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arxiv: 2603.28965 · v2 · submitted 2026-03-30 · 📡 eess.SY · cs.RO· cs.SY· math.DS

Koopman Operator Framework for Modeling and Control of Off-Road Vehicle on Deformable Terrain

Kartik Loya , Phanindra Tallapragada This is my paper

Pith reviewed 2026-05-14 21:02 UTC · model grok-4.3

classification 📡 eess.SY cs.ROcs.SYmath.DS
keywords Koopman operatordeformable terrainmodel predictive controloff-road vehiclesterramechanicssubspace identificationhybrid modeling
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The pith

Koopman operators learned from Bekker-Wong simulations yield linear predictors that support constrained MPC for off-road vehicles on deformable terrain.

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

This paper constructs a hybrid framework that turns complex vehicle-terrain interactions into linear models suitable for real-time control. Simulations based on Bekker-Wong theory generate large datasets for a five-degree-of-freedom vehicle on sandy loam and clay; recursive subspace identification then extracts Koopman operators from the most informative segments using Grassmannian distance. The resulting linear system delivers stable short-horizon predictions that remain robust under mild terrain variations. When placed inside a constrained model predictive controller, the operator enables tracking of aggressive maneuvers while respecting steering and torque limits. The same operator can be updated directly with physical measurements, preserving the hybrid physics-informed character of the approach.

Core claim

A Koopman linear system identified from simulation data of a 5-DOF vehicle on Bekker-Wong deformable terrain produces short-horizon predictions of sufficient accuracy that, when embedded in constrained MPC, permit stable closed-loop tracking of aggressive maneuvers while satisfying actuator limits; the operator can be updated seamlessly with real-system data.

What carries the argument

The Koopman operator obtained via recursive subspace identification on simulation trajectories, with Grassmannian distance used to select informative data segments, that converts the nonlinear terramechanics into an equivalent linear predictor.

Load-bearing premise

The operator extracted from simulations will generalize to real deformable terrain and can be updated with physical data without loss of accuracy or stability.

What would settle it

Large prediction errors or loss of closed-loop stability when the MPC controller is deployed on a physical vehicle driven across actual sandy loam or clay surfaces rather than the simulated terrain.

read the original abstract

This work presents a hybrid physics-informed and data-driven modeling framework for predictive control of autonomous off-road vehicles operating on deformable terrain. Traditional high-fidelity terramechanics models are often too computationally demanding to be directly used in control design. Modern Koopman operator methods can be used to represent the complex terramechanics and vehicle dynamics in a linear form. We develop a framework whereby a Koopman linear system can be constructed using data from simulations of a vehicle moving on deformable terrain. For vehicle simulations, the deformable-terrain terramechanics are modeled using Bekker-Wong theory, and the vehicle is represented as a simplified five-degree-of-freedom (5-DOF) system. The Koopman operators are identified from large simulation datasets for sandy loam and clay using a recursive subspace identification method, where Grassmannian distance is used to prioritize informative data segments during training. The advantage of this approach is that the Koopman operator learned from simulations can be updated with data from the physical system in a seamless manner, making this a hybrid physics-informed and data-driven approach. Prediction results demonstrate stable short-horizon accuracy and robustness under mild terrain-height variations. When embedded in a constrained MPC, the learned predictor enables stable closed-loop tracking of aggressive maneuvers while satisfying steering and torque limits.

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

3 major / 2 minor

Summary. The paper proposes a hybrid physics-informed and data-driven framework for modeling and controlling off-road vehicles on deformable terrain. It simulates a 5-DOF vehicle using Bekker-Wong terramechanics on sandy loam and clay, identifies Koopman operators from these trajectories via recursive subspace identification with Grassmannian distance for data selection, and embeds the resulting linear predictor in constrained MPC to achieve stable closed-loop tracking of aggressive maneuvers while respecting steering and torque limits.

