Incorporating prior knowledge of physics laws and structural properties of dynamical systems into the design of deep learning architectures has proven to be a powerful technique for improving their computational efficiency and generalization capacity. Learning accurate models of robot dynamics is critical for safe and stable control. Autonomous mobile robots, including wheeled, aerial, and underwater vehicles, can be modeled as controlled Lagrangian or Hamiltonian rigid-body systems evolving on matrix Lie groups. In this paper, we introduce a new structure-preserving deep learning architecture, the Lie group Forced Variational Integrator Network (LieFVIN), capable of learning controlled Lagrangian or Hamiltonian dynamics on Lie groups, either from position-velocity or position-only data. By design, LieFVINs preserve both the Lie group structure on which the dynamics evolve and the symplectic structure underlying the Hamiltonian or Lagrangian systems of interest. The proposed architecture learns surrogate discrete-time flow maps allowing accurate and fast prediction without numerical-integrator, neural-ODE, or adjoint techniques, which are needed for vector fields. Furthermore, the learnt discrete-time dynamics can be utilized with computationally scalable discrete-time (optimal) control strategies.

翻译：将物理定律与动力系统结构特性的先验知识融入深度学习架构设计，已被证明是提升计算效率与泛化能力的有效手段。机器人动力学精确建模对安全稳定控制至关重要。轮式、空中及水下自主移动机器人可建模为矩阵李群上演化的受控拉格朗日或哈密顿刚体系统。本文提出一种新型保结构深度学习架构——李群受迫变分积分器网络（LieFVIN），能够从位置-速度数据或纯位置数据中学习李群上的受控拉格朗日或哈密顿动力学。通过设计，LieFVIN可同时保持动力学演化的李群结构，以及哈密顿或拉格朗日系统潜在的辛结构。该架构学习得到离散时间代理流映射，可无需数值积分器、神经常微分方程或伴随技术（矢量场方法所需）即可实现精确快速预测。此外，所学得的离散时间动力学可直接应用于计算可扩展的离散时间（最优）控制策略。