Control Theory and Technology

Robot Impedance Control and Passivity Analysiswith Inner Torque and Velocity Feedback LoopsMichele Focchi 1†, Gustavo A. Medrano-Cerda 1, Thiago Boaventura 2, Marco Frigerio 1, Claudio Semini 1,

Jonas Buchli 2, Darwin G. Caldwell 1,

1. Dept. of Advanced Robotics, Istituto Italiano di Tecnologia (IIT), via Morego, 30, 16163 Genova, Italy;

2. Agile & Dexterous Robotics Lab, ETH Zurich, Tannenstr. 3, 8092 Zurich, Switzerland;

Received 21st January 2015; revised 19th October 2015; accepted 22nd October 2015

Abstract:Impedance control is a well-established technique to control interaction forces in robotics. However, real implementations of

impedance control with an inner loop may suffer from several limitations. In particular, the viable range of stable stiffness and dampingvalues can be strongly affected by the bandwidth of the inner control loops (e.g. a torque loop) as well as by the filtering and samplingfrequency. This paper provides an extensive analysis on how these aspects influence the stability region of impedance parameters aswell as the passivity of the system. This will be supported by both simulations and experimental data. Moreover, a methodology fordesigning joint impedance controllers based on an inner torque loop and a positive velocity feedback loop will be presented. The goalof the velocity feedback is to increase (given the constraints to preserve stability) the bandwidth of the torque loop without the need ofa complex controller.

Keywords: Impedance control, torque control, passivity and stability analysis

DOI 10.1007/s11768-014-****-*

1 Introduction

Until recently, the majority of legged robots employedhigh-gain (stiff) position feedback control [1]. However,this approach is unsuitable when a robot is in contactwith unstructured real-world environment, as the controllerwould try to satisfy the position goal at all costs [2]. In-stead, for such scenarios, a force/torque control in joint orend-effector space is desirable.

For a legged robot, force control can be useful in boththe swing and stance phase. During stance, it allows tocontrol the ground impact forces, with the purpose to im-prove balance capabilities. During the swing phase, it playsa crucial role in providing to the robot’s leg the compliancenecessary to negotiate unperceived obstacles, while stillensuring a good position tracking by using rigid body in-verse dynamics. Interaction forces can be regulated in twoways: passively and actively. Passive methods are thosein which physical compliant elements are included be-

tween the robot and the environment to limit the interac-tion forces (e.g. a passive spring in series elastic actuators[3], [4]). On the other hand active compliance is achievedthrough the active control of joints (position or torque) us-ing feedback measurements of joint torques [5]. This canemulate a virtual compliance both at the joint as well as atthe end-effector/foot level.

A major benefit of active compliance is its abilityto change the dynamic characteristics (e.g. stiffness anddamping) in real-time. Hence, legged robots can take ad-vantage of active compliance to adapt the leg stiffness toswing and stance phases, or to the surface properties [6].Many methods to actively control compliance at the end-effector have been developed, such as impedance control[7], operational space control [8], hybrid force-control [9],and virtual model control [10]. Impedance control, in par-ticular, allows the dynamic characteristics at the robot in-teraction port (e.g. the end-effector) to be specified by reg-

†Corresponding author.E-mail: michele.focchi@iit.it Tel.: +39 010 71781974; fax: +39 010 71781232.This work was supported by the Istituto Italiano di Tecnologia.

c© 2015 Istituto Italiano di Tecnologia, Advanced Robotics Department, Italy.

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M. Focchi et al. / Control Theory Tech, Vol. 12, No. 1, pp. 68–81, February 2014

ulating the dynamic relationship between forces and po-sitions (mechanical impedance). Despite impedance is ofprimary importance to achieve dynamically stable robot lo-comotion, only recently an exhaustive research has beencarried out, on the MIT Cheetah robot, to find whichimpedance parameters are suitable for locomotion [11].However, an analysis that investigates if these parametersare realizable is still missing.

In the past, impedance control algorithms were limitedby the controller bandwidth, which was set by the compu-tational 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 computerand actuator performance, made active compliance feasi-ble for highly-dynamic applications [13,14]. Nevertheless,many aspects, still create stability issues on impedancecontrol. For instance, the range of stable stiffness anddampings that can be virtually created (Z-width [15], whereZ stands for impedance [16]) can be limited by filtering,sampling frequency, and also by the bandwidth of innercontrol 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 the outerimpedance loop is closed, designing the inner loop tohave the highest possible bandwidth reduces the range ofimpedance parameters for which the whole system is sta-ble, as demonstrated later in this work. Therefore, a trade-off must be found between: having a high bandwidth to en-sure good torque and impedance tracking, and keeping thebandwidth low to increase the range of stable impedancevalues. Other aspects that directly influence the stabilityregion are the sampling frequency and filtering [18]. Theireffect is to introduce delays into the control loop, and theirinfluence will also be investigated in this work. To ensureclosed-loop stability during interactions with the environ-ment or other systems, the controller must be designed toensure the system behaves passively at the interaction port[19], [20]. From the passivity property, asymptotic stabilitycan always be ensured: both in free motion as well as whenthe robot is in contact with any type of environment (whichis usually passive). Physical compliant elements and rigidbodies are passive by nature. However, when the compliantbehavior is emulated by an actuator, the passivity is a func-tion of the controller gains. In this work it will be shownthat passivity can also be a restrictive condition to selectimpedance parameters.

Related works. The published literature about activecompliance is vast. A brief review on the issues that affectthe 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 bounded un-certainty. Lawrence in [23] considers the non-ideal, practi-cal effects of computation and communication delays onimpedance control and finds some stability boundaries.However, his analysis was in continuous time and it is notnecessarily valid for discrete time systems. Indeed sam-pling is not completely equivalent to time delays becausewhen sampling there are additional zeros that do not ap-pear in continuous 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 mo-tor dynamics. The torque feedback shapes the rotor inertiaof the motors to a desired value. More recently Buergerand Hogan [24] have revisited the problem of designingcontrollers for physically interactive robots. For a 1 DoFsystem, they reformulated the problem as a robust stabil-ity problem based on µ-synthesis (structured singular val-ues) and loop shaping methods. The approach provides im-provements in robot performance compared to traditionalpassive controllers. In [25] stiffness and impedance controlconcepts were used for robot-aided rehabilitation. New sta-bility conditions were proposed using Lyapunov approachand based on the relationship between the dynamics of therobot and its energy. In [26] Yasrebi et al. carried out atime-domain passivity analysis of the Z-width diagram.This led to the design of a new haptic controller whichextended the range of stable impedance parameters (Z-width) by means of an acceleration feedback. The analy-sis was carried out for one joint using passivity theory inthe frequency domain.

