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arxiv: 2605.19015 · v1 · pith:QULA32RRnew · submitted 2026-05-18 · 📡 eess.SY · cs.RO· cs.SY

Probabilistic Recursively Feasible Motion Planning Under Uncertain Environments

Hyeontae Sung , Hyeongchan Ham , Junyoung Park , Kai Ren , Heejin Ahn This is my paper

Pith reviewed 2026-05-20 08:39 UTC · model grok-4.3

classification 📡 eess.SY cs.ROcs.SY
keywords motion planningmodel predictive controlrecursive feasibilityuncertain environmentsprobabilistic constraintssafe set containmentautonomous driving
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The pith

The PRF-MPC framework builds safety constraints that keep the current safe set inside future safe sets with high probability, delivering probabilistic recursive feasibility for motion plans in uncertain environments.

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

Safe motion planning fails recursively when uncertain environments make today's feasible trajectory unsafe at the next step. The paper introduces properties an ideal predictor must satisfy to keep predicted trajectory distributions consistent across time, then uses those properties to write closed-form expressions for the means and covariances of future trajectories. From these expressions it derives explicit safety constraints that enforce, with chosen probability, that the present safe set lies inside all future safe sets. If the claim holds, planners can maintain feasibility guarantees without needing perfect future knowledge of the environment.

Core claim

By requiring a predictor to obey distributional consistency, the authors obtain closed-form means and covariances for trajectories at future time steps. These statistics are inserted into safety constraints that force the current safe set to be contained inside the safe sets at subsequent steps with high probability. The resulting PRF-MPC therefore guarantees recursive feasibility probabilistically rather than deterministically.

What carries the argument

Safety constraints that enforce probabilistic containment of the current safe set inside future safe sets, derived from closed-form trajectory statistics under distributional consistency.

Load-bearing premise

The predictor must obey the distributional consistency properties that permit closed-form calculation of future trajectory means and covariances.

What would settle it

A simulation or experiment in which the predictor violates distributional consistency and the constructed constraints no longer prevent loss of recursive feasibility at the observed probability level.

Figures

Figures reproduced from arXiv: 2605.19015 by Heejin Ahn, Hyeongchan Ham, Hyeontae Sung, Junyoung Park, Kai Ren.

Figure 1
Figure 1. Figure 1: (a) Satisfaction rate of (7) and recursively feasible [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Circular OV modeling and affine safety constraint. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The nominal MPC fails to retain feasibility during [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: For the nominal MPC, the probabilistically safe set [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 4
Figure 4. Figure 4: Safe set inclusion and recursive feasibility. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
read the original abstract

Safe motion planning in uncertain, time-varying environments is challenging because the safe region can change unpredictably across planning steps, often causing a loss of recursive feasibility. In this work, we present a Probabilistic Recursively Feasible Model Predictive Control (PRF-MPC) framework that guarantees recursive feasibility with a specified probability. We introduce properties that an ideal predictor should satisfy to ensure distributional consistency, and use these properties to derive closed-form expressions for the means and covariances of trajectories predicted at future time steps. Building on this analysis, we construct safety constraints that ensure, with high probability, that the current safe set is contained within the safe sets at future time steps, thereby probabilistically guaranteeing recursive feasibility. Simulation results on a lane-change scenario demonstrate that the proposed method significantly improves recursive feasibility.

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

2 major / 2 minor

Summary. The paper proposes a Probabilistic Recursively Feasible Model Predictive Control (PRF-MPC) framework for motion planning in uncertain, time-varying environments. It defines properties that an ideal predictor must satisfy to ensure distributional consistency, derives closed-form expressions for the means and covariances of predicted trajectories at future time steps, and constructs safety constraints that enforce, with probability at least 1-δ, that the current safe set is contained in the predicted future safe sets. This construction is claimed to probabilistically guarantee recursive feasibility. Simulation results on a lane-change scenario are reported to show improved recursive feasibility compared to baselines.

