Leg Locomotion Adaption for Quadruped Robots with Ground Compliance Estimation · 2020. 7. 18. ·...

Research ArticleLeg Locomotion Adaption for Quadruped Robots with GroundCompliance Estimation

Songyuan Zhang , Hongji Zhang, and Yili Fu

State Key Laboratory of Robotics and System, Harbin Institute of Technology, 150001, China

Correspondence should be addressed to Songyuan Zhang; [email protected]

Received 18 July 2020; Revised 25 August 2020; Accepted 8 September 2020; Published 22 September 2020

Academic Editor: Liwei Shi

Copyright © 2020 Songyuan Zhang et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Locomotion control for quadruped robots is commonly applied on rigid terrains with modelled contact dynamics. However, therobot traversing different terrains is more important for real application. In this paper, a single-leg prototype and a test platformare built. The Cartesian coordinates of the foot-end are obtained through trajectory planning, and then, the virtual polarcoordinates in the impedance control are obtained through geometric transformation. The deviation from the planned andactual virtual polar coordinates and the expected force recognized by the ground compliance identification system are sent tothe impedance controller for different compliances. At last, several experiments are carried out for evaluating the performanceincluding the ground compliance identification, the foot-end trajectory control, and the comparison between pure positioncontrol and impedance control.

1. Introduction

Currently, the main forms of locomotion robots include leg-ged robots and tracked/wheeled robots. Compared with atracked or wheeled robot, legged robots can easily adapt toand walk on rough terrain [1]. Legged robots can choose con-tact points with environment for overcoming obstacles andfinding the feasible stable region [2]. For example, BostonDynamics developed a hydraulic driven BigDog robot, whichaimed at building an unmanned legged robot with rough-terrain mobility [3]. MIT Cheetah was designed with propri-oceptive actuator for impact mitigation and high-bandwidthphysical interaction [4]. In our previous research, inspired bythe proprioceptive actuator [5], we designed a single-leg plat-form for high-speed locomotion. ANYmal used the compli-cated actuator which makes the robot impact with hightorque control accuracy [6]. Overall, the hydraulic actuatorhas the feature of naturally robust against impulsive loadswith a high-power density [7, 8]. However, legged robotswith hydraulic actuators such as Bigdog developed by BostonDynamics are difficult to be scaled down and will generatelarge noise [9]. Different from that, the electric actuator with

proprioceptive design allows the force proprioception whichcan deliver the desired force with motor current sensing [10].

For quadruped robot to achieve better performance, foottrajectory planning is crucial. Two factors should be consid-ered in the design of the foot trajectory. One is that the greatimpact should be avoided when the feet land on the ground;another is that the leg structure should have enough groundclearance for avoiding obstacles [11]. The foot trajectory alsorequires continuous velocity and acceleration for stability.There are three main methods to plan the trajectory of aquadruped robot, i.e., Bézier curve trajectory, cubic trajec-tory, and sinusoidal trajectory [11]. Among them, the Béziercurve meets the above requirements perfectly. After the foottrajectory planning, the behaviour that the robot interactswith irregular ground should be considered. Quadrupedrobots which can adjust the foot-end dynamic behaviour todeal with the unstructured terrains are important on realapplication. For example, Semini et al. implemented animpedance controller to control the electrohydraulicallydriven leg of HyQ [12]. Hyun et al. realized the virtual legcompliance of the MIT Cheetah with proprioceptive imped-ance control to deal with the external disturbance. After

HindawiApplied Bionics and BiomechanicsVolume 2020, Article ID 8854411, 15 pageshttps://doi.org/10.1155/2020/8854411

https://orcid.org/0000-0001-9837-098X

https://creativecommons.org/licenses/by/4.0/




https://doi.org/10.1155/2020/8854411

knowing the coordinates in the Cartesian coordinate system,the joint coordinates are obtained by inverse kinematic trans-formation, and the joint motion trajectory is obtained. More-over, the whole leg movement of quadruped robots will bedivided into stance phase with impedance control and swingphase with position control [13]. However, rigid groundassumption is applied for most researches. The different con-tact dynamics between the rigid assumption and soft contactwill affect the performance and stability of robots. In contrastto that, Kim et al. penalizing the contact interaction in thecost function during the design of the whole-body controller[14]. Neunert et al. used the soft contact model combinedwith MPC controller [15]. Bosworth et al. designed a control-ler which can be tuned for different ground types [16]. Differ-ent from that, in our study, the least-squares method isapplied to estimate the ground compliance parameters. Themethod is a soft terrain adaption algorithm which canachieve a transition between hard and soft ground with areal-time terrain-aware. For real leg locomotion adaption,

the estimated stiffness of ground will be used for adjustingthe parameters of impedances controller.

