SUBMITTED MANUSCRIPT FOR PEER REVIEW 1 High-Payload …

SUBMITTED MANUSCRIPT FOR PEER REVIEW 1

An Adaptive Control Algorithm for QuadrupedLocomotion with Proprioceptive Linear Legs

Bingchen Jin†, Yueheng Zhou†, Ye Zhao, Ming Liu, Chaoyang Song, Jianwen Luo∗

Abstract—Quadruped robots manifest great potential totraverse rough terrains with payload. Numerous traditionalcontrol methods for legged dynamic locomotion are model-based and exhibit high sensitivity to model uncertaintiesand payload variations. Therefore, high-performance modelparameter estimation becomes indispensable. However, theinertia parameters of payload are usually unknown anddynamically changing when the quadruped robot is deployedin versatile tasks. To address this problem, online identificationof the inertia parameters and the Center of Mass (CoM)position of the payload for the quadruped robots draw anincreasing interest. This study presents an adaptive controllerbased on the online payload identification for the high payloadcapacity (the ratio between payload and robot’s self-weight)quadruped locomotion. We name it as Adaptive Controller forQuadruped Locomotion (ACQL), which consists of a recursiveupdate law and a control law. ACQL estimates the external forcesand torques induced by the payload online. The estimation isincorporated in inverse-dynamics-based Quadratic Programming(QP) to realize a trotting gait. As such, the tracking accuracyof the robot’s CoM and orientation trajectories are improved.The proposed method, ACQL, is verified in a real quadrupedrobot platform. Experiments prove the estimation efficacy for thepayload weighing from 20 kg to 75 kg and loaded at differentlocations of the robot’s torso.

Index Terms—Quadruped robot, adaptive control, onlineidentification, payload

I. INTRODUCTION

LEGGED robots exhibit remarkable maneuveringcapability of traversing rough terrains [1]–[3]. This

capability enables the legged robots to have great potentialfor transportation with heavy payload in daily life. Designingand controlling such machines has motivated considerableresearch, and several quadruped platforms have been designedand demonstrated superior topography adaptability [4].

Maximizing the legged robot’s maneuverability is one of themost attractive research topics in the locomotion community.The well-known MIT Cheetah robot incorporates virtual legcompliance into its controller and realizes 2D running on the

Manuscript received: Month date, 20xx; This work was supported in partby National Natural Science Foundation of China under Grant 51905251.(Corresponding author: J. Luo. Email: [email protected].)

B. Jin and Y. Zhou are with Shenzhen Institute of Artificial Intelligenceand Robotics for Society (AIRS), Shenzhen 518172, China.

Y. Zhao is with the George W. Woodruff School of Mechanical Engineering,Georgia Institute of Technology, USA.

M. Liu is with Active Vision Group (AVG), Robotics and Multi-perceptionLab (RAM-LAB), Department of Electronic and Computer Engineering, HongKong University of Science and Technology, Hong Kong, China.

Y. Zhou and C. Song are with the Southern University of Science andTechnology (SUSTech), Shenzhen 518055, China.

J. Luo is with the Hong Kong University of Science and Technology(Guangzhou), Guangzhou, 510000, China.† Bingchen Jin and Yueheng Zhou are co-first authors.

Fig. 1. The quadruped robot with high payload capacity, Kirin, is used for thestudy of the unknown payload identification and adaptive control. It is builtwith prismatic legs, aiming for capacity of heavy payload carrying. There arethree degrees of freedoms (DoFs) for each leg.

treadmill up to 6 m/s with trot gait in the Sagittal plane [5].StarlETH is another representative quadruped robot developedby ETH Zurich, on which the hierarchical Operational SpaceControl [6] is adopted to separate dynamical constraintsfrom trajectory tracking tasks. This enables StarlETH to trotover rough terrains with loose and slippery obstacles [7].Along this line of research, a Whole-Body-Control methodis integrated in the locomotion controller on ETH ANYmal,which significantly improves the robot’s agility [8]. MIT miniCheetah, a lightweight version of the MIT Cheetah series, iscapable of highly dynamic gaits including 3D trotting andgalloping up to 2.5 m/s through convex model predictivecontrol [9]. Unitree A1 is another lightweight quadruped robotthat has performed dynamic locomotion over rough terrains[10]. Besides agile quadruped locomotion, equipping the robotwith a payload-carrying ability is also a critical researchtopic. Bigdog, the first field legged robot that leaves thelab, is hydraulically actuated with impressive robustness tothe external disturbances. This robot weighs about 109 kgand can carry a payload weighing more than 150 kg [11].HyQ is another hydraulic robot developed by IIT [12]. Itweights about 80 kg and the peak torque of its hydraulicjoint is around 180 Nm which provides the robot extremelylarge payload-carrying ability [13]. Different from traditionalrotation joint, baby elephant developed by Shanghai JiaotongUniversity has a parallel-leg. This electric-hydraulic drivenrobot weights 130 kg and can carry payload up to 100 kg[14]. Due to the technology limitations, including the electricmotor constraints, it is yet extensively explored in the areaof the payload-carrying on the electrically-actuated quadruped

