The reviewed record of science sign in
Pith

arxiv: 2606.09719 · v1 · pith:5HGZAUNO · submitted 2026-06-08 · cs.RO

Safe Polytope-in-Polytope Motion Planning and Control with Control Barrier Functions

Alejandro Gonzalez-Garcia , Dries Dirckx , Jan Swevers , Wilm Decr\'e This is my paper

Reviewed by Pith2026-06-27 16:33 UTCgrok-4.3pith:5HGZAUNOopen to challenge →

classification cs.RO
keywords motion planningcontrol barrier functionsmodel predictive controlpolytope containmentrobot footprintfree-space representationautonomous mobile robotssafety constraints
0
0 comments X

The pith

A polytopic robot footprint is kept inside a convex free-space region by encoding containment as discrete-time control barrier function constraints in a model predictive controller.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a motion planning method for robots with physical footprints in tight spaces by representing the free-space as a convex polytope and enforcing containment through control barrier functions. This approach integrates the safety constraints directly into a model predictive controller without needing separate obstacle detection or segmentation. The number of constraints scales with the geometry of the free-space and robot shape rather than the count of obstacles, which improves computational efficiency as obstacle numbers grow. Validation shows real-time performance at 10 Hz on embedded hardware for both simulation and physical robots using occupancy grids and LiDAR.

Core claim

The containment condition between the robot's polytopic footprint and the continuously updated convex free-space polytope is formulated as a set of discrete-time control barrier function constraints within a model predictive controller. This guarantees that the robot stays inside the free-space region, and the formulation does not require obstacle detection or segmentation steps.

What carries the argument

Discrete-time control barrier function (CBF) constraints that encode the polytope-in-polytope containment condition, incorporated into the model predictive controller (MPC).

If this is right

  • The number of safety constraints depends on free-space geometry complexity and robot shape, not the number of obstacles.
  • A comparative analysis shows up to 91 times reduction in computation time as the number of obstacles increases.
  • The method enables safe real-time motion planning and control at 10 Hz on an onboard embedded computer.
  • It handles reactive avoidance of dynamic obstacles using both occupancy grids and LiDAR sensing.
  • The approach works for autonomous surface vehicles in simulation and non-holonomic mobile robots on hardware.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Extending this to non-convex free-spaces might require decomposition into multiple polytopes while retaining the CBF encoding.
  • This could combine with global path planners for complete navigation stacks in larger environments.
  • Applications in higher-dimensional spaces like 3D robotic arms could benefit from similar scaling properties if polytopic representations are used.

Load-bearing premise

The continuously updated free-space region can be represented as a convex polytope containing the robot footprint.

What would settle it

A scenario with many obstacles where the computation time does not decrease substantially compared to obstacle-based methods or where the robot violates containment despite the constraints being active.

Figures

Figures reproduced from arXiv: 2606.09719 by Alejandro Gonzalez-Garcia, Dries Dirckx, Jan Swevers, Wilm Decr\'e.

Figure 1
Figure 1. Figure 1: Circular footprint approximations do not permit to traverse through [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Theoretical comparison of the inflation of the MPC problem [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Quantitative validation for an increasing amount of obstacles of the [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Visualization of the real-world lab setup including the AMR [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of runs for the ASV (a, b) and AMR (c, d) cases using [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Visualization of PiP-CBF’s conservatism for the environment in [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
read the original abstract

Autonomous mobile robots operating in tight environments require motion planning frameworks that account for the physical footprint of the robot. Simplifying the geometry to a point or a circle is conservative and discards information needed to successfully and safely traverse narrow passages. This work proposes a safe local motion planning and control method that guarantees that a polytopic robot footprint stays inside a continuously updated convex free-space region. The containment condition is formulated as a set of discrete-time control barrier function constraints within a model predictive controller. The number of safety constraints depends on the complexity of the local free-space geometry and the robot shape, instead of the number of obstacles. The proposed free-space formulation does not need any obstacle detection or segmentation. A comparative analysis against a polytope-based obstacle avoidance formulation confirms favorable scaling up to a reduction of 91$\times$ in computation time as the number of obstacles increases. The approach is validated in simulation with an autonomous surface vehicle and on hardware with a non-holonomic mobile robot, using both occupancy grids and LiDAR sensing. The experiments demonstrate safe real-time motion planning and control at 10~Hz on an onboard embedded computer, including reactive avoidance of dynamic obstacles.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper proposes a local motion planning and control framework for mobile robots with polytopic footprints that guarantees containment inside a continuously updated convex free-space polytope. Containment is encoded as a collection of discrete-time CBF inequalities inside an MPC; the number of inequalities scales with the number of free-space facets and robot vertices rather than the number of obstacles, and the formulation requires no explicit obstacle detection or segmentation. Comparative experiments report up to a 91× reduction in computation time relative to a polytope-based obstacle-avoidance baseline, and the method is demonstrated in simulation on an autonomous surface vehicle and on hardware with a non-holonomic robot using occupancy grids and LiDAR, achieving real-time 10 Hz operation on embedded hardware.

