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arxiv: 2606.18630 · v1 · pith:QDGMFL6Enew · submitted 2026-06-17 · 💻 cs.RO

DNN Koopman-Based Deviation Compensation for UGV Path Tracking Control on Coupled Slope and Potholed Road

Jian Zhao , Wenbo Zhou , Zhicheng Chen , Bing Zhu , Jiayi Han , Dongjian Song , Yinju Lin , Peixing Zhang This is my paper

Pith reviewed 2026-06-26 21:07 UTC · model grok-4.3

classification 💻 cs.RO
keywords UGV path trackingKoopman operatorDNN compensationevent-triggered controloff-road terrainLaguerre MPCtire stiffness estimationdeviation compensation
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The pith

A DNN Koopman deviation compensator paired with Laguerre MPC and event-triggered activation improves UGV path tracking by more than 11.5 percent on coupled slopes and potholed roads.

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

The paper establishes a control architecture that first estimates tire cornering stiffness adaptively on slopes, then applies Laguerre-based model predictive control for reliable slope tracking with lower computation, and finally adds a DNN Koopman model to correct path deviations induced by potholes. An event-triggered parallel cooperative mechanism decides when to apply the compensation and verifies its credibility so that the combined steering command stays feasible and the vehicle remains stable. Hardware-in-the-loop tests across multiple conditions show the combined strategy reduces tracking error by more than 11.5 percent, which matters for any UGV that must maintain accurate paths without constant high-fidelity terrain sensing or excessive onboard compute.

Core claim

By integrating Koopman operator theory with a deep neural network, the DNN Koopman method models pothole-induced path deviations and, when coupled to the LMPC through an event-triggered parallel cooperative mechanism based on activation criteria and credibility verification, raises path tracking accuracy on potholed roads while keeping overall steering commands feasible and vehicle dynamics stable.

What carries the argument

The DNN Koopman (DK) path deviation compensation method, which learns a lifted linear representation of nonlinear pothole effects to generate corrective actions that are selectively applied by the event-triggered mechanism.

If this is right

  • Laguerre functions embedded in the MPC reduce the number of decision variables while preserving tracking on coupled slopes.
  • The adaptive forgetting recursive least squares estimator supplies updated tire stiffness values that keep the LMPC model accurate across slope changes.
  • The credibility verification step in the EPC mechanism prevents unsafe compensation activations.
  • Overall closed-loop tracking error decreases by more than 11.5 percent relative to the baseline LMPC alone across the tested operating conditions.

Where Pith is reading between the lines

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

  • If the Koopman lifting functions prove reusable across vehicle platforms, the same DK compensator could be transferred with modest retraining.
  • The event-triggered structure suggests a natural way to layer additional disturbance-specific compensators without continuous computational overhead.
  • Real-vehicle validation on physical potholed slopes would test whether HiL results translate when actuator delays and sensor noise are present.

Load-bearing premise

The DNN Koopman model trained on the collected data will produce compensation signals that remain within stability and feasibility bounds for pothole disturbances not seen during training.

What would settle it

A hardware-in-the-loop run on a previously unseen pothole pattern in which the event-triggered DK activation produces a steering command that drives lateral error outside the LMPC feasible set or causes yaw-rate instability.

Figures

Figures reproduced from arXiv: 2606.18630 by Bing Zhu, Dongjian Song, Jian Zhao, Jiayi Han, Peixing Zhang, Wenbo Zhou, Yinju Lin, Zhicheng Chen.