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2 SUBMITTED MANUSCRIPT FOR PEER REVIEW

robots.Traditional control theories are effectively deployed on

quadruped robots, which demonstrated great robustness andagility [2], [9]. To date, most of these advanced locomotioncontrollers require accurate robot models, and the predictedcontrol torques to drive the robot heavily rely on theaccuracy of robot models, including the link inertia and CoMpositions [6], [15], [16]. However, in the real world, therobot may be commanded to carry unknown payload fortransporting goods. In such cases, the control methods thathighly depend on deterministic robot models are prone tofailures. Motivated by this problem, this paper explores anadaptive controller based on online payload identification foran electric-actuated quadruped robot to handle the unknownpayload. The contributions of this letter lie in the followingtwofold:

• An online payload identification algorithm based ona recursive formulation is devised for a high PayloadCapacity (PLC) quadruped robot. This algorithmguarantees the fast convergence of the identification.

• An adaptive controller based on the online payloadidentification is verified for PLC from 0.2 to 1.5. To ourbest knowledge, it is the first time to deploy the adaptivecontrol for a wide PLC range on an electrically actuatedquadruped robot.

The rest of the letter is organized as follows. The relatedwork is reviewed in Section II. Section III introduces themodel and dynamics control of the quadruped robot. Thepayload identification and adaptive control are proposed inSection IV. Section V shows the experimental results. Thisline of research is concluded in Section VI.

II. RELATED WORK

Over the last few decades, research about the robotparameters identification and adaptive control for robots withpayload has achieved evident improvements. The inertialparameters identification for legged locomotion has been acritical research topic [17], [18]. These identification methodsare mostly designed offline since the robot model parametersare usually deterministic. These methods have been verifiedin several robots such as UT-µ2 [19], a small-size humanoidrobot, and the quadruped robot HyQ [20]. However, for therobot’s torso, these methods play a limited role in identificationdue to the unknown payload. By far, there are two mainapproaches to solve this issue. One is to identify the torso’sparameters online, taking the robot’s torso and the payloadas a rigid body. Research such as [21] proposes an onlineinertial parameters identification for a manipulator. However, itis only appropriate for the fixed-base rigid body system. HyQovercomes the shortcomings of this method and proposes acombination of techniques that guarantee the robot locomotionstability [22]. These methods have been verified on HyQ ina static walking gait. Scalf-III, a hydraulic actuated heavy-duty quadruped robot developed by Shandong University,proposes a CoM estimation and adaptation method in dynamictrot gait and verifies it in simulation [23]. Another approachis to incorporate the adaptive control into the traditional

quadruped locomotion controller. A L1 adaptive control theoryis proposed for legged robots and verified in simulation [24],[25].

To our best knowledge of the existing work, the adaptivecontrol with online unknown payload identification is yetfully explored for high capacity quadruped locomotion.In this study, an adaptive control with high-payloadonline identification is proposed for an electrically-actuatedquadruped robot. The robot is named Kirin as shown in Fig. 1.The leg mechanism is designed to be prismatic so as to greatlyincrease the payload capacity. Experiments are conducted onKirin to verify the effectiveness of ACQL.

