Morphological-Symmetry-Equivariant
Heterogeneous Graph Neural Network
for Robotic Dynamics Learning

SOLO C2 Symmetry

A1 robot demonstrating C2 symmetry properties

Mini Cheetah K4 Symmetry

Mini-Cheetah exhibiting K4 symmetry properties

Overview of MS-HGNN

Method Overview

Method Overview

Method Overview

Overview of our MorphSym-HGNN architecture incorporating morphological symmetries for robotic dynamics learning.

Abstract

We propose MS-HGNN, a Morphological-Symmetry-Equivariant Heterogeneous Graph Neural Network for robotic dynamics learning, which integrates robotic kinematic structures and morphological symmetries into a unified graph network. By embedding these structural priors as inductive biases, MS-HGNN ensures high generalizability, sample and model efficiency. This architecture is versatile and broadly applicable to various multi-body dynamic systems and dynamics learning tasks. We prove the morphological-symmetry-equivariant property of MS-HGNN and demonstrate its effectiveness across multiple quadruped robot dynamics learning problems using real-world and simulated data.

Compare with SOTA Methods on Classification Task

Figure 2
Figure 2: Contact state detection results visualization. Left: F1 score for each leg, averaged F1 score, and 16-state contact state accuracy, averaged over 4 random runs. The number of parameters for each method is also provided. Right: Averaged F1 score for models trained using various numbers of training samples. Our MS-HGNN ( C2 & K4 ) achieve around 0.9 averaged F1-score trained with only 5% of the entire training set.
Table 1: Contact state detection performance on the real-world Mini-Cheetah dataset. This table reports the numerical results corresponding to Fig. 2-left. Metrics include the mean±std of F1 score per leg, 16-state accuracy, and the averaged F1 score across 4 runs. Bold and underlined values indicate the best and second-best results, respectively.
Model (# of Param.)Sym.Leg-LF F1 Leg-LH F1 Leg-RF F1 Leg-RH F1 State Acc Legs-Avg F1
CNN (10,855,440)-0.771 ± 0.013 0.899 ± 0.0030.884 ± 0.0140.891 ± 0.024 0.731 ± 0.0130.861 ± 0.004
CNN-Aug (10,855,440)C20.854 ± 0.009 0.896 ± 0.0220.835 ± 0.0150.906 ± 0.013 0.778 ± 0.0190.873 ± 0.007
ECNN (5,614,770)C20.884 ± 0.012 0.887 ± 0.0100.853 ± 0.0110.860 ± 0.016 0.788 ± 0.0290.871 ± 0.011
MI-HGNN (1,585,282)S40.932 ± 0.006 0.936 ± 0.0100.927 ± 0.003 0.928 ± 0.0050.870 ± 0.010 0.931 ± 0.005
MS-HGNN (2,407,810)C20.928 ± 0.013 0.933 ± 0.0110.913 ± 0.0160.937 ± 0.010 0.856 ± 0.0130.929 ± 0.009
MS-HGNN (2,144,642)K40.936 ± 0.008 0.944 ± 0.0060.930 ± 0.011 0.948 ± 0.0060.875 ± 0.012 0.939 ± 0.006
Table 2. Sample efficiency analysis on the real-world Mini-Cheetah contact dataset. The dataset includes 634.6K training and validation samples. This table presents the legs-averaged F1 scores when training on different proportions of the data. Results correspond to Fig. 2-right.
ModelSym.Training Samples (%)
2.505.0010.00 15.0021.2542.50 63.7585.00
CNN-0.745 0.7940.8310.802 0.8110.8400.850 0.836
CNN-AugC20.764 0.8510.8270.859 0.8440.8290.839 0.881
ECNNC20.840 0.8410.8510.843 0.8670.8770.785 0.881
MI-HGNNS40.872 0.9080.9260.930 0.9370.9400.932 0.931
MS-HGNNC20.760 0.8930.9100.923 0.9260.9390.935 0.939
MS-HGNNK40.869 0.8970.9130.922 0.9190.9390.935 0.942

Compare with SOTA Methods on Regression Task (GRF Estimation)

