arxiv: 2605.10738 · v1 · submitted 2026-05-11 · 🧮 math.OC · cs.MA· cs.RO· cs.SY· eess.SY

Recognition: 2 theorem links

· Lean Theorem

Decentralized Contingency MPC based on Safe Sets for Nonlinear Multi-agent Collision Avoidance

Max Studt , Georg Schildbach

Authors on Pith no claims yet

Pith reviewed 2026-05-12 03:51 UTC · model grok-4.3

classification 🧮 math.OC cs.MAcs.ROcs.SYeess.SY

keywords decentralized MPCmulti-agent collision avoidancesafe setscontingency planningnonlinear dynamicsrecursive feasibilityLyapunov convergencestate-only information

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The pith

A decentralized contingency MPC with safe sets guarantees recursive feasibility and collision avoidance for nonlinear multi-agent systems under state-only information.

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

The paper establishes a framework in which each agent independently solves a local MPC problem that pairs a nominal trajectory with a contingency certificate for a feasible backup maneuver. This is supported by a geometric decentralized mechanism for updating safe sets that preserves feasibility properties across receding-horizon steps without any exchange of planned trajectories. The approach yields collision-free motion for agents obeying nonlinear dynamics while all agents follow identical consensual rules. A reader would care because existing decentralized methods often sacrifice either safety guarantees or the ability to operate without communication, and this scheme supplies both recursive feasibility and a convergence result.

Core claim

The decentralized contingency MPC scheme based on safe sets guarantees recursive feasibility, including collision avoidance, and establishes a Lyapunov-type convergence result to an admissible safe equilibrium for nonlinear multi-agent systems operating under a state-only information pattern.

What carries the argument

The geometric and decentralized safe-set update mechanism that couples with contingency certificates in each agent's local optimization to maintain feasible backup maneuvers.

If this is right

Collision avoidance holds across time steps even when agents enter or leave the group.
The same rule set produces safe motion in both sparse and bottleneck environments without trajectory sharing.
Lyapunov-type analysis shows convergence to an admissible safe equilibrium under the decentralized updates.
Recursive feasibility is retained by the safe-set mechanism between consecutive optimization steps.

Where Pith is reading between the lines

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

The method could be tested for robustness when state observations contain noise or delays.
It opens a route to combine the safe-set construction with online adaptation for time-varying environments.
The guarantees suggest applicability to other multi-agent tasks requiring backup plans, such as coordinated transport.
Performance scaling could be examined by increasing agent count beyond the reported simulations.

Load-bearing premise

Agents can compute and update safe sets in a decentralized manner while preserving the properties required for the contingency certificates, and the nonlinear dynamics admit feasible backup maneuvers from state information alone.

What would settle it

A counter-example in which safe-set updates are applied yet recursive feasibility is lost or a collision occurs in a dense multi-agent scenario with the given nonlinear dynamics would falsify the guarantees.

Figures

Figures reproduced from arXiv: 2605.10738 by Georg Schildbach, Max Studt.

**Figure 2.** Figure 2: Loss of recursive feasibility without FoS. Although the local MPC problems are feasible at time t with disjoint active safe sets S ∗ i (t) (red), the naive update to t + yields sets S ∗ i (t +) (dashed) that are no longer pairwise disjoint. The resulting overlaps can violate the safe-set constraints and render the next-step problems infeasible. In the following, the closed-loop execution of the proposed d… view at source ↗

**Figure 5.** Figure 5: Closed-loop trajectories in the bottleneck scenario. [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

**Figure 3.** Figure 3: Representative closed-loop trajectories in the increas [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

**Figure 6.** Figure 6: Distance evolution in the bottleneck scenario. Top: pairwise distances between agents; the dashed line indicates the minimum admissible distance. Bottom: distances between agent safe sets; values close to zero indicate strong interaction and freeze-operator activation [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗

