Symmetry-Reduced Physics-Informed Learning of Tensegrity Dynamics
Authors: Jing Qin, Muhao Chen
Organizations: Department of Mathematics, University of Kentucky, Lexington, KY · Department of Mechanical and Aerospace Engineering, University of Houston, TX
Tensegrity structures possess intrinsic geometric symmetries that govern their dynamic behavior. However, most existing physics-informed neural network (PINN) approaches for tensegrity dynamics do not explicitly exploit these symmetries, leading to high computational complexity and unstable optimization. In this work, we propose a symmetry-reduced physics-informed neural network (SymPINN) framework that embeds group-theory-based symmetry directly into both the solution expression and the neural network architecture to predict tensegrity dynamics. By decomposing nodes into symmetry orbits and representing free nodal coordinates using a symmetry basis, the proposed method constructs a reduced coordinate representation that preserves geometric symmetry of the structure. The full coordinates are then recovered via symmetry transformations of the reduced solution learned by the network, ensuring that the predicted configurations automatically satisfy the symmetry constraints. In this framework, equivariance is enforced through orbit-based coordinate generation, symmetry-consistent message passing, and physics residual constraints. In addition, SymPINN improves training effectiveness by encoding initial conditions as hard constraints, incorporating Fourier feature encoding to enhance the representation of dynamic motions, and employing a two-stage optimization strategy. Extensive numerical experiments on symmetric T-bars and lander structures demonstrate significantly improved prediction accuracy and computational efficiency compared to standard physics-informed models, indicating the great potential of symmetry-aware learning for structure-preserving modeling of tensegrity dynamics.
Figures & tables
Figure 1: Pipeline of the proposed SymPINN framework. The neural network learns the dynamics in a symmetry-reduced subspace using Fourier feature embeddings, and the full free nodal trajectories are reconstructed by mapping the reduced solution back to the original space.
Dataset
T-bar
lander
bar material
wood
steel
string material
rubber band
steel string
Young’s modulus of bars
8.10×109 Pa
2.06×1011 Pa
Young’s modulus of strings
2.00×106 Pa
7.60×1010 Pa
Table 1: Simulation configurations for each dataset.
Figure 2: Initial configuration of the T-bar. The black and red lines represent bars and strings, respectively. The bar length is 1 m.
Figure 3: Trajectories of the x and y coordinates of the two orbit nodes in the T-bar structure.
Figure 4: Prediction MSE on the T-bar dataset for different training data sampling rates.
Figure 5: Prediction RE on the T-bar dataset for different training data sampling rates.
Rate
PINN
SymPINN
RE
Time (s)
RE
Time (s)
0.1
0.1850
272.4
7.40×10−2
48.01
0.2
0.1813
347.7
2.85×10−3
61.66
0.3
0.1817
392.2
2.24×10−3
81.24
0.4
0.1769
446.1
2.18×10−3
110.69
0.5
0.1768
480.2
2.03×10−3
152.24
Table 2: Prediction performance comparison between PINN and SymPINN on the T-bar dataset under various training data rates.
Figure 6: Initial configuration of the six-bar tensegrity lander. The bar length is 1 m.
Figure 7: Trajectories of the x , y and z coordinates of the two nodes in the lander structure.
Figure 8: Prediction MSE on the lander dataset for different training data sampling rates.
Figure 9: Prediction RE on the lander dataset for different training data sampling rates.
Rate
PINN
SymPINN
RE
Time (s)
RE
Time (s)
0.1
6.84×10−2
2419.48
1.79×10−2
113.13
0.2
6.77×10−2
2634.08
2.30×10−3
153.42
0.3
6.88×10−2
3068.93
2.18×10−3
226.37
0.4
6.81×10−2
3068.56
2.01×10−3
341.22
0.5
6.72×10−2
2797.26
2.10×10−3
515.33
Table 3: Prediction performance comparison between PINN and SymPINN on the lander dataset under various training data rates.
Tensegrity form-finding and physical property prediction are fundamental inverse problems in structural mechanics, which aim to determine equilibrium configurations and internal force distributions. These problems are challenging due to strong nonlinearity arising from the coupling between geometry and forces, the need to ensure structural stability, and the enforcement of constraints such as boundary conditions and symmetry. Moreover, traditional methods often lack robustness to noise and outliers. This paper proposes an energy-based learning framework for clustered tensegrity form finding and physical property prediction. The proposed approach incorporates total potential energy minimization and constitutive relations into the training objective, enabling the simultaneous prediction of equilibrium nodal configurations and associated physical quantities, including member forces and force densities. By incorporating energy-based physical losses directly into the learning process, the framework improves physical consistency, robustness, and data efficiency. Numerical experiments on tensegrity structures, including prism and lander systems, show the great potential of the proposed approach and demonstrate its capability for scalable form finding and accurate prediction of structural properties.
Jing Qin, Muhao Chen
Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA · Department of Mechanical and Aerospace Engineering, University of Houston, Houston, TX 77204, USA
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
Wenzhe Tong, Yicheng Jiang, Chi Zhang +2
Robotics Department, University of Michigan, Ann Arbor, MI 48109 USA
Physics-informed neural networks (PINNs) offer a unified framework for solving both forward and inverse problems of differential equations, yet their performance and physical consistency strongly depend on how governing laws are incorporated. In this work, we present a systematic comparison of different thermodynamic structure-informed neural networks by incorporating various thermodynamics formulations, including Newtonian, Lagrangian, and Hamiltonian mechanics for conservative systems, as well as the Onsager variational principle and extended irreversible thermodynamics for dissipative systems. Through comprehensive numerical experiments on representative ordinary and partial differential equations, we quantitatively evaluate the impact of these formulations on accuracy, physical consistency, noise robustness, and interpretability. The results show that Newtonian-residual-based PINNs can reconstruct system states but fail to reliably recover key physical and thermodynamic quantities, whereas structure-preserving formulation significantly enhances parameter identification, thermodynamic consistency, and robustness. These findings provide practical guidance for principled design of thermodynamics-consistency model, and lay the groundwork for integrating more general nonequilibrium thermodynamic structures into physics-informed machine learning.
Guojie Li, Liu Hong
School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China.