From Underactuation to Quasi‐Full Actuation: A Unifying Control Framework for Rigid and Elastic Joint Robot

The quest for animal-like performance in robots has driven the integration of elastic elements in their drive trains, sparking a revolution in robot design. Elastic robots can store and release potential energy, providing distinct advantages over traditional robots, such as enhanced safety in human-robot interaction, resilience to mechanical shocks, improved energy efficiency in cyclic tasks, and dynamic motion capabilities. Exploiting their full potential, however, necessitates novel control methods. This thesis advances the field of nonlinear control for underactuated systems and utilizes the results to push the boundaries of motion and interaction performance of elastic robots. Through real-life experiments and applications, the proposed controllers demonstrate that compliant robots hold promise as groundbreaking robotic technology. To achieve these objectives, we first derive a simultaneous phase space and input transformation that enables a specific class of underactuated Lagrangian systems to be treated as if fully actuated. These systems can be represented as the interconnection of actuated and underactuated subsystems, with the kinetic energy of each subsystem depending only on its own velocity. Elastic robots are typical representatives. We refer to the transformed system as quasi-fully actuated due to weak constraints on the new inputs. Fundamental aspects of the transforming equations are 1) the same Lagrangian function characterizes both the original and transformed systems, 2) the transformed system establishes a passive mapping between inputs and outputs, and 3) the solutions of both systems are in a one-to-one correspondence, describing the same physical reality. This correspondence allows us to study and control the behavior of the quasi-fully actuated system instead of the underactuated one. Thus, this approach unifies the control design for rigid and elastic joint robots, enabling the direct application of control results inherited from the fully-actuated case while ensuring closed-loop system stability and passivity. Unlike existing methods, the quasi-full actuation concept does not rely on inner control loops or the neglect and cancellation of dynamics. Notably, as joint stiffness values approach infinity, the control equivalent of a rigid robot is recovered. Building upon the quasi-full actuation concept, we extend energy-based control schemes such as energy shaping and damping injection, Euler-Lagrange controllers, and impedance control. Moreover, we introduce Elastic Structure Preserving (ESP) control, a passivity-based control scheme designed for robots with elastic or viscoelastic joints, guided by the principle of ``do as little as possible''. The underlying hope is that reducing the system shaping, i.e., having a closed-loop dynamics match in some way the robot's intrinsic structure, will award high performance with little control effort. By minimizing the system shaping, we obtain low-gain designs, which are favorable concerning robustness and facilitate the emergence of natural motions. A comparison with state-of-the-art controllers highlights the minimalistic nature of ESP control. Additionally, we present a synthesis method, based on purely geometric arguments, for achieving time-optimal rest-to-rest motions of an elastic joint with bounded input. Finally, we showcase the remarkable performance and robustness of the proposed ESP controllers on DLR David, an anthropomorphic robot implemented with variable impedance actuators. Experimental evidence reveals that ESP designs enable safe and compliant interaction with the environment and rigid-robot-level accuracy in free motion. Additionally, we introduce a control framework that allows DLR David to perform commercially relevant tasks, such as pick and place, teleoperation, hammer drilling into a concrete block, and unloading a dishwasher. The successful execution of these tasks provides compelling evidence that compliant robots have a promising future in commercial applications

Robust Stabilization of Elastic Joint Robots by ESP and PID Control: Theory and Experiments

This work addresses the problem of global set-point control of elastic joint robots by combining elastic structure preserving (ESP) control with non-collocated integral action. Despite the popularity and extensive research on PID control for rigid joint robots, such schemes largely evaded adoption to elastic joint robots. This is mainly due to the underactuation inherent to these systems, which impedes the direct implementation of PID schemes with non-collocated (link position) feedback. We remedy this issue by using the recently developed concept of “quasi-full actuation,” to achieve a link-side PID control structure with “delayed” integral action. The design follows the structure preserving design philosophy of ESP control and ensures global asymptotic stability and local passivity of the closed loop. A key feature of the proposed controller is the switching logic for the integral action that enables the combination of excellent positioning accuracy in free motion with compliant manipulation in contact with the environment. Its performance is evaluated on an elastic joint testbed and a compliant robot arm. The results demonstrate that elastic robots may achieve positioning accuracy comparable to rigid joint robots

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

Elastic Structure Preserving Impedance Control for Nonlinearly Coupled Tendon-Driven Systems

