horizon Estimator
  13. Conversion Procedure
  14. Simulation Results
  15. Conclusion

Key Result

Theorem 1

Let $\mathcal{C}: [t_0, t_\text{f}) \rightrightarrows \mathbb{R}^n$ be a set-valued flow defined by Eq:SafeFlow, for a continuously differentiable function $h: \mathbb{R}^n\times[t_0, t_\text{f}) \rightarrow \mathbb{R}$ such that $\frac{\partial}{\partial\mathbf{x}}h(\mathbf{x}, t) \neq \mathbf{0}$

Figures (3)

  • Figure 1: Navigation of an agent with single-integrator dynamics and ellipsoidal and polytopic geometry around obstacles with mixed ellipsoidal and polytopic geometry. The left-hand plots show trajectories from different initial positions, and the right-hand plots illustrate the respective time evolution of the control inputs and overall CBF values. The colors in the right-hand plots correspond to those of the associated trajectories on the left-hand plots. In the input plots, solid lines represent the first input component and dotted lines the second.
  • Figure 2: Safe navigation of an ellipsoidal agent with single-integrator dynamics around moving obstacles with uncertain linear dynamics. The left-hand plot in subfigure (a) illustrates the agent and obstacle trajectories, and the right-hand plots shows the respective time evolution of the control input and overall CBF values. In the input plot, the solid line represents the first input component and the dotted line the second. Subfigure (b) presents snapshots of the agent and obstacle configurations at selected time instants. The orange dotted contours illustrate the estimated CCG obstacle sets used for collision avoidance.
  • Figure 3: Navigation of an ellipsoidal agent with second-order strict-feedback dynamics around moving obstacles with uncertain linear dynamics. The left-hand plot in subfigure (a) illustrates the agent and obstacle trajectories, and the right-hand plots shows the respective time evolution of the control input and top-level CBF values. In the input plot, the solid line represents the first input component and the dotted line the second. Subfigure (b) presents snapshots of the agent and obstacle configurations at selected time instants. The orange dotted contours illustrate the estimated CCG obstacle sets used for avoidance.

Theorems & Definitions (28)

  • Definition 1: Extended Class-$\mathcal{K}$/$\mathcal{K}_\infty$ Function
  • Definition 2: Graph of a Set-Valued Flow
  • Definition 3: Forward Invariance (Set)
  • Definition 4: Forward Invariance (Set-Valued Flow)
  • Definition 5: CBF ames2016control
  • Theorem 1: Safe Controller ames2016control
  • Remark 1
  • Remark 2: Practical Scenarios
  • Remark 3: Agent Geometry
  • Remark 4: Agent Dynamics
  • ...and 18 more