Operational Space Control Framework for Torque Controlled Humanoid Robots with Joint Elasticity Jaesug Jung 1 , Donghyeon Kim 1 and Jaeheung Park 1,2 Abstract—Torque controlled robots have the capability of implementing compliant behavior with back-drivability. In practice, however, joint elasticity often prevents an accurate position tracking performance of a robot. In particular, hu- manoid robots are influenced more by elasticity because of a long kinematic chain between the feet and hands. In this paper, we present a new inverse dynamics based control approach for torque controlled humanoid robots with joint elasticity. When formulating the operational space control framework, feedback control consists of only motor-related parts with measured motor angle values, and the link dynamics is compensated by the feedforward terms. The experiment results of the proposed approach show a noticeable improvement in the position tracking performance in 6-DoF manipulator. Finally, the proposed method was applied to a torque controlled biped robot for walking. Both stiff motion control of the CoM and compliant motion control of the foot were simultaneously achieved, demonstrating the advantage of the torque controlled robot. I. INTRODUCTION Torque controlled robots can achieve compliant behavior to external disturbances [1]–[4] and provide a high band- width for contact force modulation [5], allowing torque con- trol to be a useful approach to various robots that frequently interact with the environment under uncertainty. Several torque controlled humanoids have recently been developed [6]–[9] to take advantage of torque control. Compliant motion is beneficial to humanoid locomotion on uneven terrain or in an unstructured environment. However, torque controlled humanoids still have difficulty operating in task space owing to a low position tracking performance. To avoid this problem, we developed a reactive bipedal walking method that relies solely on the current state [10]. However, the robot still requires an improved tracking performance to walk to the desired position and to create an accurate motion. The low position tracking performance of the torque controlled robot is caused by several factors, such as joint elasticity, sensor noise, communication delays, and model errors [11]. One of the most critical problems we have experienced is joint elasticity, and our previous study pointed out that joint elasticity with a communication delay is one of the most problematic issues limiting the position tracking performance by reducing the gain margin [12]. In particular, humanoid control using the operational control framework [13] is vulnerable to joint elasticity because joint elasticity is 1 J. Jung, D. Kim and J. Park is with the Graduate School of Conver- gence Science and Technology, Seoul National University, Seoul, Korea. {jjs916, kdh0429, park73}@snu.ac.kr. 2 J. Park is with Advanced Institutes of Convergence Technology (AICT), Suwon, Korea. generally not considered in the derivation of the operational space control framework. Various control strategies have been proposed to deal with the joint elasticity problem found in joint space control [14]–[18]. Most controllers require either a link-side angle measurement or torque feedback between the motor and link. However, a link-side encoder and a joint torque sensor are not only expensive but also difficult to utilize owing to the resolution and noise. Furthermore, the formulation of a task space control using these methods is not straightforward because they have only been discussed for control in the joint space. There are approaches that consider the joint elasticity in task space control [3], [19]. These approaches use joint torque sensors and attempt to control the link dynamics by substituting the link side angles with the motor angles. Despite various studies, the task space precision of torque controlled humanoids has not been discussed much for difficult tasks such as locomotion. In this paper, we propose an inverse dynamics based controller that uses motor inertia for feedback and the link dynamics for feedforward compensation to avoid adverse effects of the joint elasticity. The operational space control framework is derived through the inverse dynamics of elastic joint robots with the proper Jacobian. Therefore, we can consider joint elasticity in both the joint space and the task space. The most noticeable improvements using the proposed controller is that it is possible to increase the available feedback gains allowing a desirable position tracking perfor- mance to be achieved. This control framework is developed from a study of torque controlled humanoid robots, but it is applicable to other articulated robots with a fixed base. The remainder of this paper is organized as follows. In Section II, the dynamics of the elastic joint robot is described followed by the problem statement. The proposed joint space control framework and operation space control framework are described in detail in Section III. In Section IV, the experiment results with the proposed framework are shown using a 12 DoF biped robot. Finally, in Section V, some concluding remarks and areas of future works are described. II. BACKGROUNDS A. Robot Dynamics with Joint Elasticity Assuming that the angular kinetic energy of each motor is due to only to its own spinning, the modeling of the robot dynamics with flexible joints is described as follows [14]. A m ¨ θ + K(θ - q)= Γ (1)
