Physics-informed Deep Mixture-of-Koopmans Vehicle Dynamics Model with Dual-branch Encoder for Distributed Electric-drive Trucks
Authors: Jinyu Miao, Pu Zhang, Rujun Yan, Yifei He, Bowei Zhang, Zheng Fu, Ke Wang, Qi Song, Kun Jiang, Mengmeng Yang, Diange Yang
Organizations: School of Vehicle and Mobility, and State Key Laboratory of Intelligent Green Vehicle and Mobility, Tsinghua University, Beijing, China · KargoBot Inc., Beijing, China
Advanced autonomous driving systems require accurate vehicle dynamics modeling. However, identifying a precise dynamics model remains challenging due to strong nonlinearities and the coupled longitudinal and lateral dynamic characteristics. Previous research has employed physics-based analytical models or neural networks to construct vehicle dynamics representations. Nevertheless, these approaches often struggle to simultaneously achieve satisfactory performance in terms of system identification efficiency, modeling accuracy, and compatibility with linear control strategies. In this paper, we propose a fully data-driven dynamics modeling method tailored for complex distributed electric-drive trucks (DETs), leveraging Koopman operator theory to represent highly nonlinear dynamics in a lifted linear embedding space. To achieve high-precision modeling, we first propose a novel dual-branch encoder which encodes dynamic states and provides a powerful basis for the proposed Koopman-based methods entitled KODE. A physics-informed supervision mechanism, grounded in the geometric consistency of temporal vehicle motion, is incorporated into the training process to facilitate effective learning of both the encoder and the Koopman operator. Furthermore, to accommodate the diverse driving patterns of DETs, we extend the vanilla Koopman operator to a mixture-of-Koopman operator framework, enhancing modeling capability. Simulations conducted in a high-fidelity TruckSim environment and real-world experiments demonstrate that the proposed approach achieves state-of-the-art performance in long-term dynamics state estimation.
Figures & tables
Fig. 1: A diagram of (a) physics-based dynamics model, (b) learning-based dynamics model, and (c) the proposed Koopman-based dynamics model KODE using a dual-branch encoder and MoK operator, respectively.
Fig. 2: An overview of the proposed method. The dynamics states are encoded to dynamics embeddings through a dual-branch encoder. Specific Koopman operator is selected by given driving pattern. Then, the dynamics embeddings are linearly evolved and decoded to future dynamics states.
Fig. 3: A diagram of feature interaction by Transformer encoder and feature aggregation by Transformer decoder, respectively.
Fig. 4: A diagram of geometric consistency of the temporal vehicle motion.
Fig. 5: A diagram of five driving patterns in this work.
Pattern
train
val
Definition
Data Generation Rules
c1:FRO
154.3K
13.9K
The rear wheels are steered in the opposite direction to the front wheels, reducing the turning radius. Under high-speed steering in simulation, this pattern may induce rollover incidents where the wheels on one side lift off the ground.
T∗∗=T,δ∗f=−δ∗r=δ,δ∗m=0
c2:COR
9.2K
2.3K
This pattern, also termed the “tank turn” pattern, steers the left and right, front and rear wheels in opposite directions, aligning all steering centers at the vehicle’s CoG. It permits rotation around the vertical axis with no translational movement.
The front wheels and middle wheels are steered according to Ackerman steering geometry, whereas the rear wheels are locked with no steering input.
T∗∗=T,δ∗f=δ,δ∗m=atan(δ/2),δ∗r=0
c4:LAM
11.5K
2.3K
All wheels are steered to 90 ° relative to the vehicle’s longitudinal axis, enabling lateral movement for parallel parking or precise positioning in confined spaces.
T∗∗=T,δ∗∗=δ=π/2
c5:CRM
92.0K
18.4K
All wheels are oriented at an identical angle and direction, producing diagonal movement without altering the vehicle’s heading angle.
T∗∗=T,δ∗∗=δ<π/2
TABLE I: The definition of five driving patterns for distibuted electric drive trucks.
Config
Value
Config
Value
sprung mass
4455.0 kg
load mass
5000.0 kg
unsprung mass
1050.0 kg
track width
2.030 m
axis f-m distance
3.250 m
axis f-r distance
6.500 m
TABLE II: The configuration of the truck in the simulation.
