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

Kartik Loya; Phanindra Tallapragada

eess.SYMar 30, 2026

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

Authors: Kartik Loya, Phanindra Tallapragada

Organizations: Department of Mechanical Engineering, Clemson University, Clemson, SC 29607 USA

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.

Figures & tables

Figure 1: Longitudinal dynamics using single-track (bicycle) dynamic model

Figure 2: Vertical dynamics using half-car suspension model

Parameter	Value
Tire radius $r$	$0.33\penalty\ m$
Tire width $b$	$0.2286\penalty\ m$
Sprung mass $m$	$452\penalty\ Kg$
Unsprung mass $m_{w}$	$30\penalty\ Kg$
Wheel base $l_{f}+l_{r}$	$2.719\penalty\ m$
Suspension stiffness $k_{f},k_{r}$	$5\times 10^{3}\penalty\ N/m$

Table 1: Vehicle Parameters

Figure 3: Wheel-terrain interaction model

Parameter	Unit	Sandy Loam	Clay
Cohesion modulus $k_{c}$	$[kN/m^{n+1}]$	5.27	13.19
Frictional modulus $k_{\phi}$	$[kN/m^{n+2}]$	1515.04	692.15
Cohesion $c$	[kPa]	1.72	4.14
Friction angle $\phi$	[rad]	0.5061	0.2269
Sinkage exponent $n$		0.7	0.5
Shear deformation	$[m]$	0.025	0.01

Table 2: Soil Parameters

Panel of Figure 6 — Algorithm 1 Recursive SSID with Grassmannian-Guided Data Selection

Panel of Figure 4 — Figure 4: Sandy loam soil results for Koopman model selection and prediction: (a) RMSE versus refresh time, (b) open-loop RMSE growth with time, (c) normalized RMSE versus model order $r$ (dotted line: mean across outputs), and (d) eigenvalue spectrum of the identified Koopman $A$ matrix (unit circle shown for reference).

Figure 5: Sample trajectory prediction on sandy loam soil.

Figure 7: Sample trajectory prediction on clay soil.

Figure 8: Sample trajectory prediction on sandy loam with terrain elevation of maximum $0.1m$ height.

Figure 9: Sample trajectory prediction on clay with terrain elevation of maximum $0.1m$ height.

Panel of Figure 10 — Figure 10: (a) RMSE versus refresh time for sandy loam soil with terrain elevation, (b) RMSE versus refresh time for clay soil with terrain elevation.

Soil	$\%RMSE_{u}$	$\%RMSE_{v}$	$\%RMSE_{\dot{\psi}}$
Sandy Loam	$14.6048$	$9.4656$	$3.2462$
Clay	$15.0323$	$2.2215$	$9.8556$

Table 3: RMSE percentage increased for dataset with elevation with refresh rate

1.25s

Soil		Straight	Circle	Multisine	Slalom	Fishhook
Sandy	$u$	0.260	0.260	0.256	0.257	0.259
loam	$v$	0.292	0.285	0.287	0.287	0.290
	$\dot{\psi}$	0.064	0.063	0.064	0.064	0.064
Clay	$u$	0.296	0.311	0.315	0.310	0.303
	$v$	0.390	0.373	0.376	0.382	0.377
	$\dot{\psi}$	0.155	0.152	0.155	0.158	0.157

Table 4: RMSE on different maneuvers

Panel of Figure 11 — Figure 11: Koopman-based MPC tracking on sandyloam. Top: reference trajectory $y_{\mathrm{ref}}$ and executed trajectory under K-MPC in the $(X,Y)$ plane, and the corresponding state tracking for $u$ , $v$ , and $\dot{\psi}$ . Bottom: optimized control inputs steering $\delta$ and drive torque $\tau$ applied by the MPC.

Figure 12: RMSE versus refresh time for sandy loam Koopman operator on clay soil.

Panel of Figure 13 — Figure 13: Terrain based Koopman operator MPC tracking on clay soil. Top: reference trajectory $y_{\mathrm{ref}}$ and executed trajectory under sandy loam K-MPC (blue) and clay K-MPC (red). Bottom: optimized control inputs steering $\delta$ and drive torque $\tau$ applied by the MPC.

Figure 14: Slip ratio for sandy loam K-MPC and Clay K-MPC in rear wheel.

Terrain-Aware Robot Locomotion

Terrain-aware robot locomotion studies how robots sense, predict, and adapt movement to uneven, deformable, or otherwise challenging ground.

See all 148 papers

RK-MPC: Residual Koopman Model Predictive Control for Quadruped Locomotion in Offroad Environments
Apr 5, 2026Sriram S. K. S. Narayanan, Umesh VaidyaModel Predictive Control Quadruped Locomotion
Department of Mechanical Engineering, Clemson University, Clemson, SC 29630, USA
NeSAM: Neuro-Symbolic Kinodynamics with Soil Adaptation for Off-Road Mobility
Aug 21, 2026Chenhui Pan, Tong Xu, Francesco Cancelliere +1Terrain-Aware Robot Locomotion Robot Dynamics Learning
Department of Computer Science, George Mason University, USA · Department of Electrical, Electronic and Computer Engineering, University of Catania, Italy
Koopman operator theory: fundamentals, control, and applications
Jul 2, 2026Igor Mezić, Jorge Cortés, Karl Worthmann +2Surrogate Modeling Koopman Operator Learning
Department of Mechanical Engineering, University of California, Santa Barbara, USA · Department of Mechanical and Aerospace Engineering, University of California, San Diego, USA · Optimization-based Control Group, TU Ilmenau, Germany +2

Apr 5, 2026J/K move · Enter open · S save

RK-MPC: Residual Koopman Model Predictive Control for Quadruped Locomotion in Offroad Environments

Sriram S. K. S. Narayanan, Umesh Vaidya

Department of Mechanical Engineering, Clemson University, Clemson, SC 29630, USA

Model Predictive Control Quadruped Locomotion Residual Dynamics Learning Koopman Operator Learning Terrain-Aware Robot Locomotion

Open paper AskPDF arXiv

This paper presents Residual Koopman MPC (RK-MPC), a Koopman-based, data-driven model predictive control framework for quadruped locomotion that improves prediction fidelity while preserving real-time tractability. RK-MPC augments a nominal template model with a compact linear residual predictor learned from data in lifted coordinates, enabling systematic correction of model mismatch induced by contact variability and terrain disturbances with provable bounds on multi-step prediction error. The learned residual model is embedded within a convex quadratic-program MPC formulation, yielding a receding-horizon controller that runs onboard at 500 Hz and retains the structure and constraint-handling advantages of optimization-based control. We evaluate RK-MPC in both Gazebo simulation and Unitree Go1 hardware experiments, demonstrating reliable blind locomotion across contact disturbances, multiple gait schedules, and challenging off-road terrains including grass, gravel, snow, and ice. We further compare against Koopman/EDMD baselines using alternative observable dictionaries, including monomial and

SE(3)

-structured bases, and show that the residual correction improves multi-step prediction and closed-loop performance while reducing sensitivity to the choice of observables. Overall, RK-MPC provides a practical, hardware-validated pathway for data-driven predictive control of quadrupeds in unstructured environments. See https://sriram-2502.github.io/rk-mpc for implementation videos.

Abstract

Terrain-Aware Robot Locomotion

Explore similar work

RK-MPC: Residual Koopman Model Predictive Control for Quadruped Locomotion in Offroad Environments

NeSAM: Neuro-Symbolic Kinodynamics with Soil Adaptation for Off-Road Mobility

Koopman operator theory: fundamentals, control, and applications