Transcript
Operational Space Control Framework for Torque ControlledHumanoid Robots with Joint Elasticity
Jaesug Jung1, Donghyeon Kim1 and Jaeheung Park1,2
Abstract— Torque controlled robots have the capability ofimplementing compliant behavior with back-drivability. Inpractice, however, joint elasticity often prevents an accurateposition tracking performance of a robot. In particular, hu-manoid robots are influenced more by elasticity because of along kinematic chain between the feet and hands. In this paper,we present a new inverse dynamics based control approach fortorque controlled humanoid robots with joint elasticity. Whenformulating the operational space control framework, feedbackcontrol consists of only motor-related parts with measuredmotor angle values, and the link dynamics is compensatedby the feedforward terms. The experiment results of theproposed approach show a noticeable improvement in theposition tracking performance in 6-DoF manipulator. Finally,the proposed method was applied to a torque controlled bipedrobot for walking. Both stiff motion control of the CoMand compliant motion control of the foot were simultaneouslyachieved, demonstrating the advantage of the torque controlledrobot.
I. INTRODUCTION
Torque controlled robots can achieve compliant behaviorto external disturbances [1]–[4] and provide a high band-width for contact force modulation [5], allowing torque con-trol to be a useful approach to various robots that frequentlyinteract with the environment under uncertainty.
Several torque controlled humanoids have recently beendeveloped [6]–[9] to take advantage of torque control.Compliant motion is beneficial to humanoid locomotion onuneven terrain or in an unstructured environment. However,torque controlled humanoids still have difficulty operating intask space owing to a low position tracking performance. Toavoid this problem, we developed a reactive bipedal walkingmethod that relies solely on the current state [10]. However,the robot still requires an improved tracking performance towalk to the desired position and to create an accurate motion.
The low position tracking performance of the torquecontrolled robot is caused by several factors, such as jointelasticity, sensor noise, communication delays, and modelerrors [11]. One of the most critical problems we haveexperienced is joint elasticity, and our previous study pointedout that joint elasticity with a communication delay is oneof the most problematic issues limiting the position trackingperformance by reducing the gain margin [12]. In particular,humanoid control using the operational control framework[13] is vulnerable to joint elasticity because joint elasticity is
1J. Jung, D. Kim and J. Park is with the Graduate School of Conver-gence Science and Technology, Seoul National University, Seoul, Korea.{jjs916, kdh0429, park73}@snu.ac.kr.
2J. Park is with Advanced Institutes of Convergence Technology (AICT),Suwon, Korea.
generally not considered in the derivation of the operationalspace control framework.
Various control strategies have been proposed to deal withthe joint elasticity problem found in joint space control[14]–[18]. Most controllers require either a link-side anglemeasurement or torque feedback between the motor and link.However, a link-side encoder and a joint torque sensor arenot only expensive but also difficult to utilize owing to theresolution and noise. Furthermore, the formulation of a taskspace control using these methods is not straightforwardbecause they have only been discussed for control in thejoint space. There are approaches that consider the jointelasticity in task space control [3], [19]. These approachesuse joint torque sensors and attempt to control the linkdynamics by substituting the link side angles with the motorangles. Despite various studies, the task space precision oftorque controlled humanoids has not been discussed muchfor difficult tasks such as locomotion.
In this paper, we propose an inverse dynamics basedcontroller that uses motor inertia for feedback and the linkdynamics for feedforward compensation to avoid adverseeffects of the joint elasticity. The operational space controlframework is derived through the inverse dynamics of elasticjoint robots with the proper Jacobian. Therefore, we canconsider joint elasticity in both the joint space and the taskspace. The most noticeable improvements using the proposedcontroller is that it is possible to increase the availablefeedback gains allowing a desirable position tracking perfor-mance to be achieved. This control framework is developedfrom a study of torque controlled humanoid robots, but it isapplicable to other articulated robots with a fixed base.