Significance. If the central claim holds, the work supplies a practical, scalable alternative to conservative point/circle approximations or per-obstacle CBFs for footprint-aware navigation in tight spaces. The explicit grounding in standard polytope containment and discrete CBF theory, together with the reported hardware validation at 10 Hz, constitutes a concrete engineering contribution that could be adopted in real-time robotics pipelines.

minor comments (3)
  1. Abstract and §1: the 91× computation-time claim is central to the scaling argument; the manuscript should state the exact obstacle counts, baseline implementation details, and timing measurement protocol used to obtain this figure.
  2. The description of how the free-space polytope is extracted from occupancy grids or LiDAR (mentioned in the abstract) should include a brief algorithmic outline or reference so that the “no segmentation” claim can be reproduced.
  3. Notation: the discrete-time CBF class-K function α(·) and the MPC horizon length appear in the formulation but are not given explicit symbols in the abstract; consistent notation would aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The provided summary accurately captures the core contribution of encoding polytope containment as discrete-time CBF constraints within MPC, with scaling independent of obstacle count.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper encodes polytope-in-polytope containment directly as discrete-time CBF inequalities h(x_{k+1}) ≥ (1-α)h(x_k) applied to each robot vertex against free-space half-planes. This follows from the standard geometric definition of containment and the usual CBF form without any fitted parameters, self-citation load-bearing steps, or renaming of known results. Experiments on occupancy grids, LiDAR, and hardware provide external validation independent of the formulation itself.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on standard CBF theory and convex polytope representations from prior literature; no new entities are introduced. Possible free parameters include CBF gain values typically tuned per application.

free parameters (1)
  • CBF class-K function parameters (e.g., alpha, gamma)
    Standard in discrete-time CBF formulations; likely chosen or tuned to ensure constraint satisfaction for the specific robot and environment.
axioms (1)
  • domain assumption Local free-space can be represented and continuously updated as a single convex polytope
    Invoked in the containment formulation and free-space update step.

pith-pipeline@v0.9.1-grok · 5748 in / 1281 out tokens · 31049 ms · 2026-06-27T16:33:53.346259+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

43 extracted references

  1. [1]

    Using occupancy grids for mobile robot perception and navigation,

    A. Elfes, “Using occupancy grids for mobile robot perception and navigation,”Computer, vol. 22, no. 6, pp. 46–57, 1989

    1989

  2. [2]

    A Formal Basis for the Heuristic Determination of Minimum Cost Paths,

    P. E. Hart, N. J. Nilsson, and B. Raphael, “A Formal Basis for the Heuristic Determination of Minimum Cost Paths,”IEEE Transactions on Systems Science and Cybernetics, vol. 4, no. 2, pp. 100–107, 1968

    1968

  3. [3]

    A Note on Two Problems in Connexion with Graphs

    E. W. Dijkstra, “A Note on Two Problems in Connexion with Graphs.” Numerische Mathematik, vol. 1, pp. 269–271, 1959

    1959

  4. [4]

    The Marathon 2: A Navigation System,

    S. Macenski, F. Martin, R. White, and J. Gin ´es Clavero, “The Marathon 2: A Navigation System,” in2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2020

    2020

  5. [5]

    Probabilistic roadmaps for path planning in high-dimensional configuration spaces,

    L. Kavraki, P. Svestka, J.-C. Latombe, and M. Overmars, “Probabilistic roadmaps for path planning in high-dimensional configuration spaces,” IEEE Transactions on Robotics and Automation, vol. 12, no. 4, pp. 566–580, 1996

    1996

  6. [6]

    Kinodynamic trajectory planning for efficient uav exploration and reconstruction of unknown environments,