III. DYNAMICS CONTROL FOR QUADRUPED LOCOMOTION

The legged locomotion model and control relate to theeffects of the forces and the torques exerted on the robot,which originate from the robot and payload’s gravity, and thephysical interaction with the environment. In this study, thedesired forces and torques are computed by:

Fb = Kfp (r

db − rab ) +Kf

i

∫(rdb − rab )+

Kfd (v

db − vab ) +mab +

∑mig +mpg

Tb = Ktp log (q

db · (qab )−1)

+Kti

∫log (qdb · (qab )−1)

+Ktd log (ω

db · (ωab )−1)

+∑

ri ×mig + rp ×mpg

, (1)

where rdb , vdb , ab, qdb , ωdb represent the desired position,linear velocity, linear acceleration, rotation matrix and angularvelocity of the robot torso respectively. rab , vab , qab and ωabrepresent the actual position, linear acceleration, rotationmatrix and angular velocity of the robot torso respectively.mig represents the gravitational force acting on the body i. rirepresents the corresponding position vector from the originof the inertial frame to the CoM of the body i. m representsthe total mass of the robot, which is equal to the summationof mi. mpg is the gravity of the payload and rp meansthe corresponding position vector of mpg. Different frommig and ri that can be generated from computer-aid designsoftware directly, mpg and rp are the unknown, dynamicallychanging parameters that can not be ignored. K(·)

(·) representsthe diagonal gain matrices which need to be manually tunedin the experiment. For a matrix M ∈ SO(3), the logarithmoperation is:

log(M) =

{12 (M −M

T ) d→ 1

arccos d(M −MT )/(2√1− d2) other

, (2)

where d = (trace(M)− 1)/2.The first three components for each equation’s right side

in (1) are the PID tracking controllers and the remainingcomponents represent the feedforward controller.

In this study, an inverse-dynamics-based QuadraticProgramming is adopted to realize the quadruped locomotion,

JIN et al.: HIGH-PAYLOAD ONLINE IDENTIFICATION AND ADAPTIVE CONTROL FOR AN ELECTRICALLY-ACTUATED QUADRUPED ROBOT 3

including the trotting gait. The inverse dynamics solveroutputs the desired force Fb and torque Tb exerted onthe torso of the robot to track the desired motion. (1) isformulated as a Quadratic Programming (QP) problem, bywhich the contact force F d is computed. The QP formulationis given as:

F d = argminF

(AF −B)TQ(AF −B) + FTRF

s.t.

τmin ≤ J−1F ≤ τmax−µFiz ≤ Fix ≤ µFiz−µFiz ≤ Fiy ≤ µFizDiFi = 0

, (3)

where τmin and τmax are the robot’s minimal and maximumjoint torque. Fix, Fiy , and Fiz are the components of eachfoot’s contact force vector. µ is the coefficient of frictionbetween the contact foot and the ground. Di represents thematrix which selects the feet that dose not contact the ground.rbi is the skew-symmetric matrix defining the cross product ofthe position vector rbci from base to contact foot. F d ∈ <3N×1

is the concatenated vector of the contact forces. N representsthe number of legs that contact the ground. The diagonalmatrix Q and R are the weight matrix which needs to beadjusted via experiment as well. A and B are given as:

A =

[· · · I · · ·· · · rbi · · ·

]

B =

[Fb

Tb

] . (4)

From (1)-(4), The locomotion controller requires accuratemodel parameters, especially the unknown payloadparameters. The identification of the payload will beintroduced in detail in section IV.

IV. PAYLOAD IDENTIFICATION AND ADAPTIVE CONTROL

The high-payload identification and adaptive control forquadruped locomotion are introduced in this section. Whenquadruped robots are deployed for goods transportation, thepayload is mostly unknown and fluctuating. The variablepayload incurs significant disturbances to the robot balance.In this study, an adaptive control for quadruped locomotion(ACQL) is proposed to tackle this issue. For simplicity ofanalysis within the scope of this study, ACQL is based on thefollowing assumptions:

(i) The payload effects to the robot are identified as a forceand a torque. Although the payload is always in surface contactwith the robot, only the force and moment acting on the robotare considered in the robot’s dynamics model. Therefore, thespecific force distribution is not taken into account.

(ii) The force and the torque to be identified do not changeduring locomotion. The identification method proposed in thisstudy mainly relies on the robot’s orientation error. Duringthe dynamic gaits, orientation errors easily bring noises in

Fig. 2. Control scheme of the adaptive control with online unknown payloadidentification. The Adaptive Control for Quadruped Locomotion (ACQL)identifies the mass of the payload directly and generates an update law. Therecursive result of the update law is the moment of the payload concerningthe robot. ACQL also generates a control law by which the robot can adjustits posture. The torques of joints are calculated via inverse dynamics basedon Quadratic Programming (QP) solver.

the identification. Therefore, it is preferred to estimate theseparameters in a static pose, such as standing on the ground.

