Analysis of Position Tracking in Torque Control of Humanoid RobotsConsidering Joint Elasticity and Time Delay

Jaesug Jung1, Soonwook Hwang1, Yisoo Lee1, Jaehoon Sim1, and Jaeheung Park1,2

Abstract— This study investigates the position tracking per-formance of torque controlled humanoid robots in the presenceof joint elasticity and time delay in torque command. One of themain purposes using torque control for humanoid robots is toachieve compliant behaviors on uncertain external disturbancesuch as uneven terrain and interaction with human. On theother hand, high performance of position tracking is alsorequired to implement motion control of robots. In this study,the effects of joint elasticity and time delay in torque commandarea investigated in terms of position tracking. First, a jointmodel is derived and validated, which reflects the elasticityand time delay. Frequency response analysis is exploited totheoretically evaluate the performance of the control system forposition tracking. This joint model with the elasticity and timedelay is used to estimate the limitations in the controller designfor our torque controlled humanoid robot. Theoretical analysisand its comparison with experimental results demonstrate thatthe joint elasticity and time delay significantly affect the systemperformance.

I. INTRODUCTIONTorque control is one of the effective strategies for hu-

manoid robots to operate in complex environments, facilitat-ing their behaviors compliant with external disturbance [1],[2]. In contrast with position control that keeps each jointof the robots highly stiff, a wide range of the joint stiffnessis possible in accordance with various situations in torquecontrol. These advantages are beneficial to humanoid robotsto safely carry out both manipulation and locomotion inhuman environments, where arbitrary disturbance is causedby unexpected contact or collision [3], [4].

Implementation of torque control to humanoid robots typ-ically requires a central computer as a high level controller,and distributed motor drives as a low level controller whichcontrols joint torque of actuators [5]. The central computerhas a relatively slow control period [2], [6], due to a largeamount of the computation required to solve the torquesolution for motion and force control of the humanoid robot[7], [8]. The motor drive controls the single joint at afaster cycle than the central computer to generate the jointtorque commanded by the high level controller. Generally,there exists a time delay in communication between the twocontrollers. The inevitable processing delay in motor drivesdeteriorates the control stability [9].

For most actuators used for robots, a gear reducer isinstalled to amplify the generated torque by the actuator [10].

1J. Jung, S. Hwang, Y. Lee, J. Sim and J. Park is with the GraduateSchool of Convergence Science and Technology, Seoul National University,Seoul, Korea. {jjs916, jbs4104, howcan11, simjeh,park73}@snu.ac.kr.

2J. Park is with Advanced Institutes of Convergence Technology (AICT),Suwon, Korea.

Central Computer 𝐽𝐽𝑀𝑀

𝐽𝐽𝐿𝐿

Motor drive

Gear Reducer

Motor

LoadΓ

Γ𝑑𝑑

Encoder

𝑞𝑞

Delay

Fig. 1. Configuration of the joint system of torque controlled humanoidrobots.

Joint elasticity is induced by the deformable metallic materialof the gear reducer such as harmonic drives [11], [12]. Itis well known that the joint elasticity adversely affects theperformance of feedback control for position tracking, whichleads to oscillation [13], [14].

Fig. 1 is a configuration of the joint system of the robot,which reflects the intrinsic joint elasticity at the gear reduceras well as the time delay at motor drive. Furthermore, timedelay in communication amplifies the adverse effect of thejoint elasticity in the performance of torque control. Thetime delay usually occurs when the torque control is used,because the torque control is different from the positioncontrol. The typical position control commands referencevalue of position to the motor drive directly and the drivecalculates and generates desired torque [15]. On the otherhand, the torque control separates the torque calculation partand generation part, and the central computer and the motordrive respectively handle the calculation and generation.