The main contribution of this work, is a methodologyto analyze (based on an accurate model) stability andpassivity of a gearbox driven actuator (plus load) sys-tem. The analysis takes into account all the non-idealitiespresent in real implementation of an impedance controller,namely: actuator dynamics, discrete implementation, fil-tering, nested loops. This allows to find the impedance”stability regions” which represent the impedance param-eters that can be rendered in a stable way. Simulations andexperimental data show how the above-mentioned non-idealities influence the stability regions as well as the pas-sivity of the system. The study is carried out for the ad-

M. Focchi et al. / Control Theory Tech, Vol. 12, No. 1, pp. 68–81, February 2014

duction/adduction (HAA) joint of the HyQ [27] robot (seeFig. 1), where impedance control was implemented with aninner torque loop [28]. However, the underlying ideas arevalid for any electric actuator moving a load with a gear-box reduction. In the bigger picture, the stability regionsare the basis to develop a gain scheduler (in the low-levelcontrol layer) which is able to adapt the bandwidth of theinner torque loop according to the impedance parametersset by the user.

Fig. 1: Picture of a lateral view of the HyQ robot, the HAAjoint axis is marked in red.This paper is structured as follows: the mathematicalmodel of the system is introduced in Section 2 followedby a description of the control system implementation inSection 3. The stability issues associated with real imple-mentation of an impedance controller are analysed, both insimulations and experimentally, in Section 4. A brief as-sessment about passivity for the system is then given inSection 5. Finally, Section 6 discusses the results and fu-ture works.

2 System description and mathematicalmodel

The studies and experiments presented in this workare all conducted on HyQ [27]. HyQ is a fully torque-controlled quadruped robot with a mix of hydraulic andelectric actuation for each leg: two hydraulic joints on thesagittal plane (hip HFE and knee KFE flexion-extension)and one electric joint moving in the traversal plane (hipadduction-abduction HAA, Fig.2 on the left). This paperfocuses on modeling and control of the electric joint, whichconsists of a DC brush-less motor (Emoteq HT2301) and aharmonic drive gearbox (CSD-25-100). The leg is attached

to the gearbox output via an interface consisting of 6 par-allel pins (evenly distributed on a circle around the axis ofrotation) that enable easy dismounting (see Fig. 3). Thisinterface represents a small intermediate rotational inertia(JL1) placed before the inertia represented by the leg (JL2)in the transmission train.

Z

X

Y

KFE

HFE

HAA

Electricmotor

assembly

Electricmotor

assembly

Fig. 2: HyQ Leg. Lateral (left) and frontal (right) view. Thefigures show the definition of the joints and their angles, aswell as the coordinate frame.

Normally, this type of assembly is modelled by twosecond order differential equations coupled via the gear-box transmission flexibility [29]. However, after perform-ing several open loop tests using chirp signals, an anti-resonance was detected for the link velocity (see the fre-quency response of the link velocity to a chirp input volt-age in Fig. 5).

Rotor

Torquesensor

IN

Gearbox

Encoder

Intermediateinertia

Fig. 3: Cross-section of the mechanical assembly of theelectric joint. The intermediate inertia JL1 represents thepart that interfaces the leg with the gearbox output.

Since a model with two inertia and one spring can-not capture this behavior, a more complex model withthree inertia coupled by springs was used, as shown in theschematic in Fig. 4 where Khd and Kp are the stiffness re-lated to the gearbox and the leg flexibility, respectively.

M. Focchi et al. / Control Theory Tech, Vol. 12, No. 1, pp. 68–81, February 2014

Torque sensor

Encoder

(gravity)

Fig. 4: Motor drive system with torsional load, schematicfor the 3 mass-2 spring model.Joint position and torque are measured by an encoder anda torque sensor. Due to the topology depicted in Fig. 3 theposition encoder measures the angle of the intermediate in-ertia JL1 while the link velocity is measured by averagingfirst order differences (4 samples) of the position encoder(averaging filter). In addition a strain gage based torquesensor is mounted at the output of the harmonic drive. Nofilter is implemented on the torque signal because it wouldintroduce delays in the control action. According to thismodel, the Laplace transforms of the differential equationsthat describe the linearized dynamics of the load and of theelectric motor are:

Im =Vm

Ls+R− kwsθm

Ls+R

θm =kt Im

(Jms+Bm)s− Khd + sDhd

N(Jms+Bm)s

(θm

N−θL1

)+Tf r

θL1 =−(Kp + sDp)

(JL1s+BL1)s(θL1−θL2)+

Khd + sDhd

(JL1s+BL1)s

(θm

N−θL1

)

θL2 =Kp + sDp

(JL2s+BL2)s(θL1−θL2)−

KL2θL2

(JL2s+BL2)s+Tdist

(1)

−40

−30

−20

Magnitude[dB]

100

101−100

−50

0

Phase

[deg]

Frequency [Hz]

Fig. 5: Frequency response of the link velocity θL1 to achirp input voltage at the HAA motor, experimentally ob-tained with an unconstrained HyQ leg.

where Im, θm denote the motor current and motor posi-tion; θL1 and θL2 are the intermediate inertia and leg po-sitions; Vm is the motor voltage; Tf r is the friction torquein the harmonic drive and Tdist is an external disturbancetorque applied to the leg. All other symbols and parametervalues are given in Table 1. A block diagram that illustratesthe relationships between the state variables is depicted inFig. 6. Since the rotational inertia of the leg JL2 varies withthe configuration of the joints HFE and KFE, the table in-cludes also upper and lower bounds for the leg inertia. Inparticular the fact that the leg can retract or extend resultsin different mass distributions around the hip abduction-adduction axis. Namely, the inertia is higher when the legis extended and lower when it is retracted. Gravity has alsoan effect on the load dynamics, that, when linearized, be-haves as a rotational spring KL2. The stiffness of this ”grav-ity spring” is also dependent on the leg configuration and isreported in Table 1 for an extended and retracted leg con-figuration.

