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.

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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.

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Animals rely on the elasticity of their tendons and muscles to execute robust and efficient locomotion patterns for a vast and continuous range of velocities. Replicating such capabilities in artificial systems is a long-lasting challenge in robotics. By taking advantage of a pitch dynamics decoupling spring potential, this work aims to provide design rules and a control strategy to generate dynamic, efficient locomotion patterns in quadrupeds moving in a sagittal plane. We rely on nonlinear modal theory, which provides the tools to characterize continuous families of efficient oscillations in nonlinear mechanical systems. We provide simulations of an elastic quadruped showing that the proposed solution can robustly excite efficient locomotion patterns under non-ideal conditions.

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Implementing dynamic legged locomotion entails stabilizing oscillatory behaviors in complex mechanical systems. Whenever possible, locomotion algorithms should also exploit the improved capabilities of elastic elements added to the structure to improve efficiency and robustness. This work aims to shed some light on implementing generic dynamic locomotion by stabilizing nonlinear modes. The nonlinear modal analysis extends the linear modal theory to nonlinear systems and thus characterizes the oscillations that a robot can execute as autonomous evolutions. We execute forward hopping motions with a single segmented elastic leg as the first step towards generic modal locomotion. We propose a locomotion algorithm that exploits the modes of an extension of the SLIP model. We develop this strategy to generalize to other robotic systems, and we extensively validate it with experiments on an elastically actuated segmented leg.@en

This letter discusses an extension of the famous PD regulator implementing point to point motions with prescribed exponential rates of convergence. This is achieved by deriving a novel global exponential stability result, dealing with mechanical systems evolving on uni-dimensional invariant manifolds of the configuration space. The construction of closed loop controllers enforcing the existence of such manifolds is then discussed. Explicit upper and lower bounds of convergence are provided, and connected to the gains of the closed loop controller. Simulations are carried out, assessing the effectiveness of the controller and the tightness of the exponential bounds.

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Eigenmanifolds are two-dimensional submanifolds of the state space, which generalize linear eigenspaces to nonlinear mechanical systems. Initializing a robot on an Eigenmanifold (or driving it there by control) yields hyper-efficient and regular oscillatory behaviors, called modal oscillations. This letter investigates the possibility of transitioning between two modal oscillations without ever leaving the Eigenmanifold. This is essential to generate control inputs that decrease or increase the amplitude of the nonlinear oscillations. First, we prove that this goal can be achieved using bounded inputs only for Eigenmanifolds with unidimensional projection in configuration space. Then, we show that by allowing for impulsive control actions, the problem can be solved for all Eigenmanifolds which self-intersect when projected in configuration space.

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This letter discusses an extension of the famous PD regulator implementing point to point motions with prescribed exponential rates of convergence. This is achieved by deriving a novel global exponential stability result, dealing with mechanical systems evolving on uni-dimensional invariant manifolds of the configuration space. The construction of closed loop controllers enforcing the existence of such manifolds is then discussed. Explicit upper and lower bounds of convergence are provided, and connected to the gains of the closed loop controller. Simulations are carried out, assessing the effectiveness of the controller and the tightness of the exponential bounds.

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