This paper is structured as follows: Section 2 intro-duces the kinematic modelling method for the leg. In Sec-tion 2, the foot trajectory of the leg is designed utilizingthe Bézier curve and sinusoidal waves. The detailed imple-mentation of impedance control and ground complianceidentification will be introduced in Section 2. The experi-mental results are provided in Section 3. At last, the con-clusions are given.

2. Materials and Methods

The detailed leg design with three-joint leg structure can befound in our previous research [5]. The overall of the controldiagram is shown in Figure 1. During the movement ofrobots, the Cartesian coordinates of the foot-end areobtained through trajectory planning, and then, the virtualpolar coordinates in the impedance control are obtained

Trajectory Planner

px, p·x, py, p

·y

q1,q·1, q2, q

·2

𝜌d, 𝜌·d, 𝜃d, 𝜃

·d

𝜌, 𝜌·, 𝜃, 𝜃

·

e𝜌, e·𝜌, e𝜃, e

·𝜃Geometric

transformationImpedancecontroller

F1,F2

FG, Xf, X·f

𝜏1,𝜏2

Modeling

JT

Robot system

Geometrictransformation

Ground systemidentification State estimator

Figure 1: Overall of the control diagram.

OB

ZB

YB

XB ZBHYBH

XBHXBHipR

ZBHipR

YBHipR

XBHipP

YBHipP

ZBHipP

ZAnkle

YKnee

XKnee

YAnkle

XAnkle

YFoot

XFoot

ZFoot

Figure 2: Link coordinate systems for the right front leg of the quadruped robot.

2 Applied Bionics and Biomechanics

through geometric transformation. The deviation from theplanned and actual virtual polar coordinates and theexpected force recognized by the ground identification sys-tem is sent to the impedance controller. Finally, the desiredfoot-end force is obtained and then, the joint torque can becalculated by the change of the Jacobian matrix and sent tothe robot system. The status of the robot system is also fedback to the previous process.

2.1. One Leg Kinematic Analysis. For realizing the foot trajec-tory control of the leg, the kinematics formula should bederived first. To simplify the subsequent analysis of themotion control problems, the first step is to establish a com-plete coordinate system for a quadruped robot. Since theconfiguration of the four legs of the robot is identical, theonly difference is the position of the hip joint relative tothe centre of mass (COM) of the robot. Therefore, the kine-matics analysis of the foot-end with the hip coordinate sys-tem is consistent for all four legs. Here, only the right frontleg is selected for the analysis in this section. The followingjoint coordinate systems are established according to theDenavit-Hartenberg (D-H) method: the hip rolling coordi-nate system ∑HipR , the hip pitch coordinate system ∑HipP,the knee pitch coordinate system ∑Knee, the ankle pitchcoordinate system ∑Ankle, and the foot-end coordinate sys-tem ∑Foot are shown in Figure 2. The blue line in the figureshows the three links of the leg; the red line shows the axes ofthe leg rotating joints.

The coordinate system of each link, as well as fourjoint angles, is shown in Figure 3 in detail. The left pictureis the front view, where the XBH axis of the hip torsocoordinate system is perpendicular to the paper surface.The right picture is a schematic plan view of the rightfront leg, the rotation axes of the hip pitch, knee pitch,and ankle pitch joints which are oriented perpendicularto the paper in. In addition, according to the right-handrule, from the foot-end to the hip joint, the ankle pitchjoint θ4, knee pitch joint θ3, hip pitch joint θ2, and hiproll joint θ1 are defined.

According to the given kinematic model of the quadru-ped robot, from the hip pitch coordinate system ∑BH to thefoot-end coordinate system ∑Foot, the homogeneous trans-formation matrix between the neighbouring link coordinatesystems can be derived. Firstly, the homogeneous transfor-mation matrix of the hip rolling coordinate system relativeto the hip torso coordinate system is

BHHipRT =

1 0 0 00 cos −θ1ð Þ −sin −θ1ð Þ 00 sin −θ1ð Þ cos −θ1ð Þ 00 0 0 1

2666664

3777775: ð1Þ

The homogeneous transformation matrix of the hip pitchcoordinate system relative to the hip rolling coordinatesystem is

HipRHipPT =

cos θ2ð Þ 0 sin θ2ð Þ 00 1 0 −L1

−sin θ2ð Þ 0 cos θ2ð Þ 00 0 0 1

2666664

3777775: ð2Þ

The homogeneous transformation matrix of the kneepitch coordinate system relative to the hip pitch coordinatesystem is

HipPKneeT =

cos −θ3ð Þ 0 sin −θ3ð Þ 00 1 0 0

−sin −θ3ð Þ 0 cos −θ3ð Þ −L20 0 0 1

2666664

3777775: ð3Þ

The homogeneous transformation matrix of the anklepitch coordinate system relative to the knee pitch coordinatesystem is