Table 1: Model parameters

Name Model Parameters Value

Jm Rotor + gearbox inertia (D) 5.72 ·10−5kgm2

Khd Gearbox stiffness (I) 8.077 ·103Nm/radDhd Gearbox damping (I) 16.56 Nms/radBm Visc. frict. rotor (I) 0.0015 Nms/radJL1 Interm. inertia (I) 1 ·10−4 kgm2

BL1 Visc. frict. of inertia JL1 (I) 0 Nms/rad

JL2 Leg inertia (C)0.439 kgm2 (ext.)0.129 kgm2 (ret.)

BL2 Visc. frict. of inertia JL2 (I) 0.756 Nms/rad

KL2Linear stiffness 11.2 Nm/rad (ext.)due to gravity 7.17 Nm/rad (ret.)

Kp Leg stiffness (I) 1.923·103Nm/radDp Leg damping (I) 7.56 Nms/radL Coil inductance (D) 2.02·10−3HR Coil resistance (D) 3.32 Ω

kt Motor torque constant (D) 0.19 Nm/Akw Motor speed constant (D) 0.19 Nms/radN Gear ratio (D) 100

M. Focchi et al. / Control Theory Tech, Vol. 12, No. 1, pp. 68–81, February 2014

--

--- -

Fig. 6 Block diagram representing Eqn. (1) that describes the linearized dynamics of the electric motor and of the load. Vm is the voltageinput, Tf r the harmonic-drive disturbance torque, Tdist is an external disturbance torque coming from the load side, θm is the motorposition, θL1 and θL2 the positions of the intermediate and leg inertia, respectively.

Name State variables Unit

θm Ang. pos. of the rotor radθL1 Ang. pos. of the intermediate inertia radθL2 Ang. pos. of the leg radIm Motor current A

Name Inputs Unit

Vm Motor voltage VVmVC Vel. comp. voltage VTdist Ext. dist. torque (load side) NmTf r Frict. dist. torque (motor side) Nm

Name Outputs Unit

Tl Load torque Nm

Name Controller Gains

Pt Torque controller prop. gainIt Torque controller integral gainβ Gain of the PI torque controllerα Velocity compensation (scalar) gain

Name Transfer functions

PIt(z) PI torque controllerGt(s) TF between Vm and Tl

VCgain TF of velocity compensationGtVC(s) TF between Vm and Tl after vel. comp.

3 Controller Design

This section explains the design of the control system.The controller architecture is shown in Fig. 7 where an in-ner positive velocity feedback loop is followed by a torqueloop controller and finally an outer impedance (position)loop. Specifications for the impedance loop vary depend-ing on the gait, for example a trotting gait frequency isaround 2 Hz for HyQ that has a mass of 75 kg. The spec-ifications for the performance will depend on the type oflocomotion gait and the gains of the impedance loop will

vary in a specified range. The inner torque loop and thevelocity feedback loop must be designed to be consistentwith these requirements.

3.1 Positive feedback velocity compensation

One difficulty in the design of the torque loop controlleris that the load dynamics may introduce severe limitationsin the closed loop performance of the torque loop. Thisproblem has been largely overlooked since in many casesthe load dynamics are ignored in the analysis. In this sub-section a positive velocity feedback (velocity compensa-tion) is introduced to address these limitations and improvethe torque bandwidth. To exhibit the above-mentioned lim-itations, first of all, the system response has been consid-ered after closing the inner velocity feedback loop. Thetorque transmitted to the load is measured by the torquesensor and can be expressed as:

Tl = (Khd + sDhd)

(θm

N−θL1

)(2)

From Eqn. (1), Eqn. (2) and Fig. 7, closing the velocityloop (without considering the averaging filter for the sakeof simplicity), the transfer function from the motor voltageto the torque is given by (cf. Eq. 3.24, Ch. 4, p. 45 in [30]for complete derivation):

GtVC =kt(Dhds+Khd)p1

N(p1q1 +(q2−VCgains)q3)(3)

where:

p1 = (JL2s2 +BL2s+KL2)(JL1s2 +BL1s)

+[(JL2s2 +BL2s+KL2)+(JL1s2 +BL1s)

](Dps+Kp)

q1 = (Ls+R)(

Jms2 +Bms+(Dhds+Khd)

N2

)+ ktkws

q2 =Nkt(Ls+R)(Jms2 +Bms)+Nkws

q3 =kt

N(JL2s2 +BL2s+Dps+Kp +KL2)(Dhds+Khd)

M. Focchi et al. / Control Theory Tech, Vol. 12, No. 1, pp. 68–81, February 2014

+

Actuator+

Load dynamics

-

-

+

Torque loop

-

VelocityCompensated

system

Averaging filter

Impedance loop

Fig. 7 Block diagram of the velocity compensated system with inner torque loop (PI, orange block) outer impedance loop (PD, greenblocks). The velocity compensation term (VmVC) is added to the output (VmPI) of the torque controller.

Observe that the transmission zeros in Eqn. (3) intro-duced by the polynomial p1 depend entirely on the loaddynamics, that is the load connected at the output of theharmonic drive. When the damping coefficients BL1 andBL2 are small, as is usually the case, some of these trans-mission zeros are very close to the stability region bound-ary. In the case of continuous time systems the boundary isthe imaginary axis and in the case of discrete time systemsthis is the unit circle. Notice that the zeros may be realor complex depending on the value of the stiffness KL2.These zeros impose limitations in the achievable closedloop bandwidth when using a simple proportional and inte-gral torque controller. This is because the controller pole,located at the origin, will be attracted towards the trans-mission zeros becoming the dominant pole of the system,thus limiting the closed loop bandwidth of the torque loopunless very high gains are used in the torque controller.In most cases the torque loop gain will have a finite gainmargin and therefore the controller gain cannot be madesufficiently large. This is more pronounced in digital con-trol where the gain margin is likely to be much lower thanthe gain margin for a continuous time system. The effect ofthe velocity compensation is that these unwanted transmis-sion zeros (polynomial p1) can be cancelled if the velocityfeedback gain is chosen as VCgain = q2/s so that the termq2−VCgains in the denominator of Eq. (3) is equal to zero.