The remainder of this paper is organized as follows. InSection II, the dynamics of the elastic joint robot is describedfollowed by the problem statement. The proposed joint spacecontrol framework and operation space control frameworkare described in detail in Section III. In Section IV, theexperiment results with the proposed framework are shownusing a 12 DoF biped robot. Finally, in Section V, someconcluding remarks and areas of future works are described.
II. BACKGROUNDS
A. Robot Dynamics with Joint Elasticity
Assuming that the angular kinetic energy of each motor isdue to only to its own spinning, the modeling of the robotdynamics with flexible joints is described as follows [14].
Amθ +K(θ −q) = Γ (1)
Al(q)q+b(q, q)+g(q) = K(θ −q) (2)
Am is the diagonal motor inertia matrix, θ is the vector ofthe motor angles, and Γ is the vector of the torques generatedby the motors. The variables in (1) are expressed in the jointspace, reflecting the gear ratio for the motor inertia. K is adiagonal matrix with the stiffness of gears. q is the vector ofthe link side joint angles. Al(q), b(q, q), and g(q) denote theinertia matrix of links, the Coriolis/centrifugal torque vector,and the gravitational torque vector respectively. In this paper,(1) is referred to as the motor part of the dynamics and (2)is referred to as the link part of the dynamics.
B. Problem Statement
Despite the advantages of the torque control mentionedin Section I, its implementation to the humanoid robot isvery difficult to achieve. When joint elasticity exists, thegain margin decreases as the weight of the load becomesheavier, consequently worsening the tracking performance[12]. Humanoid robots are particularly susceptible to thisproblem at their joints near the distal end, such as the anklejoints, because of their heavy weight compared to the sizeof its ankle. This prevents increasing the feedback gains andreduces CoM tracking performance during locomotion.
Robots are usually thought to be rigid unless joint elas-ticity is intentionally added like SEA, but joint elasticityalways exists owing to the stiffness of the gear. Nevertheless,because the stiffness of the gear is very high (tens of thou-sands of Nm/rad) [16], [20], [21], robots whose elasticityis not intentionally added are usually controlled under theassumption that the effect of elasticity is small or negligible.
One method for controlling a flexible robot with highstiffness is to consider that the link side angles can beapproximated as those of motors (i.e. q ' θ ). In this case,(1) and (2) become
Γ = (Al(q)+Am)q+b(q, q)+g(q) (3)
which indicates the dynamics of a typical articulated sys-tem without joint elasticity. Therefore, we formulate theoperational space control framework in the original manner.However, although the stiffness is very high, the system maynot behave like an articulated rigid-body system without jointelasticity. This is because the robot becomes a singularlyperturbed system, which has a subsystem in addition to thelink dynamics like (4) and (5) owing to its finite stiffness[22].
Al(q)q+b(q, q)+g(q) = Γl (4)
ε2Γl = K(Am
−1(Γ−Γl)+Al(q)−1(b(q, q)+g(q)−Γl)) (5)
where Γl = K(θ −q), K = K/ε2 and 1/ε2 >> 1. Therefore,the robot may not operate as in (3).
Another way of controlling a flexible robot with highstiffness is using a flexible joint robot model and reducingthe effect of the motor dynamics. By choosing the motortorque as
Γ = AmA−1d Γd +(I−AmA−1
d )K(θ −q) (6)
where Ad is motor inertia scaling matrix and Γd is the desiredtorque calculated by PD controller. The inverse dynamicsbecomes
Γd = Ad θ +Al(q)q+b(q, q)+g(q). (7)
If Ad is sufficiently small, the robot is controlled as
Γd = Al(q)q+b(q, q)+g(q). (8)
The task Jacobian is defined as follows,
xq = Jq(q)q (9)
but motor angles θ are used instead of q in (9). A detailedexplanation is provided in [3], [4]. This method reflects thejoint elasticity to torque control, but the position trackingperformance of torque control in task space is not discussed.