KneeAnkleT =

cos θ4ð Þ 0 sin θ4ð Þ 00 1 0 0

−sin θ4ð Þ 0 cos θ4ð Þ −L30 0 0 1

2666664

3777775, ð4Þ

The homogeneous transformation matrix of the foot-endcoordinate system relative to the ankle pitch coordinatesystem is

AnkleFoot T =

1 0 0 00 1 0 00 0 1 −L40 0 0 1

2666664

3777775: ð5Þ

At last, by integrating these transformation matrixes, the(6) and (7) can be derived

BHFootT = BH

HipRT ⋅ HipRHipPT ⋅ HipPKneeT ⋅ KneeAnkleT ⋅ AnkleFoot T , ð6Þ

L1

YBH

YHipR

ZHipRZBH

L2

L3

L4

YHipPXBH

XHipP

YB

ZB

XB

YKnee

YAnkle

ZHipPZKnee

XKnee

ZAnkle

XAnkle

ZHipR𝜃1 𝜃2

𝜃3

𝜃4

Figure 3: One leg kinematic model of the quadruped robot.

3Applied Bionics and Biomechanics

BHFootT =

c2−3+4 0 s2−3+4 −L4s2−3+4 − L3s2−3 − L2s2

−s1s2−3+4 c1 s1c2−3+4 −L1c1 − L2c2 + L3c2−3 + L4c2−3+4ð Þs1−c1s2−3+4 −s1 c1c2−3+4 L1s1 − L2c2 + L3c2−3 + L4c2−3+4ð Þc1

0 0 0 1

2666664

3777775,ð7Þ

where c1 represents cos θ1, s1 represents sin θ1, c2 representscos θ2,c2−3 represents cos ðθ2 − θ3Þ,s2−3 represents sin ðθ2 −θ3Þ,c2−3+4 represents cos ðθ2 − θ3 + θ4Þ, and s2−3+4 representssin ðθ2 − θ3 + θ4Þ.

Therefore, according to the positive kinematic analysis,in the case of the known joint angle θ1, θ2, θ3, θ4, the positionand orientation of the foot-end relative to the hip torso coor-dinate system can be obtained. For calculating the inversesolution, the solvability should be considered for avoidingthe no solution or multiple solutions. Usually, solving robotkinematic equations by the inverse operation is a nonlinearproblem, and solving forward kinematic problems is to checkwhether the target point is in the working space. Therefore,for deciding the existence of the robot inverse kinematicssolution, the robot leg’s workspace should be calculated.For ensuring that the inverse kinematic is solvable, the footof the quadruped robot must be within the workspace ofthe leg joint. According to the design index θ1 ∈ ½−20o, 20o�,θ2 ∈ ½−50o, 50o�, θ3 ∈ ½−120o, 120o�, a series of coordinatepoints can be obtained with different joint angles. A pointcloud map indicting the whole working space of the leg isshown in Figures 4 and 5.

For the solution of inverse kinematic problems, there aremainly two types of closed-form solutions (analytic solu-tions) and numerical solutions. In this paper, the closed solu-tion method is used to solve the analytical solution. Becausethe quadruped robot has the same leg configuration, onlythe right front leg is considered to establish its inverse kine-matics model. From equation (7), we can get

BHP2x + BHP

2y + BHP

2z = L21 + L22 + L23 + L24 + 2L2L3 cos θ3

+ 2L3L4 cos θ4 + 2L2L4 cos θ3 − θ4ð Þ:ð8Þ

Because of the parallelogram structure, it is structurallyguaranteed that the (8) simplifies to

BHP2x + BHP

2y + BHP

2z = L21 + L22 + L23 + L24

+ 2 L2L3 + L3L4ð Þ cos θ3 + 2L2L4:ð9Þ

So, we can obtain

θ3 = θ4 = arccosBHP

2x + BHP

2y + BHP

2z − L21 − L22 − L23 − L24 − 2L2L4

2 L2L3 + L3L4ð Þ

!:

ð10Þ

From equation (7), we get

BHPy + L1 cos θ1BHPz + L1 sin θ1

= sin θ1cos θ1

: ð11Þ

Further,

−BHPzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBHP

2y + BHP

2z

q sin θ1 −−BHPyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

BHP2y + BHP

2z

q cos θ1

= −L1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBHP

2y + BHP

2z

q :

ð12Þ

So,

θ1 = arcsin −L1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBHP

2y + BHP

2z

q + arctanBHPyBHPz

: ð13Þ

From equation (7), we can also get

BHPx = − L4 + L2 + L3 cos θ3ð Þ sin θ2 + L3 sin θ3 cos θ2:ð14Þ

Since θ3 has been given by equation (10), so

θ2 = arcsinBHPxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

L4 + L2 + L3 cos θ3ð Þ2 + L3 sin θ3ð Þ2q

+ arctan L3 sin θ3L4 + L2 + L3 cos θ3

:

ð15Þ

Formula (10), formula (13), and formula (15) are theinverse kinematic equations of the leg, and the inversesolution is the only solution. Thus, if we know the coordi-nates of the foot-end of any leg in the hip joint body coor-dinate system, we can solve the requirement of each jointangle.

