IROS2019国际学术会议论文集2529

Operational Space Control Framework for Torque Controlled Humanoid Robots with Joint Elasticity Jaesug Jung1, Donghyeon Kim1and Jaeheung Park1,2 AbstractTorque 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 infl uenced 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 14 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 69 to take advantage of torque control. Compliant motion is benefi cial to humanoid locomotion on uneven terrain or in an unstructured environment. However, torque controlled humanoids still have diffi culty 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 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, park73snu.ac.kr. 2J. 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 1418. 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 diffi cult 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 diffi cult 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 fi xed 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 fl exible joints is described as follows 14. Am +K( q) = (1) 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) Macau, China, November 4-8, 2019 978-1-7281-4003-2/19/$31.00 2019 IEEE3063 Al(q) q+b(q, q)+g(q) = K( q)(2) Amis the diagonal motor inertia matrix, is the vector of the motor angles, and is the vector of the torques generated by the motors. The variables in (1) are expressed in the joint space, refl ecting the gear ratio for the motor inertia. K is a diagonal matrix with the stiffness of gears. q is the vector of the link side joint angles. Al(q), b(q, q), and g(q) denote the inertia 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 mentioned in Section I, its implementation to the humanoid robot is very diffi cult to achieve. When joint elasticity exists, the gain margin decreases as the weight of the load becomes heavier, consequently worsening the tracking performance 12. Humanoid robots are particularly susceptible to this problem at their joints near the distal end, such as the ankle joints, because of their heavy weight compared to the size of its ankle. This prevents increasing the feedback gains and reduces CoM tracking performance during locomotion. Robots are usually thought to be rigid unless joint elas- ticity is intentionally added like SEA, but joint elasticity always 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 elasticity is not intentionally added are usually controlled under the assumption that the effect of elasticity is small or negligible. One method for controlling a fl exible robot with high stiffness is to consider that the link side angles can be approximated as those of motors (i.e. q ). In this case, (1) and (2) become = (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 the operational space control framework in the original manner. However, although the stiffness is very high, the system may not behave like an articulated rigid-body system without joint elasticity. This is because the robot becomes a singularly perturbed system, which has a subsystem in addition to the link dynamics like (4) and (5) owing to its fi nite stiffness 22. Al(q) q+b(q, q)+g(q) = l(4) 2l= K(Am1(l)+Al(q)1(b(q, q)+g(q)l) (5) where l= K( q), K = K/2and 1/2 1. Therefore, the robot may not operate as in (3). Another way of controlling a fl exible robot with high stiffness is using a fl exible joint robot model and reducing the effect of the motor dynamics. By choosing the motor torque as = AmA1 d d+(IAmA1 d )K( q)(6) where Adis motor inertia scaling matrix and dis the desired torque calculated by PD controller. The inverse dynamics becomes d= Ad +Al(q) q+b(q, q)+g(q).(7) If Ad is suffi ciently small, the robot is controlled as d= Al(q) q+b(q, q)+g(q).(8) The task Jacobian is defi ned as follows, xq= Jq(q) q(9) but motor angles are used instead of q in (9). A detailed explanation is provided in 3, 4. This method refl ects the joint elasticity to torque control, but the position tracking performance 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 increasing the feedback gains. However, we observed a bandwidth limitation while using a feedback controller consisting of the link dynamics, despite the robot having high stiffness. In particular, such problems are the most serious in the ankle joints of the humanoid robot, where a heavy weight needs to be controlled. III. CONTROLFRAMEWORKFORMULATION In this approach, feedback control is constructed using only the motor part of the dynamics, and the link part of the dynamics is compensated with the feedforward terms. Fig. 1 shows a block diagram of the proposed controller. A. Joint Space Control Before formulating the operational space control frame- work, we fi rst present the joint space control. The proposed control framework uses the motor inertia without the link inertia when weighting the control input and the link part of the robot dynamics is compensated by the feedforward terms. The control law is formed as follows. Combining (1) and (2) results in = Am +Al(q) q+b(q, q)+g(q).(10) Then we design a controller as = Amu+Al q+b( q, q)+ g( q).(11) In our case, we choose control input u as u = K0pe+K0 v e, e = d.(12) dis the desired motor angle. K0pand K0 vare the proportional and derivative gain matrices. q is the estimated link side joint angle. Al, b, g are the estimated joint space inertia matrix, the Coriolis/centrifugal torque vector, and the gravitational torque vector, respectively. To formulate the feedforward compensation, the link side angles q are required. When the robot has external encoders with a suffi cient resolution, the measured q can be used to compute the compensation torque for the link part of the dynamics. By contrast, if there are no proper sensors available, the link side angles should be estimated. In our 3064 Robot Link Dynamics Estimation ( + + ) PD Controller ( + ( ) , , Feed-back Control for Motor part of dynamics Feed-forward Compensation for Link dynamics Motor Inertia Weighting Link Angle Estimation (a) Joint space control framework of the proposed controller. is the vector of the motor angles measured by the encoders. K0pand K0vare the gain matrices for the feedback control. Amis 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. dis the desired motor torque. Robot Link Dynamics Estimation ( 1 + + ) PD Controller ( + ( ) , , Feed-back Control for Motor part of dynamics Feed-forward Compensation for Link dynamics Operational Space Inertia Weighting Link Angle Estimation (b) Operational space control framework of the proposed controller. xmis the current task in the Cartesian coordinates calculated by the motor angles. K0pxand K0vxare the gain matrices for the task feedback control. is the operational space inertia matrix derived without the link inertia matrix. 