Experimental Results
Figue 1(b). Ground reaction force estimation test RMSE on simulated A1 dataset.
Table 3: Ground reaction force estimation on the simulated A1 dataset. This table provides the numerical results corresponding to Fig. 1(b). The metric is the mean±std of the test RMSE over 4 runs. The best performance is highlighted in bold.
Test Sequence1D GRF3D GRF
MI-HGNNMS-HGNN (C2) MI-HGNNMS-HGNN (K2)
Unseen Friction8.089 ± 0.1027.850 ± 0.154 6.437 ± 0.0556.355 ± 0.050
Unseen Speed9.787 ± 0.1119.733 ± 0.142 7.887 ± 0.0647.721 ± 0.048
Unseen Terrain8.826 ± 0.1448.685 ± 0.136 7.332 ± 0.0767.208 ± 0.047
Unseen All10.245 ± 0.16810.137 ± 0.084 8.708 ± 0.0528.630 ± 0.097
Total9.035 ± 0.1168.899 ± 0.079 7.388 ± 0.0567.268 ± 0.032

Compare with SOTA Methods on Regression Task (CoM Estimation)

Figure 3
Figure 3: Centroidal momentum estimation results on the synthetic Solo dataset. Left: The test linear, angular cosine similarity, and MSE of each model's prediction, averaged over 4 random runs. Right: The linear cosine similarity for models of different sizes. Our MS-HGNN ( C2 & K4 ) methods exhibit superior model efficiency without overfitting.
Table 4. Centroidal momentum estimation results on the synthetic Solo dataset. This table reports the numerical results corresponding to Fig. 3-left. Evaluation metrics include linear cosine similarity, angular cosine similarity, and test MSE (mean±std over 4 runs). Bold and underlined indicate the best and second-best results, respectively.
ModelSym.Lin. Cos. Sim. Ang. Cos. Sim. Test MSE
MLP-0.9617 ± 0.0036 0.9523 ± 0.00320.0499 ± 0.0037
MLP-AugC20.9639 ± 0.0026 0.9535 ± 0.00290.0478 ± 0.0020
MLP-AugK40.9647 ± 0.0023 0.9549 ± 0.00230.0472 ± 0.0014
EMLPC20.9610 ± 0.0039 0.9528 ± 0.00510.0503 ± 0.0053
EMLPK40.9673 ± 0.0045 0.9580 ± 0.00320.0435 ± 0.0048
MI-HGNNS40.9301 ± 0.0017 0.5173 ± 0.00160.3421 ± 0.0009
MS-HGNNC20.9903 ± 0.0001 0.9804 ± 0.00150.0161 ± 0.0006
MS-HGNNK40.9877 ± 0.0007 0.9799 ± 0.00100.0189 ± 0.0007
Table 5. Model efficiency comparison between MI-HGNN and MS-HGNN on the CoM momentum estimation task. This table shows the first part of the numerical results in Fig. 3-right, reporting linear cosine similarity with varying parameter counts.
# of Param.MI-HGNN# of Param. MS-HGNN (C2)# of Param.MS-HGNN (K4)
12,9340.886413,4780.9448 11,3660.9240
25,4780.913626,1500.9558 21,9260.9505
50,4380.921352,5500.9746 44,2300.9675
100,1020.9297102,4700.9870 85,8300.9854
199,1740.9275207,4940.9903 174,4700.9875
223,8780.9325405,6380.9945 339,5900.9935
396,8060.9319464,8380.9940 390,7260.9915
791,5580.9276824,5820.9959 692,9980.9936
Table 6. Model efficiency comparison of MLP, MLP-Aug, and EMLP on the CoM momentum estimation task. This table presents the second part of the numerical results in Fig. 3-right, showing linear cosine similarity across different architectures and symmetry configurations.
# of Param.MLPMLP-Aug (C2) MLP-Aug (K4)# of Param.EMLP (C2) EMLP (K4)
10,3100.91470.91700.9199 ---
36,9980.96310.96600.9644 36,9920.96400.9718
139,5260.97370.9802- 139,5200.98430.9868
541,1900.96430.98390.9843 541,1840.98650.9910

BibTeX


    @InProceedings{pmlr-v283-xie25a,
      title = 	 {Morphological-Symmetry-Equivariant Heterogeneous Graph Neural Network for Robotic Dynamics Learning},
      author =       {Xie, Fengze and Wei, Sizhe and Song, Yue and Yue, Yisong and Gan, Lu},
      booktitle = 	 {Proceedings of the 7th Annual Learning for Dynamics \& Control Conference},
      pages = 	 {1392--1405},
      year = 	 {2025},
      volume = 	 {283}
    }