**Figure 7.** Figure 7: Number of frozen agents over time in the [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗

**Figure 8.** Figure 8: PnP scenario with online agent insertion and removal. Left: Closed-loop trajectories in a cluttered workspace (gray obstacles). Filled circles denote initial positions and stars denote final positions; colors correspond to individual agents. Right: Activity timeline indicating when each agent is active/inactive; vertical dashed lines mark join/leave events. The proposed scheme maintains safe, collision-f… view at source ↗

read the original abstract

Decentralized collision avoidance remains challenging, particularly when agents do not communicate any information related to planned trajectories. Most existing approaches either rely on conservative coordination mechanisms or provide limited guarantees on recursive feasibility and convergence. This paper develops a decentralized contingency MPC framework for multi-agent systems with nonlinear dynamics that achieves collision-free motion under a state-only information pattern. Each agent follows the same consensual rule set, enabling safe decentralized planning without communication. Each agent solves a local optimization problem that couples a nominal trajectory with a contingency certificate ensuring a feasible backup maneuver under receding-horizon operation. A novel geometric and decentralized safe-set update mechanism prevents feasibility loss between consecutive time steps. The resulting scheme guarantees recursive feasibility, including collision avoidance, and establishes a Lyapunov-type convergence result to an admissible safe equilibrium. Simulation results demonstrate performance in both sparse and dense multi-agent environments, including cluttered bottleneck scenarios and under plug-and-play operation.

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.

Desk Editor's Note private letter to a colleague

This paper gives a decentralized contingency MPC for nonlinear agents to avoid collisions with no communication, using local geometric safe-set updates to hold recursive feasibility and Lyapunov convergence.

read the letter

The main point is a workable no-communication scheme for multi-agent collision avoidance in nonlinear systems. Each agent runs its own contingency MPC that includes a backup plan, and they all apply the same geometric safe-set update rule based only on observed states to stop feasibility from dropping between steps. This produces the claimed recursive feasibility for collision-free motion plus convergence to a safe equilibrium. Simulations cover sparse and dense cases, bottlenecks, and plug-and-play, which is useful to see in practice. The approach extends standard MPC and safe-set ideas to a setting that usually forces either communication or heavy conservatism, and the consensual rules keep agents consistent without extra messages. That combination is the actual novelty here. The soft spot sits in the geometric update itself. For nonlinear dynamics the local construction must guarantee that a feasible backup always remains available from state observations alone. If the geometry misses parts of the reachable set or lets non-convexity break invariance in tight spots, the recursive claim weakens even if the simulations look fine. The paper states the guarantees hold, but the derivations need close checking on how the update interacts with the full nonlinear evolution. This work is aimed at people doing decentralized control for robotics or autonomous transport. Readers who need concrete guarantees without communication will get the most from the framework and the sim results. It has enough technical substance and evidence to go to a serious referee rather than a desk reject.

Referee Report

2 major / 2 minor

Summary. The paper proposes a decentralized contingency MPC scheme for nonlinear multi-agent collision avoidance under a state-only information pattern. Each agent solves a local optimization coupling a nominal trajectory with a contingency certificate for a feasible backup maneuver; a novel geometric decentralized safe-set update is introduced to maintain feasibility across receding-horizon steps. The central claims are recursive feasibility (including collision avoidance) and a Lyapunov-type convergence result to an admissible safe equilibrium, supported by simulations in sparse, dense, and bottleneck scenarios with plug-and-play operation.

Significance. If the recursive-feasibility and convergence guarantees can be established rigorously for nonlinear dynamics, the work would represent a meaningful advance in communication-free multi-agent control. The combination of contingency certificates with a geometric safe-set update offers a concrete mechanism that could reduce conservatism relative to purely reactive or fully centralized approaches, and the plug-and-play simulation results suggest practical relevance for robotics applications.

major comments (2)

[Section 4 (safe-set update mechanism) and proof of recursive feasibility] The recursive feasibility guarantee (abstract and the main theorem on feasibility) rests on the claim that the decentralized geometric safe-set update preserves a feasible backup maneuver for nonlinear dynamics using only observed states. However, the construction does not appear to explicitly compute or over-approximate the reachable set under the nonlinear vector field between updates; without this, small state perturbations in dense or bottleneck configurations can eliminate the contingency certificate, violating the invariance property required for recursive feasibility.
[Theorem on convergence / Section 5] The Lyapunov-type convergence result assumes the existence of an admissible safe equilibrium that remains reachable under the consensual decentralized rule set. For nonlinear dynamics, the proof sketch does not address how the state-only information pattern and the geometric update together guarantee that the equilibrium set is non-empty and invariant when agents enter or leave the formation (plug-and-play).

minor comments (2)