    J. F. Mendes, M. Basiri, and R. Ventura, “Kinodynamic trajectory planning for efficient uav exploration and reconstruction of unknown environments,”IEEE Robotics and Automation Letters, vol. 11, no. 2, pp. 1530–1537, 2026

    2026

  7. [7]

    Real-time obstacle avoidance for manipulators and mobile robots,

    O. Khatib, “Real-time obstacle avoidance for manipulators and mobile robots,” inProceedings - IEEE International Conference on Robotics and Automation, vol. 2, 1985, pp. 500–505

    1985

  8. [8]

    Motion planning in dynamic environments using the relative velocity paradigm,

    P. Fiorini and Z. Shiller, “Motion planning in dynamic environments using the relative velocity paradigm,” in[1993] Proceedings IEEE International Conference on Robotics and Automation, 1993, pp. 560– 565 vol.1

    1993

  9. [9]

    Constrained model predictive control: Stability and optimality,

    D. Mayne, J. Rawlings, C. Rao, and P. Scokaert, “Constrained model predictive control: Stability and optimality,”Automatica, vol. 36, no. 6, pp. 789–814, 2000

    2000

  10. [10]

    Embedded nonlinear model predictive control for obstacle avoidance using panoc,

    A. Sathya, P. Sopasakis, R. Van Parys, A. Themelis, G. Pipeleers, and P. Patrinos, “Embedded nonlinear model predictive control for obstacle avoidance using panoc,” in2018 European Control Conference (ECC), 2018, pp. 1523–1528

    2018

  11. [11]

    Where to go next: Learning a subgoal recommendation policy for navigation in dynamic environments,

    B. Brito, M. Everett, J. P. How, and J. Alonso-Mora, “Where to go next: Learning a subgoal recommendation policy for navigation in dynamic environments,”IEEE Robotics and Automation Letters, vol. 6, no. 3, pp. 4616–4623, 2021

    2021

  12. [12]

    Nmpc-augmented visual navigation and safe learning control for large-scale mobile robots,

    M. H. Shahna, P. Mustalahti, and J. Mattila, “Nmpc-augmented visual navigation and safe learning control for large-scale mobile robots,” IEEE Robotics and Automation Letters, vol. 11, no. 4, pp. 5182–5189, 2026

    2026

  13. [13]

    Control barrier functions: Theory and applications,

    A. D. Ames, S. Coogan, M. Egerstedt, G. Notomista, K. Sreenath, and P. Tabuada, “Control barrier functions: Theory and applications,” in2019 18th European control conference (ECC). Ieee, 2019, pp. 3420–3431

    2019

  14. [14]

    High-order control barrier functions,

    W. Xiao and C. Belta, “High-order control barrier functions,”IEEE Transactions on Automatic Control, vol. 67, no. 7, pp. 3655–3662, 2021

    2021

  15. [15]

    Safety-critical planning and control for dynamic obstacle avoidance using control barrier functions,

    S. Liu, Y . Mao, and C. A. Belta, “Safety-critical planning and control for dynamic obstacle avoidance using control barrier functions,” in 2025 American Control Conference (ACC), 2025, pp. 348–354

    2025

  16. [16]

    Control barrier functions in dynamic uavs for kinematic obstacle avoidance: A col- lision cone approach,

    M. Tayal, R. Singh, J. Keshavan, and S. Kolathaya, “Control barrier functions in dynamic uavs for kinematic obstacle avoidance: A col- lision cone approach,” in2024 American Control Conference (ACC), 2024, pp. 3722–3727

    2024

  17. [17]

    Discrete control barrier functions for safety-critical control of discrete systems with application to bipedal robot navigation,

    A. Agrawal and K. Sreenath, “Discrete control barrier functions for safety-critical control of discrete systems with application to bipedal robot navigation,” inProceedings of Robotics: Science and Systems, 2017

    2017

  18. [18]

    Safety-critical model predictive control with discrete-time control barrier function,

    J. Zeng, B. Zhang, and K. Sreenath, “Safety-critical model predictive control with discrete-time control barrier function,” in2021 American Control Conference (ACC), 2021, pp. 3882–3889

    2021

  19. [19]

    Real- time-feasible collision-free motion planning for ellipsoidal objects,

    Y . Gao, F. Messerer, N. v. Duijkeren, B. Houska, and M. Diehl, “Real- time-feasible collision-free motion planning for ellipsoidal objects,” in 2024 IEEE 63rd Conference on Decision and Control (CDC), 2024, pp. 5108–5113

