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Iterative Convex Optimization with Control Barrier Functions for Obstacle Avoidance among Polytopes

Shuo Liu, Zhe Huang, Calin A. Belta

TL;DR

This paper constructs linear discrete-time control barrier function (DCBF) constraints by deriving supporting hyperplanes from exact closest-point computations between convex polytopes and proposes a novel iterative convex MPC-DCBF framework, where local linearization of system dynamics and robot geometry ensures convexity of the finite-horizon optimization at each iteration.

Abstract

Obstacle avoidance of polytopic obstacles by polytopic robots is a challenging problem in optimization-based control and trajectory planning. Many existing methods rely on smooth geometric approximations, such as hyperspheres or ellipsoids, which allow differentiable distance expressions but distort the true geometry and restrict the feasible set. Other approaches integrate exact polytope distances into nonlinear model predictive control (MPC), resulting in nonconvex programs that limit real-time performance. In this paper, we construct linear discrete-time control barrier function (DCBF) constraints by deriving supporting hyperplanes from exact closest-point computations between convex polytopes. We then propose a novel iterative convex MPC-DCBF framework, where local linearization of system dynamics and robot geometry ensures convexity of the finite-horizon optimization at each iteration. The resulting formulation reduces computational complexity and enables fast online implementation for safety-critical control and trajectory planning of general nonlinear dynamics. The framework extends to multi-robot and three-dimensional environments. Numerical experiments demonstrate collision-free navigation in cluttered maze scenarios with millisecond-level solve times.

Iterative Convex Optimization with Control Barrier Functions for Obstacle Avoidance among Polytopes

TL;DR

This paper constructs linear discrete-time control barrier function (DCBF) constraints by deriving supporting hyperplanes from exact closest-point computations between convex polytopes and proposes a novel iterative convex MPC-DCBF framework, where local linearization of system dynamics and robot geometry ensures convexity of the finite-horizon optimization at each iteration.

Abstract

Obstacle avoidance of polytopic obstacles by polytopic robots is a challenging problem in optimization-based control and trajectory planning. Many existing methods rely on smooth geometric approximations, such as hyperspheres or ellipsoids, which allow differentiable distance expressions but distort the true geometry and restrict the feasible set. Other approaches integrate exact polytope distances into nonlinear model predictive control (MPC), resulting in nonconvex programs that limit real-time performance. In this paper, we construct linear discrete-time control barrier function (DCBF) constraints by deriving supporting hyperplanes from exact closest-point computations between convex polytopes. We then propose a novel iterative convex MPC-DCBF framework, where local linearization of system dynamics and robot geometry ensures convexity of the finite-horizon optimization at each iteration. The resulting formulation reduces computational complexity and enables fast online implementation for safety-critical control and trajectory planning of general nonlinear dynamics. The framework extends to multi-robot and three-dimensional environments. Numerical experiments demonstrate collision-free navigation in cluttered maze scenarios with millisecond-level solve times.
Paper Structure (28 sections, 2 theorems, 25 equations, 4 figures, 2 tables)

This paper contains 28 sections, 2 theorems, 25 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Optimization Formulation using DHOCBFs
  4. Obstacle Avoidance between Polytopic Sets
  5. Problem Formulation and Approach
  6. Iterative MPC with DHOCBF constraints
  7. Linearization of Dynamics
  8. Robot and Obstacle Geometry
  9. Supporting Hyperplanes
  10. Iterative MPC with DHOCBF Constraints
  11. Sequential Multi-Robot Implementation
  12. Complexity Analysis
  13. NUMERICAL RESULTS
  14. One robot in a 2-D environment
  15. System Dynamics
  16. ...and 13 more sections

Key Result

Theorem 1

Given a DHOCBF $h(\mathbf{x})$ from Def. def:high-order-discrete-CBFs with corresponding sets $\mathcal{C}_{0}, \dots,\mathcal{C}_{m-1}$ defined by eq:high-order-safety-sets, if $\mathbf{x}_{0} \in \mathcal{C}_{0}\cap \dots \cap \mathcal{C}_{m-1},$ then any Lipschitz controller $\mathbf{u}_{t}$ that

Figures (4)

  • Figure 1: Motivating example: an L-shaped robot performing narrow-passage traversal. Simulation results are shown in Sec. \ref{['sec:3D-sim']}.
  • Figure 2: Schematic of the iterative process of solving the convex MPC at time $t$.
  • Figure 3: Autonomous navigation results for different robot geometries in 2-D environments. The green circle and the red pentagon denote the starting position and the goal positions, respectively. The orange curve represents the local reference path, while the curves in other colors correspond to the predicted local trajectories generated by the controller.
  • Figure 4: Navigation results from three viewpoints. The robot moves from the green circle (start) to the red pentagram (goal), generating collision-free trajectories in the cluttered environment.

Theorems & Definitions (5)

  • Definition 1: Relative degree liu2023iterative
  • Definition 2: DHOCBF xiong2022discrete
  • Theorem 1: Safety Guarantee xiong2022discrete
  • Proposition 1
  • Remark 1: Safety Margin under Geometry Linearization