ieee robotics and automation letters, 1 impact mitigation

IEEE ROBOTICS AND AUTOMATION LETTERS, 1

Impact Mitigation for Dynamic Legged Robots withSteel Wire Transmission Using Nonlinear Active

Compliance ControlJunjie Yang, Member, IEEE, Hao Sun, Member, IEEE, Hao An, Member, IEEE,

and Changhong Wang, Senior Member, IEEE

Abstract—Impact mitigation is crucial to the stable locomotionof legged robots, especially in high-speed dynamic locomotion.This paper presents a leg locomotion system including thenonlinear active compliance control and the active impedancecontrol for the steel wire transmission-based legged robot. Thedeveloped control system enables high-speed dynamic locomo-tion with excellent impact mitigation and leg position trackingperformance, where three strategies are applied. a) The feed-forward controller is designed according to the linear motor-leg model with the information of Coulomb friction and viscousfriction. b) Steel wire transmission model-based compensationguarantees ideal virtual spring compliance characteristics. c)Nonlinear active compliance control and active impedance controlensure better impact mitigation performance than linear schemeand guarantees position tracking performance. The proposedcontrol system is verified on a real robot named SCIT Dog andthe experiment demonstrates the ideal impact mitigation abilityin high-speed dynamic locomotion without any passive springmechanism.

Index Terms—Impact mitigation, Active compliance control,Active impedance control, Steel wire transmission, Legged robot

I. INTRODUCTION

THE animals such as dogs and horses have a strong locomotionability in the field. Based on bionics, legged robots can also

perform complex field tasks and flexible motions, theoretically.Traditionally, most robots including legged robots and manipulatorsare driven by rigid position control with large gearbox transmissionsand high-gain controllers [1], which results stiff position-controlledjoints. The method of generating free gaits [2] gives legged robots theability of static locomotion in unstructured terrains, which, however,is not suitable for dynamic locomotion. When a leg of the robotinteracts with the ground, it will be forced to execute the locomotionaccording to the original position control signal, causing a hugeimpact on the leg and potentially leading to unstable locomotionof the robot. Although legged robots controlled by position alsorealize the basic walking, they cannot achieve high-speed dynamicrunning trot or bounding. Conversely, the legs of real animals havenatural cushioning structures, such as tendons and plantar pads.Moreover, animals can also bend knees through nerve feedback torealize impact mitigation, this control naturally has a delay of fewtens of milliseconds or more [3][4].

Since the trajectory of the leg locomotion of a robot is generallypredefined in the swing phase, rigid position control cannot be effec-tively applied in a non-ideal environment with unstructured terrains.A strong ability to continuously and safely interact with the realenvironment like animals usually requires modelling the surrounding

The authors are with Space Control and Inertial Technology ResearchCenter, Harbin Institute of Technology, Harbin, 150001, P.R. China(e-mail:[email protected]).

environment and estimating robot states exactly, which, however, isdifficult for legged robots to realize. Aiming at mitigating the impacton legs, animals have the passive impact mitigation mechanism aswell as the ability to actively regulating muscle compliance andimpedance [5]. Likewise, Hogan et al. [6] found that the adjustmentof impedance is a key factor for the robot to walk on unstructuredterrains. Meanwhile, [7] showed that controlling contact force andjoint torque is crucial. For the interaction between legs and ground,Raibert [8] proposed a spring passive dynamics method based on thespring-loaded inverted pendulum model; Pratt et al. [9] proposed theseries elastic actuator (SEA). SEA connects springs in series with theactuators, so as to absorb impact and release energy in the next phase[10]. This method protects the structure of the robot from impactswhile saving energy. However, passive springs cause legs to be unableto complete specific rigid locomotion, in which case the positiontracking performance becomes worse. Moreover, the series springhas a natural vibration frequency that limits the bandwidth of forcecontrol and requires additional tuning of the system. In order to solvethe invariable stiffness of passive spring, variable stiffness actuator(VSA) is a relatively novel solution, where an additional actuatoradjusts spring stiffness. Tsagarakis et al. [11] proposed a new VSAdesign to increase the range of adjustable stiffness. Kim et al. [12]designed a hybrid variable stiffness actuator (HVSA), achieving awide range of stiffness and fast response. Unfortunately, VSAs arestill bulky, complex and often cannot absorb high-energy impacts dueto the limited size of the springs [13][14][15].

Interacting with the unstructured environment without mechanicalsprings is crucial for legged robots. By conducting a reasonable de-sign of the control system, the mechanical compliance and impedanceof the joint can be adjusted alone in the software, resulting in a virtualspring-damper on the robot leg (called active impedance/compliancecontrol). To realize the functions of the virtual spring-damper throughsoftware, especially in high dynamic performance, the fast responseof the control system is crucial. As a result, a sufficient torquecontrol frequency is necessary to effectively replace a real springwith a virtual spring for legged robots. Virtual springs simplify thedesign of robot mechanisms. At present, active compliance controlhas been widely implemented in the field of legged robots, whichenables many legged robots to interact with the ground without anyphysical springs. Wensing et al. implemented impedance control tobuild the well-known cheetah robots [16] [17]. Hyun et al. [18]used a low transmission ratio of proprioceptive impedance torquecontrol mode, achieving high-speed locomotion on the treadmill,and the Froude number reached 7.1. Semini et al. [19] implementedactive impedance control on hydraulic actuators and realized efficientflying trot on the 80 kg HYQ robot. HYQ robot has a good impactmitigation performance and position tracking performance. Hutter etal. [20] designed high-efficiency and energy-saving robotic legs bycombining passive compliance with active compliance and manufac-tures StarlETH robot by implementing operation space control. Theseresults show that the active compliance/impedance control can besuccessfully implemented on legged robots. Note that the researchesmentioned above only considered linear compliance, which can not

arX

iv:2

108.

