Dynamic Trajectory Generation for Serial Elastic Actuated Robots ⋆ Florian Petit * Dominic Lakatos * Werner Friedl * Alin Albu-Sch¨ affer * * Institute of Robotics and Mechatronics, German Aerospace Center (DLR), D-82234 Oberpfaffenhofen, Germany (e-mail: {florian.petit, dominic.lakatos, werner.friedl, alin.albu-schaeffer}@ dlr.de). Abstract: Robotic systems can benefit from the introduction of properly chosen joint elasticity. Besides their robustness against rigid impact, the energy saving capabilities may increase the system dynamics. In this paper, a method applicable for robots with serial elastic joints is presented, which embodies a desired oscillatory behavior into the hardware and thereby leads to improved performance. This is achieved by shaping the flexible joint robot as a linear one- mode system and embodying the natural frequency of the real intrinsic behavior. An algorithm is presented for shaping the one-mode property and exciting the system via a negative definite damping term in a decoupled coordinate space. The output of the approach is a dynamic trajectory resulting in a coordinated link motion and synchronized transfer of kinetic and potential energy. Furthermore, the dynamic trajectory is commanded to the real robot via a motor PD controller, where asymptotic stability for both subsystems—i.e. the trajectory generator and the controlled robot—is proven. The method is validated on a two-link serial elastic actuated robot. Both, simulation and experiment confirm the eigenmode embodiment, energy efficiency by velocity enlargement between motor and link side motion, and synchronized joint motion. Keywords: Serial elastic actuated, nonlinear oscillations, modal decoupling, singular perturbation. 1. INTRODUCTION Actuators with intrinsic compliances promise several ben- efits for a variety of robotic systems. Besides of technologi- cal considerations like the use of the known force-deflection relation to estimate joint torques (Pratt and Williamson (1995)), especially the increased mechanical robustness is a major advantage: The spring elements act as low pass filters against peak torques as they may occur during rigid impacts. This is relevant for scenarios like a robotic hand manipulating objects (Grebenstein et al. (2011) describe such a system) or a running robot in ground contact (Raibert (1986) makes use of passive elasticities in running robots). Furthermore, the energy storage properties of passive compliance are interesting and could be beneficial for solving highly dynamical tasks such as fast point-to- point movements, bipedal walking, throwing etc. Some work has been done to exploit the elastic elements to gain increased end effector velocity (Braun et al. (2011); Haddadin et al. (2012)) or tune the resulting oscillatory behaviour to a predefined trajectory (Uemura and Kawa- mura (2009); Visser et al. (2011)). The mentioned work mainly uses optimization and iterative methods to adjust the joint stiffness and torque to achieve the desired action. Furthermore, the topic of trajectory tracking has been addressed in multiple ways. Ranging from input shaping ⋆ This paper has been partly funded by the European commissions Seventh Framework Program as part of the project VIACTORS under grant No. 231554. techniques, optimal control, and optimization based tra- jectory generation to adaptive control and flatness based feed forward command generation, many approaches have been developed. However, the presented approach focuses on exploiting the natural system dynamics. This work aims to exploit the joint compliance of multiple degrees of freedom (DoF) series elastic actuated (SEA) robots by identifying and shaping intrinsic resonance prop- erties of the system. The goal is to synchronize the motions of the complete system by coordinating the motion of the single joints. Since synchronized motion induces synchro- nized transfer between potential and kinetic energy, there exist a point where the Hamiltonian energy is completely kinetic. Such a property can then be exploited while solv- ing a highly dynamical periodic task. In order to reach the desired behavior an eigenmode anal- ysis is performed on the locally linearized robot model. Then a desired system is formulated such that all eigen- modes of the resulting dynamics are the same. To achieve the desired dynamics, the robot needs to be tuned e.g. by varying the joint stiffnesses, or by an active control algorithm as it is given in the following. Furthermore, an efficient way to exploit the one-mode dynamics is presented. Motivated by linear second order systems, a negative damping coefficient is used to excite the system and obtain controlled oscillatory behaviour. The excitation acting at the natural frequency of the