    2024

  20. [20]

    Any- shape real-time replanning via swept volume sdf,

    Y . Wang, T. Zhang, M. Zhang, S. Ji, X. Li, and F. Gao, “Any- shape real-time replanning via swept volume sdf,” in2025 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2025, pp. 2068–2075

    2025

  21. [21]

    Safety-critical control and planning for obstacle avoidance between polytopes with control barrier functions,

    A. Thirugnanam, J. Zeng, and K. Sreenath, “Safety-critical control and planning for obstacle avoidance between polytopes with control barrier functions,” in2022 International Conference on Robotics and Automation (ICRA), 2022, pp. 286–292

    2022

  22. [22]

    Control barrier functions via minkowski operations for safe navigation among polytopic sets,

    Y .-H. Chen, S. Liu, W. Xiao, C. Belta, and M. Otte, “Control barrier functions via minkowski operations for safe navigation among polytopic sets,” in2025 IEEE 64th Conference on Decision and Control (CDC), 2025, pp. 4481–4488

    2025

  23. [23]

    Geometry- aware safety-critical local reactive controller for robot navigation in unknown and cluttered environments,

    Y . Li, X. Tang, K. Chen, C. Zheng, H. Liu, and J. Ma, “Geometry- aware safety-critical local reactive controller for robot navigation in unknown and cluttered environments,”IEEE Robotics and Automation Letters, vol. 9, no. 4, pp. 3419–3426, 2024

    2024

  24. [24]

    Computing large convex regions of obstacle-free space through semidefinite programming,

    R. Deits and R. Tedrake, “Computing large convex regions of obstacle-free space through semidefinite programming,” inAlgorithmic Foundations of Robotics XI: Selected Contributions of the Eleventh International Workshop on the Algorithmic Foundations of Robotics, 2015

    2015

  25. [25]

    Planning dynamically feasible trajectories for quadrotors using safe flight corridors in 3-D complex environments,

    S. Liu, M. Watterson, K. Mohta, K. Sun, S. Bhattacharya, C. J. Taylor, and V . Kumar, “Planning dynamically feasible trajectories for quadrotors using safe flight corridors in 3-D complex environments,” IEEE Robotics and Automation Letters, vol. 2, no. 3, pp. 1688–1695, 7 2017

    2017

  26. [26]

    Super- fast configuration-space convex set computation on GPUs for online motion planning,

    P. Werner, R. Cheng, T. Stewart, R. Tedrake, and D. Rus, “Super- fast configuration-space convex set computation on GPUs for online motion planning,” inProceedings of Robotics: Science and Systems, 2025

    2025

  27. [27]

    Fast Iterative Region Inflation for Computing Large 2-D/3-D Convex Regions of Obstacle-Free Space,

    Q. Wang, Z. Wang, M. Wang, J. Ji, Z. Han, T. Wu, R. Jin, Y . Gao, C. Xu, and F. Gao, “Fast Iterative Region Inflation for Computing Large 2-D/3-D Convex Regions of Obstacle-Free Space,”IEEE Trans- actions on Robotics, vol. 41, pp. 3223–3243, 2025

    2025

  28. [28]

    Efficient mixed-integer planning for UA Vs in cluttered environments,

    R. Deits and R. Tedrake, “Efficient mixed-integer planning for UA Vs in cluttered environments,” in2015 IEEE International Conference on Robotics and Automation (ICRA), 2015, pp. 42–49

    2015

  29. [29]

    Fast Path Planning Through Large Collections of Safe Boxes,

    T. Marcucci, P. Nobel, R. Tedrake, and S. Boyd, “Fast Path Planning Through Large Collections of Safe Boxes,”IEEE Transactions on Robotics, vol. 40, pp. 3795–3811, 2024

    2024

  30. [30]

    A Biconvex Method for Minimum-Time Motion Planning Through Sequences of Convex Sets,

    T. Marcucci, M. Halm, W. Yang, D. Lee, and A. D. Marchese, “A Biconvex Method for Minimum-Time Motion Planning Through Sequences of Convex Sets,” inProceedings of Robotics: Science and Systems, 2025

    2025

  31. [31]

    Sequential convex programming methods for real-time optimal trajectory plan- ning in autonomous vehicle racing,