0148

1v1

[cs

.RO

] 3

Aug

202

1


fill the demand for flexibly balancing between stable locomotionand impact mitigation according to different robot states. Therefore,nonlinear compliance should be considered to achieve a satisfyingtrade-off between the two.

In this paper, we present a nonlinear active compliance/impedancecontrol and design a leg locomotion control scheme based on thepresented control method. Nonlinear active compliance provides abetter impact mitigation performance than the existing linear scheme,which will be demonstrated in experiments. The main contributionof this paper is the design of a leg locomotion control scheme,with purely active compliance/impedance controlled legs and nomechanical springs. Nonlinear active compliance and impedancecontrol are implemented on a real robot named SCIT Dog. Thelimitation of the stability of the closed-loop control system to thetime-varying parameters of the controller is also derived. Comparedwith the linear scheme, it has a better impact mitigation performance.To the best of the authors’ knowledge, this is the first time that thenonlinear active compliance is implemented on quadruped robots. Thesteel wire transmission scheme is rarely applied to quadruped robotsfor reducing the inertia of the robot’s legs. Other works aim to solvethe friction interference and the steel wire transmission interferencefaced by the leg control system of SCIT Dog in practical application,including: 1) Coulomb friction and viscous friction are consideredin the linear motor-leg model, and the feed-forward controller isdesigned based on the model. The position tracking performanceand high-speed dynamic performance of the leg also benefit fromthe design scheme of the feed-forward controller. 2) A compensationmethod is proposed, which can effectively deal with the influence ofhigh order dynamics in steel wire transmission system on the robotstable locomotion.

This paper is structured as follows: Section 2 introduces thestructure design of SCIT Dog. The motor-leg model and frictionmodel of SCIT Dog are established, systematically. Section 3 givesthe overall leg torque control scheme of SCIT Dog based on centralpattern generator (CPG) locomotion planning. The control frameworkbased on the feed-forward control will be introduced firstly. Then, thePI torque controller for the motor-leg model and friction compensa-tion will be introduced briefly. Finally, active compliance/impedancecontrol and the influence of virtual spring and damping on foot-to-ground interaction will be discussed in detail, and the stabilitycondition of the closed-loop system is derived. Section 4 is theexperiment part. The position tracking experiments and impact ex-periments are carried out on the SCIT Dog. The impact mitigationperformance and position tracking performance of the nonlinearactive compliance/impedance control are demonstrated. Conclusionswill be given in Section 5.

II. MODELING OF SCIT DOG

The considered SCIT Dog is driven by 12 brushless DC motorsand has 12 joint degrees of freedom (DoF) and 6 body DoFs. Eachleg contains three joints, corresponding to the hip and knee jointsof a real dog. Since the hip joint has two DoFs, namely, span andswing, the function of the hip joint should be realized by two motors.The DoF of the knee joint is driven by steel wire transmission, steelwire is wound around two fixed transmission wheels. One wheel isconnected to the knee joint, and the other wheel is connected to themotor. The control motor for the knee is designed in the position ofthe hip joint, above the motor that controls the swing DoF of the hipjoint (Figure 1 (a)). Only the degree of freedom of the knee jointis controlled by steel wire, and the two degrees of freedom of thehip joint are rigidly connected. (Although the load distribution of therobot leg joint is related to the joint configuration, the knee jointbears the main impact during the robot locomotion.) This design hastwo advantages as follows: (1) The motor designed in the hip jointdrives the motion of the knee joint through steel wire, which greatlyreduces the inertia of legs. (2) In some extreme cases, if the robotleg suffers from excessive force, the steel wire will break to protectother structures of the leg from being damaged. No passive springs aredesigned on SCIT Dog, and the desired compliance and impedance

of the legs come from the control scheme. However, steel wiretransmission causes additional problems to joint control, primarily thenonlinear relationship between the motor and joint velocity, whichwill be considered and compensated in Section 3. Each robot leghas three segments, which are thigh, leg, and foot respectively. Thefoot is connected with the thigh through connecting rods and keepsparallel with the thigh. This design can realize three segments of legscontrolled by using only two motors. SCIT Dog is shown in Figure1.

(a) SCIT Dog.

传动轮1

传动轮2

电机1

电机2

电机3

并联传动杆

机器人足

(b) Design of leg.

Fig. 1. Illustration of the considered SCIT Dog. 12 motors are designed atthe hip joint and the knee joints are transmitted by steel wire, which caneffectively reduce the inertia of the leg.

A. Motor-leg modelAn accurate motor-leg model is critical to implementing active

compliance and impedance control. Essentially, the motor-leg modelis equivalent to a motor-load model. The input of the motor modelis torque and the output is angular velocity. The input and outputof the load model are all both angular velocities. The steel wire canbe considered as a linear spring with large stiffness. Considering thatSCIT dog’s encoders are installed to measure the output of the reducergearboxes, and the reducer ratio of the gearboxes can be ignored inthe model. Therefore, the dynamic model of the load has naturalvelocity feedback [21], rather than a simple rigid model, which isformulated as

τl = Kw(ωm − ωl) (1)

where τl is the torque exerted on the load of the leg, Kw is thestiffness coefficient of steel wire, ωm is the angular velocity of themotor, and ωl is the angular velocity of the crus. In fact, the torquethat the external force maps to the joint space has an effect on τl.The whole motor-leg model is shown in Figure 2.