VCgain =Nkt(Ls+R)(Jms+Bm)+Nkw (4)

The implementation of the compensator requires deriva-tives of the velocity signal that is often prone to quantiza-tion errors. Since this derivative is likely to be noisy thecompensator has to be approximated by adding suitablefilters which would add delay. An alternative solution is touse a simplified velocity compensation as presented in [31]which is obtained by discarding the derivative terms fromEq. (4):

VCgain =αNkt

(RBm + ktkw) α > 0 (5)

This simplified velocity compensation is obtained bysetting L = 0 and Jm = 0. This means we are neglect-ing both the electrical dynamics and the acceleration term,which would introduce noise in the system. α is introducedas an adjustable parameter. Therefore, with Eq. (5), an ex-act cancellation of the transmission zeros p1 is generallynot possible. Nevertheless, even though an exact cancel-lation is not possible, an improvement in the closed looptorque bandwidth can be achieved. To understand whenand how this is possible, consider the velocity compensa-tion given by Eq. (5). For the parameter values given inTable 1 and setting KL2 = 0, the polynomial p1 has fourreal roots z4 < z3 < z2 < z1 = 0. The transfer function (3)also has a pole at zero. This is an unobservable pole andtherefore it cancels out with the zero z1. The second zeroz2 is the closest to the imaginary axis and limits the torquebandwidth that can be achieved with a PI torque controller.Indeed, as α increases one real pole in Eq. (3) moves to-wards the stability boundary along the real axis and at somepoint it will become identical to z2. The value of the gainα for which this happens, is the ideal value required forcancelling out unwanted zero z2. In this particular case, z2

varies as a function of the leg inertia. Hence, it may bedifficult to completely cancel out this zero for all leg con-figurations with a fixed value for α . However, as long asthe pole in Eq. (3) is placed to the right of the zero z2 thenthe bandwidth limitation introduced by this zero is avoided(cf. Sec. 3.5.3, Ch. 4, p. 46 in [30]). When KL2 6= 0 thenthe roots of p1, which are closest to the imaginary axis, arecomplex (z1 and z2 are complex conjugate). As α increasestwo poles in Eq. (3) will move towards the imaginary axisas a complex conjugate pair but there is no value of α thatwill completely cancel out the unwanted zeros z1 and z2. Inthis case the velocity compensation will not be as effective

M. Focchi et al. / Control Theory Tech, Vol. 12, No. 1, pp. 68–81, February 2014

as for the case where KL2 = 0. Nevertheless, it still resultsin an improvement of the achievable closed loop torquebandwidth as shown in Fig. 8.

[th]

0

0.25

0.5

0.75

1

TorqueTl[N

m]

0 1 2 3 4 50

0.5

1

1.5

2

TorqueTl[N

m]

Time [s]

0

0.25

0.5

0.75

1

TorqueTl[N

m]

0 1 2 3 4 50

0.5

1

1.5

2

TorqueTl[N

m]

Time [s]

Fig. 8: Simulation. Unit torque step responses with KL2 = 0(upper plot) and KL2 = 11.2Nm/rad (lower plot) for differ-ent velocity compensation gains α = 0.94 (Solid line) andα = 0 (dashed line). An extended leg inertia (JL2 = 0.439kgm2) has been considered. The use of velocity compensa-tion significantly reduces the response time.

Similar results can be established after discretizationof the continuous time system. A final remark is that analternative to overcome the limitations imposed by theunwanted zeros is to consider more complex controllers,for example a double integrator PII controller, or designsbased on loop shaping control which introduce additionalpoles and zeros. The disadvantage is that tuning these com-plex controllers cannot be done without a good model forthe system.

3.2 Torque controller design

In this section the design of a PI torque controller is con-sidered. An integrator is included in the controller structureto remove steady state errors when tracking constant torqueinputs or when constant disturbances arise. The integratoris implemented in discrete time using the backward eulerapproximation. This controller is a lag compensator thathas one zero and one pole. The analysis and design are car-ried out in discrete time because sampling introduces no-ticeable differences in comparison to the continuous timecase. For example the system can become non-minimumphase even if the underlying continuous time is of mini-mum phase (non-minimum phase systems are more diffi-cult to control). The equation of the PI controller is given

by:

PIt(z) = Pt + ItzTs

z−1= (Pt + ItTs)

(z− Pt

Pt + ItTs

)z−1

(6)

where z is the Z-transform variable and Ts = 1 ms isthe sampling time interval; Pt and It are the proportionaland integral gains to be determined. We remark that smallsampling time intervals, on one hand, improve the distur-bance rejection properties of the closed loop system. How-ever, as the sampling time interval decreases, the effectsof quantization noise in the encoders become more promi-nent, especially when computing velocities from positionmeasurements using simple first order differences. In ad-dition, small sampling time intervals can introduce non-minimum phase behavior in the sampled system which ismore difficult to control. The selected sampling time of 1ms is a trade-off among all these aspects. At the actual en-coder resolution (80000 count/rev), the smallest velocitythat can be measured with 1 ms sampling time interval, is0.0125 rad/s.

Traditionally, the design of an inner loop controller iscarried out with the aim of maximizing the closed loopbandwidth of the inner loop. However, one of the first diffi-culties is how to measure the bandwidth of the torque loop.The closed loop torque bandwidth for an unconstrainedsystem, for example when the leg is moving freely in theair, is very different from the case when the system is incontact with the ground and depends on the ground stiff-ness (soft versus hard). In fact, assuming that gravity isfully compensated (this means KL2 = 0), the free leg mo-tion with the torque loop closed is not internally stable (aswas shown in the previous section there is a pole-zero can-cellation on the stability boundary). Further, it is not ob-vious that maximizing the bandwidth of the torque loopis always consistent with the specifications for the outerimpedance loop. In the approach presented in this paperit was decided to design the controller gains so that thetorque loop gain has a phase margin larger than 30 anda gain margin larger than 12 dB for the upper and lowerbounds of JL2 and KL2 and with a velocity compensationgain α = 0.94. This would result in a satisfactory response.In addition, the closed loop torque response was requiredto be stable for all values of the velocity compensation gainα between zero and one (but the gain and phase marginscan be less than 12 dB or 30 degrees, respectively). A set ofcontroller gains satisfying the given specifications is found

M. Focchi et al. / Control Theory Tech, Vol. 12, No. 1, pp. 68–81, February 2014

by using the Matlab SISOtool.