The above methods focus on the control of link dy-namics, and precise control can be achieved by increasingthe feedback gains. However, we observed a bandwidthlimitation while using a feedback controller consisting ofthe link dynamics, despite the robot having high stiffness. Inparticular, such problems are the most serious in the anklejoints of the humanoid robot, where a heavy weight needsto be controlled.
III. CONTROL FRAMEWORK FORMULATION
In this approach, feedback control is constructed usingonly the motor part of the dynamics, and the link part of thedynamics is compensated with the feedforward terms. Fig. 1shows a block diagram of the proposed controller.
A. Joint Space Control
Before formulating the operational space control frame-work, we first present the joint space control. The proposedcontrol framework uses the motor inertia without the linkinertia when weighting the control input and the link partof the robot dynamics is compensated by the feedforwardterms. The control law is formed as follows.
Combining (1) and (2) results in
Γ = Amθ +Al(q)q+b(q, q)+g(q). (10)
Then we design a controller as
Γ = Amu+ Al ¨q+ b(q, ˙q)+ g(q). (11)
In our case, we choose control input u as
u = K′pe+K′ve, e = θd−θ . (12)
θd is the desired motor angle. K′p and K′v are the proportionaland derivative gain matrices. q is the estimated link side jointangle. Al , b, g are the estimated joint space inertia matrix,the Coriolis/centrifugal torque vector, and the gravitationaltorque vector, respectively.
To formulate the feedforward compensation, the link sideangles q are required. When the robot has external encoderswith a sufficient resolution, the measured q can be usedto compute the compensation torque for the link part ofthe dynamics. By contrast, if there are no proper sensorsavailable, the link side angles should be estimated. In our
Robot
Link Dynamics Estimation
( 𝐴𝑙 𝑞 + 𝑏 + 𝑔)
𝐴𝑚PD Controller
(𝐾𝑝′ 𝜃𝑑 − 𝜃 + 𝐾𝑣′( 𝜃𝑑 − 𝜃))
𝜃𝑑, 𝜃𝑑
𝜃, 𝜃
Γ𝑑
𝑢
Feed-back Control for Motor part of dynamics
Feed-forward Compensation for Link dynamics
Motor Inertia Weighting
Link Angle Estimation
𝑞
(a) Joint space control framework of the proposed controller. θ is the vector of the motor angles measured by the encoders. K′p andK′v are the gain matrices for the feedback control. Am is the motor inertia matrix. Al , b, g, and q are the estimated link inertia matrix,Coriolis/centrifugal torque, gravitational torque, and link side angles, respectively. Γd is the desired motor torque.
Robot
Link Dynamics Estimation
(Λ𝐽𝑚𝐴𝑚−1 𝐴𝑙 𝑞 + 𝜇 + 𝑝)
ΛPD Controller
(𝐾𝑝𝑥′ 𝑥𝑑 − 𝑥𝑚 + 𝐾𝑣𝑥′ ( 𝑥𝑑 − 𝑥𝑚))
𝑥𝑑, 𝑥𝑑
𝑥𝑚, 𝑥𝑚
Feed-back Control for Motor part of dynamics
Feed-forward Compensation for Link dynamics
Operational Space Inertia Weighting
𝐽𝑚𝑇
Γ𝑑
Link Angle Estimation
𝑢
𝑞
(b) Operational space control framework of the proposed controller. xm is the current task in the Cartesian coordinates calculated by themotor angles. K′px and K′vx are the gain matrices for the task feedback control. Λ is the operational space inertia matrix derived withoutthe link inertia matrix. µ and p are the estimated Coriolis/centrifugal force and gravitational force in the operational space, respectively.q is estimated in the same way as in the joint space control framework. The desired motor torque for the desired task force is computedby JT
mF .