-

- --

-

Fig. 2. Motor-leg model diagram.

Further, define notations as follows. τout is the output torque oftorque controller, τcf is the static Coulomb friction, Jm is the inertiaof the motor, Bm is the viscosity coefficient of the motor, Jl isthe inertia of the load, Bl is viscosity coefficient of the load. As aresult, the transfer function of the model is obtained according to thediagram.


ωl(s)

τout(s)=

Kw

(Kw + Jls2 +Bls)(Jms+Bm) +Kw(Jls+Bl)(2)

The output torque of the motor is proportional to the current ofthe motor, while there is a complex electrical dynamic relationshipbetween motor voltage and current. Therefore, a control method formotor current rather than voltage is implemented on SCIT dog toreduce the controller complexity. A PI controller is designed byimplementing motor current feedback so that current can be directlyapplied, and torque can be directly applied to the input of the motor-leg model. This will be explained in detail in Section 3.

B. Friction model and load torqueFriction exists in both motor and load. Friction consists of two

parts, Coulomb friction and viscous friction [22]. For motors, totalfriction can be expressed as

τfm = τcfm +Bmωm (3)

where τfm is the total friction of the motor and τcfm is Coulombfriction of the motor. For loads, friction can also be expressed as

τfl = τcfl +Blωl (4)

where τfl is total friction of load and τcfl is Coulomb friction of load.So the total friction torque of the motor-leg model can be obtained.

τf = τcfm +Bmωm + τcfl +Blωl (5)

According to the motor-leg model, the actual torque applied to theload can be obtained.

τl = τout − τfm − Jmωm (6)

The angular acceleration of the joint may be erroneous throughsimple numerical differentiation from an optical encoder signal [23].There is also such difficulty in measuring acceleration for SCIT Dog,so the joint angular acceleration is designed to be obtained by inputbased on the CPG locomotion signal, which avoids the measurementrequirement. The generation of the reference trajectories for the feetis inspired by the CPGs of animals [24]. The trajectories of CPG hasintuitive parameters such as step length and step height, instead of an-gular joint displacement. The feed-forward torque control frameworkis implemented on SCIT dog, which efficiently applies the angularacceleration as input. The control framework will be explained inSection 3.

III. LEG LOCOMOTION CONTROL FRAMEWORK

For CPG signals, the trajectories of the robot feet are planned, andboth the desired velocity and acceleration are contained in the plannedtrajectories. The desired position and velocity are implemented byactive compliance and impedance torque control, while the desiredacceleration of the robot system is implemented on SCIT Dog byfeed-forward torque control. The overall control block diagram isshown in Figure 3.

In Figure 3, αref is the desired angular acceleration, τα is theoutput of the feed-forward controller, ωref and θref are the referencejoint angular and joint angle velocity calculated by the desired foottrajectory, τref is the reference torque for the PI controller. Inputsof the control framework are αref , ωref and θref , and outputsof the control framework are ωl and θl. The output of the feed-forward controller is the torque corresponding to the desired jointangular acceleration plus the total friction torque. Torque controllersare divided into the inner loop and outer loop. In the inner loop, thehigh-frequency PI torque controller can effectively control the outputtorque of the motor to follow the desired torque. Meanwhile, in theouter loop, the active compliance/impedance controller is the key torealize both the impact mitigation performance and position tracking

Motor-leg

Model

Active

Compliance/Impedance

Controller

PI

Controller

Feed-forward

Controller

-

+

Fig. 3. Leg locomotive control framework.

performance. In the rest of this section, we will first introduce thedesign of the PI torque controller and the realization of the frictioncompensation method briefly, and then, take focus on the design ofthe proposed novel active compliance/impedance controller.

A. Inner loop PI torque controllerThe direct input of the motor system is voltage, while the output

torque is linear with the motor current. The motor voltage and currenthave a complex motor dynamic relationship, and voltage controlcommonly causes torque ripple. To guarantee the control frequency,a PI controller for current is implemented on SCIT Dog, which issimple and effective. Here, the feedback information of the controlleris the current sensor. As we all know, the Gaussian noise of thecurrent sensor is huge. For the commonly used PID controller, thedifferential term will reduce the performance of the controller. Theintegral term plays an important role in reducing the static error of thesystem, so compared with P, PD, PID controller, it is most reasonableto use PI controller for torque control. Although the characteristics ofthe PI controller produces non-ideal oscillation in the motor current.Nevertheless, the current control system is in the inner loop of thewhole control system, and practical experiments demonstrate that theoscillation of the internal control loop has little effect on the externalcontrol loop. Behind the PI controller, a filter link and a saturationlink are added. The internal loop of torque control for motors isexpressed in Figure 4.

Low Pass

Filter-Saturation KIKi

Kp

+

PI Controller

Fig. 4. PI current controller.

Further, define notations as follows. τout is an input of the motor-leg system. KI is the torque coefficient of the motor. Iin and Ioutare desired current and actual current, Kp and Ki are parameters ofthe PI controller.

B. Friction torque compensationAccording to the friction torque model introduced above, the actual

angular velocity of the motor and load is indispensable. The feed-forward controller is described by

τα = (Jm+Jl)θref +τcfm+Bmωm+τcfl+Blωl+τcor+τg (7)

where τcor and τg are torque due to Coriolis and gravity respectively.The angular velocity of the motor is obtained directly from the en-coder information. However, the actual angular velocity informationof the load needs additional sensors. To avoid additional sensors, the


dynamic model of the load can be used to estimate the load angularvelocity. Because there are many nonlinear factors in the actualfriction torque, friction compensation uses the estimated velocity ofthe load. The main reason for the nonlinear relationship between loadvelocity and motor velocity is caused by the compliance of the steelwire transmission, which is reflected in the knee joint. Therefore, forthe hip joint, the motor angular velocity and load angular velocityis linear. According to the load model, the transfer function of loadspeed is obtained.