2.2. Velocity Jacobian Matrix. In this section, the velocityJacobian matrix is derived which is useful for further leg con-trol such as the transformation of forces and torques from thefoot-end to the joints. The definition of the Jacobian matrix isas follows:

J =

∂BHPx

∂θ1

∂BHPx

∂θ2

∂BHPx

∂θ3∂BHPy

∂θ1

∂BHPy

∂θ2

∂BHPy

∂θ3∂BHPz

∂θ1

∂BHPz

∂θ2

∂BHPz

∂θ3

26666666664

37777777775: ð16Þ


Then, according to the equation of positive kinematics,the Jacobian matrix of the velocity from the leg joint coordi-nates to the foot can be obtained as

J =0 l2c2 + l3c23 l3c23

l1s1 + l2c1c2 + l3c1c23 −s1 l3s23 + l2s2ð Þ −l3s1s23−l1c1 + l2s1c2 + l3s1c23 c1 l3s23 + l2s2ð Þ l3c1s23

26643775:ð17Þ

2.3. Foot-End Trajectory Planning. For obtaining better per-formance of the robot, we should design the foot-end tra-jectory properly. During the swing phase, the leg is notaffected by contact force, so a higher speed and accelera-tion can be achieved. During the stance phase, the exten-

sive loadings will be added on the legs; it will cause agreat rigid impact with only position control. Since theswing phase and the stance phase have different dynamiccharacteristics, the trajectories of the two phases aredesigned individually.

Later, for tracking the trajectory, two control methodswill be compared which are position control and impedancecontrol.

The swing-phase trajectories are designed from a Béziercurve defined by twelve control points, and stance-phase tra-jectories are designed as part of sinusoidal wave which has agood performance in smoothness and was also used in otherrobots [17].

2.3.1. Trajectory Design for the Swing Phase. The swing-phase trajectory design should guarantee enough groundclearance for avoiding obstacles and reduce energy losses

0.3

0.2

0.1

0

−0.1Z

(m)

Y(m)X (m)

−0.2

−0.3

−0.4

−0.50.1

−0.1−0.15

−0.2−0.25 −0.5

0.5

−0.4

0.4

−0.3

0.3

−0.2

0.2

−0.10.10

0.05

−0.050

Figure 4: 3D workspace point map of one leg.

–0.5 –0.4 –0.3 –0.2 –0.1 0 0.1X(m)

Z(m

)

0.2 0.3 0.4 0.5–0.5

–0.4

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

Figure 5: XHipRZHipR-plane workspace point map.

Table 1: The twelve control points of the Bézier curve.

x (mm) y (mm)

c0 -170 460

c1 -280.5 460

c2 -300 361.1

c3 -300 361.1

c4 -300 361.1

c5 0 361.1

c6 0 361.1

c7 0 321.4

c8 303.2 321.4

c9 303.2 321.4

c10 282.6 460

c11 170 460


during touchdown motion [18]. The design of the swing-phase trajectory should not only approximate the naturalbehaviour of the leg but also satisfy ground clearance,

which can avoid obstacles in the swing phase. The Béziercurve formula determined by the normalization parameterSSWðtÞ ∈ ½0, 1� is

psw tð Þ = psw Ssw tð Þð Þ = 〠n

k=0Ckn 1 − Sswð Þn−k Sswð Þkck,

vswt = dpsw

dSswdSsw

dt= dpsw

dSsw1Tsw

,ð18Þ

where Ckn represents the number of unordered collection

in which k elements are taken from n elements, (n + 1)is the number of control points, ck is a kth two-dimensional control point where k ∈ f0,⋯⋯ , 11g. Thecurve can be generated by twelve control points as shownin Table 1, and the trajectory shape is shown in Figure 6.

2.3.2. Trajectory Design for the Stance Phase. The stance-phase control of each leg will affect the performance ofquadruped locomotion via interaction with the ground.Therefore, the planned trajectory should not only considerthe motion requirements but also the interactive forcerequirements.

The stance-phase trajectory is proposed to simply as asinusoidal wave with two parameters: the half of the strokelength Lspan, and the amplitude variable δ. As with the swing

600–300 –200

c1 c0

c5,c6

X axis

Y axis

c11c10

c8,c9

c2,c3,c4

c7

–100 0 100 200 300

500

400

300

Y (m

m)

X (mm)

200

100

0

–100

Figure 6: The desired trajectory decided by 12 control points of the Bézier curve.