Based on the assumptions, this section introduces thequadruped dynamics model, ACQL and demonstrates thestability proof. ACQL is based on the quadruped dynamicsmodel and consists of update law and control law.

A. Quadruped Robot Model

The dynamic model of a quadruped robot is given by:mrab =

∑Fi −

∑mig −mpg

d

dt(Iωab ) =

∑rbci × Fi + rp ×mpg

+∑

ri ×mig

, (5)

where I ∈ <3×3 is the inertial of the robot. Fi represents thecontact force of each foot i. The left component of the secondequation of (5) can be extended as:

d

dt(Iωab ) = Iωab + ωab × (Iωab ). (6)

With the aforementioned assumption (ii), when the robotstands on the ground with all four legs, the robot’s angularvelocity is small, and the precession and nutation of ωab×(Iωab )of (6) contribute little to the dynamics of the robot. Thus, thiscomponent can be discarded, and the second equation of (5)can be changed to be:

Iωab =∑

rbci × Fi + rp ×mpg +∑

ri ×mig. (7)

B. Identification of Payload Mass and Moment

The scheme of ACQL for the identification of the payload’smass and moment is shown in Fig. 2. In the proposed ACQL,the payload’s estimated mass is given by an analytical solutionwhich is computationally efficient. A control law is proposedto adjust the robot’s posture while an update law is used toidentify the moment of the payload concurrently.

The first equation of (5) can be rewritten and the estimatedmass of the payload mp is given by:


mp =1

g

∑Fi −

∑mi −

mrabg. (8)

To estimate the moment of the payload, (7) can berewritten with some substitution and the robot dynamics canbe formulated in a standard state-space form:

x1 = x2

x2 = Bu+ d+ k

˙d = w

, (9)

where x1 represents the robot orientation. x2 represents theangular velocity of the robot. B represents I−1. u represents∑rbci × Fi. k represents

∑ri ×mig. d represents I−1rp ×

mpg which is the parameter to be identified. d represents theestimated value of d. w represents the update law ˙

d.

In the quadruped robot system, there are multiple variablesthat can be used as the observed quantity such as orientation,angular velocity of torso and rotation matrix. In this study,the robot orientation is selected as the tracking object. Thetracking error can be written as x = x1−x1d and the first andsecond derivative of the tracking error are ˙x = x1 = x2 and¨x = x2 respectively. Consider the function:

s = ˙x+ λx, (10)

where λ is a positive definite matrix. Combined with (9), thederivative of s is given by:

s = ¨x+ λ ˙x

= x2 + λx2

= Bu+ d+ k + λx2.

(11)

Based on the derivation, the control law is devised as:

u = −B−1(d+ k + cs+ λx2), (12)

where c is a positive definite matrix. Put (12) in (11) we have:

s = −cs+ d, (13)

where d = d − d. From (13), it can be concluded that s willasymptotically converge to zero if d asymptotically convergesto zero, and in this case s = 0. The proof of this conclusionwill be demonstrated in the next subsection.

The update law ˙d is devised based on the control law and

the aforementioned assumptions. Consider the manifold:

M = {(x1, x2, d) ∈ <3×3 | d− d+ β(x1, x2) = 0}, (14)

where β(x1, x2) is the estimation error function to bedesigned. As such, the problem is reformulated into the designof an appropriate function β(x1, x2) to ensure the manifold tobe invariant and attractive. Define the manifold as:

z = d− d+ β(x1, x2). (15)

The derivative of z is:

z =˙d+ β(x1, x2). (16)

Algorithm 1 ACQL algorithmInput: qdb , ωd

b , rdb , v

db ,mi, ri, q

ab , ω

ab , qjoint, qjoint, τjoint

Output: F d

1: Bri,B vi,

B ri ⇐ Forward kinematics (qjoint, qjoint)

2: Brab ,B vab ,

B rab ⇐B ri,B vi,

B ri

3: R⇐ qdb4: rab , v

ab , r

ab ⇐ RBrab , R

Bvab , RB rab

5: Fa ⇐ τjoint

6: mp ⇐ mi, rab , Fa

7: eqb ⇐ log (qdb · (qab )−1)

8: while eqb > ethreshold do9: u⇐ −B−1(d+ k + cs+ λx2) (control law)