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 computed by JT mF. Fig. 1: Block diagram of the proposed control framework. As the main point of this control framework, the feedback control only 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 represented as q = +K1(Am ).(13) The q obtained through (13) is q, and q and q are obtained through a differentiation. The joint stiffness K is used with reference to the harmonic drive data sheet. The motor torque is estimated from the motor current and the acceleration of the motor angles are obtained through the differentiation of the encoders. A similar method for calculating the link side angle is used in 23. B. Operational Space Control We defi ne a new task Cartesian coordinates (xm) with instead of q. The Jacobian is then defi ned as xm= Jm().(14) Using this Jacobian Jm, the operational space dynamics is derived as follows. By multiplying JmAm1to both sides of (10) and applying = JT mF, xm+(q, q,)+ p(q,)+JmAm1Al q = F.(15) where = (JmAm1JmT)1,(16) = (JmAm1bJm),(17) p = JmAm1g.(18) The vector F is the operational force. The matrix is the operational space inertia, which, unlike the operational space inertia matrix in rigid-body dynamics, consists of only the motor side information. The vector is the operational space Coriolis/centrifugal force, and the vector p is the operational space gravity force. The control law for the task in the operational space is formed as F = u+ + p+JmAm1Al q.(19) 3065 00.20.40.60.81 time(sec) -0.6 -0.4 -0.2 0 joint angle(rad) Joint 1 Desired trajectory Rigid-body dynamics Controller Proposed Controller 00.20.40.60.81 time(sec) -0.6 -0.4 -0.2 0 joint angle(rad) Joint 4 Desired trajectory Rigid-body dynamics Controller Proposed Controller 00.20.40.60.81 time(sec) 0 0.2 0.4 0.6 joint angle(rad) Joint 2 Desired trajectory Rigid-body dynamics Controller Proposed Controller 00.20.40.60.81 time(sec) -0.6 -0.4 -0.2 0 joint angle(rad) Joint 5 Desired trajectory Rigid-body dynamics Controller Proposed Controller 00.20.40.60.8 time(sec) -0.6 -0.4 -0.2 0 joint angle(rad) Joint 6 Desired trajectory Rigid-body dynamics Controller Proposed Controller 00.20.40.60.8 time(sec) 0 0.2 0.4 0.6 joint angle(rad) Joint 3 Desired trajectory Rigid-body dynamics Controller Proposed Controller 00.20.40.60.81 time(sec) -0.6 -0.4 -0.2 0 joint angle(rad) Joint 1 Desired trajectory Rigid-body dynamics Controller Proposed Controller 00.20.40.60.81 time(sec) -0.6 -0.4 -0.2 0 joint angle(rad) Joint 4 Desired trajectory Rigid-body dynamics Controller Proposed Controller 00.20.40.60.81 time(sec) 0 0.2 0.4 0.6 joint angle(rad) Joint 2 Desired trajectory Rigid-body dynamics Controller Proposed Controller 00.20.40.60.81 time(sec) -0.6 -0.4 -0.2 0 joint angle(rad) Joint 5 Desired trajectory Rigid-body dynamics Controller Proposed Controller 00.20.40.60.81 time(sec) -0.6 -0.4 -0.2 0 joint angle(rad) Joint 6 Desired trajectory Rigid-body dynamics Controller Proposed Controller 00.20.40.60.81 time(sec) 0 0.2 0.4 0.6 joint angle(rad) Joint 3 Desired trajectory Rigid-body dynamics Controller Proposed Controller 00.20.40.60.81 time(sec) -0.6 -0.4 -0.2 0 Joint 1 Desired trajectory Rigid-body dynamics Controller Proposed Controller 00.20.40.60.81 time(sec) -0.6 -0.4 -0.2 0 Joint 4 Desired trajectory Rigid-body dynamics Controller Proposed Controller 00.20.40.60.81 time(sec) 0 0.2 0.4 0.6 joint angle(rad) Joint 2 Desired trajectory Rigid-body dynamics Controller Proposed Controller 00.20.40.60.81 time(sec) -0.6 -0.4 -0.2 0 joint angle(rad) Joint 5 Desired trajectory Rigid-body dynamics Controller Proposed Controller 00.20.40.60.81 time(sec) -0.6 -0.4 -0.2 0 joint angle(rad) Joint 6 Desired trajectory Rigid-body dynamics Controller Proposed Controller 00.20.40.60.81 time(sec) 0 0.2 0.4 0.6 joint angle(rad) Joint 3 Desired trajectory Rigid-body dynamics Controller Proposed Controller 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-dotted line is the result of rigid-body dynamics controller, and the red dashed line is the result of the proposed controller. TABLE I: Gains used in joint space experiments. Rigid-body dynamics ControllerProposed Controller Joint 1Joint 2Joint 3Joint 4Joint 5Joint 6Joint 1Joint 2Joint 3Joint 4Joint 5Joint 6 K0p(s2)361400256552.25552.25552.25225001562522500156251562515625 K0v(s1)384032464646300250300250250250 where the control input is chosen as u = (K0pxex+K0 vx ex). (20) ex= xdxmand xdis the desired task. K0pxand K0 vx are the proportional gain matrix and derivative gain matrix of the task controller. , p are the estimated operational space Coriolis/centrifugal force, and the gravitational force respec- tively. The desired torque input is generated as = JTF. This operational space control can also be considered as a combination of the feedback control of the motor part and the feedforward compensation of the link part in the same way as the joint space controller in (11) and (12). The feed- back control term (K0pxex+K0 vx ex) is only affected by the motor information and the remaining terms are feedforward compensation torques which are computed from the link part of the dynamics. When we compute the feedforward compensation torques, the same q as the one in joint space control is used