𝜃virtual

𝜌virtual

Figure 7: Virtual impedance model of the leg.


phase, the stance-phase trajectory equation is also deter-mined by the normalized parameters, Sst∈[0,1]

pstx tð Þ = Lspan 1 − 2Sst tð Þ� �+ P0,x,

psty tð Þ = δ cos π

2Lspanpstx tð Þ

!+ P0,y,

vstx tð Þ = dpstxdSst

dSst

dt= −

2LspanTst

,

vsty tð Þ = dpstydpstx

dpstxdt

= δπ

Tstsin π

2Lspanpstx tð Þ

!:

ð19Þ

Considering that the robot’s leg will touch the groundand bear an impact, therefore, an impedance control shouldbe used to resist the impact during the stance phase and willbe introduced in the next part.

2.4. Impedance Controller Design. In the previous section, wediscussed the planning for robot’s foot-end trajectory. If therobot has no contact with the external environment, puremotion control is enough for trajectory tracking. However,quadruped robots are high dynamic robots that their feet willcontact the ground frequently during the movement. In thiscase, the space constraint brought by the environment willhinder the tracking movement of the robot end effector.Therefore, for ensuring the compliance during the move-ment, the impedance control is used where the leg will beimitating mass, spring, and damper properties. Based on this,the robot will present virtual mass, stiffness, and dampingcharacteristics during movement [4].

A schematic diagram of a one-dimensional mass-damping-spring model is shown in Figure 7. Impedance isused to describe the behavior of a robot. Different impedanceparameters can be set to give different dynamic characteris-tics of the robot.

The dynamics for a one-dof robot rendering an imped-ance can be written

m€x + b _x + kx = f , ð20Þ

where x is the position, m is the mass, b is the damping, k isthe stiffness, and f is the force applied by the user. If b or k inthe impedance coefficient is set to be large, it is called highimpedance; if b or k is set small, it becomes low impedance.In this paper, virtual leg impedance is created in the polarcoordinate as shown in Figure 7.

The control formula can be derived as

f control − f est =m €xd − €xð Þ + b _xd − _xð Þ + k xd − xð Þ, ð21Þ

where f control is the control force sent to the controller; f est isthe estimated ground reaction force; €xd , _xd , xd are desiredacceleration, velocity, and position; and €x, _x, x are actualacceleration, velocity, and position. Considering that the vir-tual mass has no significant effect on the impedance effect ofthe robot’s legs, no virtual mass term is added to the controlalgorithm in this paper.

The Jacobian from the hip/shoulder to foot-end in thepolar coordinate system is obtained by the transformation.The position relationship between the Cartesian coordinatesystem and the polar coordinate system is

ρvirtual

θvirtual

" #=

ffiffiffiffiffiffiffiffiffiffiffiffiffix2 + y2

parctan x/yð Þ

" #: ð22Þ

Actualtrajectory

–

+

Polartransformation PD controller

𝜏polar 𝜏1, 𝜏2

Desiredtrajectory

Joints ofleg

px

py

px

desired

Jpolar (q)T

py

Figure 8: Block diagram for impedance controller.

J = eTe = (FG – [Xf Xf]𝜃)T(FG – [Xf Xf]𝜃)

𝜃 = ([Xf Xf]T [Xf Xf])–1 [Xf Xf]TFG

= –2 [Xf Xf]T(FG – [Xf Xf]𝜃)𝜕J

𝜕𝜃

𝜕J

𝜕𝜃= 0

Figure 9: Derivation block diagram of the least-squares method.


So the Jacobian from the Cartesian coordinate system tothe polar coordinate system is

Jpolar x, yð Þ =

xffiffiffiffiffiffiffiffiffiffiffiffiffix2 + y2

p yffiffiffiffiffiffiffiffiffiffiffiffiffix2 + y2

p1

y x2/y2 + 1ð Þ−x

y2 x2/y2 + 1ð Þ

2666437775: ð23Þ

Then, impedance control law can be derived as

τ1

τ2

" #= Jpolar qð ÞT

kρeρ + bρ _eρ

kθeθ + bθ _eθ

" #, ð24Þ

where eρ, _eρ, eθ, and _eθ are radial position error, radial veloc-ity error, angular position error, and angular velocity errorbetween the actual trajectory with designed trajectory,respectively. JpolarðqÞ is the Jacobian from hip/shoulder tofoot-end in the polar coordinate system, and JpolarðqÞ =

Jpolarðx, yÞJCartesianðqÞ. And the block diagram for the imped-ance controller can be shown in Figure 8.

2.5. Ground Compliance Estimation. For the above analysis,we assumed that the ground is completely rigid, but for a realapplication, the assumption is limited. For example, if therobot is moving on a concrete floor or an asphalt road, wecan think that the assumption is completely valid. But whenthe robot moves in an environment like grass, marshes, andsnow, there will be a big difference. Therefore, the identifica-tion of ground compliance is important. Through the systemidentification method, we can get the stiffness and dampingcharacteristics of the contact ground, and then, further oper-ations to achieve the corresponding impedance characteris-tics can be implemented.

