Feedback controlled adaptive time-stepping for energy preserving variational integration.

Area of Science:

• Physics
• Robotics
• Computational Science

Background:

• High-fidelity physical simulation is essential for embodied intelligence.
• Conventional numerical methods face challenges with stability-computation trade-offs in complex systems.
• Explicit integrators cause energy drift, while symplectic variational integrators (VIs) are computationally expensive.

Purpose of the Study:

• To develop a numerical integration method that overcomes the stability-efficiency trade-off in simulating high-dimensional nonlinear multi-body systems.
• To propose a Control-Theory-Inspired Adaptive Time-step Variational Integrator (C-ATSVI) for robust Hamiltonian dynamics analysis.

Main Methods:

• Reformulated step-size adaptation as a closed-loop feedback control process.
• Designed a Lyapunov-based controller for energy error stability.
• Incorporated the Internal Model Principle (IMP) to mitigate numerical truncation errors.

Main Results:

• C-ATSVI demonstrated energy fidelity comparable to existing adaptive schemes.
• The method achieved computational efficiency similar to explicit fixed-step methods.
• Simulations were validated on 2-DoF and 7-DoF pendulums and a 7-DoF manipulator.

Conclusions:

• C-ATSVI offers a robust and efficient solution for simulating complex Hamiltonian dynamics.
• The proposed control-theory-inspired approach effectively bridges the gap between numerical stability and computational cost.
• This method advances the field of high-fidelity physical simulation for embodied intelligence.