To understand the limitation of torque control performancein terms of position tracking, a system that takes into accountof the joint elasticity is theoretically analyzed. Modeling ofthe system also reflects the adverse influence of time delayin communication, of which the presence is experimentallyverified. By combining the feedback controller for positiontracking with the system model, the control performance isinvestigated through frequency response analysis. Validationof the analysis is achieved through experiments with botha single joint and a biped robot with high degrees offreedom (DoF). Through this study, we present a usefulmodel to design controllers for torque controlled robots.By using presented model, we can explain the limitationof PD control. The PD control was chosen as an examplein the paper to demonstrate the effectiveness of the modelin the controller design, because it is commonly used forimplementing position control. However, the proposed jointmodel is useful to design other control methods.

The remainder of this paper is organized as follows. InSection II, modeling of the system is discussed with takingaccount of both joint elasticity and time delay. Based on

the system model that includes the feedback controller, fre-quency response analysis is described in Section III, provid-ing the information of the control performance. Experimentresults are provided with feedback gains that assure stablecontrol and the resultant tracking accuracy in Section IV, todemonstrate the model that includes joint elasticity and timedelay. In Section V, conclusion is remarked with summaryand future works.

II. SYSTEM MODELING

In order to analyze the control performance of humanoidrobots, characterization of each joint, which is the basisof operation, should be preceded. Modeling of the jointsystem is conducted to take account of both time delay incommunication and joint elasticity. Especially, these occurwhen each motor drive locally controls the motor and thetorque input is remotely computed in the central computer.

The testbed is set up to verify the issues. It consists of abrushless DC (BLDC) electric motor from Parker HannifinCorp., a harmonic drive with the gear ratio of 100:1, and alink of which length is 0.6 m and weight is 3.9 kg. A Linux-OS based computer with i7-3770 CPU and a motor drivefrom Elmo Motion Control Ltd. are used as the high leveland low level controllers, respectively. Xenomai is utilizedas a software framework for the real-time capability. TheEtherCAT is utilized for the communication between thecomputer and the motor drive. Current control is enabledusing CANopen protocols. Assuming that the input current ofthe motor is proportional to the output torque, torque controlis performed using the current controller of the motor drive.The testbed utilized in the experiment is shown in Fig. 2.

A. Joint Elasticity

Considering only inertias of the motor and the load, therelationship between the input torque Γ and output positionP is as follows.

P =1

(JM + JL)s2 Γ, (1)

where JM is the motor inertia and JL is the load inertiaconnected to the motor.

In the system described in (1), the positions of motor(PM) and load (PL) are always the same (P = PM = PL). Ifthere exists the joint elasticity, the relationship between thepositions changes. Fig. 3 shows the block diagram of theelastic coupling between the motor and the load. Due tothe elasticity, the motor and the load react differently andthe differences of position and velocity occur. The positiondifference between the PM and the PL and the differencebetween the motor velocity (VM) and load velocity (VL) affectthe input torque. The transfer function from the input torqueΓ and output motor position PM can be described as (2),where K and B are coefficients of the spring and damper,respectively [16].

PM

Γ=

1(JM + JL)s2

JLs2 +Bs+K( JMJL

JM+JL)s2 +Bs+K

(2)

(a) (b)

0.6 m

(c)

Motor with Harmonic drive

Motor drive

Fig. 2. Snapshots of the 1-DoF testbed system for the experiments. Thetestbed is composed of a motor with harmonic drive, a motor drive, and alink. (a) Overall view. (b) Side view. (c) Top view.

1𝐽𝐽𝑀𝑀

1𝑠𝑠

𝐾

𝐵𝐵

-

-

-+

+

+ 𝑃𝑃𝑀𝑀

𝑃𝑃𝐿𝐿

𝑉𝑉𝑀𝑀

𝑉𝑉𝐿𝐿

++

Γ 1𝑠𝑠

1𝐽𝐽𝐿𝐿

1𝑠𝑠

1𝑠𝑠

Fig. 3. Block diagram for a joint with the elasticity.

The joint with elasticity and without elasticity is compara-tively analyzed with Bode plot depicted in Fig. 4. Resonanceand anti-resonance occur due to the elasticity.

