Lie Group Forced Variational Integrator Networks for Learning and Control of Robot Systems

The candidate solution reproduces the paper’s forced discrete Euler–Lagrange equations on SE(3) (rotational, translational, and kinematic updates) and the Cayley-based solver exactly as derived in Appendix A and Appendix B. In particular, the Lagrangian-form updates (eqs. (75)–(77) in the paper) and the Cayley reduction to the vector equation φ(z)=a+a×z+z(a^T z)−2Jz with Jacobian ∇φ(z)=S(a)+(a^T z)I+z a^T−2J match verbatim (see Appendix A and B, and Section 4.2) . The structure-preserving statements also agree with the standard discrete variational framework summarized in the paper’s preliminaries on variational and forced variational integrators .