Traditionally, most of the nonlinear control techniques for elastic robotic systems focused on achieving a desired closed-loop behavior by modifying heavily the intrinsic properties of the plant. This is also the case of elastic tendon-driven systems, where the highly nonlinear couplings lead to several control challenges. Following the current philosophy of exploiting the mechanical compliance rather than fighting it, this letter proposes an Elastic Structure Preserving impedance (ESPi) control for systems with coupled elastic tendinous transmissions. Our strategy achieves a globally asymptotically stable closed-loop system that minimally shapes the intrinsic inertial and elastic structure. %to add desired stiffness and damping on the link side. It further allows to impose a desired link-side impedance behavior. Simulations performed on the tendon-driven index finger of the DLR robot David show satisfactory results of link-side interaction behavior and set-point regulation

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

Simultaneous Motion Tracking and Joint Stiffness Control of Bidirectional Antagonistic Variable-Stiffness Actuators

Since safe human-robot interaction is naturally linked to compliance in these robots, this requirement presents a challenge for the positioning accuracy. The class of variable- stiffness robots features intrinsically soft contact behavior where the physical stiffness can even be altered during operation. Here we present a control scheme for bidirectional, antagonistic variable-stiffness actuators that achieve high-precision link-side trajectory tracking while simultaneously ensuring compliance during physical contact. Furthermore, the approach enables to regulate the pretension in the antagonism. The theoretical claims are confirmed by formal analyses of passivity during physical interaction and the proof of uniform asymptotic stability of the desired link-side trajectories. Experiments on the forearm joint of the DLR robot David verify the proposed approach

Joint-Level Control of the DLR Lightweight Robot SARA

Lightweight robots are known to be intrinsically elastic in their joints. The established classical approaches to control such systems are mostly based on motor-side coordinates since the joints are comparatively stiff. However, that inevitably introduces errors in the coordinates that actually matter: the ones on the link side. Here we present a new joint-torque controller that uses feedback of the link-side positions. Passivity during interaction with the environment is formally shown as well as asymptotic stability of the desired equilibrium in the regulation case. The performance of the control approach is experimentally validated on DLR’s new generation of lightweight robots, namely the SARA robot, which enables this step from motor-side-based to link-sided-based control due to sensors with higher resolution and improved sampling rate

From underactuation to quasi-full actuation: Aiming at a unifying control framework for articulated soft robots

We establish a structure preserving state and input transformation that allows a class of underactuated Euler Lagrange systems to be treated as “quasi-fully” actuated. In this equivalent quasi-fully actuated form, the system is characterized by the same Lagrangian structure as the original one. This facilitates the design of control approaches that take into account the underlying physics of the system and that shape the system dynamics to a minimum extent. Due to smoothness constraints on the new input vector that acts directly on the noncollocated coordinates, we coin the term quasi-fully actuated. The class of Euler–Lagrange systems we consider is the class of articulated soft robots with nonlinear spring characteristics that are modeled with a block diagonal inertia matrix. We illustrate how the quasi-fully actuated form enables the direct transfer of control concepts that have been derived for fully actuated manipulators. We adopt the popular energy-shaping and two passivity-based concepts. The exemplary adoptions of the PD+ and Slotine and Li controllers allow us to solve the task-space tracking problem for highly elastic joint robots with nonlinear spring characteristics. These control schemes allow compliant behavior of the robot's TCP to be specified with respect to a reference trajectory. A key aspect of the presented framework is that it enables the adoption of rigid joint controllers as well as concepts underlying the original stability analysis. We believe that our framework presents an important step toward unifying the control design for rigid and articulated soft robots

Tele Running - Energy Efficient Locomotion for Elastic Joint Robots by Imitation Learning

This thesis presents an imitation learning approach to energy-efficient trajectory generation for elastic, legged robots. The trajectories are generated by teleoperation with force feedback. The presented framework allows an operator to achieve locomotion on an one-leg hopper by controlling its foot tip. The force feedback is designed to assist the operator to find gaits which exploit the natural harmonics of the hopper and thus improve energy efficiency. The resulting trajectory is approximated, parameterized, and replayed on the robot. The operator achieves a cost of transport of 0.25 at 0.63 m/s, considering the mechanical energy. Black-box optimization is used to keep this value with varying hardware parameters, such as different foot-tip stiffness. A reinforcement learning algorithm stabilizes lateral movement by active balance in simulation. Learning on hardware shows an improvement in stability. The concept is extended to multi-legged robots by teleoperating the two feet of the biped DLR C-Runner in simulation. The force feedback assists the operator to find stable gaits where the center of mass does not leave the support polygon of the feet. On both systems, the presented teleoperation framework utilizes the human's capability of estimating the properties of non-linear dynamics by designing appropriate haptic feedback