Fig. 1: Block diagram of the proposed control framework. As the main point of this control framework, the feedback controlonly has motor side information in both the joint space control and operational space control.
case, we obtain q from (1). In addition, q can be representedas
q = θ +K−1(Amθ −Γ). (13)
The q obtained through (13) is q, and ˙q and ¨q are obtainedthrough a differentiation. The joint stiffness K is used withreference to the harmonic drive data sheet. The motor torqueΓ is estimated from the motor current and the accelerationof the motor angles are obtained through the differentiationof the encoders. A similar method for calculating the linkside angle is used in [23].
B. Operational Space Control
We define a new task Cartesian coordinates (xm) with θ
instead of q. The Jacobian is then defined as
xm = Jm(θ)θ . (14)
Using this Jacobian Jm, the operational space dynamics isderived as follows. By multiplying JmAm
−1 to both sides of(10) and applying Γ = JT
mF ,
Λxm +µ(q, q,θ , θ)+ p(q,θ)+ΛJmAm−1Al q = F. (15)
where
Λ = (JmAm−1Jm
T )−1, (16)
µ = Λ(JmAm−1b− Jmθ), (17)
p = ΛJmAm−1g. (18)
The vector F is the operational force. The matrix Λ is theoperational space inertia, which, unlike the operational spaceinertia matrix in rigid-body dynamics, consists of only themotor side information. The vector µ is the operational spaceCoriolis/centrifugal force, and the vector p is the operationalspace gravity force.
The control law for the task in the operational space isformed as
Fig. 2: Experiment results of joint space control. Joints 1 through 6 represent Hip Yaw, Hip Roll, Hip Pitch, Knee Pitch,Ankle Pitch, and Ankle Roll of the biped robot respectively. The blue solid line is the desired trajectory, the green dash-dottedline is the result of rigid-body dynamics controller, and the red dashed line is the result of the proposed controller.
ex = xd − xm and xd is the desired task. K′px and K′vx arethe proportional gain matrix and derivative gain matrix ofthe task controller. µ , p are the estimated operational spaceCoriolis/centrifugal force, and the gravitational force respec-tively. The desired torque input is generated as Γ = JT F .
This operational space control can also be considered asa combination of the feedback control of the motor part andthe feedforward compensation of the link part in the sameway as the joint space controller in (11) and (12). The feed-back control term Λ(K′pxex +K′vxex) is only affected by themotor information and the remaining terms are feedforwardcompensation torques which are computed from the linkpart of the dynamics. When we compute the feedforwardcompensation torques, the same q as the one in joint spacecontrol is used. This framework can be extended for floatingbase robots such as humanoids. Contact consistent controlframework [24] can be adapted to formulate the constraineddynamics in the operational space and build the control law.
IV. EXPERIMENT RESULTS
In this section, the tracking performance of the proposedcontroller is shown by comparing it with a controller using(3). The experiments were conducted with a 12-DoF bipedrobot [6]. When implementing the proposed controller, weapply a first-order low-pass filter (cut-off frequency of 20 Hz)to θ , ˙q, and ¨q to remove the noise. Overall, the experiment
Fig. 3: Joint space experiment using biped robot hanging ona fixed frame.
results showed that the proposed controller increases thegain limit that comes from the elasticity and communicationdelays described in our previous study [12], allowing formore precise control.
A. Joint Space Control
In the joint space control experiment, the controller using(3) and the proposed controller (11) were compared. Theformer one is named as rigid-body dynamics controller forconvenience and assumes q' θ because of the high stiffnessof the joints. Fig. 2 shows the results of the joint spacecontrol experiments. We hung the biped robot on a fixedframe and then controlled one 6-DoF leg (Fig.3). All sixjoints were controlled simultaneously to move 30 degreesduring a period of 1 second. Table I shows the loop gains,
Fig. 4: Operational space control experiment results. The blue solid line is the desired trajectory, the green dash-dottedline is the results of rigid-body dynamics controller, and the red dashed line is the results of the proposed controller. Eachdirection corresponds to the coordinates in Fig. 5.
TABLE II: Gains used in operational space experiments: Fixed Base.