Pt = 0.382β It = 18β 25 > β > 0 (7)

Changing the gain β only affects the gain of the controllerbut the controller zero remains fixed. This gives amplefreedom to investigate the effects of increasing the torqueloop gain β and hence increasing the closed loop torquebandwidth when the outer loop specifications are consid-ered. We must remark that, when defining the torque band-width, we consider that the system is free to move in theair. Then the bandwidth is the frequency where the torqueamplitude decreases by -3 dB respect to the reference.

3.2.1 Harmonic drive torque ripple compensation

One drawback of using a harmonic drive gearbox is thatit introduces torque ripples (Tf r in (1)). The problem is re-lated to the working principle of the gearbox that is basedon the motion of an elliptic element (wave generator). Thismotion creates torque fluctuation with a fundamental fre-quency which is twice the wave generator angular velocity.While this disturbance is normally neglected in positioncontrol schemes, because it is passively filtered out by theinertia of the system, conversely it has a detrimental ef-fect on torque control and creates vibrations and wearingof the components. A way to mitigate this problem is toadd a lead/lag compensator (notch) in series to the PI con-troller in order to add enough phase lead at the resonanceof the transfer function between Tf r and Tl where the rippleis more prominent, as illustrated in Sec. 3.7, Ch. 4, p. 50in [30].

3.3 Impedance control

An impedance controller is added as an outer loop asshown in Fig. 7. The controller gains Pgain and Dgain rep-resent the stiffness and damping for the joint. The outputTlre f of the controller provides the reference torque for theinner loop:

Tlre f = Pgain(θL1re f −θL1)−DgainθL1 (8)

while θL1re f is the desired trajectory for the joint position.The term involving the link velocity feedback is imple-mented using an averaging filter to reduce the effects ofencoder quantization. TID (see Fig. 7) is an external com-pensation torque (e.g. inverse dynamics) that can be addedto remove the effects of gravity and inertia (and thus reduceposition tracking errors):

TID = (JL1 + JL2)θL2re f +mglcomsin(θL2) (9)

where lcom is the distance of the leg center of mass from thejoint axis. For HyQ, a range of values for the impedanceloop gains that is considered to be sufficient for walking,trotting and running tasks is Pgain ∈ [1,2000]Nm/rad andDgain ∈ [1,50]Nms/rad.

4 Stability regions

The analysis here is limited to the abduction-adductionelectric joint of HyQ with a variable load inertia that de-pends on the configuration of the leg joints. In particular,given a range of impedance parameters, Pgain and Dgain,this analysis will assess how the region of closed loop sta-bility is affected by varying the torque controller gain (β ),the velocity compensation gain (α), the number of samplesused in the averaging filter (Nav) and the sampling time(Ts).

The analysis has been performed by varying the stiff-ness Pgain between 1 and 20000 Nm/rad and the dampingDgain between 1 and 50. The upper-bound for the stiffnesswas chosen such that we could determine the boundaryfor the stability and passivity regions and is way beyondthe maximum value used in locomotion. The stability ofthe overall system is determined by computing the closedloop eigenvalues and checking that they are inside the unitcircle. In addition, when closed loop stability is attained,the region where the phase margin is less than 30 degreescan be determined. These calculations were carried out inMatlab using the mathematical model presented in Sec-tion 2. The results are displayed in Figs. 9, 10, 11 and12 where the white area corresponds to the stable region;light grey is a stable region with a phase margin of lessthan 30 degrees and the dark area is the unstable region.In the analysis all the regions have been computed for theleg in stretched configuration JL2 = 0.439 kgm2 unless itis otherwise stated. Fig. 9 shows that as the torque con-troller gain increases, the unstable region for low stiffnessand damping decreases but the unstable region for largestiffness and/or large damping increases. This clearly il-lustrates that increasing the torque loop bandwidth may notbe consistent with the (stability) requirements for the outerimpedance loop. This can be explained if we consider thatfor any given system and controller architecture, there is alimit on the maximum loop gain thatcan be achieved, be-yond which stability is not ensured and performance de-grades. In a nested architecture both loops contribute tothe loop gain. If the loop gain contribution from the torqueloop increases (e.g. the gain β increases), then the contri-bution from the impedance loop should reduce otherwise

M. Focchi et al. / Control Theory Tech, Vol. 12, No. 1, pp. 68–81, February 2014

0

0.5

1

1.5

2x 104

5 10 15 20 25 30 35 40 45 50D [Nms/rad]

K[N

m/rad]

(a) β = 1

0

0.5

1

1.5

2x 104

5 10 15 20 25 30 35 40 45 50D [Nms/rad]

K[N

m/rad]

(b) β = 2

0

0.5

1

1.5

2x 104

5 10 15 20 25 30 35 40 45 50D [Nms/rad]

K[N

m/rad]

(c) β = 4

0

0.5

1

1.5

2x 104

5 10 15 20 25 30 35 40 45 50D [Nms/rad]

K[N

m/rad]

(d) β = 6

Fig. 9 Stability regions varying the torque controller gain β (with α = 0.94, Nav = 4 and Ts = 1ms). White area corresponds to thestable region; light grey is a stable region with a phase margin of less than 30 degrees and the dark area is the unstable region. Crossesand squares denote unstable and stable experimental points respectively.

0

0.5

1

1.5

2x 104

5 10 15 20 25 30 35 40 45 50D [Nms/rad]

K[N

m/rad]

(a) α = 0

0

0.5

1

1.5

2x 104

5 10 15 20 25 30 35 40 45 50D [Nms/rad]

K[N

m/rad]

(b) α = 0.5

0

0.5

1

1.5

2x 104

5 10 15 20 25 30 35 40 45 50D [Nms/rad]

K[N

m/rad]

(c) α = 0.94

0

0.5

1

1.5

2x 104

5 10 15 20 25 30 35 40 45 50D [Nms/rad]

K[N

m/rad]

(d) α = 1.2

Fig. 10 Stability regions varying the velocity compensation gain α (with β = 1, Nav = 4 and Ts = 1ms). White area corresponds to thestable region; light grey is a stable region with a phase margin of less than 30 degrees and the dark area is the unstable region. Crossesand squares denote unstable and stable experimental points respectively.