[Section 3] Notation for the contingency certificate and the safe-set update should be introduced with explicit definitions of all sets and operators before their first use in the algorithm description.
[Section 6] Simulation figures would benefit from explicit reporting of the number of Monte-Carlo runs, the distribution of initial conditions, and quantitative metrics (e.g., minimum inter-agent distance over time) rather than qualitative descriptions of “dense” and “bottleneck” scenarios.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. The comments highlight important aspects of the recursive feasibility and convergence proofs for nonlinear systems, which we will address in the revision to strengthen the rigor of our claims.

read point-by-point responses

Referee: [Section 4 (safe-set update mechanism) and proof of recursive feasibility] The recursive feasibility guarantee (abstract and the main theorem on feasibility) rests on the claim that the decentralized geometric safe-set update preserves a feasible backup maneuver for nonlinear dynamics using only observed states. However, the construction does not appear to explicitly compute or over-approximate the reachable set under the nonlinear vector field between updates; without this, small state perturbations in dense or bottleneck configurations can eliminate the contingency certificate, violating the invariance property required for recursive feasibility.

Authors: We are grateful for this detailed observation. Our geometric safe-set update is constructed to ensure that the contingency certificate remains feasible by updating the safe sets based on observed states in a decentralized manner that accounts for the nonlinear dynamics through conservative geometric operations. However, we acknowledge that the current manuscript does not explicitly include an over-approximation of the reachable set. To address this, we will revise Section 4 to include a new lemma that derives a reachable set over-approximation using the local Lipschitz constant of the vector field and the geometry of the safe sets. This will demonstrate that the update preserves the invariance property even under small perturbations, particularly in dense and bottleneck scenarios. We believe this addition will clarify the proof. revision: partial
Referee: [Theorem on convergence / Section 5] The Lyapunov-type convergence result assumes the existence of an admissible safe equilibrium that remains reachable under the consensual decentralized rule set. For nonlinear dynamics, the proof sketch does not address how the state-only information pattern and the geometric update together guarantee that the equilibrium set is non-empty and invariant when agents enter or leave the formation (plug-and-play).

Authors: Thank you for this comment. The convergence result relies on the existence of an admissible safe equilibrium, which is guaranteed by the initial feasibility and the properties of the safe sets. Regarding plug-and-play, the geometric update and consensual rules are intended to maintain the non-emptiness of the equilibrium set by allowing agents to adjust their safe sets upon detecting new or departing agents via state observations. Nevertheless, we agree that the current proof sketch is brief on this point for nonlinear dynamics. In the revised manuscript, we will expand Section 5 with a detailed proof of invariance of the equilibrium set under plug-and-play, showing that the state-only information and geometric updates ensure the set remains non-empty and that the Lyapunov function continues to decrease towards it. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained with no circular reductions

full rationale

The paper derives recursive feasibility (including collision avoidance) and Lyapunov-type convergence from the proposed decentralized geometric safe-set update mechanism applied to contingency MPC. This construction is presented as novel and is used to establish the invariance and feasibility properties between receding-horizon steps. No steps reduce by construction to fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations whose validity depends on the current result. The framework builds on standard MPC and safe-set concepts but the central guarantees follow from the explicit update rule and certificate structure without circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The approach relies on standard assumptions in MPC for nonlinear systems and introduces new concepts like the safe-set update and contingency certificates without external validation in the abstract.

axioms (2)

domain assumption Nonlinear dynamics of agents allow for feasible contingency maneuvers
Assumed for the MPC optimization to have solutions.
ad hoc to paper Safe sets can be updated in a decentralized geometric manner without losing feasibility
Central to the novel mechanism.

invented entities (2)

Contingency certificate no independent evidence
purpose: Ensuring feasible backup maneuver
Introduced as part of the local optimization problem.
Decentralized safe-set update mechanism no independent evidence
purpose: Preventing feasibility loss between time steps
Novel geometric mechanism proposed.

pith-pipeline@v0.9.0 · 5461 in / 1470 out tokens · 49493 ms · 2026-05-12T03:51:32.416848+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

novel geometric and decentralized safe-set update mechanism prevents feasibility loss... FoS update rule... pairwise disjointness of the active safe sets
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Lyapunov-type convergence result... Jc_i(t) ≤ Ĵc_i(t) ... shifted-tail bound

