Learning Forced Multibody Dynamics on Lie Groups

How can we accurately learn the dynamics of mechanical systems—such as rigid bodies or multibody systems—from observational data without relying on noisy velocity measurements or restrictive Euclidean coordinate systems that suffer from singularities?

The authors propose a framework called LieDFLNN (Lie Group Discrete Forced Euler-Lagrange Neural Network). Instead of mapping dynamics to Euclidean space, the model operates directly on Lie groups (e.g., SO(3) for rotations, SE(3) for rigid body poses). By parameterizing the discrete Lagrangian and external forces, the model learns to satisfy the discrete forced Euler-Lagrange equations. This formulation uses only position observations, effectively bypassing the need for numerical differentiation, which is a common source of noise in dynamical system identification.

The proposed architecture naturally respects the geometric constraints of the configuration space, ensuring that predicted trajectories remain on the manifold (e.g., rotation matrices remain orthogonal). Because the model is built on variational principles, it inherently preserves physical invariants and conservation laws. Experiments on both synthetic and real-world multibody datasets demonstrate that the model achieves high predictive accuracy and robustness, even in the presence of external control inputs, without the discontinuities often found in Euler-angle-based models.

This work provides a principled way to integrate geometric deep learning with classical mechanics. By avoiding Euclidean embeddings, the model is better suited for robotics and aerospace applications where systems naturally evolve on nonlinear manifolds. The ability to learn from position-only data makes it highly practical for real-world scenarios where velocity sensors are unavailable, unreliable, or prone to high-frequency noise.