We identify the nonlinear normal modes spawning from the stable equilibrium of a double pendulum under gravity, and we establish their connection to homoclinic orbits through the unstable upright position as energy increases. This result is exploited to devise an efficient swing-up strategy for a double pendulum with weak, saturating actuators. Our approach involves stabilizing the system onto periodic orbits associated with the nonlinear modes while gradually injecting energy. Since these modes are autonomous system evolutions, the required control effort for stabilization is minimal. Even with actuator limitations of less than 1% of the maximum gravitational torque, the proposed method accomplishes the swing-up of the double pendulum by allowing sufficient time.

Continuum soft robots are nonlinear mechanical systems with theoretically infinite degrees of freedom (DoFs) that exhibit complex behaviors. Achieving motor intelligence under dynamic conditions necessitates the development of control-oriented reduced-order models (ROMs), which employ as few DoFs as possible while still accurately capturing the core characteristics of the theoretically infinite-dimensional dynamics. However, there is no quantitative way to measure if the ROM of a soft robot has succeeded in this task. In other fields, like structural dynamics or flexible link robotics, linear normal modes are routinely used to this end. Yet, this theory is not applicable to soft robots due to their nonlinearities. In this work, we propose to use the recent nonlinear extension in modal theory -- called eigenmanifolds -- as a means to evaluate control-oriented models for soft robots and compare them. To achieve this, we propose three similarity metrics relying on the projection of the nonlinear modes of the system into a task space of interest. We use this approach to compare quantitatively, for the first time, ROMs of increasing order generated under the piecewise constant curvature (PCC) hypothesis with a high-dimensional finite element (FE)-like model of a soft arm. Results show that by increasing the order of the discretization, the eigenmanifolds of the PCC model converge to those of the FE model.

Inspired by the vertebrate branch of the animal kingdom, articulated soft robots are robotic systems embedding elastic elements into a classic rigid (skeleton-like) structure. Leveraging on their bodies elasticity, soft robots promise to push their limits far beyond the barriers that affect their rigid counterparts. However, existing control strategies aiming at achieving this goal are either tailored on specific examples, or rely on model cancellations -- thus defeating the purpose of introducing elasticity in the first place. In a series of recent works, we proposed to implement efficient oscillatory motions in robots subject to a potential field, aimed at solving these issues. A main component of this theory are Eigenmanifolds, that we defined as nonlinear continuations of the classic linear eigenspaces. When the soft robot is initialized on one of these manifolds, it evolves autonomously while presenting regular -- and thus practically useful -- evolutions, called normal modes. In addition to that, we proposed a control strategy making modal manifolds attractors for the system, and acting on the total energy of the soft robot to move from a modal evolution to the other. In this way, a large class of autonomous behaviors can be excited, which are direct expression of the embodied intelligence of the soft robot. Despite the fact that the idea behind our work comes from physical intuition and preliminary experimental validations, the formulation that we have provided so far is however rather theoretical, and very much in need of an experimental validation. The aim of this paper is to provide such an experimental validation using as testbed the articulated soft leg. We will introduce a simplified control strategy, and we will test its effectiveness on this system, to implement swing-like oscillations. We plan to extend this validation with a soft quadruped.

Abstract-- This paper proposes a Nonlinear Model-Predictive Control (NMPC) method capable of finding and converging to energy-efficient regular oscillations, which require no control action to be sustained. The approach builds up on the recently developed Eigenmanifold theory, which defines the sets of lineshaped oscillations of a robot as an invariant two-dimensional submanifold of its state space. By defining the control problem as a nonlinear program (NLP), the controller is able to deal with constraints in the state and control variables and be energyefficient not only in its final trajectory but also during the convergence phase. An initial implementation of this approach is proposed, analyzed, and tested in simulation. In the last three decades, numerous roboticists have devoted their effort to generating energy-efficient robot motion [1], [2].

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.