💻 computer science

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

This paper proposes a Deep Neural Network (DNN) Koopman-based deviation compensation strategy integrated with Laguerre Model Predictive Control and an event-triggered parallel cooperative mechanism to significantly enhance unmanned ground vehicle path tracking accuracy and stability on complex coupled slope and potholed road terrains.

Original authors: Jian Zhao, Wenbo Zhou, Zhicheng Chen, Bing Zhu, Jiayi Han, Dongjian Song, Yinju Lin, Peixing Zhang

Published 2026-06-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jian Zhao, Wenbo Zhou, Zhicheng Chen, Bing Zhu, Jiayi Han, Dongjian Song, Yinju Lin, Peixing Zhang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are driving a rugged, off-road vehicle (an Unmanned Ground Vehicle or UGV) through a chaotic landscape filled with steep hills, side slopes, and deep potholes. Your goal is to follow a specific line drawn on the ground perfectly.

In a perfect world, the car would just follow the line. But in reality, the terrain is a nightmare. The hills push the car sideways, and the potholes jolt the tires, making the car's grip on the road change instantly. If you try to steer using a standard computer program, it might get confused, overreact, or simply be too slow to calculate the right move, causing the car to drift off course.

This paper presents a new "smart driving system" designed to keep the UGV on track in these messy conditions. Here is how it works, broken down into three simple parts:

1. The "Smart Calculator" (LMPC)

First, the system needs a baseline plan. The researchers built a controller called LMPC (Laguerre Model Predictive Control).

The Problem: Traditional controllers are like a student trying to solve a complex math problem by writing out every single step on paper. It's accurate, but it takes too long, and the car moves faster than the math can be solved.
The Solution: The LMPC is like a student who has memorized a shortcut formula. Instead of solving the whole problem from scratch every time, it uses a special mathematical "template" (Laguerre functions) to guess the best steering angle almost instantly.
The Result: It handles the coupled slopes (when the road is both tilted sideways and up/down) very well, keeping the car steady without getting bogged down in slow calculations.

2. The "Intuition Engine" (DNN Koopman)

However, when the car hits a pothole, the math gets messy. The tires lose grip in weird, unpredictable ways that standard formulas can't predict.

The Problem: Standard math models are like a map that only shows paved roads. When you hit a pothole, the map is useless.
The Solution: The researchers added a "Intuition Engine" using Deep Neural Networks (DNN) and something called the Koopman Operator.
- Think of the Koopman Operator as a translator. It takes the chaotic, non-linear jolts of a pothole (which are hard to understand) and translates them into a simple, linear language that a computer can easily predict.
- The Deep Neural Network is the student who learns this translation. It watches thousands of examples of the car hitting potholes and learns the best way to translate those jolts into a steering correction.
The Result: When the car hits a pothole, this engine predicts exactly how much the car will slide and suggests a tiny, precise steering tweak to cancel it out.

3. The "Traffic Cop" (Event-Triggered Parallel Cooperative Mechanism)

Here is the most critical part: You can't just let the "Intuition Engine" take over completely. If it gets excited and suggests a wild steering move, it could break the car's steering system or make the car spin out.

The Problem: The "Smart Calculator" (LMPC) is reliable but slow to react to potholes. The "Intuition Engine" (DNN) is fast and smart but can be a bit reckless and lacks clear rules.
The Solution: The researchers built a Traffic Cop (an Event-Triggered mechanism) to manage the two.
- The Trigger: The cop watches the "Load Transfer Rate" (LTR). Imagine this as a "wobble meter." If the car is driving smoothly on a slope, the meter is low, and the cop ignores the Intuition Engine, letting the Smart Calculator do the work.
- The Handoff: If the car hits a big pothole and the wobble meter spikes, the cop wakes up the Intuition Engine.
- The Credibility Check: Before the Intuition Engine's suggestion is used, the cop checks if the suggestion is "believable." It asks: "Is this steering angle so huge that it would snap the steering wheel?"
  - If the suggestion is safe, the cop adds it to the Smart Calculator's plan.
  - If the suggestion is too wild, the cop ignores it to keep the car safe.
The Result: The car gets the best of both worlds: the steady reliability of the calculator for normal driving, and the quick, smart reflexes of the intuition engine for potholes, all while staying within safe physical limits.

The Proof

The team tested this system on a Hardware-in-the-Loop (HiL) platform. This is like a flight simulator for cars, where the computer code talks to a real physical steering motor and a high-fidelity simulation of the car.

The Results:

Speed: The new system calculated steering commands 98.89% faster than the traditional method. It was lightning-fast.
Accuracy: On rough, potholed roads, the new system reduced the error (how far the car drifted from the line) by more than 45% compared to the standard method.
Stability: It kept the car stable even when the road was a mess of slopes and holes, whereas older methods made the car wobble and drift.

In short, this paper describes a system that combines a fast, reliable calculator with a smart, learning-based reflex, managed by a safety-conscious supervisor, to keep an off-road robot vehicle driving straight on the roughest roads imaginable.

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

Problem Statement
Unmanned Ground Vehicles (UGVs) operating in off-road environments face significant challenges in path tracking due to complex terrain disturbances. Unlike structured urban roads, off-road scenarios involve coupled slopes (combinations of longitudinal and lateral grades) and short-wavelength unevenness such as potholes. These conditions induce variations in vehicle parameters, including vertical tire forces, lateral forces, and tire cornering stiffness, leading to reduced driving stability and tracking accuracy.

