Tensegrity robots possess lightweight and resilient structures but present significant challenges for state estimation due to compliant and distributed ground contacts. This paper introduces a symmetry-aware heterogeneous graph neural network (Sym-HGNN) that infers contact states directly from proprioceptive measurements, including IMU and cable-length histories, without dedicated contact sensors. The network incorporates the robot's dihedral symmetry D3 into the message-passing process to enhance sample efficiency and generalization. The predicted contacts are integrated into a state-of-the-art contact-aided invariant extended Kalman filter (InEKF) for improved pose estimation. Simulation results demonstrate that the proposed method achieves up to 15% higher accuracy and 5% higher F1-score using only 20% of the training data compared to the CNN and MI-HGNN baselines, while maintaining low-drift and physically consistent state estimation results comparable to ground truth contacts. This work highlights the potential of fully proprioceptive sensing for accurate and robust state estimation in tensegrity robots. Code available at: https://github.com/Jonathan-Twz/Tensegrity-Sym-HGNN
Figures & tables
Fig. 1: The proposed symmetry-aware heterogeneous graph neural network (Sym-HGNN) represents a tensegrity robot as a D3 -symmetric node graph, where rods, tendons, and endcaps form typed connections for structure-aware message passing and ground-contact inference.
Fig. 2: Structural composition and symmetry properties of the 3-bar tensegrity robot. The structure consists of rigid rods, tensile tendons, and six endcaps, exhibiting 120° rotational symmetry about longitudianl axis r and reflection symmetry across vertical plane f.
Fig. 3: Overview of the proposed Sym-HGNN contact prediction pipeline. IMU and motor encoders provide proprioceptive inputs, which are propagated through a heterogeneous graph with typed edges. Message passing is performed across layers under six D3 -equivariant transformations {e,r,r2,f,fr,fr2} , enabling symmetry-aware contact inference.
Fig. 4: Simulation datasets with diverse motion primitives (left) and turning radii (right). The left plot illustrates six motion directions, while the right plot shows forward-right (FR) gaits with different turning radii (1.0, 0.6, and 0.2).
Samples
CNN
MI-HGNN
Sym-HGNN
All primitives (F, B, FL, FR, BL, BR)
137k (full)
0.943 / 0.982
0.913 / 0.973
0.953 / 0.984
30k
0.075 / 0.511
0.795 / 0.938
0.931 / 0.978
30k (aug)
0.097 / 0.522
0.916 / 0.975
—
Three primitives (F, FR, FL)
83k (full)
0.836 / 0.923
0.810 / 0.936
0.942 / 0.980
TABLE I: Cross-motion generalization in simulation. Comparison of CNN, MI-HGNN, and Sym-HGNN trained on varying motion primitive subsets and sample sizes. Accuracy / F1 reported on the mixed test set.
Model
Training Data ↓
Avg. Accuracy ↑
Avg. F1 ↑
CNN
30k
0.06
0.53
CNN (full)
143k
0.80
0.93
MI-HGNN
30k
0.44
0.77
MI-HGNN (full)
143k
0.70
0.89
Sym-HGNN
30k
0.77
0.91
Sym-HGNN (full)
143k
0.85
0.94
TABLE II: Generalization across turning radius in simulation: evaluation on unseen-radius sequence.
#Layers
History Length
Train Acc. ↑
Test Acc. ↑
Test F1 ↑
Avg. Inference Time ↓
10
100
0.9931
0.9313
0.9776
40.83 ms
8
100
0.9966
0.9306
0.9777
34.76 ms
6
100
0.9970
0.9190
0.9737
28.28 ms
4
100
0.9805
0.8612
0.9586
22.41 ms
8
200
0.9980
0.9184
0.9802
34.33 ms
8
100
0.9966
0.9306
0.9777
34.76 ms
TABLE III: Ablation study of network depth and history data length on contact estimation performance
Fig. 5: Estimated pose of the 3-bar tensegrity robot using the contact-aided Invariant Extended Kalman Filter (InEKF). The Sym-HGNN contact predictions provide observation updates that maintain consistency with ground-truth trajectories.