Hamiltonian-based Neural ODE Networks on the SE(3) Manifold For Dynamics Learning and Control

Abstract

Accurate models of robot dynamics are critical for safe and stable control and generalization to novel operational conditions. Hand-designed models, however, may be insufficiently accurate, even after careful parameter tuning. This motivates the use of machine learning techniques to approximate the robot dynamics over a training set of state-control trajectories. The dynamics of many robots, including ground, aerial, and underwater vehicles, are described in terms of their SE(3) pose and generalized velocity, and satisfy conservation of energy principles. This paper proposes a Hamiltonian formulation over the SE(3) manifold of the structure of a neural ordinary differential equation (ODE) network to approximate the dynamics of a rigid body. In contrast to a black-box ODE network, our formulation guarantees total energy conservation by construction. We develop energy shaping and damping injection control for the learned, potentially under-actuated SE(3) Hamiltonian dynamics to enable a unified approach for stabilization and trajectory tracking with various platforms, including pendulum, rigid-body, and quadrotor systems.

Safe Autonomous Navigation for Systems with Learned SE(3) Hamiltonian Dynamics

“…the input matrix B θ (q) is invertible. The controller in (9) exists and the resulting closed-loop error dynamics becomes (Duong and Atanasov, 2021b):…”

“…Solving the matching conditions, as described in Duong and Atanasov (2021b), leads to a controller, consisting of an energy-shaping term u ES and a damping-injection term u DI :…”

“…based on Gaussian processes (Deisenroth et al, 2015;Kabzan et al, 2019) or neural networks (Raissi et al, 2018;Chua et al, 2018). For physical systems, recent works (Lutter et al, 2019;Zhong et al, 2019;Duong and Atanasov, 2021b) design the model architecture to encode Lagrangian or Hamiltonian formulation of robot dynamics (Lurie, 2013;Holm, 2008), which a black-box model might struggle to infer. For Hamiltonian formulation, Zhong et al (2019) use a differentiable neural ODE solver (Chen et al, 2018) to generate predicted state trajectory.…”

“…A loss function is back-propagated through the ODE solver to update its parameters. Duong and Atanasov (2021b) extend this approach by imposing both Hamiltonian formulation and the SE(3) constraints on the ODE structure. A Hamiltonianbased model architecture also simplifies the design of a stable regulation or tracking controller by energy shaping (Zhong et al, 2019;Duong and Atanasov, 2021a,b).…”

“…distances to obstacles from a depth sensor. We first learn a Hamiltonian model of the system dynamics using a physics-guided neural ODE networks (Duong and Atanasov, 2021b). As the robot dynamics are equivariant to translation, we offset the trajectories to start from the origin and train a translation-equivariant Hamiltonian neural ODE model.…”