Figure 1
Figure 1. Figure 1: Schematic diagram of UGV driving on coupled slope [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic diagram of vehicle and tire force 𝑅𝑂→𝑜 𝑝 = ⎡ ⎢ ⎢ ⎣ cos 𝜓𝑝 sin 𝜓𝑝 0 − sin 𝜓𝑝 cos 𝜓𝑝 0 0 0 1 ⎤ ⎥ ⎥ ⎦ , (3) where 𝑅𝑜 𝑝→𝑜 is the coordinate transformation matrix from the vehicle projected coordinate frame to the vehicle coordinate frame, 𝑅𝑂→𝑜 𝑝 is the coordinate transformation matrix from the global coordinate frame to the vehicle projected coordinate frame, 𝜓𝑝 is the projected heading angle of the … view at source ↗
Figure 3
Figure 3. Figure 3: Path tracking kinematics model 2.2. Path tracking kinematics model The schematic of vehicle path tracking is shown in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Path tracking control architecture on complex off-road scenario 3. Basic path tracking control strategy This paper proposes a path tracking control architecture for complex off-road scenarios, as shown in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: DK method learning framework and decoder (De) are both composed of four fully connected layers, and the numbers of neurons in the 𝑖-th layer of the encoder and decoder are 𝑛𝑒,𝑖 and 𝑛𝑑,𝑖, respectively. Rectified linear unit (ReLU) activation functions are used in the hidden layers to balance the accuracy of nonlinear mapping and computational efficiency, and no activation function is used in the output laye… view at source ↗
Figure 6
Figure 6. Figure 6: Credibility coefficient Then, the EPC compensation mechanism is designed. When the vehicle traverses large pothole surfaces and satisfies the activation criterion in (60), the DNN Koopman method generates a compensatory steering angle based on the vehicle states. Subsequently, the generated compensatory steering angle 𝛿comp is fed back to the LMPC module for credibility verification, to determine the credi… view at source ↗
Figure 7
Figure 7. Figure 7: HiL platform of path tracking [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: DLC on a coupled slope with potholes [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Complex off-road scenario the coupled slope and the potholed road [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Estimation result of tire cornering force [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Performance of UGV path tracking under Case 1 [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Performance of UGV path tracking under Case 2 KDMD–LMPC: The Koopman operator is approximated in a finite-dimensional form by implicitly constructing an infinite-dimensional feature space via the kernel trick, thereby compensating the UGV path deviation without manually constructing the lifting vector. The Gaussian radial basis function in (63) is selected as the kernel function. DK–LMPC: The lifting func… view at source ↗
Figure 13
Figure 13. Figure 13: Actual computation time of different strategies in the two scenarios, respectively. This further demonstrates the superiority of the proposed CDK–LMPC. In addition, compared with CDK–LMPC, DK–LMPC produces compensatory steering angles with larger fluctuations and insufficient confidence, making it difficult to satisfy the physical constraints of the steering system. The superimposed overall steering angle… view at source ↗
read the original abstract

Unmanned ground vehicles (UGVs) operating in off-road scenarios are confronted with complex terrain disturbances that can substantially degrade path tracking performance. To address this challenge, this paper proposes a deep neural network (DNN) Koopman-based deviation compensation strategy for UGV path tracking control. Firstly, based on the vehicle dynamic function on coupled slope, an adaptive forgetting recursive least squares method with decoupled error terms is designed to estimate tire cornering stiffness. On this basis, a Laguerre model predictive control (LMPC) path tracking control strategy is designed by incorporating Laguerre functions, which can reduce computational resource usage while maintaining reliable tracking performance across different coupled slope scenarios. Then, by integrating Koopman operator theory with DNN, a DNN Koopman (DK) path deviation compensation method is proposed, which significantly improves the path tracking accuracy of UGV under potholed road disturbances. Furthermore, an event-triggered parallel cooperative (EPC) compensation mechanism that couples LMPC with DK is established based on compensation activation criteria and credibility verification. This mechanism improves path tracking accuracy on potholed road while ensuring the feasibility of overall steering command and stability of vehicle after DK compensation. Finally, a hardware-in-the-loop (HiL) experimental platform is constructed for validation. Experimental results demonstrate that the proposed UGV path tracking strategy improves tracking performance by more than 11.5% across multiple operating conditions.