ωl(s)

ωm(s)=

Kw

Kw + Jls2 +Bls(8)

The estimated load velocity is obtained by solving Laplace inversetransformation, and the input of the system is regarded as a fixedvalue in each control cycle. Because of the stiffness of the steel wireis relatively large, the nonlinear motor-load velocity difference will begenerated only when the legs interact with the ground. Therefore, thenonlinear motor-load velocity difference generated in the stance phasewill be compensated quickly in the swing phase. This characteristicmakes the motor-load velocities to be linear at the beginning of eachstance phase. The initial value of load velocity is calculated by motorvelocity.

C. Nonlinear active compliance and impedance torque controlThe impact between the feet and the ground is the most important

issue discussed in this work, which may damage the structure ofthe leg and some fragile connections. Even if the structure is notdamaged, it will become less compact after a long-term operation,enlarging the dead zone of the transmission. Such an impact cancause an undesirable shake in the center of mass (COM) of therobot, resulting in instability locomotion even leading the body tofall. To mitigate the impacts, robots must have the ability to controlforces that interact with the ground, which requires adjustable com-pliance/impedance of the robot’s leg joints. When quadruped animalsrun and walk, they can actively regulate leg compliance by changingthe stiffness of leg muscles. Robots need to imitate this ability toactively adjust compliance so that the impact can be effectivelymitigated, and compliance needs to be gradually reduced to ensurethe accuracy of foot position tracking after the impact. There are twokinds of leg compliance, which are passive and active respectively.Passive compliance is achieved by adding flexible elements such assprings between actuators and loads. Active compliance means thatcompliance control is added to the robot’s torque control algorithm,and the coefficients can be adjusted by software in real-time sothat the robot leg has the same potential as the dog leg to controlthe interaction force. Active compliance torque is implemented byposition and velocity feedback, which is described by

τ ref = JT [Kvs,m(xdes − x) +Kd,m(xdes − x)] + τα (9)

where J is the foot Jacobian matrix, Kvs,m ∈ R3×3 is stiffnesscoefficient matrix, Kd,m ∈ R3×3 is damping coefficient matrix,xdes,x ∈ R3 is the expected and actual position of robot foot.Because compliance control does not need to guarantee zero steady-state error, the integral term I is not needed in the controller.

This method is equivalent to realizing a virtual spring-damper onthe robot leg. The proportional term can be regarded as a spring andthe differential term can be regarded as a damper. Kvs is the elasticcoefficient of the virtual spring and Kd is the damping coefficient ofthe damper. Note that Kvs and Kd are the parameters of the virtualspring, the diagonal terms of the stiffness and damping matricesKvs,m and Kd,m, respectively. The virtual spring-damper elementis shown in Figure 5.

According to the motor-leg model, the transmission wire has non-negligible compliance, which will lead to additional interference inthe design of Kvs function. This disturbance means that there isanother spring element in the designed virtual spring that is causedby the compliance of the steel wire so that the designed virtual spring

Fig. 5. Leg and virtual spring-damper.

is not an ideal functional characteristic. The transmission wire iscompensated by fine-tuning theKvs curve. The nesting of two springsis not a simple linear superposition of two elastic coefficients. Assumethat the springs with two stiffness coefficients Kdes and Kw are inseries, and the total stiffness coefficient is Kvs. This assumption isnot perfect because the forces acting on the springs are not in thesame direction. But in practice, this assumption is reasonable becausethe stiffness coefficient of steel wire cannot be accurately measured(the accuracy is the order of magnitude). The deformations of thesprings with stiffness coefficients Kdes and Kw are 4x1 and 4x2respectively, and the total deformation is4x = 4x1+4x2. ThroughKvs4 x = Kdes4 x1 = Kw 4 x2, the compensated value of Kvs

is approximately described as

Kvs =KdesKw

Kw −Kdes(10)

where Kdes is the desired elastic coefficient. The denominator of Kvs

is always positive because the stiffness of the steel wire is much largerthan the desired spring stiffness.

In the actual leg, the characteristics of the virtual spring have beenverified as follows. SCIT Dog’s single leg is manually compressed.Given the desired foot position, the position deviation can be obtainedby joint angle and leg kinematics, and the output force is measured.The virtual spring characteristics of the legs are obtained by repeatedexperiments, which are shown in Figure 6.

After compensating, the desired spring characteristics are obtained.Figure 6(a) and Figure 6(b) show the characteristics of a linearspring and an exponential nonlinear spring, respectively. The abscissacoordinates are radial position deviation and the ordinate coordinatesare output force. The set spring elastic coefficient in the experimentis relatively small because smaller spring characteristics can amplifyand prominently show the tracking error. The virtual springs showideal coefficient tracking characteristics.

