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.