The paper addresses the static balancing and walking of elastically actuated legged robots. The control is realized by commanding the motor positions only and exploiting the bijective relation between motor and link positions at equilibrium under static external forces. The approach is formulated in a quite general framework first. The main implementation contribution is the definition of a body coordinate system and of an appropriate set of constraints, which leads to a fully determined system of equations. In addition to the desired COM and the vertical foot positions, which are defined by the walking task and the terrain, the imposed constraints are related to distances between individual legs. The controller is experimentally validated on a compliantly actuated quadruped.

Since the spread of robotic systems embedding in their mechanics purposefully designed elastic elements, the interest in characterizing and exploiting non-linear oscillatory behaviors has progressively grown. However, few works so far looked at the problem from the point of view of modal analysis. This is particularly surprising if considered the central role that modal theory had in the development of classic results in analysis and control of linear mechanical systems. With the aim of making a step toward translating and extending this powerful tool to the robotic field, we present the complete modal characterization of a simple yet representative non-linear elastic robot: the 2D planar mass-spring-damper system. Generic non-linear elastic forces and dissipative effects are considered. We provide here exact descriptions of the two non-linear normal modes of the system. We then extend the analysis to generic combinations of the modes in conservative case and for small damping. Simulations are provided to illustrate the theoretical results. This is one of the very firsts applications of normal mode theory to dynamically coupled non-linear systems, and the first exact result in the field.

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.

Motion planning for fast locomotion of compliantly actuated robotic legs is generally considered to be a challenging issue, posing considerable real-time problems. This is at least the case if time-continuous trajectories need to be generated online. In this paper we take advantage of a simple controller structure, which reduces the motion planning to a discrete-time planning problem, in which only a small set of input parameters need to be determined for each step. We show that for a planar leg with serial elastic actuation, hopping on a ground with stairs of irregular length and height can be planned online, based on a parameter mapping which has been learned in a data-driven manner by performing hopping trials with an adaptive exploration algorithm to evenly sample the parameter space. Experiments on a planar hopping leg prototype validate the approach.

Physical compliance can be considered one of the key technical properties a robot should exhibit to increase its mechanical robustness. In addition, the accompanying temporal energy-storing capabilities enable explosive and energy efficient cyclic motions. But these advantages come at a price, as compliance introduces unwanted intrinsic oscillatory dynamics, underactuation, and reduces the natural frequency of the plant. These aspects make control of the link configuration variables a challenging task. This paper presents two novel control methods for implementing link-side motion tracking capabilities and injecting a desired damping characteristic to suppress link vibrations along the reference trajectory for compliantly actuated robots with nonlinear elastic characteristics. We prove their uniform global asymptotic stability by invoking a theorem by Matrosov. Both approaches, namely elastic structure preserving (ESP) and ESP+, have in common that they preserve the link-side inertial properties and the elastic structure of the original plant dynamics, hence the name ESP control. Apart from that, ESP control focuses on preserving the inertial properties of motor dynamics. While ESP+ control aims at minimizing the dynamic shaping on the motor side. The performance of the feedback control laws have been evaluated on the Hand Arm System from the German Aerospace Center (DLR), a variable stiffness robot arm, where the stiffness in each of its joints is highly nonlinear. To the best of our knowledge, this is the first experimentally validated tracking controller for compliantly actuated, multijoint robots with nonlinear elastic elements.

This paper presents a control method to implement trajectory tracking and disturbance rejection characteristics for the link-side dynamics of compliantly actuated robots with nonlinear spring characteristics. This is achieved by introducing new motor coordinates reflecting the damping and feedforward terms and shaping the dynamics of the motor such that it structurally equals the dynamics in the original coordinates. Thus, the approach achieves the control goal while changing the original plant dynamics only to a minimum extent. Passivity, stability, and convergence properties of the closed loop dynamics are proven. The performance of the control approach has been experimentally evaluated on the variable stiffness robot arm DLR Hand Arm System, where the stiffness in each of the joints is highly nonlinear. To our best knowledge, this is the first experimentally validated tracking controller for compliantly actuated robots with nonlinear elastic elements.

We address the operation of robotic legs with intrinsic elasticity in hopping cycles determined by the mechanical resonant properties of the system. This ensures energy efficiency and high jumping velocity and distance. Recently, we have shown in simulation that a simple, biologically inspired bang-bang controller operating in the local coordinate of the first resonant mode leads to limit cycles which are robust with respect to leg model uncertainties and ground properties. In this paper we address the velocity control of the hopping and the planning of the bang-bang control parameters for the case that the systems should not move at steady state velocity, but should have different step lengths and heights. We exploit the discrete structure and the small number of parameters of the controller to develop a fast optimization procedure for generating an arbitrary sequence of steps. This approach can provide high motion performance, robustness and substantial computational time saving compared to continuous trajectory and controller gain planning. The stationary and the aperiodic hopping is validated by experiments on a new planar elastic leg.

A feedback control to generate jumping motions for compliantly actuated multilegged robots is proposed. The method allows to specify the direction of the jumping motion. This is achieved by a constraint that defines a one-dimensional submanifold and a bang-bang control which generates a limit cycle on this submanifold. The approach is based on classical impedance control with the difference that the stiffness on the submanifold and the force to preserve a predefined nominal body configuration result from the intrinsic mechanical springs in the joints. Furthermore, we propose two controller implementations: the first implementation does not require to detect the contact state, while the second implementation requires contact state detection, but accounts in addition for Coulomb friction constraints. The controller is validated in simulation with a compliantly actuated quadruped.

The elastic energy storage in biologically inspired variable impedance actuators (VIA) offer the capability of executing cyclic and/or explosive multi-degree of freedom (DoF) motions efficiently. This paper studies the generation of cyclic motions for strongly nonlinear underactuated multi-DoF serial robotic arms. By experimental observations of human motor control, a simple and robust 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 oscillatory behavior of human and robotic arms is analyzed in simulations and experiments. It is found that the existence of easily excitable oscillation modes strongly depends on the damping properties of the plant. If the intrinsic damping properties are such that oscillations excited in the undesired modes decay faster than in the desired mode, then multi-DoF oscillations are easily excitable. Simulations and experiments reveal that serially-structured elastic multibody systems such as VIA or human arms with approximately equal joint damping, fulfill these requirements.

This paper presents a control approach to stabilize limit cycle motions along a mechanical mode of variable stiffness actuated (VSA) robots. Thereby, first a PD controller with gravity and Coriolis/centrifugal compensation shapes a desired dynamics, which is decoupled in terms of modal coordinates. Then an asymptotically stable limit cycle is generated on the link side dynamics for a selected mode. Finally, the modal control approach first introduced for rigid robots is extended to the VSA case. This is done by a joint torque controller, which decouples the torque dynamics from the link side dynamics. Stability and convergence are proven for the dynamics resulting from each feedback control. Furthermore, the energy efficiency of the proposed approach is verified by simulation and experiments on the VSA robotic arm DLR Hand Arm System.

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.

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.

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.