0

0.5

1

1.5

2x 104

5 10 15 20 25 30 35 40 45 50D [Nms/rad]

K[N

m/rad]

(a) Nav = 1

0

0.5

1

1.5

2x 104

5 10 15 20 25 30 35 40 45 50D [Nms/rad]

K[N

m/rad]

(b) Nav = 10

0

0.5

1

1.5

2x 104

5 10 15 20 25 30 35 40 45 50D [Nms/rad]

K[N

m/rad]

(c) Nav = 20

0

0.5

1

1.5

2x 104

5 10 15 20 25 30 35 40 45 50D [Nms/rad]

K[N

m/rad]

(d) Nav = 50

Fig. 11 Stability regions varying number of samples Nav of the link velocity filter (with β = 1, α = 0.94 and Ts = 1ms). White areacorresponds to the stable region; light grey is a stable region with a phase margin of less than 30 degrees and the dark area is the unstableregion. Crosses and squares denote unstable and stable experimental points respectively.

closed loop stability would be lost. From Figs. 10 (a), (b)and (c) it is clear that increasing the velocity compensationgain results in an increasing instability region. Once morethis gives further evidence that as the torque loop band-width increases, the outer loop may become unstable. Theeffects of increasing the number of samples (Nav) in thevelocity averaging filter are shown in Fig. 11. Averaging alarge number of samples enlarges the instability region forlow stiffness values but the unstable region for high stiff-ness and low damping decreases in size. Figure 12 clearly

shows that for a large sampling time interval the region forinstability is the largest both for low and high damping andstiffness. The region where the phase margin is less than30 degrees also increases as the sampling time increases.For low leg inertia configuration in Fig. 12 (d) the instabil-ity region for low damping and low stiffness increases butthe unstable region decreases for low damping and largestiffness. The region with a phase margin smaller than 30degrees is also larger for the low inertia configuration.

To determine the stability regions experimentally is not

M. Focchi et al. / Control Theory Tech, Vol. 12, No. 1, pp. 68–81, February 2014

0

0.5

1

1.5

2x 104

5 10 15 20 25 30 35 40 45 50D [Nms/rad]

K[N

m/rad]

(a) Ts = 1ms

0

0.5

1

1.5

2x 104

5 10 15 20 25 30 35 40 45 50D [Nms/rad]

K[N

m/rad]

(b) Ts = 6ms

0

0.5

1

1.5

2x 104

5 10 15 20 25 30 35 40 45 50D [Nms/rad]

K[N

m/rad]

(c) Ts = 8ms

0

0.5

1

1.5

2x 104

5 10 15 20 25 30 35 40 45 50D [Nms/rad]

K[N

m/rad]

(d) JL2 = 0.129 kgm2

Fig. 12 (a),(b),(c) Stability regions varying the sampling times Ts (with β = 1, Nav = 4, α = 0.94). (d) Stability region for retractedleg (lower inertia) JL2 = 0.129 kgm2 (with β = 1, α = 0.94, Nav = 4 and Ts = 1ms). White area corresponds to the stable region; lightgrey is a stable region with a phase margin of less than 30 degrees and the dark area is the unstable region. Crosses and squares denoteunstable and stable experimental points respectively.

an easy task because it would involve a vey large numberof experiments. Even only finding the stability boundariesis not a simple task. Several experimental tests were car-ried out to validate the analytical results predicted by themodel. The adopted methodology was to start from valuesPgain and Dgain well inside the stable region and change theparameters in small steps until the instability was triggered.This enabled us to get a rough idea of the boundary of thestability region. The experimental results are displayed inFig. 9, 10, 11 and 12, where crosses and squares denoteunstable and stable points respectively. With the exceptionof Fig. 9(c) and 9 (d), overall the experimental tests areconsistent with the analytical calculations. Fig. 9 (b), 10(a), 10 (b) and 11 (b) are fully in agreement with the theo-retical results. For test points near the stability boundariesinconsistencies are present suggesting that the model lacksaccuracy from a quantitative point of view but qualitativelyit is correct. The experimental results shown in Fig. 9 (c)and 9 (d) are quite different from the expected outcome.Even though a precise explanation is currently not avail-able, the authors have several possible hypothesis for thesediscrepancies: the experimental set-up might have reachedsome limiting conditions, which invalidate the linear anal-ysis (for example by generating motor voltages that exceedthe capabilities of the motor drive electronics, or by gener-ating reference torques that exceed the range of the torquesensor). Their investigation is part of future work.

5 Passivity analysis

Another question of interest is whether the closed loopsystem remains stable when it interacts with a passive en-vironment. It is well-known that a strictly passive system,connected to any passive environment, is necessarily sta-ble [32]. Thus, since most terrain surfaces are passive, to

ensure a stable contact with the environment also the robotjoints have to be passive. This section therefore analyzesthe main factors that influence passivity. Stating that a sys-tem is passive, is equivalent to saying that the system isintrinsically dissipative. This is not always the case whencompliance is obtained actively, where the compliant be-havior is emulated by controlled actuators. In this case, thecontroller gains can destroy passivity. The requirement toensure this type of stability for the robot interacting withthe environment is the following: the port of interactionbetween the system and the environment, i.e. the drivingport impedance, has to be passive. For linear time invariantsystems this is a necessary and sufficient condition, but it isonly a sufficient condition for nonlinear systems. Let Z(s)denote the driving port impedance transfer function. ThenZ(s) is passive if and only if it is positive real [33]. In [16]and [34] it has been shown that this is equivalent to:1. Z(s) has no poles in ℜ(s)> 0;2. the phase of Z(s) lies between -90 and 90 degrees.For sampled data control systems, Colgate [16] has sug-gested an approximate method based on computing thecorresponding discrete time transfer function Z(z), assum-ing that the port of interaction is also sampled. The phaseof Z(z) is computed and corrected by subtracting ωTs/2rad, where Ts is the sampling time interval. Although manystudies have been carried out for analyzing the passivity ofsampled-data systems [35], there is still a lack of informa-tion about the influence of the closed-loop torque controlbandwidth on the combinations of stiffness and dampingthat can be passively rendered (in the field of haptics calledZ-width) [16]. Therefore, this section will show that thetorque loop performance plays an important role in deter-mining the range of passively achievable impedance val-ues. The discrete time transfer function Z(z) (impedance)has been computed from the link velocity θL1 to the load