10: d⇐ −M(Bu+ d+ β(x1, x2) + k) (update law)11: dk+1 = d∆t+ dk

12: Tb ⇐ Iu

13: Fb ⇐ mi, mP , rab

14: end while15: eqr ⇐ rab , r

db

16: eqr ⇐ vab , vdb

17: eqb ⇐ log (ωdb · (ωa

b )−1)

18: Fb ⇐ eqr, eqr,mi, mp, ab

19: Tb ⇐ eqb, eqb, ri,mi, rp, mp

20: B ⇐ Fb, Tb

21: i⇐ rab , ri22: F d ⇐ arg min

F(AF −B)TQ(AF −B) + FTRF

Substitute the robot dynamic system (9) into (16):

z = w +∂β

∂x1x2 +

∂β

∂x2x2

= w +∂β

∂x1x2 +

∂β

∂x2(Bu+ d+ k)

= w +∂β

∂x1x2 +

∂β

∂x2(Bu+ d− z + β(x1, x2) + k).

(17)

The update law ˙d and manifold z can thus be designed as:{

˙d = − ∂β

∂x1x2 − ∂β

∂x2(Bu+ d+ β(x1, x2) + k)

z = − ∂β∂x2

z. (18)

In order to make sure the system z is Lyapunov stableand reduce the computation complexity, the estimation errorfunction β(x1, x2) is designed as:

β(x1, x2) =

k1ωabxk2ωaby

k3ωabz

, (19)

where k1, k2, k3 are all greater then zero. ωabx, ωaby, ω

abz are the

components of vector ωab .

The update law ˙d and the function z can thus be written as:

{˙d = −M(Bu+ d+ β(x1, x2) + k)

z = −Mz, (20)


where M is given as:

M =

k1 0 00 k2 00 0 k3

. (21)

The ACQL algorithm is shown in Algorithm 1. In termsof Lyapunov’s second method for stability, it proves that (20)asymptotically converges to zero. The moment of the payloadconcerning the robot can be identified with (12) and (20). Itis noteworthy that the control law of (12) is the aggregatemoment of all four foot-end forces. This value is redistributedto each foot with the QP solver.C. Stability Proof of ACQL

The proof of the convergence of s in the identification of thepayload moment section is shown in this subsection. Firstly,it is proven that there exists s = 0. Substitute (12) into (20):

˙d = −M(−cs− λx2 + β)

=M [c( ˙x+ λx) + λx2 −Mx2]

=M [cλ(x1 − x1d) + (c+ λ−M)Mx2].

(22)

It can be summarized by (22) that the update law ˙d depends

on the robot orientation error and its angular velocity. Bychoosing appropriate positive definite matrices of c, λ, andM , the robot is able to adjust its orientation until its errorapproaches zero, which implies the function s will eventuallyapproach zero.

Secondly, it is proved that (13) asymptotically converge tozero. since d is designed to asymptotically converge to zero asdescribed above, only s = −cs needs to be considered. Definethe Lyapunov function as:

V (s) =1

2s2. (23)

It can be seen in (23) that V = 0 if and only if s = 0 andV > 0 when s 6= 0. The derivative of V is:

V (s) = ss = −cs2 ≤ 0. (24)

According to the second method of Lyapunov, the systemas described in (13) is proven to asymptotically converges tozero. Therefore, ACQL is proven to be Lyapunov stable.

V. EXPERIMENT

A. Experiment Platform

The experiments to verify the verification of theeffectiveness of the proposed method are conducted on thequadruped robot, Kirin. The Kirin is an electrically actuatedquadruped robot developed for high payload capacity. Kirinhas one hip roll joint, one hip pitch joint, and one knee jointon each leg (as shown in Fig. 3). The total weight of Kirinis around 50 kg and payload capacity can reach up to 2.0(with at least 100 kg payload). The hip roll and pitch jointsare revolute joints, while the knee joint is designed as theprismatic joint. The controller of Kirin consists of Nvidia TX2,on which RT-Linux is installed. The high-level forward andinverse dynamics, online payload identification, and adaptivecontrol algorithms are running on RT-Linux. The size of Kirin

Fig. 3. The quadruped robot for high payload capacity locomotion. It is anelectrically-actuated quadruped robot with 12 degrees-of-freedom (DoFs) andis named as Kirin. The knee joint is designed to be prismatic to enhance thepayload-carrying capacity in dynamic locomotion.

