One-Dimensional Solution Families of Nonlinear Systems Characterized by Scalar Functions on Riemannian Manifolds

For the study of highly nonlinear, conservative dynamic systems, finding special periodic solutions which can be seen as generalization of the well-known normal modes of linear systems is very attractive. However, the study of low-dimensional invariant manifolds in the form of nonlinear normal modes is rather a niche topic, treated mainly in the context of structural mechanics for systems with Euclidean metrics, i.e., for point masses connected by nonlinear springs. Newest results emphasize, however, that a very rich structure of periodic and low-dimensional solutions exist also within nonlinear systems such as elastic multi-body systems encountered in the biomechanics of humans and animals or of humanoid and quadruped robots, which are characterized by a non-constant metric tensor. This paper discusses different generalizations of linear oscillation modes to nonlinear systems and proposes a definition of strict nonlinear normal modes, which matches most of the relevant properties of the linear modes. The main contributions are a theorem providing necessary and sufficient conditions for the existence of strict oscillation modes on systems endowed with a Riemannian metric and a potential field as well as a constructive example of designing such modes in the case of an elastic double pendulum

Visco-Elastic Structure Preserving Impedance (VESPi) Control for Compliantly Actuated Robots

In this paper we consider the control of robots that feature visco-elastic actuators with adjustable physical damping. Considering the link variables of the robot as output, the corresponding system dynamics has a relative degree of 3. We present a novel control approach that allows to realize a torque interface on the link side, while preserving the intrinsic visco-elastic structure and the inertial properties of the system. By means of this joint torque interface one can implement link-side position tracking and impedance tasks. For this case, we provide a stability and passivity analysis. The control approach has been verified by experiments with a visco-elastic joint testbed

Elastic Structure Preserving Impedance (ESPi) Control for Compliantly Actuated Robots

We present a new approach for Cartesian impedance control of compliantly actuated robots with possibly nonlinear spring characteristics. It reveals a remarkable stiffness and damping range in the experimental evaluation. The most interesting contribution, is the way the desired closed-loop dynamics is designed. Our control concept allows to add a desired stiffness and damping directly on the end-effector, while leaving the system structure intact. The intrinsic inertial and elastic properties of the system are preserved. This is achieved by introducing new motor coordinates that reflect the desired spring and damper terms. Theoretically, by means of additional motor inertia shaping it is possible to make the end-effector interaction behavior with respect to external loads approach, arbitrarily close, the interaction behavior that is achievable by classical Cartesian impedance control on rigid robots. The physically motivated design approach allows for an intuitive understanding of the resulting closed-loop dynamics. We perform a passivity and stability analysis on the basis of al physically motivated storage and Lyapunov function

The Grasp Perturbator: Calibrating human grasp stiffness during a graded force task

In this paper we present a novel and simple handheld device for measuring in vivo human grasp impedance. The measurement method is based on a static identification method and intrinsic impedance is identified inbetween 25 ms. Using this device it is possbile to develop continuous grasp impedance measurement methods as it is an active research topic in physiology as well as in robotics, especially since nowadays (bio-inspired) robotics can be impedance-controlled. Potential applications of human impedance estimation range from impedance-controlled telesurgery to limb prosthetics and rehabilitation robotics. We validate the device through a physiological experiment in which the device is used to show a linear relationship between finger stiffness and grip force

Nonlinear Oscillations for Cyclic Movements in Variable Impedance Actuated Robotic Arms

Biologically inspired Variable Impedance Actuators (VIA) offer the capability to execute cyclic and/or explosive multi degree of freedom (DoF) motions efficiently by storing elastic energy. This paper studies the preconditions which allow to induce robust cyclic motions for strongly nonlinear, underactuated multi DoF robotic arms. By experimental observations of human motor control, a simple control law is deduced. This controller achieves intrinsic oscillatory motions by switching the motor position triggered by a joint torque threshold. Using the derived controller, the periodic behavior of the robotic arm is analyzed in simulations. It is found that a modal analysis of the linearized system at the equilibrium point allows to qualitatively predict the periodic behavior of this type of strongly nonlinear systems. The central statement of this paper is that cyclic motions can be induced easily in VIA systems, if the eigenfrequencies and modal damping values of the linearized system are well separated. Validation is given by simulation and experiments, where a human controls a simulated robotic arm, and the developed regulator controls a robotic arm in simulation and experiments

Modal Matching: An Approach to Natural Compliant Jumping Control

This letter derives the basic concept of modal matching-an approach to natural motion control. Modal matching exploits the nonlinearity of the rigid multi-body dynamics (and the variability of the elastic transmissions) as degree of freedom to fit the natural plant dynamics to the desired dynamics of the task. Modal matching achieves a desired intrinsic oscillation behavior, which is locally equivalent to the dynamics of the basic spring-loaded inverted pendulum or pogo-stick model (both implementing a linear inertia acting on a linear spring), well established in locomotion analysis and control. Using the concept of modal matching, an efficient and effective methodology to natural jumping control is introduced

Switching Based Limit Cycle Control for Compliantly Actuated Second-Order Systems

This paper derives a stability statement for a novel, switching based limit cycle control. The stability proof is based on multiple Lyapunov functions and a new interpretation of contraction analysis. By showing that the dissipated energy on the cycle increases with increasing velocity, while the injected energy is constant, the emergence of an attractive limit cycle is shown. The approach applies for general, nonlinear, and compliantly actuated second-order systems, with positive definite plant parameters and non-aperiodic solutions. An analysis of the controller parameters reveals, that for the majority of parameters, global attractiveness of the limit cycle can be guaranteed

A Modally Adaptive Control for Multi-Contact Cyclic Motions in Compliantly Actuated Robotic Systems

Compliant actuators in robotic systems improve robustness against rigid impacts and increase the performance and efficiency of periodic motions such as hitting, jumping and running. However, in the case of rigid impacts, as they can occur during hitting or running, the system behavior is changed compared to free motions which turns the control into a challenging task. We introduce a controller that excites periodic motions along the direction of an intrinsic mechanical oscillation mode. The controller requires no model knowledge and adapts to a modal excitation by means of measurement of the states. We experimentally show that the controller is able to stabilize a hitting motion on the variable stiffness robot DLR Hand Arm System. Further, we demonstrate by simulation that the approach applies for legged robotic systems with compliantly actuated joints. The controlled system can approach different modes of motion such as jumping, hopping and running, and thereby, it is able to handle the repeated occurrence of robot-ground contacts