For linear and nonlinear virtual spring characteristics, impactmitigation performance and position tracking performance shouldbe realized simultaneously. The leg locomotion is divided into twophases: the stance phase and the swing phase. The impact happensin the stance phase and should be effectively mitigated, and thefoot of the robot should have ideal position tracking performance inthe swing phase. Although linear spring compliance also has impactmitigation capability [18] [25], nonlinear virtual spring compliancehas better performance, based on which we presented nonlinearcompliance control on quadruped robots. Considering the leg as aspring-damper mechanism, the dynamic process after the impact issimilar to a second-order system impact response. The knee load(similar to the force of spring) reaches a peak firstly, then falls backand fluctuates, and eventually stabilizes at the gravity load. Impactmitigation characteristic requires that the impact peak of the load


-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

Spring Displacement,x [m]

0

5

10

15

20

25

30

For

ce,f

[N]

Actual forceExpected force

(a) Linear spring characteristic.

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

Spring Displacement,x [m]

0

5

10

15

20

25

30

For

ce,f

[N]

Actual forceExpected force

(b) Nonlinear spring characteristic.

Fig. 6. Virtual spring characteristics. Given the virtual linear spring Kvs =75N/m and the nonlinear spring Kvs = 125(e|x

rdes−x

r| − 1)N/m. (xrmeans the radial distance of the position)

should be small, also the change of the robot’s COM caused by theimpact should be small.

Reducing the stiffness of the virtual spring Kvs means a smallerimpact peak and fewer load fluctuations. However, low stiffnessincreases the position tracking error and causes the COM position tochange more. Inversely, high stiffness reduces the position trackingerror, and the COM position change with a greater impact peakand more load fluctuations. After the impact occurs, Kvs should beactively reduced to alleviate the impact peak, and after the impactpeak, Kvs should be actively increased to maintain the stability ofCOM. High Kvs should be kept in the swing phase to guarantee theposition tracking performance. A feasible scheme can be expressedas

Kvs = Kvs1 +2Kvs2

exp(Kcv(xrdes − xr)) + exp(−Kcv(xrdes − xr))(11)

where Kcv is a constant. Kvs1 and Kvs2 are constants that describethe range of Kvs, Kvs varies between Kvs1 and (Kvs1+Kvs2). Afterthe impact occurs, the radial position deviation (xrdes−xr) increasessignificantly, Kvs is actively reduced. After the impact peak, theCOM position begins to recover and Kvs is actively increased. Inthe swing phase (xrdes − xr) is small, and high Kvs is maintainedautomatically.

Leg locomotion state is divided into compression state and exten-sion state after the impact. The leg is compressed to alleviate theimpact peak and then extended to restore the COM position. Theradial velocity error (xrdes− xr) increases greatly in the compressionstate, after which the controller will produce a large reverse torqueτref to compensate for this deviation. Therefore, Kd should beactively reduced in the compression state, which means a smallerimpact peak. Kd should be actively increased in the extension state

to restore the COM position. The adjustment of Kd is related to leglocomotion state, a feasible scheme is based on the error function

Kd = Kd1 +Kd21 + erf(−Kcd(x

rdes − xr))

2(12)

where erf(x) = 2√π

∫ x0e−η

2

dη and Kcd is a constant. Kd1 and Kd2

are constants that describe the range of Kd, Kd varies between Kd1

and (Kd1 +Kd2). In the swing phase Kd ≈ Kd1 +Kd2/2.Although the load distribution of the robot leg joint is related to the

joint configuration, the knee joint bears the main impact during therobot locomotion. The active adjustment of parameter Kd is basedon velocity error (xrdes − xr), which is effective in one direction.Therefore, the control scheme described in this subsection can alsobe applied to the swing hip joint, but it can not be directly appliedto the span hip joint. For the span hip joint, the impact directionmay come from left or right directions, not from one direction, sothe parameter Kd should be a constant. The active adjustment of theparameter Kvs is independent of the impact direction. The schemecan be directly applied to the span hip joint.

D. Stability analysisThe stability of the closed-loop control system incorporating the

active compliance/impedance controller is investigated. Although themotor-leg model is a third-order linear system, the active compli-ance/impedance controller is nonlinear and time-varying. Kvs variesbetween Kvs1 and (Kvs1 + Kvs2), while Kd varies between Kd1

and (Kd1 + Kd2). The stability analysis plays a guiding role forparameters Kvs1,Kvs2,Kd1,Kd2. For the combination of nonlin-ear controller and linear system, the Popov stability criterion is areasonable method. Assuming that Kd is a constant and Kvs is anonlinear gain, the stability limit of Kvs with respect to Kd canbe found. Firstly, the closed-loop control system incorporating theactive compliance/impedance controller is restructured. By changingthe output, the constant Kd and the derivative element are addedto the motor-leg model (eq. (2)) and become a new linear transferfunction P (s), which is described by

P (s) =

Kw(Kw+Jls

2+Bls)(Jms+Bm)+Kw(Jls+Bl)

s[1 +Kd

Kw(Kw+Jls

2+Bls)(Jms+Bm)+Kw(Jls+Bl)

]=

Kw

as4 + bs3 + cs2 + ds(13)

where a = JlJmb = JmBl + JlBmc = JmKw + JlKw +BlBmd = KwBm +KwBl +KdKw

(14)

The Popov plot Re [P (jω)] related to ωIm [P (jω)] of P (jω) isused to analyse the stability, where ω is the frequency, Re [P (jω)]is the real part of P (jω) and Im [P (jω)] is the imaginary partof P (jω). The Popov stability criterion is a sufficient condition:if the Popov plot of P (jω) lies to the right of the Popov linecompletely, the closed-loop system is global asymptotic stability.The Popov line passes through (−K−1

vs,max, j0), Kvs,max is themaximum Kvs that guarantees the stability of the closed-loop system(Kvs,max > Kvs > 0). Re [P (jω)] and ωIm [P (jω)] are calculatedby {