Among the system identification theory, the least-squaresmethod is widely used, and the effect is also excellent. There-fore, we use the least-squares method to estimate the groundcompliance parameters.

(a) (b)

Figure 10: Experimental platform. (a) One leg experimental platform. (b) Trajectory measurement with the NDI Optotrak Certus sensor.

Apply external force

Apply external counterforce

Actu

al to

rque

(N⁎

m)

Time (sec)

Actu

al p

ositi

on (d

eg)

10

5

0

–5

–100 2 4 6 8 10 12 14 16 18

–20

–10

0

10

20

Figure 11: The actual torque and position of the hip joint change after applying force.


Taking into account the storage performance and com-putational performance of industrial controllers, we estimatethe ground parameters using the least squares method in lim-ited memory, which limit the estimated range. The groundreaction force (GRF) is described as

f G = kGxf + bG _xf , ð25Þ

where kG and bG are ground stiffness and damping coeffi-cients; that is, the values we need to identify by the systemidentification xf and f G are the depth and the ground reac-tion force. Then, we can estimate the ground parameters bythe least squares method in limited memory.

At every time instant n, we gather samples from the pre-vious k time instances and compute the ground parameters.By choosing the appropriate k value, i.e., the defined range,

a good parameter estimate can be obtained and the data sat-uration phenomenon can be effectively improved.

To implement the estimation of parameters, we constructthe following dataset

FG = f G nð Þ f G n − 1ð Þ⋯ f G n − kð Þ½ �T ,Xf = xf nð Þxf n − 1ð Þ⋯ xf n − kð Þ� �T ,_Xf = _xf nð Þ _xf n − 1ð Þ⋯ _xf n − kð Þ� �T

:

ð26Þ

By estimating bθ and F̂G as the optimal estimate of θ =½kG bG�T and FG, we can get

F̂G = Xf_Xf

h ibθ: ð27Þ

Then, by defining e = FG − F̂G and J = eTe, and from that,the least-squares estimation requires that the sum of thesquares of the residuals be the smallest; we can obtain theparameters as shown in Figure 9.

Moreover, considering that recent samples make moregreat impact on the results, we use a weighting matrix Q ∈ℝk×k to penalize the error on most recent sample comparedto the less recent ones and thus, giving more importance tothe most recent samples. And the result is

bθ = Xf_Xf

h iTQ Xf

_Xf

h i� �−1Xf

_Xf

h iTFG:

ð28Þ

According to the estimates bθ = k∧G b∧G½ �T , we canacquire

f̂ G = k̂Gxf + b̂G _xf : ð29Þ

Actu

al to

rque

(N·m

)

Actual position (deg)

6

4

2

0

–2

–4

–6

1050–5–10

Actual torque vs. Actual positionlinear fitting

Figure 12: The actual torque change with the position of the hip joint and the fitting straight line.

Figure 13: Diagram of free dropping apparatus of the single-legprototype.


Then, in the stance phase, for getting better impedancecharacteristics for the robot, we apply this force to the imped-ance control formula (21), and we can get

f control − f̂ G =m €xd − €xð Þ + b _xd − _xð Þ + k xd − xð Þ: ð30Þ

By the least-squares estimationmethod with limitedmem-ory and fading memory for ground parameters, we reasonablyintroduced the estimated value of GRF. Applying the esti-mated GRF to the f desired impedance controller, the value willchange with different ground conditions, allowing the robot toadapt to different ground conditions with suitable compliance.

3. Experimental Results

3.1. Foot-End Trajectory Experiment. For ensuring that therobot can well perform the planned motion characteristics,an experiment was carried out to verify the robot’s perfor-mance of foot-end trajectory tracking.

The experimental platform was constructed as shown inFigure 10. The NDI Optotrak Certus was used to trace thepresticked marker on the endpoint of the leg, and the actualtrajectories can be obtained. The accuracy that NDI OptotrakCertus can achieve is 0.01mm and its sampling frequency is100Hz, which is sufficient for our trajectory following theacquisition.

Free dropStop under external force

Current variation

Position variation

Time (s)

Time (s)

Curr

ent (

A)

Posit

ion

(deg

)

5

0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.5 1 1.5 2 2.5 3 3.5 4

0–5

–10

25

30

35

Figure 14: The current and position variation of the knee motor under pure position control.

Free drop

Current variation

Position variation

Time (s)

Curr

ent (

A)

Posit

ion

(deg

)

0 0.5 1 1.5 2 2.5 3 3.5 4

Time (s)0 0.5 1 1.5 2 2.5 3 3.5 4

–4

30

40

50

–2

0

Figure 15: The current and position variation of the knee motor under impedance control.

