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Long-Horizon Geometry-Aware Navigation among Polytopes via MILP-MPC and Minkowski-Based CBFs

Yi-Hsuan Chen, Salman Ghori, Ania Adil, Eric Feron, Calin Belta

Abstract

Autonomous navigation in complex, non-convex environments remains challenging when robot dynamics, control limits, and exact robot geometry must all be taken into account. In this paper, we propose a hierarchical planning and control framework that bridges long-horizon guidance and geometry-aware safety guarantees for a polytopic robot navigating among polytopic obstacles. At the high level, Mixed-Integer Linear Programming (MILP) is embedded within a Model Predictive Control (MPC) framework to generate a nominal trajectory around polytopic obstacles while modeling the robot as a point mass for computational tractability. At the low level, we employ a control barrier function (CBF) based on the exact signed distance in the Minkowski-difference space as a safety filter to explicitly enforce the geometric constraints of the robot shape, and further extend its formulation to a high-order CBF (HOCBF). We demonstrate the proposed framework in U-shaped and maze-like environments under single- and double-integrator dynamics. The results show that the proposed architecture mitigates the topology-induced local-minimum behavior of purely reactive CBF-based navigation while enabling safe, real-time, geometry-aware navigation.

Long-Horizon Geometry-Aware Navigation among Polytopes via MILP-MPC and Minkowski-Based CBFs

Abstract

Autonomous navigation in complex, non-convex environments remains challenging when robot dynamics, control limits, and exact robot geometry must all be taken into account. In this paper, we propose a hierarchical planning and control framework that bridges long-horizon guidance and geometry-aware safety guarantees for a polytopic robot navigating among polytopic obstacles. At the high level, Mixed-Integer Linear Programming (MILP) is embedded within a Model Predictive Control (MPC) framework to generate a nominal trajectory around polytopic obstacles while modeling the robot as a point mass for computational tractability. At the low level, we employ a control barrier function (CBF) based on the exact signed distance in the Minkowski-difference space as a safety filter to explicitly enforce the geometric constraints of the robot shape, and further extend its formulation to a high-order CBF (HOCBF). We demonstrate the proposed framework in U-shaped and maze-like environments under single- and double-integrator dynamics. The results show that the proposed architecture mitigates the topology-induced local-minimum behavior of purely reactive CBF-based navigation while enabling safe, real-time, geometry-aware navigation.

Paper Structure

This paper contains 17 sections, 4 theorems, 30 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Configuration Obstacles via Minkowski Operation
  4. High-Order Control Barrier Functions (HOCBF)
  5. Differentiable Optimization
  6. Problem Formulation
  7. Multi-rate Planning Architecture
  8. High-level MPC planner via MILP
  9. Low-level safety filter via Minkowski-CBF
  10. Extension to HOCBF for double-integrator dynamics
  11. Simulation Results
  12. System Dynamics
  13. Long-Horizon Predictive Guidance
  14. Comparative Evaluation
  15. Discussion and Limitations
  16. ...and 2 more sections

Key Result

Theorem 1

Given a HOCBF $h(\bm{x})$ defined by def: hocbf with the associated sets $C_1,\cdots,C_m$ defined by eq: Cs_hocbf, if $\bm{x}(0)\in C_1\cap\cdots\cap C_m$, then any Lipschitz continuous controller $\bm{u}(t)\in \mathcal{U}$ that satisfies eq: def_hocbf_constr, $\forall t\geq 0$ renders $C_1\cap\cdot

Figures (3)

  • Figure 3: Top: The proposed multi-rate framework with a MILP-MPC planner and a geometry-aware Minkowski-CBF safety filter. Bottom: A triangular robot with double-integrator dynamics navigating in a polytopic maze-like environment. It collides with an obstacle when using MILP-MPC alone with a point-mass model (pink), and becomes trapped in a local minimum using a CLF-CBF-QP approach (orange) due to its inherent reactive nature. In contrast, the proposed framework (blue) successfully guides the robot to the goal without a reference path or waypoints from an external planner.
  • Figure 4: Long-horizon nominal predictions (dashed cyan) generated by the high-level MILP-MPC planner under (a) single- (Top) and (b) double-integrator (Bottom) dynamics, navigating the robot to the goal. Left: The actual closed-loop trajectory (solid blue) tracks the MPC predicted states (brown squares) as the robot (blue) avoids obstacles (gray). Right: Control inputs $\bm{u}$ and CBF values, verifying that strict safety $h(\bm{x})\geq0$ is guaranteed with respect to the real robot's geometry over time.
  • Figure 5: Top-row: Comparison of the proposed MILP-MPC-CBF framework in a U-shaped environment under single- (Left column) and double-integrator (Right column) dynamics. In each case, the left compares the proposed framework with MILP-MPC only, where the green segment indicates the interval during which the geometry-aware Minkowski-(HO)CBF safety filter is active, while the right compares it with the reactive CLF-(HO)CBF-QP. Bottom-row: Time histories of control inputs $\bm{u}$ and CBFs $h$ for MILP-MPC-CBF, showing that the full-geometry robot remains in the safe set throughout the navigation.

Theorems & Definitions (13)

  • Definition 1: Configuration Obstacle (CO) LaValle.Planning.06
  • Definition 2: Forward invariant set Ames.etal.ECC19
  • Definition 3: HOCBF Xiao.Calin.TAC21
  • Theorem 1: Safety Guarantee Xiao.Calin.TAC21
  • Theorem 2: Sensitivity of the optimal value of the objective function with respect to the parameter Castillo.etal.EngOpt6
  • Remark 1
  • Theorem 3: CBF Gradient with respect to Position
  • proof
  • Remark 2
  • Remark 3
  • ...and 3 more