When the load-to-motor inertia ratio increases, the effectof elasticity also increases. This phenomenon is explainedby the Bode plot (Fig. 5). The Bode plot according to theload-to-motor inertia ratio is shown for three cases of theratio: 0.1, 1, and 16. The motor inertia (1.0×10−4 kg ·m2)and the spring-damper coefficients (1.0×104 Nm/rad & 1.0Nm · s/rad) are the same in each case. As the load inertiaincreases, the magnitude difference between the region oflow and high frequency becomes larger. Regardless of theload inertia, the magnitude of the high frequency range isdetermined by the inertia of the motor. The gain marginis dependent only on the motor inertia because the cross-over frequency is typically above the resonance frequency.

-200

-100

0

100

200M

ag

nitu

de

(d

B) w/ elasticity

w/o elasticity

100

101

102

103

104

105

-180

0

Ph

ase

(d

eg

)

Frequency (rad/s)

Fig. 4. Bode plot of the joint with and without the elasticity.

-200

-100

0

100

200

Magnitude (

dB

) Motor : Load = 1 : 0.1

Motor : Load = 1 : 1

Motor : Load = 1 : 16

100

101

102

103

104

105

-180

0

Phase (

deg)

Frequency (rad/s)

Fig. 5. Bode plot of joint with different load inertias.

Therefore, the feedback gain increasing is heavily affectedby the motor inertia and that the influence of the load inertiais small.

B. Time Delay

Time delay is an inevitable phenomenon in communica-tion, due to the interpretation of digital data to analog signals.When we use the torque control, the delay of 4∼6 ticks ispresent. That is, when the robot is controlled at 2 kHz, thedelay of about 2∼3 ms occurs. The motor drives from othercompanies will also have the delay, but the size of delaymay vary. The delay is identified by utilizing project dataobjects (PDO) of torque signals including the target torque,the demand torque, and the actual torque. The target torqueis the torque we command to the motor drive, the demandtorque is the desired torque that the motor drive will create,and the actual torque is the measured torque calculated bythe measured current that the drive actually generates. Fig. 6shows the delay between the target torque (Γt ), the demandtorque (Γd), and the actual torque (Γa) of the motor drive.Each value was recorded at 0.5 ms intervals.

The time delay of T seconds is represented as e−T s. Bymultiplying e−T s to (2), we can obtain the transfer functionwhich considers both the elasticity and the time delay. Thetime delay causes a phase lag, and the effect of the phase lagcan be seen in the Bode plot (Fig. 7). As seen in Fig. 7, thegain margin is reduced because of the phase lag. Therefore,it can be seen that the time delay has a negative effect onthe magnitude of feedback gains.

0.498 0.499 0.5 0.501 0.502 0.503 0.504 0.505 0.506

Time (sec)

0

2

4

6

8

10

12

To

rqu

e (

Nm

)

Target Torque

Demand Torque

Actual Torque

Fig. 6. Target input torque, demand torque and actual torque from Elmomotor drive.

-100

-50

0

50

Ma

gn

itu

de

(d

B) w/ delay

w/o delay

100

101

102

103

104

105

-720

-540

-360

-180

0

Ph

ase

(d

eg

)

Frequency (rad/s)

Fig. 7. Bode plot of joint with time delay and without time delay.

C. Modeling of a Joint Control System

Both elasticity and time delay affect the feedback gain, soboth effects must be considered when modeling the singlejoint control system. A joint control system configurationfor tracking the position is the following block diagram inFig. 8. In this system, joint is controlled to track the desiredposition (Pd) with a position controller (D). The positioncontroller can be any type of controller (e.g, PD, PID) andit calculates target torque for the joint control. The demandtorque is transferred to the current controller C after the timedelay T . The current controller is modeled approximately asseen below,

C =ω2

c

s2 +2ζ ωcs+ω2c. (3)

The M is a model which describe the motor with the loadincluding the elasticity. This system also reflects disturbanced and the measurement noise n.

III. CHARACTERISTICS OF FEEDBACK SYSTEM

This section presents the analysis of PD control forposition tracking and the analysis in operational space. Byanalyzing the controller, the adverse effects of the time delayand the elasticity on controllers will be shown. The analysisin operational space will show the relationship betweenthe joint space feedback gains and the operational spacefeedback gains.

