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arxiv: 2605.02361 · v1 · submitted 2026-05-04 · 💻 cs.RO · cs.SY· eess.SY

Feedback Motion Planning for Stochastic Nonlinear Systems with Signal Temporal Logic Specifications

Liqian Ma , Zishun Liu , Glen Chou , Yongxin Chen This is my paper

Pith reviewed 2026-05-08 18:26 UTC · model grok-4.3

classification 💻 cs.RO cs.SYeess.SY
keywords signal temporal logicstochastic motion planningfeedback controlprobabilistic reachable setscontraction theorychance-constrained optimizationnonlinear systemsrobotics
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The pith

Stochastic nonlinear systems satisfy signal temporal logic rules with high probability by eroding predicates using probabilistic reachable tube bounds from contraction theory.

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

The paper develops a complete pipeline for feedback motion planning of continuous-time stochastic nonlinear systems under signal temporal logic specifications. It converts the original chance-constrained stochastic problem into a deterministic trajectory optimization problem whose STL constraints are tightened by a computed amount. The tightening, called predicate erosion, is sized exactly by a probabilistic reachable tube that upper-bounds the deviation between any realized stochastic trajectory and an associated nominal trajectory. The tube itself is obtained from contraction theory together with explicitly designed tracking controllers. When the eroded deterministic problem is solved, the resulting feedback policy guarantees that the original stochastic closed-loop system meets the STL specification with a user-specified high probability such as 99.99 percent.

Core claim

The central claim is that a probabilistic reachable tube computed via contraction theory and feedback tracking controllers supplies a tight bound on trajectory deviation; eroding the STL predicates by this bound converts the intractable stochastic chance-constrained planning problem into a standard deterministic STL trajectory optimization whose solution, when applied in closed loop, ensures the stochastic system satisfies the original specification with high probability.

What carries the argument

The predicate erosion strategy driven by a probabilistic reachable tube (PRT) that bounds the deviation between stochastic trajectories and nominal ones under contraction-based tracking controllers.

If this is right

  • The resulting deterministic optimization can be solved numerically to produce implementable feedback control policies for robotic systems.
  • The same pipeline applies to any continuous-time stochastic nonlinear system for which contraction metrics and tracking controllers can be designed.
  • Simulations on multiple robotic platforms and hardware experiments on a quadrupedal robot achieve higher STL satisfaction probability than representative baseline methods.
  • The approach is reported to be less conservative than direct stochastic or sampling-based alternatives.

Where Pith is reading between the lines

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

  • The method could be used for online replanning if the reachable-tube computation is made fast enough to run repeatedly.
  • Similar erosion ideas might apply to other temporal logic specifications or to systems whose uncertainty is described by different distributions.
  • If the contraction metric is chosen adaptively, the same framework could handle time-varying noise statistics without redesigning the entire pipeline.

Load-bearing premise

The probabilistic reachable tube computed from contraction theory and the chosen tracking controllers must be a valid and sufficiently tight upper bound on the actual deviation between stochastic and nominal trajectories.

What would settle it

Run the closed-loop stochastic system under the computed policy many times and count the fraction of trajectories that satisfy the STL formula; if this empirical probability falls below the target (for example 99.99 percent) on repeated trials, the erosion bound is insufficient.

Figures

Figures reproduced from arXiv: 2605.02361 by Glen Chou, Liqian Ma, Yongxin Chen, Zishun Liu.

Figure 1
Figure 1. Figure 1: Overview. Top: Problem setup. We consider a stochastic nonlinear system subject to a chance-constrained STL specifi￾cation. Middle: Predicate erosion strategy. We characterize the stochastic deviation between the closed-loop stochastic trajec￾tory and its associated deterministic trajectory by a PRT, and use this tube to erode the obstacle and goal predicates. This converts the original chance-constrained … view at source ↗
Figure 2
Figure 2. Figure 2: Nominal plans and stochastic rollouts for the simulated benchmarks. The top row shows the nominal trajectories together with the original and eroded predicates used in planning. The bottom row shows stochastic closed-loop rollouts under the synthesized feedback controller. Across all four systems, the stochastic trajectories satisfy the original STL tasks. A. Benchmarks a) Double Integrator (DI): We first … view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of chance-constraint tightening methods on a linear double-integrator benchmark. (a) Erosion radius versus time under different numbers of discretization steps N. Our method provides a continuous-time bound and is therefore independent of N, whereas both discrete-time baselines become more conservative as N increases. (b) Erosion radius versus time under different risk levels δ. All methods requ… view at source ↗
Figure 4
Figure 4. Figure 4: Quadrupedal robot experiments on the Go1 reach-avoid task. (a) High-level abstraction of the quadruped by a kinematic unicycle model with forward-speed and yaw-rate commands. (b) Nominal plan generated on the unicycle abstraction under the eroded STL specification. (c) Stochastic closed-loop rollouts in simulation under the feedback controller, showing 5000 trajectories of the quadruped base projected onto… view at source ↗
Figure 5
Figure 5. Figure 5: Real Go1 pass-before trajectories. The figure shows 50 hardware executions of the pass-before task under the feedback controller. The corresponding nominal plan and keyframe overlay are shown in view at source ↗
read the original abstract

We study feedback motion planning for continuous-time stochastic nonlinear systems under signal temporal logic (STL) specifications. We propose a framework that synthesizes control policies for chance-constrained STL trajectory optimization problems, with the goal of ensuring that the closed-loop stochastic system satisfies a given STL formula with high probability (e.g., 99.99\%). Our approach is based on a predicate erosion strategy that transforms the intractable stochastic problem into a deterministic STL trajectory optimization problem with tightened STL formula constraints. The amount of erosion is determined by a probabilistic reachable tube (PRT) that bounds the deviation between the stochastic trajectory and an associated nominal trajectory. To compute such bounds, we leverage contraction theory and feedback design, and develop several tracking controllers. This yields a complete feedback motion planning pipeline which can be implemented by numerical optimizations. We demonstrate the efficacy and versatility of the proposed framework through simulations on several robotic systems and through experiments on a real-world quadrupedal robot, and show that it is less conservative and achieves higher specification satisfaction probability than representative baselines.