Rigid-body dynamics Controller Proposed ControllerX Y Z Rx Ry Rz X Y Z Rx Ry Rz
which were the values set as large as possible. The mostnotable change when using the proposed controller is thewide range of control gains. When the rigid-body dynamicscontroller is used, the proportional gains are limited ata lower level (under 600 s−2). In contrast, the proposedcontroller can use proportional gains of over 10000 s−2. Evenconsidering the weighted inertia matrix, the gain used in eachjoint is much larger in the proposed method.
Comparing the error of the rigid-body dynamics controllerand that of the proposed controller, it is verified that thetracking performance was significantly improved. In mostcases, the errors of the rigid-body dynamics controller duringmotion are over 0.015 rad (except Joint 2), whereas the errorsof the proposed controller are under 0.002 rad. In particular,the tracking performance of joints 3, 4, and 5, which rotate inthe same axial direction and have a large coupling with eachother, were improved more than those of the other joints.
B. Task Space Control : Fixed Base
Fig. 4 shows the results of the operational space controlexperiments. The robot was also hung on a fixed frame, sim-ilar to the joint space experiment (Fig. 5). In this experiment,we controlled one leg in the Cartesian coordinates by usingthe operational space control framework. The controller tobe compared with the proposed operational space controlleris the one derived from (3). Because (3) assumes q ' θ
owing to the high stiffness of the joints, the operationalspace control framework is derived with respect to the rigid-body dynamics (3) and the Jacobian (J) that satisfies x = Jθ .
Fig. 5: Task space control experiment using a biped robothanging on a fixed frame.
The robot moved in all directions at the same time. It wascontrolled to move 6 cm in each direction for 1 second (atotal of 10.4 cm) while maintaining its orientation. Table IIshows the largest possible gains used in the experiments.
The usable control gain range was significantly expandedwhen the proposed method was used. The limits of theproportional gains were 30 times larger and the derivativegains were 6 to 7 times larger than those of the rigid-body dy-namics controller. With the rigid-body dynamics controller,the error of each Cartesian direction was approximately2 ∼ 3 mm, whereas the error of the proposed controllerwas under 0.4 mm. A notable result is the improvement inthe orientation control. In particular, rotations about the xand y axis were controlled much better when the proposedcontroller was used. The errors of Rx and Ry were improved
Fig. 6: The operational space control framework on a biped robot with the proposed controller. CoM was controlled withhigh gains while the foot was controlled with relatively low gains.
Fig. 7: CoM trajectory tracking errors and orientation control errors of biped robot motion. Each direction corresponds tothe direction shown in Fig. 6
TABLE III: Gains used in operational space experiments: Floating Base.
Rigid-body dynamics Controller Proposed ControllerX Y Z Rx Ry Rz X Y Z Rx Ry Rz
from approximately 0.02 rad to under 4×10−4 rad. Rz errordecreased from 5×10−4 rad to 3×10−4 rad.
C. Task Space Control : Floating Base
Fig. 6 shows the snapshots of the experiment using theoperational space control framework on a biped robot. Weexploited control gains for CoM control as much as possibleand applied relatively low control gains for the foot control.Table III shows gains used in this experiment. K′p com, K′d comare the proportional and derivative gain matrices of CoMand K′p f oot , K′d f oot are the proportional and derivative gainmatrices of the swing foot. Because of the weighting matrix
Λ, gains of CoM were much larger than those of the foot. Toconsider the upper-body weight of a humanoid, steel platesof 40kg were loaded onto the body part of the robot (the totalweight of the robot including the steel plates was 96kg). Therobot stepped on a wood stick (2.5 cm) during the steppingmotion. As shown in Fig. 7, the proposed controller reducedthe position error by half compared to the error when usingthe rigid dynamics controller. The robot was commandedto maintain its orientation during the motion and its errorsabout y, z axis were reduced significantly. There was animprovement of the tracking performance, but due to theheavy weight, the improvement of the tracking performance
was less than that of the fixed base case. In both fixedbase and floating base experiments, the magnitude of thetorque for compensating the link acceleration, the last termin (19), was much smaller than that of the torque computedby feedback control because the control gains were large.