1𝐽𝐽𝑀𝑀

1𝑠𝑠

𝐵𝐵

-

-

-+

+

+ 𝑃𝑃𝑀𝑀

𝑃𝑃𝐿𝐿

𝑉𝑉𝑀𝑀

𝑉𝑉𝐿𝐿

++

1𝑠𝑠

1𝐽𝐽𝐿𝐿

1𝑠𝑠

1𝑠𝑠

𝑃𝑃𝑑𝑑 𝐷𝐷 𝐶𝐶𝑒𝑒−𝑇𝑇𝑇𝑇+

-

𝑑𝑑

𝑛𝑛

𝑀𝑀

𝐾𝐾

Γ𝑡𝑡 Γ𝑑𝑑 Γ𝑎𝑎

Fig. 8. Block diagram of overall system for a joint position tracking withtorque control.

A. Frequency Analysis on PD control

With the joint model in Section. II-C and the PD controller(D = Kp + sKv), the position of motor is represented asfollows.

PM =CM(Kp + sKv)

1+CM(Kp + sKv)Pd−

CM(Kp + sKv)

1+CM(Kp + sKv)n

+CM

1+CM(Kp + sKv)d (4)

There are limitations in increasing the feedback gainsdue to the time delay and the elasticity. This limitationcan be known by the gain margin at the Bode plot. Whenthe feedback gain becomes larger, the system will becomeunstable because there is lack of gain margin. Fig. 9 is theopen loop Bode plot of block diagram of a joint system withthe PD controller.

The effect of the elasticity and time delay on the gainmargin can be seen in denoted points (a), (b), and (c) in Fig.9. The elasticity reduces the gain margin from 52.1 dB to27.5 dB, which is seen in Fig. 9 (a). As seen in Fig. 9 (b),the time delay affects to the gain margin to reduce from 52.1dB to 31 dB. Fig. 9 (c) shows that the gain margin is reducedfrom 52.1 dB to 6.2 dB due to the both elasticity and timedelay. It can be seen that the gain margin is greatly reducedwhen time delay and elasticity are present together.

As seen in Fig. 5, the magnitude of the low frequencydecreases as the inertia of the load increases. Therefore, thefeedback gain of the joint with the link of large inertia shouldbe higher than the joint with the link of small inertia forsimilar control performance at the low frequency. However,due to the time delay and the elasticity, it is difficult toincrease the feedback gain for the joint control sufficiently asthe load inertia increases. Therefore, the joint with a largeinertia load (e.g, ankle joint of the supporting leg of thehumanoid robots) cannot achieve high tracking performancewith a PD control scheme.

B. Analysis in Operational Space

Each joint of torque controlled humanoid robots has limi-tation because of the time delay and the elasticity. If multiplejoints are coupled together, it is difficult to assume theperformance limit, because of the cross coupled effect. Whenthe robot is controlled in operational space, the feedback

-100

-50

0

50

Ma

gn

itu

de

(d

B)

w/ elasticity & delay w/ elasticity w/ delay w/o elasticity & delay

100

101

102

103

104

105

-900

-720

-540

-360

-180

0

180

Ph

ase

(d

eg

)

Frequency (rad/s)

(a)(c)

(b)

Fig. 9. Bode plot of the joint control system with PD controller. Blueline describes result of control system with both elasticity and time delay.Gain Margin is 6.2 dB at 620 rad/sec. Red dash line describes result ofcontrol system only with elasticity. Gain Margin is 27.5 dB at 3970 rad/sec.Yellow dash line describes result of control system only with time delay.Gain Margin is 31 dB at 619 rad/sec. Purple dot line describes result ofcontrol system without both elasticity and delay. Gain Margin is 52.1 dB at3970 rad/sec

gains of the task space control can be understood by itsrelationship with the feedback gains in joint space.