TABLE ISPECIFICATIONS OF THE QUADRUPED ROBOT KIRIN

Property ParametersDimensions (L×W×H) (mm) 700 × 240 × 600Active DoF number 12Total weight (kg) 50Maximum payload *(kg) 100+Motion range of hip roll joint (◦) 280Motion range of hip pitch (◦) 360Motion range of knee joint (mm) 300Electric actuator Customized QDDComputing board Nvidia Jetson TX2Joint peak torque (Nm) 200Joint peak speed (rpm) 150Motor driver G-SOLWH120/100EES* Due to the safety consideration, 100kg is the experiment result at the

current stage. More payload is to be tested.

is 700 mm × 240 mm × 600 mm. The peak joint torque andvelocity are around 200 Nm and 150 rpm respectively. Theactuator specifications of all the joints are uniform due to thecommercial constraints. The joint power and the mechanismdesign are capable of supporting the heavy payload carryingand dynamic gait. The specifications of the quadruped robotKirin are listed in Table. I.

B. ACQL Test with Varied Payload

In this section, several experiments were carried out withthe Kirin that verifies the efficacy of the proposed ACQL

Fig. 4. The experiment scenario of the payload mass identification. (a) isthe initial posture of the robot. (b) is the posture of the robot when theidentification ends.


Fig. 5. The mass of the payload identified in this experiment. Payload variesfrom 20kg to 75kg. Payload varies from 20kg to 75kg, which correspondsto PLC (Payload Capacity) from 0.4 to 1.5. The identification starts from 6s (red region) and ends at around 6.5 s (blue region)

Fig. 6. The estimated d and the estimated moment of payload. Since theinertial matrix of the robot is a diagonal matrix, the estimated d and theestimated moment of payload have the same shape with the different values.The recursive process undergoes in the red region and converges in the blueregion.

Fig. 7. The norm of the robot position error and orientation error. Eachcorresponds to the payload arrangements in Fig. 8(a) and (b) respectively.The recursive process undergoes in the red region and converges in the blueregion.

algorithm. As shown in Fig. 4, the robot initially standson the ground and carries unknown payloads (sandbags).A predefined control torques are applied to drive the robotto reach the desired height with four legs on the groundsupporting the main body. Influenced by the unknownpayloads, there are both significant errors in the position andorientation of the robot’s torso. Then, the robot will not stopthe identification of the mass and the moment concerning therobot generated by the payload and adjusting its posture usingthe adaptive control proposed above until the errors of therobot’s orientation reduce to the predefined thresholds. In theexperiments, the thresholds for the orientation convergence areset to 0.01 rad in each rotational direction. The parameters

Fig. 8. The top view of the quadruped robot in the experiment of identifyingthe moment with respect to the robot induced by payload. The payload istwo sandbags, each of which weighs about 25 kg. They are placed on thedifferent location on the robot’s back.

Fig. 9. The experiment scenario of different PLC. Payload of different massare placed in the front on the robot back. Each sandbag weights 25kg whileeach dumbbell weights 5kg.

c, λ, and M in ACQL are set to 0.7I, 0.7I , and 1.3Irespectively. These parameters govern the recursive rate of thepayload moment identification.

The results of the payload mass identification are shown inFig. 5. In the experiment, payload varies from 20 kg to 75 kg,which implies that PLC varies from 0.4 to 1.5. Benefiting fromour previous work [26] and ACQL, the maximum estimatederror of the payload mass is about 3 kg, which accounts foronly 6% of the robot’s mass. The error has little influence onthe robot dynamic locomotion.

The results of the moment concerning the robot generatedby the payload are depicted as shown in Fig. 6 to Fig. 7. Asshown in Fig. 8 (a), two sandbags (each weighs around 25 kg)are loaded on the back of Kirin. These two sandbags are placedin the front and left part on the back of Kirin, away from therobot’s original CoM, to verify the effectiveness of ACQL. Theresult of the derivative of d is shown in Fig. 6. These valuesconverge to less than 0.005 Nm/(kg ·m3 ·s) within 2 s whenthe orientation error reduces below the threshold. The resultof the estimated d and the moment of the payload are shownin Fig. 6. Since the inertial matrix of the robot, generatedfrom CAD, is a diagonal matrix, the estimated d and theestimated moment of payload have the same structure yet with


Fig. 10. The experiment screenshots of Kirin trotting with payload weighing around 50kg. Two sandbags are placed on the back of Kirin, each of whichweighs around 25kg.