V. CONCLUSIONSIn this study, we discussed how to utilize the inverse
dynamics of robots with elastic joints to improve the positiontracking performance of model-based torque control. In theproposed control framework, feedback control consists ofonly the motor part of the dynamics whereas the link dynam-ics is compensated by the feedforward terms. By implement-ing this approach to the operational space control framework,we can construct a task space feedback controller using onlythe motor side information. Unlike other approaches, thismethod can also be used for elastic robots with no link sideencoders or torque sensors.
The proposed control framework results in a significantimprovement in performance. Both joint space and oper-ational space experiments show that this framework canachieve a high position tracking performance. It also widensthe range of usable gain helping the robot move under variousstrategies. This is particularly advantageous in complicatedsituations such as humanoid locomotion where stiff CoMcontrol with compliant foot control is simultaneously re-quired, as shown in the final experiment.
There are some aspects to be closely examined. First,we need more analysis on why the range of usable gainis expanded. We will then analyze the change in the controlperformance according to the magnitude of the motor inertiaand joint stiffness.
ACKNOWLEDGMENTThis 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). Theauthors would like to thank Jinoh Lee for his valuablediscussion.
REFERENCES
[1] A. Albu-Schaffer, C. Ott, and G. Hirzinger, “A unified passivity-based control framework for position, torque and impedance control offlexible joint robots,” The international journal of robotics research,vol. 26, no. 1, pp. 23–39, 2007.
[2] G. Stepan and T. Insperger, “Stability of time-periodic and delayedsystemsa route to act-and-wait control,” Annual Reviews in Control,vol. 30, no. 2, pp. 159–168, 2006.
[3] C. Ott, A. Albu-Schaffer, A. Kugi, S. Stamigioli, and G. Hirzinger, “Apassivity based cartesian impedance controller for flexible joint robots-part i: Torque feedback and gravity compensation,” in Robotics andAutomation, 2004. Proceedings. ICRA’04. 2004 IEEE InternationalConference on, vol. 3. IEEE, 2004, pp. 2659–2665.
[4] A. Albu-Schaffer, C. Ott, and G. Hirzinger, “A passivity based carte-sian impedance controller for flexible joint robots-part ii: Full statefeedback, impedance design and experiments,” in IEEE InternationalConference on Robotics and Automation, 2004. Proceedings. ICRA’04.2004, vol. 3. IEEE, 2004, pp. 2666–2672.
[5] S. Katsura, Y. Matsumoto, and K. Ohnishi, “Modeling of forcesensing and validation of disturbance observer for force control,” IEEETransactions on Industrial Electronics, vol. 54, no. 1, pp. 530–538,2007.
[6] M. Schwartz, S. Hwang, Y. Lee, J. Won, S. Kim, and J. Park,“Aesthetic design and development of humanoid legged robot,” inHumanoid Robots (Humanoids), 2014 14th IEEE-RAS InternationalConference on. IEEE, 2014, pp. 13–19.
[7] J. Englsberger, A. Werner, C. Ott, B. Henze, M. A. Roa, G. Garofalo,R. Burger, A. Beyer, O. Eiberger, K. Schmid et al., “Overview of thetorque-controlled humanoid robot toro,” in Humanoid Robots (Hu-manoids), 2014 14th IEEE-RAS International Conference on. IEEE,2014, pp. 916–923.
[8] N. A. Radford, P. Strawser, K. Hambuchen, J. S. Mehling, W. K.Verdeyen, A. S. Donnan, J. Holley, J. Sanchez, V. Nguyen, L. Bridg-water et al., “Valkyrie: Nasa’s first bipedal humanoid robot,” Journalof Field Robotics, vol. 32, no. 3, pp. 397–419, 2015.
[9] N. G. Tsagarakis, D. G. Caldwell, F. Negrello, W. Choi, L. Baccelliere,V. Loc, J. Noorden, L. Muratore, A. Margan, A. Cardellino et al.,“Walk-man: A high-performance humanoid platform for realistic en-vironments,” Journal of Field Robotics, vol. 34, no. 7, pp. 1225–1259,2017.