Re [P (jω)] = Kw(aω4−cω2)

(aω4−cω2)2+ω2(d−bω2)2

ωIm [P (jω)] = −Kwω2(d−bω2)

(aω4−cω2)2+ω2(d−bω2)2

(15)

Let ωIm [P (jω0)]=0, the Popov stability criterion is described by


ω0 =

√db

Re [P (jω0)] =Kw

ad2

b2− cd

b

Kvs,max = − 1ReP (jω0)

(16)

From eq. (16), Kvs,max is calculated by

Kvs,max =(JmKw + JlKw +BlBm)(Bm +Bl +Kd)

JmBl + JlBm

− JmJlKw(Bm +Bl +Kd)2

(JmBl + JlBm)2(17)

where Kd > 0,Kvs,max > Kvs > 0.In fact, the same method can be used to calculate the rela-

tionship between the limitation of Kd and Kvs (assuming thatKvs is a constant and Kd is a nonlinear gain), but its form ismuch more complex. Equation (17) shows the functional relationshipbetween Kvs,max and Kd, and the parameters of the controller(Kvs1,Kvs2,Kd1,Kd2) can be selected reasonably based on thisrelation. In practical application, due to the inner loop PI controllerand various unmodeled characteristics, the parameters selection ofactive compliance/impedance controller should be more conservativethan the theoretical value. The relation between leg stiffness andanimal running performance is studied by biologists [26].

IV. EXPERIMENTS

A series of experiments are performed with the quadruped robotSCIT Dog that used only active compliance and impedance. Threeexperimental results will be shown to illustrate the performance ofthe proposed controller after the friction coefficient is determinedexperimentally. The experiment shows that SCIT Dog has a goodability to interact with the ground. Nvidia Xavier onboard processoris the main control unit. The data is transmitted through the UDP/IPnetwork and CAN bus to implement the motor control. Supportedby hardware, a sufficient torque control frequency is realized. Theparameters and the experiment hardware of the robot are shown inTable 1.

TABLE IROBOT EXPERIMENT PARAMETERS.

Parameters Definition Value

L×W ×H The robot size 76cm× 26cm× 35cml1 Thigh equivalent length 22.5cml2 Lower leg equivalent length 12.5cmM Total mass of robot 15kg

Perception sensors IMUOnboard computer NvidiaXavier

maximum joint torque 33Nm

A. PI torque controller performanceThe actual control frequency of the PI controller in the inner loop

of the system reaches 5 kHz, which is 5 times the control frequencyof the outer loop controller (1 kHz). Each motor is equipped with amicro-controller as the processing core of the PI controller and thecontrol core of the motor encoder. The structure of the PI controlleris shown in Figure 4. The cutoff frequency of the first-order low-passfilter is 20Hz. To demonstrate the performance of the PI controller,the robot joint is fixed in this experiment. Since the torque sensormeasures the external torque of the joint, the relationship betweenthe expected torque and the actual torque cannot be compared duringjoint rotation. Given an expected sinusoidal torque signal of a specificfrequency, the experiment result is shown in Figure 7.

Although the output torque has oscillation, the PI torque controllerstill shows good tracking performance, and the oscillation of the PIcontroller in the inner loop hardly affects the outer loop controller.

0 1 2 3 4 5 6 7 8 9 10-0.2

-0.1

0

0.1

0.2

0.3

0.4 Actual torqueExpected torque

6.5 7 7.5 8-0.05

0

0.05

Fig. 7. PI torque controller performance.

B. Position tracking experimentAccording to the analysis in Section 3, Subsection C, the perfor-

mance of position tracking needs to be concerned in the swing phase.In this experiment, the robot leg is suspended, meaning that there areonly gravity and no ground reaction force. (The control indicatorsto be concerned in the stance phase will be discussed in Section 4,Subsection C.) A desired radial sinusoidal position signal is applied tothe foot and the actual position of the foot is measured. The positiontracking performances are shown in Figure 8. Blue curves are theexpected position and the red curves are the actual position.

The experimental results show that the nonlinear spring (Kvs1 =500N/m,Kvs2 = 500N/m) has better position tracking perfor-mance than the linear spring (Kvs = 750N/m). The positiontracking error of the linear spring is ±4.4mm and the positiontracking error of the nonlinear spring is ±3.6mm (Figure 8 (c)).In the swing phase, high Kvs of nonlinear spring is maintainedautomatically.

C. Impact mitigation experimentsIn the experiments, the robot stands with all four legs on the

ground. A vertical impulse (200 N peak, 500 N peak, 800 N peakand 200 ms duration) is then applied to the robot through the feed-forward controller. The robot jumps up and then impacts on theground. When all four feet touch the ground, it is regarded as thebeginning of impact. The COM position is estimated through the legkinematics. The experiment results are shown in Figure 9-11.

