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

A1 robot demonstrating \(\mathbb{C}_2\) symmetry properties

Mini-Cheetah exhibiting \(\mathbb{K}_4\) symmetry properties

Overview of MS-HGNN

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: 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 ( \(\mathbb{C}_2\) & \(\mathbb{K}_4\) ) 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\(\pm\)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 \(\uparrow\)	Leg-LH F1 \(\uparrow\)	Leg-RF F1 \(\uparrow\)	Leg-RH F1 \(\uparrow\)	State Acc \(\uparrow\)	Legs-Avg F1 \(\uparrow\)
CNN (10,855,440)	-	0.771 \(\pm\) 0.013	0.899 \(\pm\) 0.003	0.884 \(\pm\) 0.014	0.891 \(\pm\) 0.024	0.731 \(\pm\) 0.013	0.861 \(\pm\) 0.004
CNN-Aug (10,855,440)	\(\mathbb{C}_2\)	0.854 \(\pm\) 0.009	0.896 \(\pm\) 0.022	0.835 \(\pm\) 0.015	0.906 \(\pm\) 0.013	0.778 \(\pm\) 0.019	0.873 \(\pm\) 0.007
ECNN (5,614,770)	\(\mathbb{C}_2\)	0.884 \(\pm\) 0.012	0.887 \(\pm\) 0.010	0.853 \(\pm\) 0.011	0.860 \(\pm\) 0.016	0.788 \(\pm\) 0.029	0.871 \(\pm\) 0.011
MI-HGNN (1,585,282)	\(\mathbb{S}_4\)	0.932 \(\pm\) 0.006	0.936 \(\pm\) 0.010	0.927 \(\pm\) 0.003	0.928 \(\pm\) 0.005	0.870 \(\pm\) 0.010	0.931 \(\pm\) 0.005
MS-HGNN (2,407,810)	\(\mathbb{C}_2\)	0.928 \(\pm\) 0.013	0.933 \(\pm\) 0.011	0.913 \(\pm\) 0.016	0.937 \(\pm\) 0.010	0.856 \(\pm\) 0.013	0.929 \(\pm\) 0.009
MS-HGNN (2,144,642)	\(\mathbb{K}_4\)	0.936 \(\pm\) 0.008	0.944 \(\pm\) 0.006	0.930 \(\pm\) 0.011	0.948 \(\pm\) 0.006	0.875 \(\pm\) 0.012	0.939 \(\pm\) 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.

Model	Sym.	Training Samples (%)
Model	Sym.	2.50	5.00	10.00	15.00	21.25	42.50	63.75	85.00
CNN	-	0.745	0.794	0.831	0.802	0.811	0.840	0.850	0.836
CNN-Aug	\(\mathbb{C}_2\)	0.764	0.851	0.827	0.859	0.844	0.829	0.839	0.881
ECNN	\(\mathbb{C}_2\)	0.840	0.841	0.851	0.843	0.867	0.877	0.785	0.881
MI-HGNN	\(\mathbb{S}_4\)	0.872	0.908	0.926	0.930	0.937	0.940	0.932	0.931
MS-HGNN	\(\mathbb{C}_2\)	0.760	0.893	0.910	0.923	0.926	0.939	0.935	0.939
MS-HGNN	\(\mathbb{K}_4\)	0.869	0.897	0.913	0.922	0.919	0.939	0.935	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\(\pm\)std of the test RMSE over 4 runs. The best performance is highlighted in bold.

Test Sequence	1D GRF		3D GRF
Test Sequence	MI-HGNN	MS-HGNN (\(\mathbb{C}_2\))	MI-HGNN	MS-HGNN (\(\mathbb{K}_2\))
Unseen Friction	8.089 ± 0.102	7.850 ± 0.154	6.437 ± 0.055	6.355 ± 0.050
Unseen Speed	9.787 ± 0.111	9.733 ± 0.142	7.887 ± 0.064	7.721 ± 0.048
Unseen Terrain	8.826 ± 0.144	8.685 ± 0.136	7.332 ± 0.076	7.208 ± 0.047
Unseen All	10.245 ± 0.168	10.137 ± 0.084	8.708 ± 0.052	8.630 ± 0.097
Total	9.035 ± 0.116	8.899 ± 0.079	7.388 ± 0.056	7.268 ± 0.032

Compare with SOTA Methods on Regression Task (CoM Estimation)

**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 ( \(\mathbb{C}_2\) & \(\mathbb{K}_4\) ) 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\(\pm\)std over 4 runs). Bold and underlined indicate the best and second-best results, respectively.

Model	Sym.	Lin. Cos. Sim. \(\uparrow\)	Ang. Cos. Sim. \(\uparrow\)	Test MSE \(\downarrow\)
MLP	-	0.9617 \(\pm\) 0.0036	0.9523 \(\pm\) 0.0032	0.0499 \(\pm\) 0.0037
MLP-Aug	\(\mathbb{C}_2\)	0.9639 \(\pm\) 0.0026	0.9535 \(\pm\) 0.0029	0.0478 \(\pm\) 0.0020
MLP-Aug	\(\mathbb{K}_4\)	0.9647 \(\pm\) 0.0023	0.9549 \(\pm\) 0.0023	0.0472 \(\pm\) 0.0014
EMLP	\(\mathbb{C}_2\)	0.9610 \(\pm\) 0.0039	0.9528 \(\pm\) 0.0051	0.0503 \(\pm\) 0.0053
EMLP	\(\mathbb{K}_4\)	0.9673 \(\pm\) 0.0045	0.9580 \(\pm\) 0.0032	0.0435 \(\pm\) 0.0048
MI-HGNN	\(\mathbb{S}_4\)	0.9301 \(\pm\) 0.0017	0.5173 \(\pm\) 0.0016	0.3421 \(\pm\) 0.0009
MS-HGNN	\(\mathbb{C}_2\)	0.9903 \(\pm\) 0.0001	0.9804 \(\pm\) 0.0015	0.0161 \(\pm\) 0.0006
MS-HGNN	\(\mathbb{K}_4\)	0.9877 \(\pm\) 0.0007	0.9799 \(\pm\) 0.0010	0.0189 \(\pm\) 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 (\(\mathbb{C}_2\))	# of Param.	MS-HGNN (\(\mathbb{K}_4\))
12,934	0.8864	13,478	0.9448	11,366	0.9240
25,478	0.9136	26,150	0.9558	21,926	0.9505
50,438	0.9213	52,550	0.9746	44,230	0.9675
100,102	0.9297	102,470	0.9870	85,830	0.9854
199,174	0.9275	207,494	0.9903	174,470	0.9875
223,878	0.9325	405,638	0.9945	339,590	0.9935
396,806	0.9319	464,838	0.9940	390,726	0.9915
791,558	0.9276	824,582	0.9959	692,998	0.9936

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.	MLP	MLP-Aug (\(\mathbb{C}_2\))	MLP-Aug (\(\mathbb{K}_4\))	# of Param.	EMLP (\(\mathbb{C}_2\))	EMLP (\(\mathbb{K}_4\))
10,310	0.9147	0.9170	0.9199	-	-	-
36,998	0.9631	0.9660	0.9644	36,992	0.9640	0.9718
139,526	0.9737	0.9802	-	139,520	0.9843	0.9868
541,190	0.9643	0.9839	0.9843	541,184	0.9865	0.9910

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}
    }

\(\mathbb{M}\)orphological-\(\mathbb{S}\)ymmetry-Equivariant\(\mathbb{H}\)eterogeneous \(\mathbb{G}\)raph \(\mathbb{N}\)eural \(\mathbb{N}\)etwork for Robotic Dynamics Learning