Fig. 11. The convergence time and the RMSE of the tracking errors for aseries of PLCs with ACQL. ACQL is verified to converge within a wide rangeof payload with PLC up to 1.5.

different values. With the measured mass of the payload andthe dimensions of the robot depicted in table I, it can be provedthat the estimated moment of the payload is close to its truevalue. The norm of the robot position and orientation trackingerrors is shown in Fig. 7, which are usually used to illustratethe identification performance [27]. As shown in Fig. 7, afterthe payload identification, the robot adjusts its body with theposition and orientation tracking error at around 3 mm and0.008 rad respectively. To furtherly verify the effectivenessof ACQL, both two sandbags (50 kg) are placed in the frontof the robot as shown in Fig. 8 (b). The robot has the sametracking performance as shown in the right subfigure of Fig.7.

The convergence time of the proposed ACQL is tested onKirin with a wide range of PLC. Payload varies from 20 kg to75 kg which represents PLC from 0.4 to 1.5 as shown in Fig.9. These payloads are placed at the same place, in the front ofthe robot, which will generate moment along the pitch axis.Each test for a PLC is repeated five times. The experimentresults are depicted as shown in Fig. 11. As shown in thefigure, when PLC is low, less than 0.4 in this experiment, theidentification is able to converge fast. In fact, due to the robot’sinertia, the payload has limited influence on the robot. Evenwith the open-loop force control method, the robot can achievesatisfactory performance. However, given a large PLC, whichis yet fully tested on electrically-actuated quadruped robots,

Fig. 12. The robot position and orientation in the world coordinate duringtrotting with the payload.

Fig. 13. The information of the right front leg of the robot. The left figureis the actual joint position and the right figure is the actual joint torque.

the influence of the payload on the robot can not be ignored.It will take a few seconds before the identification converges.As shown in Fig. 11, for different PLC, the convergence timeis consistently around 2 s. The convergence rate can be fasterby manually tuning c, λ, and M , in this study the performanceis constrained within a safe range, i.e. 2 s, to prevent possibleovershooting for each robot joint and the potential overturningfor the robot.


C. Payload-carrying Trotting Test with ACQL

In this subsection, the robot is designed to trot with apayload of 50 kg, as heavy as the robot’s own weight. Theexperiment screenshots is shown in Fig. 10. The predefinedtime of the swing phase and the stance phase are both 0.5 s.The results of the experiment are shown in Fig. 12 and Fig.13. The robot is designed to trot in place at a desired robotCoM position (0, 0, 0.41) with the desired orientation (3.25,-0.01, 0), representing the yaw, pitch, and roll. As shown inFig. 12, the robot trotting with the position deviation around0.01 m in each direction and the orientation less than 0.03 radin each axis. It can be derived that the robot is able to trot withthe desired position and orientation even with the payload asheavy as itself. The locomotion performance is not discountedby the payload. Usually, the torque of the knee joint will betwo times that of the hip pitch joint if all the robot jointsare designed as the articulated ones [28]. Benefiting from theprismatic knee joint in our quadruped robot Kirin, the actualtorque of the knee joint is similar to that of the hip pitch jointas shown in Fig. 13. This means that, with the prismatic kneejoint instead of the traditional rotation joint, the robot’s abilityof payload-carrying is greatly improved.

VI. CONCLUSION

This letter presents an online high-payload identificationand adaptive control for an electrically-actuated quadrupedrobot. By the aid of the identified mass and moment ofthe unknown payload, the quadruped locomotion is able toadapt to the dynamically changing high-payload. In this study,an electrically-actuated quadruped robot for heavy payload-carrying, Kirin, is used the support the tests to verify theeffectiveness of the proposed method, ACQL. ACQL isvalidated for a wide range of PLC from 0.2 to 1.5. Theeffectiveness of ACQL is also verified in the trotting gaitwith a payload as heavy as the robot itself. Statistic resultsshow that ACQL is able to converge fast and run efficientlyonline, which demonstrates that the proposed method is validfor the payload with no matter unknown weight or unknownlocation. However, at the current stage, ACQL is still limitedfor the payload with static weight. Therefore, the futurework will include dynamic quadruped locomotion control withdynamically changing payload.

VII. ACKNOWLEDGEMENT

We would like to thank Mr. Juntong Su and Mr. ShushengYe for their assistance in the experiments.

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