The performances of linear spring (four kinds) and nonlinearspring are compared. Linear springs are: 1) Kvs = 1000N/m,Kd = 50Ns/m (high stiffness and low damping, red curves) 2)Kvs = 1000N/m, Kd = 150Ns/m (high stiffness and highdamping, green curves) 3) Kvs = 500N/m, Kd = 50Ns/m(low stiffness and low damping, blue curves) 4) Kvs = 500N/m,Kd = 150Ns/m (low stiffness and high damping, yellow curves).Nonlinear spring is : Kvs = variable(500 ∼ 1000N/m),Kd = variable(50 ∼ 150Ns/m) (the black curves). From Figure9(b),10(b),11(b), it can be seen that a higher joint stiffness Kvs

reduces the steady-state error of the COM vertical position (createdby gravity forces). But a higher joint stiffness Kvs reduces a largerknee load (Figure 9(a),10(a),11(a)). The experiment results showthat the nonlinear spring has a better impact mitigation perfor-mance (smaller knee load and smaller COM change). From Figure9(c),10(c),11(c) and Figure 9(d),10(d),11(d), after the impact occurs,the active compliance/impedance controller reduces Kvs and Kd toalleviate the impact peak. After the impact peak, Kvs and Kd isactively increased to maintain the stability of COM. Various linearsprings show disadvantages, such as larger knee loads or COMdeviations. High linear Kvs and Kd cause larger average knee load,while low linear Kvs and Kd cause more COM change in vertical


0 0.5 1 1.5 2 2.5 3

Time,t [s]

250

260

270

280

290

300

310

320

330

340

Pos

ition

,xr [m

m]

Expected positionActual position

0.6 0.8 1

275

280

285

(a) Linear spring position tracking.

0 0.5 1 1.5 2 2.5 3

Time,t [s]

250

260

270

280

290

300

310

320

330

340

Pos

ition

,xr [m

m]

Expected positionActual position

0.6 0.8 1

275

280

285

290

(b) Nonlinear spring position tracking.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time,t [s]

-2

-1

0

1

2

3

4

5

Pos

ition

dev

iatio

n,xr [m

m]

Nonlinear springLinear spring

(c) Position tracking deviation.

Fig. 8. Position tracking experiment.

direction. The experimental results confirm the analysis in Section3 and demonstrate that the active compliance/impedance controllerproposed in this paper has a better mitigation capability for impact.

V. CONCLUSIONS

A new nonlinear active compliance control is applied to quadrupedrobots and achieves better interaction capability than linear com-pliance control. Experiments show that SCIT Dog’s legs have theability to cope with large and impulsive impact forces in runningand jumping without any passive compliance mechanism. This workalso shows a reasonable control scheme for leg dynamic locomotion.A PI controller is used to directly control the motor current to solvethe nonlinear characteristics of motor voltage input and torque output.The model-based compensation method is successfully implementedon SCIT Dog. Active compliance control and active impedancecontrol have been applied, and experiments show both ideal impactmitigation performance and position tracking performance. The pro-posed scheme in this paper can be extended to all legged robots.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time,t [s]

2

3

4

5

6

7

8

Ave

rage

Kne

e lo

ad ,

[Nm

]

Kvs

=1000,Kd=50

Kvs

=1000,Kd=150

Kvs

=500,Kd=50

Kvs

=500,Kd=150

Kvs

=variable,Kd=variable

(a) Average knee load.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time,t [s]

-3

-2

-1

0

CO

M d

evia

tion,

x [c

m]

Kvs

=1000,Kd=50

Kvs

=1000,Kd=150

Kvs

=500,Kd=50

Kvs

=500,Kd=150

Kvs


(b) Vertical direction COM deviation.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time,t [s]

700

800

900

1000K

vs [N

/m]

(c) Time series of Kvs.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time,t [s]

50

100

150

Kd [N

s/m

]

(d) Time series of Kd.

Fig. 9. Impact mitigation experiments (Feed-forward vertical impulse 200 Npeak and 200 ms duration).

REFERENCES

[1] K. Hirai, “Current and future perspective of honda humanoid robot,”in IEEE/RSJ International Conference on Intelligent Robots & Systems,2002.

[2] J. Estremera and P. G. D. Santos, “Generating continuous free crabgaits for quadruped robots on irregular terrain,” IEEE Transactions onRobotics, vol. 21, no. 6, pp. 1067–1076, 2005.

[3] E. Kandel, J. Schwartz, T. Jessell, S. Siegelbaum, and A. J. Hudspeth,Principles of Neural Science, Fifth Edition, 2013.

[4] H. Geyer and H. M. Herr, “A muscle-reflex model that encodes principlesof legged mechanics produces human walking dynamics and muscleactivities,” IEEE transactions on neural systems and rehabilitationengineering: a publication of the IEEE Engineering in Medicine andBiology Society, vol. 18, no. 3, pp. 263–273, 2010.

[5] E. Burdet, R. Osu, D. W. Franklin, T. E. Milner, and M. Kawato, “Thecentral nervous system stabilizes unstable dynamics by learning optimalimpedance,” Nature, vol. 414, no. 6862, pp. 446–449, 2001.

[6] N. Hogan, “Adaptive control of mechanical impedance by coactivation ofantagonist muscles,” IEEE Transactions on Automatic Control, vol. 29,no. 8, pp. 681–690, 1984.

[7] N. Hogan, “Impedance control: An approach to manipulation. part i, ii,iii,” in American Control Conference, 2009.

[8] M. H. Raibert and E. R. Tello, Legged robots that balance, 1986.[9] G. Pratt and M. M. Williamson, “Series elastic actuators,” in Interna-

tional Conference on Intelligent Robots & Systems, 1995.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time,t [s]

2

4

6

8

10

12

Ave

rage

Kne

e lo

ad ,

[Nm

]

Kvs

=1000,Kd=50

Kvs

=1000,Kd=150

Kvs

=500,Kd=50

Kvs

=500,Kd=150

Kvs



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time,t [s]

-5

-4

-3

-2

-1

0

CO

M d

evia

tion,

x [c

m]

Kvs

=1000,Kd=50

Kvs

=1000,Kd=150

Kvs

=500,Kd=50

Kvs

=500,Kd=150

Kvs



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time,t [s]

500

1000

Kvs

[N/m

]


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time,t [s]

50

100

150

Kd [N

s/m

]


Fig. 10. Impact mitigation experiments (Feed-forward vertical impulse 500N peak and 200 ms duration).