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disturbance Tdist . The transfer function has been computedfor two cases: one for the torque loop (with velocity com-pensation) and one for the system after closing the outerimpedance loop. Then the above-mentioned phase correc-tion is applied to include the fact that the system is sam-pled. In the approach presented in this paper the analysisof passivity has been done by varying several parametersto have a better understanding of their influence: the gainof the PI torque controller (by varying the gain β in Eq.(7)), the velocity compensation gain α , the sampling timeTs, and the number of samples Nav of the averaging filterof the link velocity. For each set of parameters the analysiswas performed by first checking the stability of Z(z) andthen verifying that the corrected phase of Z(z) was in therange ±90 for frequencies up to the Nyquist frequency. Ifthese conditions are not satisfied then Z(z) is not passivefor the particular set of parameter values. The results ofthis analysis are summarized in Table 2.

Table 2: Passivity

Torque Imp. loop Imp. looploop Pgain = 200 P = 20000

Dgain = 10 D = 50β = 1 No Yes Yesβ = 0.5 No Yes Noβ = 2 No Yes Noβ = 4 No Yes Unstableβ = 6 No Yes Unstableα = 0 Yes Yes Yesα = 0.5 Yes Yes YesTs = 4 ·10−3[s] No Yes NoTs = 2 ·10−3[s] No Yes NoTs = 0.5 ·10−3[s] No Yes YesAverag. Nav = 1 No Yes YesAverag. Nav = 10 No Yes YesAverag.Nav = 20 No Yes YesAverag. Nav = 50 No Yes YesLow leg Inertia(ret.) JL2 = 0.129kgm2

No Yes Yes

In this table the nominal set of parameters are: α = 0.94,Nav = 4 samples, leg inertia JL2 = 0.439 kgm2, Pt = 0.38and It = 18. The overall (impedance + torque loop) systemis always passive for low impedances Pgain = 200 Dgain =

10 while passivity might be destroyed when the torquecontroller gain β (and so the torque bandwidth) increasesor the sampling frequency decreases. In particular it can benoticed that when the torque gain is larger than or equal

to 4 the closed loop system with the impedance loop be-comes unstable. This is a clear indication that increasingthe bandwidth of the torque loop is not always consistentwith the requirements of the position loop. When only thetorque loop is closed the system is almost never passive ex-cept for low values of α . The table shows that the velocitycompensation is the key parameter affecting the passivityif the torque loop alone is considered. In particular whenthe amount of velocity compensation increases, the innertorque loop becomes not passive and therefore the torquecontrol system alone can become unstable when the leg in-teracts with some environments. Further analysis showedthat the system (without closing the impedance loop) be-comes unstable if the leg is in contact with an environ-ment with a stiffness KL2 between 72 Nm/rad and 3500Nm/rad. This has been verified with experimental tests us-ing the test setup depicted in Fig. 13 by commanding a legmotion to have an impact against a physical spring. Thisspring is positioned in order to create a certain stiffnessKL2.

Fig. 13: Experimental setup for passivity tests. The ro-tational stiffness KL2 is obtained by positioning a linearspring at a certain distance from the HAA axis.Figure 14 shows the phase of Z(z) when only the torqueloop is closed (β = 1 and α = 0.94) and when both thetorque and the impedance loops are closed (Pgain = 200,Dgain = 10). The curves illustrate that, in the case of thetorque loop alone, the phase of Z(z) exceeds 90 degreesbetween 10 and 50 rad/s, indicating the loss of passivity,while, when the outer loop is closed, the phase always re-mains within±90 degrees, demonstrating that the passivityproperty is ensured as indicated in Table 2.

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100

101

102−90

−55

−20

15

50

85

120

6Z(z)[deg]

Frequency [rad/s]

Torque loopImpedance loop

Fig. 14: Simulation. Phase plot of the driving portimpedance Z(z) when only the torque loop is closed (β = 1and α = 0.94) (red plot) and when both the torque and theimpedance loops are closed (Pgain = 200, Dgain = 10) (blueplot). The black line shows the limit of 90 degrees.

6 Conclusions and future work

This paper presented a methodology for designing jointimpedance controllers based on an inner torque loop and apositive velocity feedback loop. In particular, it was shownthat the positive velocity feedback can be used to increasethe closed loop bandwidth of the torque loop without theneed of a complex controller. It has been demonstratedthat besides the sampling frequency and filtering, the band-width of the torque loop has a strong influence on the rangeof impedance parameters (Z-width) that exhibit a passiveand/or stable behavior. Indeed, larger inner loop bandwidthcan be beneficial for disturbance rejection and to improvethe tracking of the impedance (enlarge the range of fre-quencies for which the desired impedance is emulated bythe system) but, at the same time, it can reduce the regionof stable impedance parameters. This fact can limit the per-formance and versatility of a robot. Thus, the highest pos-sible bandwidth for the torque loop might not be the bestsolution for all situations. It is therefore important to find abalance between the torque loop requirements (e.g. to havea good torque tracking) and the stability/passivity specifi-cations of the overall system. Furthermore, it has becomeevident that, even for simple controllers, the design prob-lem is challenging and that there are competing trade-offsto consider when selecting the controller gains.

This suggests that there is a need for a problem for-mulation that can encompass the design objectives in amore systematic and generic framework. There are a num-

ber of areas that need further research and are left for fu-ture work. Adaptive schemes (e.g. gain scheduling) canbe developed to modify the torque controller gains to sat-isfy the constraints of stability/passivity given by the de-sired impedance parameters specified by the system de-signer. The torque gains can also be modified dependingon the load inertia variations that is changing with the legconfiguration. In addition, varying the location of the PItorque controller zero can provide improvements in per-formance. The torque controller architecture can also beenhanced to reduce the effects of torque ripples arising inthe gear transmission system (drive jitter). Since increas-ing the torque controller gain has been shown to be detri-mental, this option for mitigating the drive jitter can be dis-carded and more different solutions must be found. Finally,there is a need to develop strategies to quantify the range ofimpedances that are required for specific tasks. At presentthere are some guidelines that only provide qualitative re-sults, for example a high stiffness (Pgain) is specified whenthere is contact with a compliant environment and the po-sitioning accuracy is important. On the other hand, a lowstiffness is used to maintain small contact forces or whenthe environment is stiff. Similarly, large damping values(Dgain) are needed to reduce vibrations or to quickly dissi-pate energy. Future work also encompasses an extension ofthe methodology presented in this paper to include multi-degree of freedom systems.