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

[1]

Alsterda, Matthew Brown, and J

John P. Alsterda, Matthew Brown, and J. Christian Gerdes. Contingency model predictive control for automated vehicles. InProceedings of the American Control Conference, 2019

work page 2019
[2]

Control Barrier Functions: Theory and Applications

Aaron D. Ames, Samuel Coogan, Magnus Egerstedt, Gennaro Notomista, Koushil Sreenath, and Paulo Tabuada. Control barrier functions: Theory and applications. arXiv preprint arXiv:1903.11199, 2019

work page Pith review arXiv 1903
[3]

Stability and robustness of distributed suboptimal model predictive control.IF AC-PapersOnLine, 56(2):5115–5120, 2023

Giuseppe Belgioioso, Dominic Liao-McPherson, Mathias Hudoba de Badyn, Nicolas Pelzmann, John Lygeros, and Florian D¨ orfler. Stability and robustness of distributed suboptimal model predictive control.IF AC-PapersOnLine, 56(2):5115–5120, 2023. 22nd IFAC World Congress

work page 2023
[4]

Chen and F

H. Chen and F. Allg¨ ower. A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica, 34(10):1205–1217, 1998

work page 1998
[5]

Motion planning in dynamic environments using velocity obstacles.The International Journal of Robotics Research, 17(7):760–772, 1998

Paolo Fiorini and Zvi Shiller. Motion planning in dynamic environments using velocity obstacles.The International Journal of Robotics Research, 17(7):760–772, 1998

work page 1998
[6]

Matveev, and Andrey V

Michael Hoy, Alexey S. Matveev, and Andrey V. Savkin. Collision free cooperative navigation of multiple wheeled robots in unknown cluttered environments.Robotics and Autonomous Systems, 60(10):1253–1266, 2012

work page 2012
[7]

Distributed optimization using aladin for mpc in smart grids.IEEE Transactions on Control Systems Technology, 29(5):2142–2152, 2021

Yuning Jiang, Philipp Sauerteig, Boris Houska, and Karl Worthmann. Distributed optimization using aladin for mpc in smart grids.IEEE Transactions on Control Systems Technology, 29(5):2142–2152, 2021

work page 2021
[8]

M¨ uller, and Frank Allg¨ ower

Johannes K¨ ohler, Matthias A. M¨ uller, and Frank Allg¨ ower. Distributed model predictive control—recursive feasibility under inexact dual optimization.Automatica, 102:1–9, 2019

work page 2019
[9]

Nonlinear MPC for collision-free and deadlock-free navigation of multiple nonholonomic mobile robots.Robotics and Autonomous Systems, 141:103774, 2021

Amir Salimi Lafmejani and Spring Berman. Nonlinear MPC for collision-free and deadlock-free navigation of multiple nonholonomic mobile robots.Robotics and Autonomous Systems, 141:103774, 2021

work page 2021
[10]

Merry-go-round: Safe control of decentralized multi-robot systems with deadlock prevention

Wonjong Lee, Joonyeol Sim, Joonkyung Kim, Siwon Jo, Wenhao Luo, and Changjoo Nam. Merry-go-round: Safe control of decentralized multi-robot systems with deadlock prevention. In2025 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 4589–4595, 2025

work page 2025
[11]

Contract- based predictive control of distributed systems with plug and play capabilities.IF AC-PapersOnLine, 48(23):205–211,

Sergio Lucia, Markus K¨ ogel, and Rolf Findeisen. Contract- based predictive control of distributed systems with plug and play capabilities.IF AC-PapersOnLine, 48(23):205–211,

work page
[12]

5th IFAC Conference on Nonlinear Model Predictive Control NMPC 2015

work page 2015
[13]

Bookmundo Direct (Edition MoRa), 2 edition, 2022

Jan Lunze.Networked Control of Multi-Agent Systems. Bookmundo Direct (Edition MoRa), 2 edition, 2022

work page 2022
[14]

Maestre, D

J.M. Maestre, D. Mu˜ noz de la Pe˜ na, E.F. Camacho, and T. Alamo. Distributed model predictive control based on agent negotiation.Journal of Process Control, 21(5):685–697,

work page
[15]

Special Issue on Hierarchical and Distributed Model Predictive Control

work page
[16]