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

1 major / 2 minor

Summary. The manuscript claims to develop a feedback motion planning framework for continuous-time stochastic nonlinear systems subject to signal temporal logic (STL) specifications. It transforms the intractable stochastic chance-constrained STL problem into a deterministic trajectory optimization problem via a predicate erosion strategy, where the erosion amount is set by a probabilistic reachable tube (PRT) that bounds deviation between the stochastic trajectory and a nominal one. The PRTs are computed using contraction theory together with several designed tracking controllers. The resulting pipeline is implemented numerically and validated in simulations on robotic systems plus hardware experiments on a quadrupedal robot, with claims of lower conservatism and higher STL satisfaction probability than baselines.

Significance. If the PRT bounds hold with the stated tightness for optimized nominal trajectories, the work supplies a practical, implementable method for guaranteeing high-probability STL satisfaction under stochastic disturbances in nonlinear robotic systems. The combination of contraction-theoretic bounds with STL planning, plus real-robot validation, is a concrete strength that could influence feedback design under temporal logic constraints.

major comments (1)
  1. The predicate-erosion reduction requires that the PRT radius r(t) computed from contraction theory remains a valid, sufficiently tight bound on sup_t ||x(t) - x_nom(t)|| for the closed-loop stochastic system when x_nom is the output of the deterministic STL optimizer. The manuscript must explicitly verify that the contraction metric (or differential Lyapunov function) and incremental stability condition are independent of the particular optimized nominal trajectory and feedback law, or else derive a bound that accounts for any such dependence; otherwise the high-probability STL guarantee does not transfer. This step is load-bearing for the central claim.
minor comments (2)
  1. The abstract states that 'several tracking controllers' are developed; the results section should include a direct comparison of their achieved PRT radii, computational cost, and achieved satisfaction probabilities on the same benchmarks.
  2. Clarify the precise class of stochastic disturbances (e.g., bounded support, Gaussian, or otherwise) under which the PRT concentration arguments hold, and state any additional assumptions needed for the tube-propagation step.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive and detailed review. The major comment raises an important point about ensuring the transfer of the PRT bounds to optimized nominal trajectories. We address it below and agree that explicit clarification strengthens the central claim.

read point-by-point responses

  1. Referee: The predicate-erosion reduction requires that the PRT radius r(t) computed from contraction theory remains a valid, sufficiently tight bound on sup_t ||x(t) - x_nom(t)|| for the closed-loop stochastic system when x_nom is the output of the deterministic STL optimizer. The manuscript must explicitly verify that the contraction metric (or differential Lyapunov function) and incremental stability condition are independent of the particular optimized nominal trajectory and feedback law, or else derive a bound that accounts for any such dependence; otherwise the high-probability STL guarantee does not transfer. This step is load-bearing for the central claim.

    Authors: We agree this verification is essential for the soundness of the predicate-erosion approach. In the current manuscript, the contraction metrics and differential Lyapunov functions are derived from the system dynamics and the structure of the tracking controllers (Section IV), which are designed to satisfy the contraction condition uniformly over a compact operating region independent of any particular nominal trajectory. The incremental stability holds for the closed-loop system under these controllers for any sufficiently smooth reference trajectory within the domain, including those generated by the deterministic STL optimizer. The PRT radius r(t) is therefore a valid bound for the deviation under the stochastic dynamics. To address the referee's request explicitly, we will revise the manuscript by adding a dedicated paragraph in Section IV-C (or a short appendix) that states the independence of the metric from the optimized nominal trajectory, recalls the uniform contraction condition, and sketches why the high-probability STL satisfaction transfers directly to the closed-loop stochastic system. This addition does not alter the technical results but makes the load-bearing step fully transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses independent contraction-theoretic bounds

full rationale

The central reduction (stochastic STL satisfaction to deterministic trajectory optimization via predicate erosion) is obtained by computing a PRT bound from contraction theory and separately designed tracking controllers. This bound is then applied to tighten predicates; the construction does not define the PRT radius from the STL optimizer output, nor does it rename a fitted quantity as a prediction. No self-citation is shown to be load-bearing for the uniqueness or validity of the contraction metric, and the abstract presents the tracking controllers as developed within the paper rather than smuggled via prior self-work. The derivation chain therefore remains self-contained against external contraction-theory results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the existence of contraction metrics and feedback controllers that yield computable PRT bounds; these are drawn from prior contraction theory literature rather than newly postulated here. No explicit free parameters or invented entities are named in the abstract.

axioms (1)
  • domain assumption Contraction theory provides differential bounds on trajectory deviation for the stochastic nonlinear system under the designed feedback controllers.
    Invoked to compute the PRT that determines erosion amount.

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