Optimal Potential Shaping on SE(3) via Neural ODEs on Lie Groups
This work addresses control optimization for dynamic systems on Lie groups, such as rigid bodies, but is incremental as it adapts existing neural ODE methods to this domain.
The paper tackles the optimization of dynamic systems on finite-dimensional Lie groups by formulating them as neural ODEs and presenting a scalable gradient descent algorithm, applied to optimal potential energy shaping for rigid body control on SE(3), with validation on a state-regulation task.
This work presents a novel approach for the optimization of dynamic systems on finite-dimensional Lie groups. We rephrase dynamic systems as so-called neural ordinary differential equations (neural ODEs), and formulate the optimization problem on Lie groups. A gradient descent optimization algorithm is presented to tackle the optimization numerically. Our algorithm is scalable, and applicable to any finite dimensional Lie group, including matrix Lie groups. By representing the system at the Lie algebra level, we reduce the computational cost of the gradient computation. In an extensive example, optimal potential energy shaping for control of a rigid body is treated. The optimal control problem is phrased as an optimization of a neural ODE on the Lie group SE(3), and the controller is iteratively optimized. The final controller is validated on a state-regulation task.