    P. Scheffe, T. M. Henneken, M. Kloock, and B. Alrifaee, “Sequential convex programming methods for real-time optimal trajectory plan- ning in autonomous vehicle racing,”IEEE Transactions on Intelligent Vehicles, vol. 8, no. 1, pp. 661–672, 2023

    2023

  32. [32]

    Towards Op- timizing a Convex Cover of Collision-Free Space for Trajectory Generation,

    Y . Wu, I. Spasojevic, P. Chaudhari, and V . Kumar, “Towards Op- timizing a Convex Cover of Collision-Free Space for Trajectory Generation,”IEEE Robotics and Automation Letters, vol. 10, no. 5, pp. 4762–4769, 2025

    2025

  33. [33]

    Corridor-based adaptive control barrier & lyapunov functions for safe mobile robot navigation,

    N. Mohammad and N. Bezzo, “Corridor-based adaptive control barrier & lyapunov functions for safe mobile robot navigation,” in2025 IEEE 64th Conference on Decision and Control (CDC), 2025, pp. 3107– 3114

    2025

  34. [34]

    Discrete-time control barrier function: High-order case and adaptive case,

    Y . Xiong, D.-H. Zhai, M. Tavakoli, and Y . Xia, “Discrete-time control barrier function: High-order case and adaptive case,”IEEE Transac- tions on Cybernetics, vol. 53, no. 5, pp. 3231–3239, 2023

    2023

  35. [35]

    On the implementation of an interior- point filter line-search algorithm for large-scale nonlinear program- ming,

    A. W ¨achter and L. T. Biegler, “On the implementation of an interior- point filter line-search algorithm for large-scale nonlinear program- ming,”Mathematical programming, vol. 106, pp. 25–57, 2006

    2006

  36. [36]

    Fatrop: A fast constrained optimal control problem solver for robot trajectory optimization and control,

    L. Vanroye, A. Sathya, J. De Schutter, and W. Decr ´e, “Fatrop: A fast constrained optimal control problem solver for robot trajectory optimization and control,” in2023 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2023

    2023

  37. [37]

    Nocedal and S

    J. Nocedal and S. Wright,Numerical Optimization, 2nd ed., ser. Springer Series in Operations Research and Financial Engineering. New York, NY: Springer New York, 2006

    2006

  38. [38]

    CasADi – A software framework for nonlinear optimization and optimal control,

    J. A. E. Andersson, J. Gillis, G. Horn, J. B. Rawlings, and M. Diehl, “CasADi – A software framework for nonlinear optimization and optimal control,”Mathematical Programming Computation, vol. 11, no. 1, pp. 1–36, 2019

    2019

  39. [39]

    Design, modeling, and nonlinear model predictive tracking control of a novel autonomous surface vehicle,

    W. Wang, L. Mateos, S. Park, P. Leoni, B. Gheneti, F. Duarte, C. Ratti, and D. Rus, “Design, modeling, and nonlinear model predictive tracking control of a novel autonomous surface vehicle,” inProc. 2018 IEEE Int. Conf. Robot. Autom, 2018, pp. 6189–6196

    2018

  40. [40]

    ROS: an open-source robot operating system,

    M. Quigley, B. Gerkey, K. Conley, J. Faust, T. Foote, J. Leibs, E. Berger, R. Wheeler, and A. Ng, “ROS: an open-source robot operating system,” inProc. of the IEEE Intl. Conf. on Robotics and Automation (ICRA) Workshop on Open Source Robotics, 2009

    2009

  41. [41]

    Design and use paradigms for gazebo, an open-source multi-robot simulator,

    N. Koenig and A. Howard, “Design and use paradigms for gazebo, an open-source multi-robot simulator,” in2004 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) (IEEE Cat. No.04CH37566), vol. 3, 2004, pp. 2149–2154 vol.3

    2004

  42. [42]

    A receding horizon multi-objective planner for autonomous surface vehicles in urban waterways,

    T. Shan, W. Wang, B. Englot, C. Ratti, and D. Rus, “A receding horizon multi-objective planner for autonomous surface vehicles in urban waterways,” in2020 59th IEEE Conference on Decision and Control (CDC), 2020, pp. 4085–4092

    2020

  43. [43]

    Robot operating system 2: Design, architecture, and uses in the wild,

    S. Macenski, T. Foote, B. Gerkey, C. Lalancette, and W. Woodall, “Robot operating system 2: Design, architecture, and uses in the wild,” Science robotics, vol. 7, no. 66, 2022

    2022