Table 2: Comparison between pure position control and impedance control.

Control mode Stability Position steady-state error (deg) Peak current (A) Compliance

Impedance control Stable 11.4 4.309 Soft

Position control Unstable — 11.03 Hard


Because the data collected by the NDI Optotrak Certus isthree-dimensional coordinates of a point on its body, and thetrajectory points we plan are the two-dimensional coordi-nates with the axis of the hip joint motor as the origin; it isnecessary to compare the trajectories after the coordinatetransformation.

The first step is to find the homogeneous transformationmatrix OTA from the NDI Optotrak Certus coordinate sys-tem A to the robot coordinate system O.

In the coordinate system O, four scattered coordinatepoints were selected that the foot-end of the robot can reach.Suppose the leg motion is strictly in a plane, and z = 0, thenassume that the coordinates of the selected four points areðOx1, Oy1, Oz1Þ, ðOx2, Oy2, Oz2Þ, ðOx3, Oy3, Oz3Þ, and ðOx4,Oy4, Oz4Þ.

Simultaneously, the four points are collected by the NDIOptotrak Certus, and the coordinates under frame A areðAx1, Ay1, Az1Þ, ðAx2, Ay2, Az2Þ, ðAx3, Ay3, Az3Þ, and ðAx4, Ay4,Az4Þ.

So

Ox1Oy1Oz1

1

2666664

3777775 = OTA

Ax1Ay1Az1

1

2666664

3777775: ð31Þ

After making the same transformation of all four coordi-nates and transforming the final matrix equation appropri-ately, we can get

OTA =

Ox1Ox2

Ox3Ox4

Oy1Oy2

Oy3Oy4

Oz1Oz2

Oz3Oz4

1 1 1 1

2666664

3777775 ·

Ax1Ax2

Ax3Ax4

Ay1Ay2

Ay3Ay4

Az1Az2

Az3Az4

1 1 1 1

2666664

3777775−1

:

ð32Þ

After finding the homogeneous transformation matrixOTA, the points collected by the NDI Optotrak Certus arehomogeneously transformed to obtain the actual trajectoryin the O coordinate system and compared with the plannedtrajectory.

3.2. Impedance Control Experiment. For the experiment, theimpedance control is based on the control in the torquemode. Therefore, the controller should first be switched tothe torque mode at first. In our study, the controller(C6015, Beckhoff) was chosen with EtherCAT connectionto each motor.

We observed the relationship between the real-time posi-tion, the real-time torque, and time by applying external tor-que to the hip joint, as shown in Figure 11. Then, we fit thetorque position and get the result as shown in Figure 12.

And the fitting equation is

y = −0:501x − 0:0003333: ð33Þ

3.3. Comparison between Pure Position Control andImpedance Control. As we know, the essence of impedancecontrol is to control the dynamic relationship between forceand position. The pure position control is mainly to achievean accurate position and does not consider the interactionwith the outside world. To better understand impedance con-trol and pure position control, we have done experiments tocompare the characteristics of the two control methods.According to the characteristics of the two control methods,we can determine which way is more suitable in the locomo-tion of the quadruped robot.

In the previous analysis, we have also seen that in theswing phase, since the foot-end point of the single-leg proto-type has no contact with the ground, the effects of impedancecontrol and pure position control are similar. But in thestance phase, the foot-end point of the single-leg prototypecontacts the ground, and there will be a large impact at themoment of contact. If the control strategy does not have acertain cushioning effect, it will be harmful to the motor.Therefore, it is necessary to test and analyse the control effectof pure position control and impedance control.

As shown in Figure 13, the experimental procedure is toset a fixed desired position through the impedance controland pure position control in the single-leg prototype in thesuspended state (here, the desired force in the impedancecontrol is set to 0). Then, at a certain height, the single-legprototype was dropped freely, and the current and positionchange of the single-leg prototype motor were observed.

Materials withdifferent stiffness

Force sensor

Figure 16: Diagram of the identification of various groundmaterials.


00 0

10

20

30

40

50

100

150

Estim

ated

stiff

ness

(N/m

m)

Gro

und

reac

tion

forc

e var

iatio

n (N

)

200

10 20 30 40

K1 = 110.77

50Number of data points Penetration variation (mm)

60 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 4

Experimental data pointsStraight line fitted by least square method

(a)

0

20

40

60

80

100

Estim

ated

stiff

ness

(N/m

m)

0

10

20

30

40G

roun

d re

actio

n fo

rce v

aria

tion

(N)

K2 = 85.98

Penetration variation (mm)0 10 20 30 40 50

Number of data points60 0 0.1 0.2 0.3 0.4


(b)

0

1

2

3

4

5

6

7

8

Estim

ated

stiff

ness

(N/m

m)

0

0.5

1

1.5

2

2.5

3

Gro

und

reac

tion

forc

e var

iatio

n (N

)

K3 = 0.69

Penetration variation (mm)Number of data points


0 50 100 150 200 0 1 2 3 4

(c)

Figure 17: Continued.