[10] Y. Lee and J. Park, “Reactive bipedal walking method for torquecontrolled robot,” in 2018 IEEE International Conference on Roboticsand Automation (ICRA). IEEE, 2018, pp. 395–402.
[11] J. Englsberger, G. Mesesan, A. Werner, and C. Ott, “Torque-baseddynamic walking - a long way from simulation to experiment,” inRobotics and Automation (ICRA), 2018 IEEE International Conferenceon. IEEE, 2018, pp. 440–447.
[12] J. Jung, S. Hwang, Y. Lee, J. Sim, and J. Park, “Analysis of posi-tion tracking in torque control of humanoid robots considering jointelasticity and time delay,” in Humanoid Robotics (Humanoids), 2017IEEE-RAS 17th International Conference on. IEEE, 2017, pp. 515–521.
[13] O. Khatib, “A unified approach for motion and force control of robotmanipulators: The operational space formulation,” IEEE Journal onRobotics and Automation, vol. 3, no. 1, pp. 43–53, 1987.
[14] M. W. Spong, “Modeling and control of elastic joint robots,” Journalof dynamic systems, measurement, and control, vol. 109, no. 4, pp.310–318, 1987.
[15] A. De Luca and F. Flacco, “A pd-type regulator with exact gravitycancellation for robots with flexible joints,” in Robotics and Automa-tion (ICRA), 2011 IEEE International Conference on. IEEE, 2011,pp. 317–323.
[16] N. Paine, J. S. Mehling, J. Holley, N. A. Radford, G. Johnson, C.-L. Fok, and L. Sentis, “Actuator control for the nasa-jsc valkyriehumanoid robot: A decoupled dynamics approach for torque controlof series elastic robots,” Journal of Field Robotics, vol. 32, no. 3, pp.378–396, 2015.
[17] Q. Zhang and G. Liu, “Precise control of elastic joint robot using aninterconnection and damping assignment passivity-based approach,”IEEE/ASME Transactions on Mechatronics, vol. 21, no. 6, pp. 2728–2736, 2016.
[18] M. Jin, J. Lee, and N. G. Tsagarakis, “Model-free robust adaptivecontrol of humanoid robots with flexible joints,” IEEE Transactionson Industrial Electronics, vol. 64, no. 2, pp. 1706–1715, 2017.
[19] A. Dietrich, T. Wimbock, and A. Albu-Schaffer, “Dynamic whole-body mobile manipulation with a torque controlled humanoid robot viaimpedance control laws,” in 2011 IEEE/RSJ International Conferenceon Intelligent Robots and Systems. IEEE, 2011, pp. 3199–3206.
[20] A. Parmiggiani, G. Metta, and N. Tsagarakis, “The mechatronic designof the new legs of the icub robot,” in Humanoid Robots (Humanoids),2012 12th IEEE-RAS International Conference on. IEEE, 2012, pp.481–486.
[21] N. G. Tsagarakis, S. Morfey, G. M. Cerda, L. Zhibin, and D. G.Caldwell, “Compliant humanoid coman: Optimal joint stiffness tuningfor modal frequency control,” in Robotics and Automation (ICRA),2013 IEEE International Conference on. IEEE, 2013, pp. 673–678.
[22] A. De Luca and W. Book, “Robots with flexible elements,” in SpringerHandbook of Robotics. Springer, 2008, pp. 287–319.
[23] J. Kim, M. Kim, and J. Park, “Improvement of humanoid walkingcontrol by compensating actuator elasticity,” in Humanoid Robots(Humanoids), 2016 IEEE-RAS 16th International Conference on.IEEE, 2016, pp. 29–34.
[24] J. Park and O. Khatib, “Contact consistent control framework forhumanoid robots,” in Robotics and Automation, 2006. ICRA 2006.Proceedings 2006 IEEE International Conference on. IEEE, 2006,pp. 1963–1969.