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 / 1 minor

Summary. The manuscript proposes a DNN Koopman-based deviation compensation strategy for UGV path tracking control on coupled slope and potholed roads. It designs an adaptive forgetting recursive least squares method for tire cornering stiffness estimation, a Laguerre model predictive control (LMPC) strategy, a DNN Koopman (DK) compensation method, and an event-triggered parallel cooperative (EPC) mechanism that couples LMPC with DK using compensation activation criteria and credibility verification. Validation is performed on a hardware-in-the-loop platform, claiming more than 11.5% improvement in tracking performance across multiple conditions.

Significance. If the claims hold, particularly the generalization of the DK model and the stability guarantees under the EPC mechanism, this work could contribute to robust control strategies for UGVs in challenging off-road environments by combining data-driven Koopman operators with model predictive control and event-triggering to handle disturbances while preserving feasibility. The use of HiL experiments provides practical validation.

major comments (2)
  1. [Abstract] Abstract: The headline claim of more than 11.5% improvement depends on the DK compensator producing corrections that the LMPC can absorb without violating steering feasibility or vehicle stability when the EPC trigger fires. However, no quantitative bounds on the approximation error of the Koopman lift or explicit stability certificate for the switched system are provided.
  2. [DNN Koopman and EPC mechanism] DNN Koopman compensation and EPC sections: No description of the training distribution of pothole geometries is given, which is necessary to assess generalization to unseen disturbances. The credibility verification is mentioned but lacks details on how it prevents the compensation from pushing the LMPC outside its feasible set.
minor comments (1)
  1. [Abstract] The abstract could briefly specify the operating conditions and disturbance types tested in the HiL experiments to strengthen the cross-condition claim.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below, indicating where revisions will be made and where we maintain our original approach based on the empirical focus of the work.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The headline claim of more than 11.5% improvement depends on the DK compensator producing corrections that the LMPC can absorb without violating steering feasibility or vehicle stability when the EPC trigger fires. However, no quantitative bounds on the approximation error of the Koopman lift or explicit stability certificate for the switched system are provided.

    Authors: The 11.5% improvement is an empirical result obtained from hardware-in-the-loop experiments across multiple operating conditions on coupled slope and potholed roads. The EPC mechanism, with its activation criteria and credibility verification, is intended to ensure that DK corrections remain compatible with LMPC feasibility and vehicle stability, as validated in the HiL tests. We acknowledge that the manuscript does not derive quantitative bounds on the Koopman approximation error or provide an explicit stability certificate for the switched system. Such theoretical analysis lies outside the practical and implementation-focused scope of the current work. revision: no

  2. Referee: [DNN Koopman and EPC mechanism] DNN Koopman compensation and EPC sections: No description of the training distribution of pothole geometries is given, which is necessary to assess generalization to unseen disturbances. The credibility verification is mentioned but lacks details on how it prevents the compensation from pushing the LMPC outside its feasible set.

    Authors: We will revise the DNN Koopman and EPC sections to include a description of the pothole geometries and road disturbance scenarios used in training the DNN Koopman model. We will also expand the explanation of the credibility verification process, detailing the specific checks and thresholds employed to ensure that activated compensations do not drive the LMPC outside its feasible set. revision: yes

standing simulated objections not resolved
  • Quantitative bounds on the approximation error of the Koopman lift or explicit stability certificate for the switched system

Circularity Check

0 steps flagged

No circularity: methods and validation remain independent of fitted inputs

full rationale

The abstract and description outline a sequence of distinct steps—adaptive RLS stiffness estimation, Laguerre MPC, DNN-Koopman deviation compensation, and EPC mechanism—followed by HiL experimental validation reporting >11.5% improvement. No equations, self-citations, or training details are supplied that would allow any claimed prediction or result to reduce by construction to its own inputs. The performance metric is presented as an external experimental outcome rather than a fitted or renamed quantity. Without explicit reduction (e.g., Eq. X = fitted parameter), the derivation chain is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies insufficient technical detail to enumerate free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5804 in / 1065 out tokens · 27424 ms · 2026-06-26T21:07:37.552681+00:00 · methodology

discussion (0)

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