It should be noted that the drop height under the imped-ance control is 44mm. Therefore, in the pure position con-trol, the drop height was also set to 44mm at first, but it isimpossible to collect appropriate experimental data with highimpact. At last, the drop height was determined to be 10mm.In addition, through the data measurement and analysis ofthe two joint motors, the variation of current and positionvalues of the hip motor during the falling process is not par-ticularly obvious. Therefore, we did not collect the data onhip joint motor, but only current and position acquisitionof the knee motor.

By analyzing the experimental data in Figure 14, we canknow that in the pure position control, even if it drops froma height of 10mm, the position and current value of themotor show a tendency to disperse, that is, an unstable state.And the peak current during the whole process is 11.03A asshown in Table 2. In the final stage, to avoid serious damage

to the motor and the outside world, it is necessary to addexternal force to force the single-leg prototype to stopbeating.

In the impedance control as shown in Figure 15, we cansee that although the drop height is 44mm, which is 4.4 timesin the above example, the peak current during the whole dropprocess is only 4.309A as shown in Table 2, which effectivelysolves the impact on the motor during the dropping process.

3.4. Ground Compliance Identification. Through the previousanalysis, we set up the corresponding experiments to identifythe stiffness of the ground of various materials.

A portion of the experimental setup is similar to the pre-vious motion control experiment where the motion capturesystem is used to capture the depth of the foot into the mate-rial. Since the stiffness is defined as the force acting on theunit displacement, it is also necessary to place a force sensor

–5

0

5

10

15

20

25Es

timat

ed st

iffne

ss (N

/mm

)

Gro

und

reac

tion

forc

e var

iatio

n (N

)

K4 = 1.35

Penetration variation (mm)0 10 20 30 40 50

Number of data points60 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

0.2

0.4

0.6

0.8


(d)

Estim

ated

stiff

ness

(N/m

m)

0

0.5

1

1.5G

roun

d re

actio

n fo

rce v

aria

tion

(N)

K5 = 1.94

Penetration variation (mm)0

–20

–15

–10

–5

0

5

10 20 30 40 50Number of data points

60 70 0 0.2 0.4 0.6 0.8


(e)

Figure 17: The recursive change process of stiffness and the fitting straight line. (a) Medium hard rubber. (b) High durometer rubber. (c) Lowhardness sponge. (d) Medium hardness sponge. (e) High hardness sponge.


as shown in Figure 16 at the foot-end to measure the groundreaction force when the foot-end is in contact with theground.

The experimental procedure consists of designing a con-trol program that allows the foot-end point of the single-legprototype to move vertically downward from above theground material and penetrate the material, and the NDIOptotrak Certus and the force sensor collect the displace-ment and force information of the foot-end in real-time.Then, by observing the data of the collected foot-end force,we can judge the moment when the foot-end is in contact withthe material, and then based on the displacement informationat that moment, the penetration value data of the foot-end isobtained. With the penetration value and the foot-end forceinformation, the stiffness of various ground materials can beidentified by the recursive least squares method.

In the experiment, we selected five different materials,namely, material 1 is medium hardness rubber, material 2is reinforced rubber, material 3 is low hardness sponge, mate-rial 4 is medium hardness sponge, and material 5 is highhardness sponge. The stiffness of the above five materialswas obtained by the above experimental method, as shownin Figure 17.

4. Conclusion

Locomotion adaption for quadruped robots with groundcompliance estimation is important, especially for high-speed locomotion. In this paper, the Cartesian coordinatesof the foot-end are obtained through trajectory planning,and then the virtual polar coordinates in the impedance con-trol are obtained through geometric transformation. Thedeviation from the planned and actual virtual polar coordi-nates and the expected force recognized by the ground iden-tification system is sent to the impedance controller. Thedesired foot-end force is obtained and then the joint torqueis obtained by the change of the Jacobian matrix and sentto the robot system. The status of the robot system is alsofed back to the previous process. At last, several experimentsare carried out for evaluating the performance including theground compliance identification, the motion control andthe comparison between pure position control and imped-ance control. In the future, the prosed control framework willbe used on our quadruped robots for testing the robust whentraversing on a wide variety of terrains.

Data Availability

All data were experimentally obtained.

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper.

Acknowledgments

The research was funded by the National Science Foundationof China (Grant No. 61703124), the Self-Planned Task of

State Key Laboratory of Robotics and System (HIT)(SKLRS201801A02), and the Innovative Research Groupsof the National Natural Science Foundation of China(51521003).

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https://arxiv.org/abs/1901.08100

Leg Locomotion Adaption for Quadruped Robots with Ground Compliance Estimation · 2020. 7. 18. ·...

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