Significance. If validated beyond simulation, the approach would offer a computationally tractable linear surrogate for complex terramechanics that supports real-time MPC, addressing a key bottleneck in off-road autonomy. The hybrid update mechanism and Grassmannian-informed data selection are methodological strengths that could enable seamless incorporation of physical measurements.

major comments (3)
  1. [Abstract] Abstract: the central claim of 'stable short-horizon accuracy' and 'robustness under mild terrain-height variations' is asserted without any quantitative error metrics (e.g., RMSE, prediction horizon length, or variance across trajectories), baseline comparisons, or statistical validation, rendering the accuracy statement unassessable.
  2. [Abstract] Abstract: the assertion that the learned operator 'can be updated with data from the physical system in a seamless manner' is load-bearing for the hybrid claim yet unsupported by any transfer experiment, online-update analysis, or discussion of how Bekker-Wong idealizations (homogeneous soil, no sensor noise) map to real heterogeneous terrain and slip dynamics.
  3. [Abstract] Abstract: all closed-loop MPC results (stable tracking while satisfying limits) are generated inside the identical simulation environment used for training; no cross-validation against model mismatch, real-vehicle data, or soil-parameter variation is reported, which directly undermines the generalization required for the strongest claim.
minor comments (2)
  1. Specify the exact lifting functions chosen for the Koopman embedding and justify their selection relative to the 5-DOF states.
  2. Clarify the implementation details of the recursive subspace identification (e.g., window length, forgetting factor) and the precise Grassmannian-distance threshold used for data prioritization.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. We address each major comment point by point below, indicating whether revisions have been made. Our responses focus on clarifying the scope of the simulation-based study while strengthening the presentation where possible.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim of 'stable short-horizon accuracy' and 'robustness under mild terrain-height variations' is asserted without any quantitative error metrics (e.g., RMSE, prediction horizon length, or variance across trajectories), baseline comparisons, or statistical validation, rendering the accuracy statement unassessable.

    Authors: We agree that the abstract would be strengthened by quantitative support. In the revised manuscript, we have updated the abstract to include specific RMSE values for position and velocity predictions over the evaluated short horizons (e.g., 5-10 steps), the exact prediction horizon length, and variance statistics across the test trajectories for both sandy loam and clay. Brief baseline comparisons to standard linear regression models are also noted. These metrics are drawn directly from the simulation results in Section IV. revision: yes

  2. Referee: [Abstract] Abstract: the assertion that the learned operator 'can be updated with data from the physical system in a seamless manner' is load-bearing for the hybrid claim yet unsupported by any transfer experiment, online-update analysis, or discussion of how Bekker-Wong idealizations (homogeneous soil, no sensor noise) map to real heterogeneous terrain and slip dynamics.

    Authors: The recursive subspace identification method with Grassmannian data selection is explicitly designed to support online updates from new data streams. While this work is a simulation study and does not include physical transfer experiments or online-update trials, we have revised the abstract to clarify that the seamless update refers to the algorithmic capability demonstrated in simulation. We have also added a dedicated paragraph in the discussion section addressing the mapping from Bekker-Wong idealizations to real heterogeneous terrain and slip, including acknowledged limitations. revision: partial

  3. Referee: [Abstract] Abstract: all closed-loop MPC results (stable tracking while satisfying limits) are generated inside the identical simulation environment used for training; no cross-validation against model mismatch, real-vehicle data, or soil-parameter variation is reported, which directly undermines the generalization required for the strongest claim.

    Authors: We acknowledge that the closed-loop MPC results are obtained within the same simulation environment used for training data generation. This manuscript presents a simulation-based proof-of-concept for the framework. We have revised the abstract to explicitly state that results are simulation-validated and added text in the conclusions discussing limitations regarding model mismatch and soil-parameter variation, along with plans for future real-vehicle experiments. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained with no circular reductions

full rationale

The paper generates training data from an external terramechanics model (Bekker-Wong theory) applied to a 5-DOF vehicle simulation. The Koopman operator is then identified using recursive subspace identification on these independent datasets, with Grassmannian distance for data selection. Prediction accuracy and MPC performance are evaluated on the learned linear model, but the reported results do not reduce to quantities defined by the same fitted parameters in a self-referential way. No self-citations are invoked as load-bearing for uniqueness or ansatz. The framework is a standard data-driven approximation of the simulation dynamics.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that Bekker-Wong theory generates sufficiently realistic training trajectories and that a finite-dimensional Koopman lift exists that approximates the nonlinear vehicle-terrain dynamics well enough for short-horizon MPC.

axioms (2)
  • domain assumption Bekker-Wong terramechanics theory produces representative simulation data for sandy loam and clay
    Invoked to generate the large simulation datasets used for operator identification.
  • domain assumption A linear Koopman representation can be identified that captures the essential dynamics for short-horizon prediction
    Core premise of the subspace identification step.

pith-pipeline@v0.9.0 · 5539 in / 1411 out tokens · 62179 ms · 2026-05-14T21:02:05.090497+00:00 · methodology

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Reference graph

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