Mayne, James B

David Q. Mayne, James B. Rawlings, Christopher V. Rao, and Pierre O. M. Scokaert. Constrained model predictive control: Stability and optimality.Automatica, 36(6):789–814, 2000

work page 2000
[17]

Negenborn, Bart De Schutter, and Jacob Hellendoorn

Rudy R. Negenborn, Bart De Schutter, and Jacob Hellendoorn. Multi-agent model predictive control: A survey. arXiv preprint arXiv:0908.1076, 2009

work page arXiv 2009
[18]

Richards, Tom Schouwenaars, Jonathan P

Arthur G. Richards, Tom Schouwenaars, Jonathan P. How, and Eric Feron. Spacecraft trajectory planning with avoidance constraints using mixed-integer linear programming.Journal of Guidance, Control, and Dynamics, 25(4):755–764, 2002

work page 2002
[19]

Plug-and-play decentralized model predictive control for linear systems.IEEE Transactions on Automatic Control, 58(10):2608–2614, 2013

Stefano Riverso, Marcello Farina, and Giancarlo Ferrari- Trecate. Plug-and-play decentralized model predictive control for linear systems.IEEE Transactions on Automatic Control, 58(10):2608–2614, 2013

work page 2013
[20]

Zeilinger, and Andrea Carron

Danilo Saccani, Lorenzo Fagiano, Melanie N. Zeilinger, and Andrea Carron. Model predictive control for multi- agent systems under limited communication and time-varying network topology. arXiv preprint arXiv:2304.01649, 2023

work page arXiv 2023
[21]

Architecture of distributed and hierarchical model predictive control—a review.Journal of Process Control, 19(5):723–731, 2009

Riccardo Scattolini. Architecture of distributed and hierarchical model predictive control—a review.Journal of Process Control, 19(5):723–731, 2009

work page 2009
[22]

Contingency Model-based Control (CMC) for Communicationless Cooperative Collision Avoidance in Robot Swarms, 2025

Georg Schildbach. Contingency Model-based Control (CMC) for Communicationless Cooperative Collision Avoidance in Robot Swarms, 2025. arXiv preprint arXiv:2512.20391

work page arXiv 2025
[23]

Tom Schouwenaars, Bart De Moor, Eric Feron, and Jonathan P. How. Mixed integer programming for multi- vehicle path planning. InProceedings of the European Control Conference (ECC), pages 2603–2608, 2001

work page 2001
[24]

Valenti, Eric Feron, and Jonathan P

Tom Schouwenaars, Mario J. Valenti, Eric Feron, and Jonathan P. How. Implementation and flight test results of MILP-based UAV guidance.2005 IEEE Aerospace Conference, pages 1–13, 2005

work page 2005
[25]

Stewart, Aswin N

Brett T. Stewart, Aswin N. Venkat, James B. Rawlings, Stephen J. Wright, and Gabriele Pannocchia. Cooperative distributed model predictive control.Systems & Control Letters, 59(8):460–469, 2010

work page 2010
[26]

Plug & play control: Control technology towards new challenges.European Journal of Control, 15(3– 4):311–330, 2009

Jakob Stoustrup. Plug & play control: Control technology towards new challenges.European Journal of Control, 15(3– 4):311–330, 2009

work page 2009
[27]

Guy, Ming Lin, and Dinesh Manocha

Jur van den Berg, Stephen J. Guy, Ming Lin, and Dinesh Manocha. Reciprocal n-body collision avoidance.The International Journal of Robotics Research, 30(4):371–383, 2011

work page 2011
[28]

Venkat, James B

Aswin N. Venkat, James B. Rawlings, and Stephen J. Wright. Distributed model predictive control of large-scale systems. InAssessment and Future Directions of Nonlinear Model Predictive Control, volume 358 ofLecture Notes in Control and Information Sciences. Springer, 2007

work page 2007
[29]

Zeilinger

Kim Peter Wabersich and Melanie N. Zeilinger. A predictive safety filter for learning-based control of constrained nonlinear dynamical systems.Automatica, 129:109597, 2021

work page 2021
[30]

W¨ achter and L

A. W¨ achter and L. T. Biegler. On the implementation of an interior-point filter line-search algorithm for large- scale nonlinear programming.Mathematical Programming, 106(1):25–57, 2006. 17

work page 2006