Optimal Potential Shaping on SE(3) via Neural ODEs on Lie Groups

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.

Trajectory Generation, Control, and Safety with Denoising Diffusion Probabilistic Models

We present a framework for safety-critical optimal control of physical systems based on denoising diffusion probabilistic models (DDPMs). The technology of control barrier functions (CBFs), encoding desired safety constraints, is used in combination with DDPMs to plan actions by iteratively denoising trajectories through a CBF-based guided sampling procedure. At the same time, the generated trajectories are also guided to maximize a future cumulative reward representing a specific task to be optimally executed. The proposed scheme can be seen as an offline and model-based reinforcement learning algorithm resembling in its functionalities a model-predictive control optimization scheme with receding horizon in which the selected actions lead to optimal and safe trajectories.

Passivizing learned policies and learning passive policies with virtual energy tanks in robotics

Within a robotic context, we merge the techniques of passivity-based control (PBC) and reinforcement learning (RL) with the goal of eliminating some of their reciprocal weaknesses, as well as inducing novel promising features in the resulting framework. We frame our contribution in a scenario where PBC is implemented by means of virtual energy tanks, a control technique developed to achieve closed-loop passivity for any arbitrary control input. Albeit the latter result is heavily used, we discuss why its practical application at its current stage remains rather limited, which makes contact with the highly debated claim that passivity-based techniques are associated to a loss of performance. The use of RL allows to learn a control policy which can be passivized using the energy tank architecture, combining the versatility of learning approaches and the system theoretic properties which can be inferred due to the energy tanks. Simulations show the validity of the approach, as well as novel interesting research directions in energy-aware robotics.

Discovering Efficient Periodic Behaviours in Mechanical Systems via Neural Approximators

It is well known that conservative mechanical systems exhibit local oscillatory behaviours due to their elastic and gravitational potentials, which completely characterise these periodic motions together with the inertial properties of the system. The classification of these periodic behaviours and their geometric characterisation are in an on-going secular debate, which recently led to the so-called eigenmanifold theory. The eigenmanifold characterises nonlinear oscillations as a generalisation of linear eigenspaces. With the motivation of performing periodic tasks efficiently, we use tools coming from this theory to construct an optimization problem aimed at inducing desired closed-loop oscillations through a state feedback law. We solve the constructed optimization problem via gradient-descent methods involving neural networks. Extensive simulations show the validity of the approach.

Optimal Energy Shaping via Neural Approximators

We introduce optimal energy shaping as an enhancement of classical passivity-based control methods. A promising feature of passivity theory, alongside stability, has traditionally been claimed to be intuitive performance tuning along the execution of a given task. However, a systematic approach to adjust performance within a passive control framework has yet to be developed, as each method relies on few and problem-specific practical insights. Here, we cast the classic energy-shaping control design process in an optimal control framework; once a task-dependent performance metric is defined, an optimal solution is systematically obtained through an iterative procedure relying on neural networks and gradient-based optimization. The proposed method is validated on state-regulation tasks.

Port-Hamiltonian Approach to Neural Network Training

Neural networks are discrete entities: subdivided into discrete layers and parametrized by weights which are iteratively optimized via difference equations. Recent work proposes networks with layer outputs which are no longer quantized but are solutions of an ordinary differential equation (ODE); however, these networks are still optimized via discrete methods (e.g. gradient descent). In this paper, we explore a different direction: namely, we propose a novel framework for learning in which the parameters themselves are solutions of ODEs. By viewing the optimization process as the evolution of a port-Hamiltonian system, we can ensure convergence to a minimum of the objective function. Numerical experiments have been performed to show the validity and effectiveness of the proposed methods.