Model
c1: FRO
c2: COR
c3: RWL
c4: LAM
c5: CRM
All
MDE ↓
FDE ↓
MDE ↓
FDE ↓
MDE ↓
FDE ↓
MDE ↓
FDE ↓
MDE ↓
FDE ↓
MDE ↓
FDE ↓
Physics-based Dynamics methods
6 DoF model (GT)
0.0064 !
0.0150 !
failed
0.0068
0.0162
failed
0.0040
0.0097
0.0057
0.0135
6 DoF model (BP)
0.0376
0.0977
0.0808
0.2426
0.0425
0.1119
0.7946
2.1898
0.0145
0.0378
0.0648
0.1755
linear model (LS)
0.9167
1.8189
0.5379
1.0405
1.2429
2.4641
0.1373
0.2730
0.6020
1.2061
0.8724
1.7331
linear model (BP)
0.5570
1.0847
1.8098
3.6884
1.1240
2.1366
0.0647
0.1564
0.1233
0.3139
0.6329
1.2478
TABLE III: The comparison of 100-step dynamics state estimation with different methods in terms of trajectory errors (meter).
Model
c1: FRO
c2: COR
c3: RWL
c4: LAM
c5: CRM
All
MAE ↓
FAE ↓
MAE ↓
FAE ↓
MAE ↓
FAE ↓
MAE ↓
FAE ↓
MAE ↓
FAE ↓
MAE ↓
FAE ↓
Physics-based Dynamics methods
6 DoF model (GT)
0.0379 !
0.0827 !
failed
0.0434
0.0956
failed
0.0149
0.0331
0.0315
0.0693
6 DoF model (BP)
0.1016
0.2786
9.5936
28.4310
0.1134
0.3257
16.5352
45.7721
0.0361
0.1053
1.1613
3.2976
linear model (BP)
0.3863
1.2790
2.0805
4.0713
0.2675
0.6080
0.1357
0.2626
0.4337
1.4458
0.4226
1.1852
Pure Neural Network-based Dynamics methods
TABLE IV: The comparison of 100-step dynamics state estimation with different methods in terms of heading angle errors (degree).
Fig. 6: The average trajectory errors for 100-step estimation by our method, DeepEDMD [ 20 ] , DeepEDMD* [ 20 ] , and EDMD [ 19 ] . The value of each point in the trajectory is the average trajectory error of the estimation at this point. We also visualize the average trajectory error of our method and DeepEDMD* [ 20 ] in each timestamp for clear visualization.
Encoder
D
Hp
Params.
MDE ↓
FDE ↓
Invquad [ 19 ]
16
0
0.0001 M
0.5357
1.1353
MLP [ 20 ]
16
0
0.5845 M
0.0153
0.0391
KAN [ 48 ]
16
0
0.6010 M
0.0252
0.0686
(a)
16
0
0.5503 M
0.0139
0.0343
(b)
16
0
0.0022 M
0.0708
0.1919
Ours (a+b)
16
0
0.5525 M
0.0125
0.0323
TABLE V: The 100-step dynamics state estmation performance of Koopman-based methods with different encodes (meter).
Model
PiKo
MoK
MDE@1s ↓
FDE@1s ↓
MAE@1s ↓
FAE@1s ↓
MDE@2s ↓
MAE@2s ↓
MDE@3s ↓
MAE@3s ↓
MLP [ 20 ]
✘
✘
0.0153 m
0.0391 m
0.2883 °
0.6519 °
0.0542 m
0.7256 °
0.1430 m
1.3227 °
MLP [ 20 ]
✓
✘
0.0126 m
0.0348 m
0.2651 °
0.6481 °
0.0528 m
0.7331 °
0.1551 m
1.2593 °
17.6% ↓
11.0% ↓
8.0% ↓
0.6% ↓
2.6% ↓
1.0% ↑
8.5% ↑
4.8% ↓
KAN [ 48 ]
✘
✘
0.0252 m
0.0686 m
0.2626 °
0.6017 °
0.1100 m
0.7290 °
0.3464 m
1.6376 °
KAN [ 48 ]
✓
✘
0.0217 m
0.0587 m
0.1614 °
0.3828 °
0.0894 m
0.5209 °
0.2411 m
1.2328 °
17.6% ↓
13.9% ↓
14.4% ↓
36.4% ↓
18.7% ↓
28.5% ↓
30.4% ↓
24.7% ↓
TABLE VI: The ablation experiments on dynamics state estimation about physics-informed loss and mixture-of-Koopman.