Significance. If the probabilistic recursive-feasibility guarantee can be rigorously established, the result would be significant for safety-critical MPC applications in dynamic uncertain settings, where loss of feasibility is a common failure mode. The derivation of closed-form trajectory statistics under the stated predictor properties is a clear technical strength that could support efficient online implementation. The approach also supplies a concrete mechanism (probabilistic safe-set containment) that could be adapted to other receding-horizon planners.

major comments (2)
  1. [§4] §4 (Safety Constraint Construction) and the paragraph following Eq. (12): the per-step probabilistic containment events are asserted to yield a horizon-wide recursive-feasibility guarantee, yet no explicit chaining argument, union-bound analysis, or inductive step is supplied that controls the accumulated failure probability across repeated invocations of the containment constraint. Without such a bound the overall probability claim does not follow directly from the single-step result.
  2. [§3.2] §3.2 (Predictor Properties) and the derivation of closed-form means/covariances: the distributional-consistency properties are used to obtain tractable expressions, but the manuscript does not quantify how violations of these properties (e.g., due to model mismatch or non-stationary uncertainty) propagate into the containment probability; a sensitivity or robustness analysis would strengthen the central claim.
minor comments (2)
  1. [Abstract and §4] Notation for the safe-set containment probability (1-δ) is introduced without an explicit statement of whether δ is a user-specified design parameter or derived from other quantities; a clarifying sentence would improve readability.
  2. [§5] The simulation section reports qualitative improvement in recursive feasibility but does not tabulate the empirical frequency of feasibility loss or the observed containment probability; adding these quantitative metrics would allow direct comparison with the theoretical 1-δ guarantee.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback on our manuscript. We address each of the major comments below, indicating where revisions will be made to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [§4] §4 (Safety Constraint Construction) and the paragraph following Eq. (12): the per-step probabilistic containment events are asserted to yield a horizon-wide recursive-feasibility guarantee, yet no explicit chaining argument, union-bound analysis, or inductive step is supplied that controls the accumulated failure probability across repeated invocations of the containment constraint. Without such a bound the overall probability claim does not follow directly from the single-step result.

    Authors: We agree with the referee that an explicit argument is required to connect the per-step probabilistic containment to the overall recursive feasibility guarantee. In the revised manuscript, we will introduce a new proposition in Section 4 that uses induction over the planning horizon to show that if each containment event holds with probability at least 1-δ, then the recursive feasibility is preserved with probability at least 1 - Tδ, where T is the horizon length, via a union bound. This will make the probability claim rigorous. revision: yes

  2. Referee: [§3.2] §3.2 (Predictor Properties) and the derivation of closed-form means/covariances: the distributional-consistency properties are used to obtain tractable expressions, but the manuscript does not quantify how violations of these properties (e.g., due to model mismatch or non-stationary uncertainty) propagate into the containment probability; a sensitivity or robustness analysis would strengthen the central claim.

    Authors: The distributional consistency properties are presented as ideal conditions that the predictor must satisfy for the closed-form derivations to hold exactly. We acknowledge that in practice, model mismatch could affect the containment probability. We will add a subsection discussing this limitation and include numerical sensitivity experiments in the simulations section to illustrate the effect of small violations on the observed feasibility rates. A complete theoretical sensitivity analysis is left for future work as it would require additional assumptions on the nature of the mismatch. revision: partial

Circularity Check

0 steps flagged

Derivation self-contained; no circular reductions identified

full rationale

The paper introduces predictor properties for distributional consistency, derives closed-form mean/covariance expressions from those properties, and then builds per-step probabilistic containment constraints on the resulting safe-set predictions. These steps form a forward chain from stated assumptions to the recursive-feasibility claim without any quoted equation or construction that reduces the target guarantee to a fitted parameter, self-definition, or prior self-citation by construction. The abstract and described framework treat the overall probability bound as following from the containment constraints and predictor assumptions rather than presupposing the result. No load-bearing self-citation, ansatz smuggling, or renaming of known results appears in the provided derivation outline.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central construction rests on the domain assumption that a predictor satisfying distributional consistency exists and can be used to obtain closed-form statistics; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption An ideal predictor satisfies properties that ensure distributional consistency across planning steps.
    Invoked to derive closed-form means and covariances of future trajectories.

pith-pipeline@v0.9.0 · 5674 in / 1238 out tokens · 58995 ms · 2026-05-20T08:39:51.941794+00:00 · methodology

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