In joint space, if the Coriolis, centrifugal, and gravity forceare perfectly compensated, the PD control of the multiplejoints can be written as

Γ = Aq = A(K′pqqe +K′vqqe), qe = qd−q

= Kpqqe +Kvqqe(5)

Kpq = AK′pq

Kvq = AK′vq(6)

A is the joint space inertia matrix, K′pq is the proportionalgain for the joint space control, and K′vq is the derivativegain. The term Γ is the torque vector, qd is the desired jointposition and q is the actual joint position. If the robot iscontrolled in operational space [17], the control torque canbe composed as follows, also assuming that the Coriolis,centrifugal, and gravity force are compensated.

Γ = JT F

= JTΛ(K′pxxe +K′vxxe), xe = xd− x

= JT (Kpxxe +Kvxxe)

(7)

Kpx = ΛK′px

Kvx = ΛK′vx(8)

Where F represents the operational space force, xd is adesired task position, and x is an actual task position. Theterm J is the Jacobian for the tasks and Λ is the operationalspace inertia matrix. The operational gain matrices are writ-ten as K′p and K′v. If xd−x is small, xd−x' J(qd−q). Then,the relation between the gains in joint space and operationalspace can be approximated as follows.

Fig. 10. Snapshot of DYROS-Red. Left figure is front view and right figureis side view of the robot.

Γ' JTΛ(K′pJqe +K′vJqe)

' (JTΛK′pJ)qe +(JT

ΛK′vJ)qe(9)

Kpq = AK′pq ' JTΛK′pxJ

Kvq = AK′vq ' JTΛK′vxJ

(10)

Therefore, we can estimate the limits on the gains ofthe operational space control using this relationship if thelimits in the joint space control are known. However, thelimits in the joint space control are not easily computedbecause there are dynamic coupling terms. The gain matricescan be approximated to be diagonal using 1-DOF modelof each joint but this approximation may not be accurateenough to represent the real system. Experimental resultsand discussion will be presented in the following section.

IV. EXPREIMENT RESULT

Following experiments were carried out to demonstratethe system model that reflects the joint elasticity and thetime delay. Firstly, the system model is validated using anexperimental setup with a single joint testbed. Then controlperformance of the robot with high DoFs is investigated withoperational space control.

A. Setup

For the experiments in both joint and operational spacecontrol, the testbed with single joint in Section. II is usedto verify the theoretical feedback gain limit and the actuallimit of the feedback gain of torque control with positionfeedback. A 12-DoF biped robot DYROS-Red [18] in Fig.10 is utilized to measure the limitation of the performanceof the operational space control framework, when there existthe joint elasticity and time delay.

2.8 2.9 3 3.1 3.2 3.3 3.4 3.5

Time (sec)

-0.51

-0.5

-0.49

-0.48

-0.47

-0.46

-0.45

-0.44

An

gle

(ra

d)

Desired position

Kp : Kd = 1600 : 80

Kp : Kd = 1600 : 70

Fig. 11. Trajectory tracking experiment results of single joint. Green dotline is the reference trajectory. Red line is control result with Kv over theavailable value. Blue line is control result with Kv lower than the availablevalue.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.05

0

0.05

X d

ire

ctio

n(m

) Desired CoM Position Actual CoM Position

Time(sec)

(a)

Desired CoM Position Actual CoM Position

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.05

0.1

Y d

irection(m

)

Time(sec)

(b)

Desired CoM Position Actual CoM Position

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time(sec)

0.8

0.85

0.9

Z d

ire

ctio

n(m

)

(c)

Fig. 12. Trajectory tracking experiment results of center of mass with PDcontroller. (a) is a X direction tracking, (b) is a Y direction tracking, (c) isa Z direction tracking.

B. Single Joint Control

This experiment uses a PD controller to show the lim-itation of torque control with position feedback. The test-bed uses harmonic gear for the transmission and its springconstant is measured as 1.0× 104 Nm/rad. It is measuredby fixing the output of the joint and applying torques. Theinertia of motor is 0.2×10−4 kg ·m2 and the load-to-motorinertia ratio was 4. Theoretically, the feedback gains canbe set as Kp = 1600 Nm/rad and Kv = 77 Nm · s/rad. Inthis condition, gain margin is 76.8×10−3 dB at 494 rad/s.Fig. 11 shows the result of the experiment. The experimentcompares two cases. One is the case that feedback gain wasset as Kp = 1600 Nm/rad and Kv = 80 Nm · s/rad, Kv ishigher than the theoretically set. Another case is Kp = 1600Nm/rad and Kv = 70 Nm · s/rad which is lower than thetheoretically set.