Acknowledgements

This research has been funded by the Fondazione IstitutoItaliano di Tecnologia. J. Buchli is supported by a SwissNational Science Foundation professorship.

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Michele Focchi is currently a Post-Doc re-searcher at the Advanced Robotics department ofIstituto Italiano di Tecnologia. He received boththe B.Sc. and the M.Sc. in Control System Engi-neering from Politecnico di Milano in 2004 and2007, respectively. Until 2009 he worked for theR&D department of Indesit company. In 2009 he

M. Focchi et al. / Control Theory Tech, Vol. 12, No. 1, pp. 68–81, February 2014

joined the Advanced Robotic Department (ADVR) at Istituto Italiano diTecnologia, (Genova, Italy) developing a prototype of a novel conceptof air-pressure driven micro-turbine for power generation for which heobtained an international patent and several awards. In 2013 he got aPh.D. degree in robotics, getting involved in the Hydraulically ActuatedQuadruped Robot project where he developing low-level controllers forlocomotion purposes. Currently his research interests range from plan-ning of dynamic motions to whole body control.

G.A. Medrano-Cerda received the B.Sc. de-gree in electro-mechanical engineering from theUniversidad Nacional Autonoma de Mexico in1977, and the M.Sc. and Ph.D. degrees in controlsystems from Imperial College, London, in 1979and 1982, respectively. From 1982 to 1985 he was

an associate professor at the Division de Estudios de Postgrado, Fac-ultad de Ingenieria, Universidad Nacional Autonoma de Mexico. From1985 to 1986 he was a research fellow at the Department of Engineer-ing, University of Warwick. From 1986 to 2002 he was a lecturer at theDepartment of Electronic and Electrical Engineering at the University ofSalford. During this period he set up the Advanced control and robot lo-comotion laboratory at the University of Salford. In 1999 he became acontrol systems consultant at Las Cumbres Observatory (formerly Tele-scope Technologies Ltd) and later in 2002 he joined the company as asenior control engineer pioneering work in H-infinity control system de-sign and implementation for astronomical telescopes. Since 2009 he hasbeen a senior research scientist at the Advanced Robotics Department,Istituto Italiano di Tecnologia. His research interests are in the areas ofrobust control, adaptive control, modelling and identification, fuzzy sys-tems and advanced robotic applications, in particular to walking robots.

Thiago Boaventura received his B.Sc. andM.Sc. degrees in mechatronic engineering fromFederal University of Santa Catarina, Florianop-olis, Brazil, in 2009 and the Ph.D. degree inrobotics, cognition, and interaction technologiesfrom a partnership between Istituto Italiano di Tec-nologia and University of Genoa, Genova, Italy,

in 2013. He is a Postdoctoral Researcher with Agile and DexterousRobotics Laboratory, ETH Zurich, Zurich, Switzerland. He is mainly in-volved in the EU FP7 BALANCE project with a focus in the collabo-rative impedance control of exoskeletons. His research interests includeimpedance and admittance control, model-based control, legged robotics,optimal and learning control, and wearable robotics.

Marco Frigerio received his B.Sc. and M.Sc.degrees in computer science from the Universityof Milano Bicocca, respectively in 2006 and 2008,and the Ph.D. degree in Robotics from the Isti-tuto Italiano di Tecnologia (IIT) in 2013. He is

currently a Post-Doc researcher at the DynamicLegged Systems lab of IIT, where he is involved in

the development of hydraulic legged robots. His research interests includesoftware for robotics, software architectures, kinematics and dynamics ofmechanisms.

Claudio Semini is the head of the DynamicLegged Systems lab of the Department of Ad-vanced Robotics at Istituto Italiano di Tecnologia(IIT). He holds an MSc degree from ETH Zurichin electrical engineering and information technol-ogy (2005). From 2004 to 2006 he first visited theHirose Lab at the Tokyo Institute of Technology,

followed by work on mobile service robotics at the Toshiba R&D cen-ter in Kawasaki, Japan. During his doctorate from 2007-2010 at the IIThe designed and constructed the quadruped robot HyQ and worked on itscontrol. After a post-doc in the same department, in 2012, he became thehead of the Dynamic Legged Systems lab. His research focus lies on theconstruction and control of highly dynamic and versatile legged robots inreal-world environments.

Jonas Buchli received the Diploma degree inelectrical engineering from ETH Zurich, Zurich,Switzerland, in 2003 and the Ph.D. degree fromEcole Polytechnique Federale de Lausanne, Lau-sanne, Switzerland, in 2007. From 2007 to 2010,he was a Postdoctoral Researcher with the Compu-tational Learning and Motor Control Laboratory,

University of Southern California (USC), where he was the Team Leaderof the USC Team for the DARPA Learning Locomotion challenge. From2010 to 2012, he was a Team Leader with the Advanced Robotics De-partment, Istituto Italiano di Tecnologia, Genova, Italy. He is currentlyan Assistant Professor with the Institute of Robotics and Intelligent Sys-tems, ETH Zurich, Zurich, Switzerland, and the Director of the Agile andDexterous Robotics Lab. Dr. Buchli has received a Prospective and anAdvanced Researcher Fellowship from the Swiss National Science Foun-dation (SNF). In 2012, he received a Professorship Award from the SNF.

Darwin G Caldwell received the B.Sc. andPh.D. degrees in robotics from University of Hull,Hull, U.K., in 1986 and 1990, respectively, and theM.Sc. degree in management from University ofSalford, Salford, U.K., in 1996. He is the Direc-tor of Robotics with Istituto Italiano di Tecnologia,Genova, Italy. He is a Visiting/Honorary/Emeritus

Professor with University of Sheffield, the University of Manchester, andUniversity of Wales, Bangor. His research interests include innovative ac-tuators and sensors, haptic feedback, force augmentation exoskeletons,dexterous manipulators, humanoid robotics, biomimetic systems, reha-bilitation robotics, and telepresence and teleoperation procedures.