Robot Impedance Control and Passivity Analysis with .buchlij@ethz.ch, tboaventura@ethz.ch Impedance

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Transcript of Robot Impedance Control and Passivity Analysis with .buchlij@ethz.ch, tboaventura@ethz.ch Impedance

  • Robot Impedance Control and Passivity Analysiswith Inner Torque and Velocity Feedback Loops

    Michele Focchi , Gustavo A. Medrano-Cerda , Thiago Boaventura, Marco Frigerio, Claudio Semini

    Jonas Buchli, Darwin G. Caldwell

    Dept. of Advanced Robotics,Istituto Italiano di Tecnologia (IIT),

    via Morego, 30, 16163 Genova.@iit.it

    Agile & Dexterous Robotics Lab,ETH Zurich,

    Tannenstr. 3, 8092 Zurichbuchlij@ethz.ch, tboaventura@ethz.ch

    Impedance control is a well-established technique tocontrol interaction forces in robotics. However, real imple-mentations of impedance control with an inner loop may suf-fer from several limitations. In particular, the viable range ofstable stiffness and damping values can be strongly affectedby the bandwidth of the inner control loops (e.g. a torqueloop) as well as by the filtering and sampling frequency. Thispaper provides an extensive analysis on how these aspects in-fluence the stability region of impedance parameters as wellas the passivity of the system. This will be supported by bothsimulations and experimental data. Moreover, a methodol-ogy for designing joint impedance controllers based on aninner torque loop and a positive velocity feedback loop willbe presented. The goal of the velocity feedback is to increase(given the constraints to preserve stability) the bandwidth ofthe torque loop without the need of a complex controller.Keywords Impedance control, torque control, passivity andstability analysis;

    1 IntroductionUntil recently, the majority of legged robots employed

    high-gain (stiff) position feedback control [1]. However, thisapproach is unsuitable when a robot is in contact with un-structured real-world environment, as the controller wouldtry to satisfy the position goal at all costs [2]. Instead, forsuch scenarios, a force/torque control in joint or end-effectorspace is desirable.

    For a legged robot, force control can be useful in boththe swing and stance phase. During stance, it allows to con-trol the ground impact forces, with the purpose to improvebalance capabilities. During the swing phase, it plays a cru-cial role in providing to the robots leg the compliance neces-sary to negotiate unperceived obstacles, while still ensuring agood position tracking by using rigid body inverse dynamics.Interaction forces can be regulated in two ways: passively

    and actively. Passive methods are those in which physicalcompliant elements are included between the robot and theenvironment to limit the interaction forces (e.g. a passivespring in series elastic actuators [3], [4]). On the other handactive compliance is achieved through the active control ofjoints (position or torque) using feedback measurements ofjoint torques [5]. This can emulate a virtual compliance bothat the joint as well as at the end-effector/foot level.

    A major benefit of active compliance is its ability tochange the dynamic characteristics (e.g. stiffness and damp-ing) in real-time. Hence, legged robots can take advan-tage of active compliance to adapt the leg stiffness to swingand stance phases, or to the surface properties [6]. Manymethods to actively control compliance at the end-effectorhave been developed, such as impedance control [7], oper-ational space control [8], hybrid force-control [9], and vir-tual model control [10]. Impedance control, in particular,allows the dynamic characteristics at the robot interactionport (e.g. the end-effector) to be specified by regulating thedynamic relationship between forces and positions (mechan-ical impedance). Despite impedance is of primary impor-tance to achieve dynamically stable robot locomotion, onlyrecently an exhaustive research has been carried out, on theMIT Cheetah robot, to find which impedance parameters aresuitable for locomotion [11]. However, an analysis that in-vestigates if these parameters are realizable is still missing.

    In the past, impedance control algorithms were limitedby the controller bandwidth, which was set by the computa-tional power and actuator dynamics. That was one of thereasons for the introduction of passive elements in serieswith the actuator [12], which have intrinsically unlimitedbandwidth. However, recent advances in both computer andactuator performance, made active compliance feasible forhighly-dynamic applications [13, 14]. Nevertheless, manyaspects, still create stability issues on impedance control.For instance, the range of stable stiffness and dampings thatcan be virtually created (Z-width [15], where Z stands for

  • impedance [16]) can be limited by filtering, sampling fre-quency, and also by the bandwidth of inner control loops (e.g.a torque loop).