[10] M. Laffranchi, L. Chen, N. Kashiri, J. Lee, N. G. Tsagarakis, and D. G.Caldwell, “Development and control of a series elastic actuator equippedwith a semi active friction damper for human friendly robots,” Roboticsand Autonomous Systems, vol. 62, no. 12, pp. 1827–1836, 2014.

[11] N. G. Tsagarakis, I. Sardellitti, and D. G. Caldwell, “A new variablestiffness actuator (compact-vsa): Design and modelling,” in IEEE/RSJInternational Conference on Intelligent Robots & Systems, 2011.

[12] B. S. Kim and J. B. Song, “Design and control of a variable stiffnessactuator based on adjustable moment arm,” IEEE Transactions onRobotics, vol. 28, no. 5, pp. 1145–1151, 2012.

[13] L. Liu, S. Leonhardt, and B. J. E. Misgeld, “Experimental validation of atorque-controlled variable stiffness actuator tuned by gain scheduling,”IEEE/ASME Transactions on Mechatronics, vol. 23, no. 5, pp. 2109–2120, 2018.

[14] J. Sun, Z. Guo, Y. Zhang, X. Xiao, and J. Tan, “A novel design of serialvariable stiffness actuator based on an archimedean spiral relocationmechanism,” IEEE/ASME Transactions on Mechatronics, vol. 23, no. 5,pp. 2121–2131, 2018.

[15] L. Liu, Z. Hong, B. Penzlin, B. Misgeld, C. Ngo, L. Bergmann,and S. Leonhardt, “Low impedance-guaranteed gain-scheduled geso fortorque-controlled vsa, with application of exoskeleton-assisted sit-to-stand,” IEEE/ASME Transactions on Mechatronics, pp. 1–1, 2020.

[16] P. M. Wensing, A. Wang, S. Seok, D. Otten, and S. Kim, “Propri-oceptive actuator design in the mit cheetah: Impact mitigation andhigh-bandwidth physical interaction for dynamic legged robots,” IEEETransactions on Robotics, vol. 33, no. 3, pp. 509–522, 2017.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time,t [s]

0

5

10

15

20

Ave

rage

Kne

e lo

ad ,

[Nm

]

Kvs

=1000,Kd=50

Kvs

=1000,Kd=150

Kvs

=500,Kd=50

Kvs

=500,Kd=150

Kvs



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time,t [s]

-8

-6

-4

-2

0

CO

M d

evia

tion,

x [c

m]

Kvs

=1000,Kd=50

Kvs

=1000,Kd=150

Kvs

=500,Kd=50

Kvs

=500,Kd=150

Kvs



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time,t [s]

500

1000

Kvs

[N/m

]


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time,t [s]

50

100

150

Kd [N

s/m

]


Fig. 11. Impact mitigation experiments (Feed-forward vertical impulse 800N peak and 200 ms duration).

[17] S. Seok, A. Wang, D. Otten, and S. Kim, “Actuator design for highforce proprioceptive control in fast legged locomotion,” in IEEE/RSJInternational Conference on Intelligent Robots & Systems, 2012.

[18] J. H. Dong, O. S. Sang, J. Lee, and S. Kim, “High speed trot-running: Implementation of a hierarchical controller using proprioceptiveimpedance control on the mit cheetah,” International Journal of RoboticsResearch, vol. 33, no. 11, pp. 1417–1445, 2014.

[19] C. Semini, V. Barasuol, T. Boaventura, M. Frigerio, and J. Buchli, “Isactive impedance the key to a breakthrough for legged robots?” RoboticsResearch: The 16th International Symposium ISRR, pp. 3–19, 2016.

[20] M. Hutter, C. D. Remy, M. A. Hoepflinger, and R. Siegwart, “Efficientand versatile locomotion with highly compliant legs,” IEEE/ASMETransactions on Mechatronics, vol. 18, no. 2, pp. 449–458, 2013.

[21] T. Boaventura, J. Buchli, C. Semini, and D. G. Caldwell, “Model-basedhydraulic impedance control for dynamic robots,” IEEE Transactions onRobotics, vol. 31, no. 6, pp. 1324–1336, 2017.

[22] A. Tzes, P. Y. Peng, and J. Guthy, “Genetic-based fuzzy clustering fordc-motor friction identification and compensation,” IEEE Transactionson Control Systems Technology, vol. 6, no. 4, pp. 462–472, 1998.

[23] S. H. Lee and J. B. Song, “Acceleration estimator for low-velocity andlow-acceleration regions based on encoder position data,” IEEE/ASMETransactions on Mechatronics, vol. 6, no. 1, pp. 58–64, 2001.

[24] A. J. Ijspeert, “Central pattern generators for locomotion control inanimals and robots: a review,” Neural Networks, vol. 21, 2008.

[25] C. Semini, V. Barasuol, T. Boaventura, M. Frigerio, and J. Buchli,“Towards versatile legged robots through active impedance control,”


International Journal of Robotics Research, vol. 34, no. 7, pp. 1003–1020, 2015.

[26] D. V. Lee, M. R. Isaacs, T. E. Higgins, A. A. Biewener, and C. P.Mcgowan, “Scaling of the spring in the leg during bouncing gaits ofmammals,” Integrative and Comparative Biology, vol. 54, no. 6, pp.1099–1108, 2014.

ieee robotics and automation letters, 1 impact mitigation

Documents