Model
c1: FRO
c2: COR
c3: RWL
c4: LAM
c5: CRM
All
MDE ↓
FDE ↓
MDE ↓
FDE ↓
MDE ↓
FDE ↓
MDE ↓
FDE ↓
MDE ↓
FDE ↓
MDE ↓
FDE ↓
vanilla
0.0126
0.0324
0.0156
0.0318
0.0122
0.0318
0.0080
0.0166
0.0070
0.0187
0.0105
0.0269
MoK
0.0120
0.0323
0.0186
0.0331
0.0089
0.0254
0.0051
0.0113
0.0055
0.0133
0.0088
0.0228
TABLE VII: The ablation experiments on dynamics state estimation about MoK operator.
Method
MDE ↓
FDE ↓
MAE ↓
FAE ↓
Linear model (LS)
2.0039
3.9822
23.2635
46.0574
Linear model (BP)
2.1753
4.3873
0.4035
1.1294
EDMDInvquad∗ [ 19 ]
1.2098
2.0335
2.0702
4.4623
DeepEDMD* [ 20 ]
0.2774
0.4937
5.7466
10.3690
KODE (Ours)
0.2508
0.4991
0.1025
0.2125
KODE-S2R (Ours)
0.0722
0.1206
0.1001
0.2011
TABLE VIII: The dynamics state estimation performance of different methods in real-world vehicle case testing.
Fig. 7: The average trajectory errors for 100-step estimation by our method, both with and without sim-to-real adaptation strategy, across a 45-second clip. The value of each point in the trajectory is the average trajectory error of the estimation at this point. We also visualize the average trajectory error in each timestamp for clear visualization.
Modeling and forecasting nonlinear dynamics under distribution shifts is essential for robust decision-making in real-world systems. In this work, we propose MetaKoopman, a Bayesian meta-learning framework for modeling nonlinear dynamics through linear latent representations. MetaKoopman learns a Matrix Normal-Inverse Wishart (MNIW) prior over the Koopman operator, enabling closed-form Bayesian updates conditioned on recent trajectory segments. Moreover, it provides a closed-form posterior predictive distribution over future state trajectories, capturing both epistemic and aleatoric uncertainty in the learned dynamics. We evaluate MetaKoopman on a full-scale autonomous truck and trailer system across a wide range of adverse winter scenarios, including snow, ice, and mixed-friction conditions, as well as in simulated control tasks with diverse distribution shifts. MetaKoopman consistently outperforms prior approaches in multi-step prediction accuracy, uncertainty calibration, and robustness to distributional shifts. Field experiments further demonstrate its effectiveness in dynamically feasible motion planning, particularly during evasive maneuvers and operation at the limits of traction. Project website: https://mahmoud-selim.github.io/MetaKoopman/
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.
Kartik Loya, Phanindra Tallapragada
Department of Mechanical Engineering, Clemson University, Clemson, SC 29607 USA
The dominance of traditional rule-based methods in autonomous driving has gradually been replaced by learning-based approaches. While learning-based planners have achieved considerable success in passenger vehicles, their performance on heavy-duty trucks, particularly modern distributed electric-drive trucks (DETs), remains largely unexplored. To facilitate research and application of learning-based planners in DETs, this letter presents the first high-fidelity benchmark, called nuTruck, designed to support large-scale neural network training and closed-loop evaluation. Given the complex dynamics and high rollover susceptibility of DETs, we first incorporate a highly accurate nonlinear truck dynamical model into the simulation, which enables independent driving and steering of all wheels and captures dynamic load transfer caused by acceleration, deceleration, and cornering, thereby allowing quantitative assessment of rollover risk in closed-loop simulation. Second, we adapt several rule-based and learning-based planners as baselines for DETs and evaluate their performance in closed-loop simulation. Finally, using real-world driving scenarios from the nuPlan dataset, we conduct extensive closed-loop evaluations, analyzing not only conventional collision-free planning performance, but also the dynamical safety of the planned trajectories. The proposed nuTruck benchmark is expected to serve as a new standard for fair and realistic evaluation of autonomous driving planners on DETs.
Jinyu Miao, Pu Zhang, Yifei He +5
School of Vehicle and Mobility, Tsinghua University, Beijing, China · KargoBot.AI Inc., Beijing, China