    A common practice in designing nested loop controlsystems is to maximize the bandwidth of the innermostloop [17]. However, maximizing the inner loop controllerbandwidth is not always the best strategy. When theouter impedance loop is closed, designing the inner loop tohave the highest possible bandwidth reduces the range ofimpedance parameters for which the whole system is stable,as demonstrated later in this work. Therefore, a trade-offmust be found between: having a high bandwidth to ensuregood torque and impedance tracking, and keeping the band-width low to increase the range of stable impedance values.Other aspects that directly influence the stability region arethe sampling frequency and filtering [18]. Their effect is tointroduce delays into the control loop, and their influencewill also be investigated in this work. To ensure closed-loopstability during interactions with the environment or othersystems, the controller must be designed to ensure the sys-tem behaves passively at the interaction port [19], [20]. Fromthe passivity property, asymptotic stability can always be en-sured: both in free motion as well as when the robot is incontact with any type of environment (which is usually pas-sive). Physical compliant elements and rigid bodies are pas-sive by nature. However, when the compliant behavior isemulated by an actuator, the controller gains set the systempassivity. In this work it will be shown that passivity can alsobe a restrictive condition to select impedance parameters.

    Related works. The published literature about activecompliance is vast. A brief review on the issues that af-fect the performance of force controlled robots can be foundin [21]. Stability analysis and performance specificationsfor compliance control was first introduced by Kazerooniet al. [22] for a manipulator whose model had boundeduncertainty. Lawrence in [23] considers the non-ideal,practical effects of computation and communication delayson impedance control and finds some stability boundaries.However, his analysis was in continuous time and it is notnecessarily valid for discrete time systems. Indeed samplingis not completely equivalent to time delays because whensampling there are additional zeros that do not appear in con-tinuous time.

    Regarding controllers based on passivity, Albu-Schafferet al. in [20] implemented a full state controller for jointor Cartesian impedance with passive capabilities. The con-troller is not passive itself but it is together with the motordynamics. The torque feedback shapes the rotor inertia ofthe motors to a desired value. More recently Buerger andHogan [24] have revisited the problem of designing con-trollers for physically interactive robots. For a 1 DoF sys-tem, they reformulated the problem as a robust stability prob-lem based on mu-synthesis (structured singular values) andloop shaping methods. The approach provides improvementsin robot performance compared to traditional passive con-trollers. In [25] stiffness and impedance control conceptswere used for robot-aided rehabilitation. New stability con-ditions were proposed using Lyapunov approach and based

    on the relationship between the dynamics of the robot andits energy. In [26] Yasrebi et al. carried out a time-domainpassivity analysis of the Z-width diagram. This led to thedesign of a new haptic controller which extended the rangeof stable impedance parameters (Z- width) by means of anacceleration feedback. The analysis was carried out for onejoint using passivity theory in the frequency domain.

    The main contribution of this work, is a methodology toanalyze (based on an accurate model) stability and passivityof a gearbox driven actuator (plus load) system. The analysistakes into account all the non-idealities present in real im-plementation of an impedance controller, namely: actuatordynamics, discrete implementation, filtering, nested loops.This allows to find the impedance stability regions whichrepresent the impedance parameters that can be rendered ina stable way. Simulations and experimental data show howthe above-mentioned non-idealities influence the stability re-gions as well as the passivity of the system. The study iscarried out for the adduction/adduction (HAA) joint of theHyQ [27] robot (see Fig. 1), where impedance control wasimplemented with an inner torque loop [28]. However, theunderlying ideas are valid for any electric actuator movinga load with a gearbox reduction. In the bigger picture, thestability regions are the basis to develop a gain scheduler (inthe low-level control layer) which is able to adapt the band-width of the inner torque loop according to the impedanceparameters set by the user.

    Fig. 1: HyQ robot

    This paper is structured as follows: the mathematicalmodel of the system is introduced in Section 2 followed by adescription of the control system implementation in Section3. The stability issues associated with real implementationof an impedance controller are analysed, both in simulationsand experimentally, in Section 4. A brief assessment aboutpassivity for the system is then given in Section 5. Finally,Section 6 discusses the results and future works.

  • 2 System description and mathematical modelThe studies