When the actual Kv exceeds theoretical gain, the joint

TABLE ITHE LARGEST JOINT SPACE FEEDBACK GAINS CONVERTED FROM OPERATIONAL SPACE FEEDBACK GAINS DURING THE PD CONTROL EXPERIMENT.

Left Leg Right LegHip Yaw Hip Roll Hip Pitch Knee Pitch Ankle Pitch Ankle Roll Hip Yaw Hip Roll Hip Pitch Knee Pitch Ankle Pitch Ankle Roll

Kpq 41.80 109.4 421.6 1228 1234 813.2 43.70 132.5 488.9 1199 1270 742.7Kvq 2.26 4.73 19.1 57.6 16.8 40.9 2.40 6.20 22.5 56.0 22.5 36.6

+𝑃𝑃𝑑𝑑 𝐾𝐾𝑝𝑝 𝐶𝐶𝑒𝑒−𝑇𝑇𝑇𝑇+

-

𝑑𝑑

𝑛𝑛

𝑀𝑀𝑃𝑃𝑀𝑀

𝑠𝑠

𝐾𝐾𝑣𝑣+-

𝐾𝐾𝑖𝑖𝑠𝑠

Γ𝑡𝑡 Γ𝑑𝑑 Γ𝑎𝑎

Fig. 13. Block diagram of P/PI controller.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.05

0

0.05

X d

ire

ctio

n(m

) Desired CoM Position Actual CoM Position

Time(sec)

(a)

Desired CoM Position Actual CoM Position

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.05

0.1

Y d

irection(m

)

Time(sec)

(b)

Desired CoM Position Actual CoM Position

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time(sec)

0.8

0.85

0.9

Z d

irection(m

)

(c)

Fig. 14. Trajectory tracking experiment results of center of mass with P/PIcontroller. (a) is a X direction tracking, (b) is a Y direction tracking, (c) isa Z direction tracking.

starts vibration, which means the system is considered unsta-ble. However, when the Kv is lower than the theoretical gain,the motor is stable. This experiment verifies that the feedbackgain over the limit causes instability and the theoreticalestimation of the limitation with our model is reasonable.

C. Operational Space Control of Biped Robot

DYROS-Red uses different motors and has different shaftdesign from 1-DOF test-bed, so the gains can be set higher.When each joint is controlled independently, gains of jointscan be set to around Kpq = 1.0×104 Nm/rad and Kvq = 300Nm · s/rad.

The task of the robot is controlled in operational space,using the operational space based whole-body control frame-work [19] with PD control. In this experiment, the position of

the center of mass (CoM) is controlled to track the trajectorygenerated with cubic spline. Fig. 12 and 14 show the trackingperformance when the CoM of the robot is controlled totranslate 6.0 cm in x, y, and z directions simultaneously (total10.4 cm) in 1.0 sec.

The gains used in operational space are K′px = 50 s−2/m2

and K′vx = 1 s−1/m2 for x direction, K′px = 90 s−2/m2 andK′vx = 5 s−1/m2 for y direction, and K′px = 200 s−2/m2

and K′vx = 15 s−1/m2 for z direction. The gains are set tothe values that are stable and can maximize performanceof tracking. These gains are empirically derived from therepetitive experiments. These gains (K′px and K′vx) can beconverted to the joint space gains (Kpq and Kvq) through themethod in Section. III-B. The diagonal elements of convertedresult are shown in the Table I. From the gains in Table I,we can speculate the gain limit of joints. Especially, D gainsof Knee Pitch and Ankle roll of both legs are much largerthan other joints. This shows that these two joints of eachleg are problematic in control stability.