Existing Model Predictive Control (MPC) strategies, while effective for multi-constraint problems, suffer from high computational costs due to quadratic programming (QP) solving processes, limiting real-time performance. Furthermore, standard model-based approaches often neglect the influence of long-wavelength road unevenness (grade) on lateral dynamics or rely on linearized models that fail to capture the strong nonlinearities and discontinuities caused by large pothole disturbances. Conversely, purely model-free methods (e.g., reinforcement learning) often lack interpretability and explicit physical constraints, risking actuator saturation and instability when control inputs are superimposed.

Methodology
The paper proposes a hybrid control architecture integrating a Laguerre Model Predictive Control (LMPC) baseline with a Deep Neural Network (DNN) Koopman-based deviation compensation strategy, managed by an Event-Triggered Parallel Cooperative (EPC) mechanism.

Vehicle Modeling and Parameter Estimation:
- A dynamic model for UGVs on coupled slopes is established, incorporating gravitational force components and yaw moments induced by slope coupling.
- An Adaptive Forgetting Recursive Least Squares (AFRLS) method with decoupled error terms is designed to estimate time-varying tire cornering stiffness. This estimator uses an adaptive filter to ensure convergence even when the vehicle travels straight (where state and input vectors are near zero), addressing the limitations of conventional RLS.
Baseline Control (LMPC):
- To reduce computational burden, the QP solving process in traditional MPC is replaced by a fitting and differentiation procedure using Laguerre functions.
- This approach generates a baseline steering angle ( $\delta_{bas}$ ) and an additional yaw moment ( $M_z$ ) that maintain reliable tracking performance across various coupled slope scenarios while significantly lowering resource consumption.
Deviation Compensation (DNN Koopman - DK):
- To address pothole-induced nonlinearities, the paper integrates Koopman operator theory with DNN.
- A DNN with an autoencoder structure (encoder-decoder) is trained offline to learn the optimal "lifting function" ( $\psi$ ). This function maps the nonlinear UGV path tracking system (including compensatory steering angles and vertical loads) into a finite-dimensional linear space.
- This allows for the construction of a finite-dimensional approximation matrix of the Koopman operator, enabling the prediction of a compensatory steering angle ( $\delta_{comp}$ ) to correct path deviations on potholed roads.
Event-Triggered Parallel Cooperative (EPC) Mechanism:
- To ensure the feasibility and stability of the combined control input, an EPC mechanism couples the LMPC and DK modules.
- Activation: Compensation is triggered only when the Load Transfer Rate (LTR) exceeds a specific threshold, indicating significant pothole disturbance.
- Credibility Verification: A sigmoid-based credibility coefficient ( $\lambda$ ) is calculated based on the magnitude of the compensatory angle relative to the vehicle's physical steering limits. This ensures the total steering angle ( $\delta_{tot} = \delta_{bas} + \lambda\delta_{comp}$ ) does not violate actuator constraints, preventing instability caused by uninterpretable model-free outputs.

Key Contributions

LMPC for Coupled Slopes: Development of a path tracking strategy using Laguerre functions to replace QP, reducing computational load while maintaining accuracy on coupled slopes. An AFRLS estimator is simultaneously designed to handle time-varying tire cornering stiffness.
DNN Koopman (DK) Compensation: Proposal of a DK method that uses an autoencoder-based DNN to learn optimal lifting functions. This enhances the Koopman operator's ability to represent the nonlinear dynamics of UGVs on potholed roads, overcoming the limitations of manually selected basis functions in traditional Extended Dynamic Mode Decomposition (EDMD).
EPC Compensation Mechanism: Establishment of a parallel cooperative mechanism that integrates LMPC and DK. By utilizing LTR-based activation and sigmoid-based credibility verification, the system ensures that path deviation compensation improves accuracy without compromising vehicle stability or violating steering constraints.

Experimental Results
The proposed strategy, termed CDK-LMPC, was validated using a Hardware-in-the-Loop (HiL) platform with a D-Class SUV model under two scenarios: a double lane change on a coupled slope with potholes and a complex off-road scenario.

Estimation Accuracy: The AFRLS method demonstrated high accuracy in estimating lateral tire forces and cornering stiffness variations, even under transient pothole excitations, outperforming standard RLS.
Tracking Performance: In both test cases, CDK-LMPC achieved the lowest lateral errors compared to MPC, LMPC, and other Koopman-based variants (EDMD-LMPC, KDMD-LMPC, DK-LMPC).
- In the complex off-road scenario (Case 2), CDK-LMPC reduced the Root Mean Square (RMS) lateral error by 37.3% and the maximum lateral error by 59.7% compared to the KDMD-LMPC baseline.
- Overall, the strategy improved path tracking performance by more than 11.5% across multiple operating conditions compared to existing methods.
Computational Efficiency: By replacing QP with Laguerre function fitting, the LMPC-based strategies reduced computation time by over 98.89% compared to traditional MPC (average time reduced from ~11.4 ms to ~0.12 ms), significantly enhancing real-time feasibility.

Significance and Claims
The paper claims that the proposed CDK-LMPC strategy effectively bridges the gap between model-based interpretability and model-free nonlinear approximation capabilities. By explicitly addressing the coupled effects of slope and potholes, and by introducing a credibility-driven mechanism to handle the superposition of control inputs, the strategy ensures both high tracking accuracy and vehicle stability. The authors assert that this approach substantially enhances operational efficiency and path tracking performance for UGVs in unstructured off-road environments, providing a robust solution for complex terrain navigation without the excessive computational cost of traditional MPC or the instability risks of unconstrained model-free control.

1. The "Smart Calculator" (LMPC)

2. The "Intuition Engine" (DNN Koopman)

3. The "Traffic Cop" (Event-Triggered Parallel Cooperative Mechanism)

The Proof

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