However, the values in Table I are smaller than the gainlimit of each joint. The reason for this gap may be due to thedynamic coupling that comes from the off-diagonal elementsin Kpq. When the operational space gains are convertedinto the joint space gain, gains are multiplied by inertiamatrix and Jacobian. So, even if K′px is a diagonal matrix,Kpq has off-diagonal elements. Therefore, we believe thatthis coupling effect can be the reason why the diagonalcomponents of the gains cannot be set as high as the gainsfor the case of single joint control.

In this experiment, P/PI cascade controller was designedas an example to improve the performance of the operationalspace control. The structure of P/PI controller is shown inFig. 13. Fig. 14 shows the result when P/PI controller isapplied to perform the same task with the case of using PDcontroller. P/PI used the gains as Kp = 40 s−1, Kv = 1 m−1/s,and Ki = 1 s−1 for x direction, Kp = 20 s−1, Kv = 5 m−1/s,and Ki = 0.1 s−1 for y direction, Kp = 100 s−1, Kv = 5 m−1/s,and Ki = 1.0 s−1 for z direction. Gains are set based onthe PD control gains. The steady state errors and trackingerrors in all directions were reduced, compared to those ofPD controller. Especially, x direction steady state error wasreduced from 14.0 mm to 5.6 mm. In y direction, the steadystate error was reduced from 5.1 mm to 2.9 mm and thez direction steady state error was reduced from 13.0 mm to1.0 mm. Because of the integral term, P/PI can reduce steadystate error. The integral term compensates the low frequencymagnitude which is caused by the load.

There are several limitations in the results, even using P/PIcontroller. Firstly, it is challenging to obtain the consistent re-

sponse depending on the task directions. The effect of frictionis different depending on the direction. Also, depending onthe direction of movement, the inertia continues to change,and hence the gains applied to each joint always change.

Secondly, although the steady-state error is reduced bythe influence of the integral term in P/PI controller, thereis relatively less decrease in transient error compared tosteady-state error. Also there still exists a fluctuation duringthe transition. This fluctuation may be due to the stick-slipphenomenon induced by the nonlinear joint friction in lowspeed [20].

V. CONCLUSIONSThe objective of this study is to understand the limitations

of torque controlled humanoid robots in two factors, thejoint elasticity and time delay. The model with the jointelasticity and time delay is useful to understand and designthe controller for humanoid robots. To enhance the positiontracking performance, the feedback gain should be increasedto the high level, but the elasticity and the time delay limit thefeedback gain. Influence of the joint elasticity and the timedelay in torque control are verified with theoretical analysisas well as experimental validation. It is notable that timedelay amplifies the adverse effect of joint elasticity in termsof the reduction in gain margin. Without one of these factors,the adverse effects are far less than those when both factorsare present.

Our study is expected to be utilized to analyze anddetermine the available control gains for the tasks of robotswith high DoFs, where the joint elasticity and the time delayexist. Since the gain tuning of most humanoid robots isperformed through trial and error, it takes much time andeffort in the process. Our research will allow the selectionof the reasonable feedback gains in high-DoF robots andreduce the time required for gain tuning.

The humanoid robot changes the contact state while per-forming complex tasks including walking. As the contactstate changes, the load-to-motor inertia ratio of each jointchanges. Typically, the load-to-motor inertia ratio of theankle joint in the supporting leg is significantly increased,compared to the double support phase. Due to this reason,the limited performance caused by the elasticity and the timedelay makes it challenging for torque controlled humanoidrobots in a single support phase, which is essential for thebipedal walking control. Therefore, in order to achieve thedesirable control performance, it is required to determine thefeedback gain according to the contact state of the robot.

In the future, the estimation of the gains in the operationalspace from those in joint space, considering the cross-coupling effect, will be investigated. Then, we will be ableto develop a real-time gain determination method online,according to the state of the robot. Also, we would liketo investigate advanced controllers that can achieve desiredperformance.

ACKNOWLEDGEMENTThis work was supported by the National Research Foun-

dation of Korea (NRF) grant funded by the Korea govern-

ment (MSIP) (No. NRF2015R1A2A1A10055798) and theTechnology Innovation Program (No. 10060081) funded bythe Ministry of Trade, Industry & Energy (MI, Korea).

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