Online Gain Switching Algorithm for Joint Position Control of a Hydraulic Humanoid Robot

Transcription

1 Online Gain Switching Algorithm for Joint Position Control of a Hydraulic Humanoid Robot Jung-Yup Kim *, Christopher G. Ateson *, Jessica K. Hodgins *, Darrin C. Bentivegna *,** and Sung Ju Cho * * Robotics Institute, Carnegie Mellon University, Pittsburgh, PA 53, USA ** ATR Computational Neuroscience Laboratories, Dept. of Humanoid Robotics and Computational Neuroscience, Kyoto, JAPAN {jungyup, cga and jh}@cs.cmu.edu, darrin@{cmu.edu or atr.jp}, sunjuc@andrew.cmu.edu Abstract This paper proposes a gain switching algorithm for joint position control of a hydraulic humanoid robot. Accurate position control of the lower body is one of the basic requirements for robust balance and waling control. Joint position control is more difficult for hydraulic robots than it is for electric robots because of a slower actuator time constant and the bac-drivability of hydraulic joints. Bacdrivability causes external forces and torques to have a large effect on the position of the joints. External ground reaction forces therefore prevent a simple proportional-derivative (PD) controller from realizing accurate joint position control. We propose a state feedbac controller for joint position control of the lower body, define three modes of state feedbac gains, and switch the gains according to the Zero Moment Point (ZMP) using linear interpolation. The performance of the algorithm is evaluated with a dynamic simulation of a hydraulic humanoid. Index Terms Gain switching, Hydraulic humanoid, Joint position control I. INTRODUCTION At present, most human-sized biped humanoid robots are powered by electric motors [-3]. As small electric motors produce relatively small torques, they are generally used together with reduction gears to increase the maximum torque. There are a number of advantages to this design: the hardware is simple and compact, the torque can be easily increased with reduction gears, the motors do not limit the angular position, and accurate joint position control can be realized because the reduction gear prevents the transfer of external torques to the electric motor. Electric robots have accurate position control but, force control is difficult because of the stiffness created by the reduction gear. The electric motor has a velocity limit due to bac EMF and friction of commutation system, so the maximum torque is limited as well. In particular, the baclashless harmonic reduction gears that are commonly used in humanoid robots have lower torque limits than general planetary reduction gears. Therefore, electric motors and reduction gears may not be suitable for an adult-sized biped humanoid robot that must be robust to disturbances and uneven terrain. An alternative approach is to use hydraulics for biped humanoid robots [4-8]. Hydraulic actuators for biped humanoid robots can produce a large torque, have efficient power distribution, easy force control and bacdrivability, and have a low rotational inertia due to the absence of a reduction gear. However, the hydraulics requires an oil compressor, maing a self-contained system difficult, and external oil lines generate disturbances. In addition, the rotational ranges of the joints are limited due to the linages required for linear actuators. Most importantly, position control is difficult because of bacdrivable joints and the large ground reaction forces seen during waling. We propose an online gain switching algorithm for accurate position control of a hydraulic humanoid robot. Some researchers have studied online gain switching of robot manipulators for robust and accurate position control [9,], but online gain switching for legged robots has not yet been explored. As a first step, we applied the algorithm to a simulation of the lower body of the Sarcos Primus System (Fig. ) because the control performance of the lower body is essential for biped waling. Joint dynamic models were derived from frequency response testing and simple mass damper models. We then designed a state feedbac controller based on a Linear Quadratic Regulator (LQR) as a joint position controller. Three gain modes were chosen, and the gains are linearly interpolated online according to the location of the ZMP. The algorithm was verified with a dynamic simulation which demonstrated that all joints of the lower body could trac the desired trajectories with good accuracy. Pitch Yaw Fig. Sarcos Primus System and simulator []. z Roll y x /7/$5. 7 IEEE 3 HUMANOIDS 7 Authorized licensed use limited to: Seoul National University of Technology. Downloaded on June 3, 9 at 6: from IEEE Xplore. Restrictions apply.

2 II. HYDRAULIC BIPED HUMANOID The target robot is a hydraulic humanoid robot developed by Sarcos (Fig. ). It is an adult-sized humanoid robot whose height and weight are.65 m and 9 g. There are 5 degrees of freedom. All actuators are hydraulic except for the eyes (electric) and fingers (pneumatic). The control architecture is distributed with communication occurring via Ethernet. The main computer communicates with the local joint controllers at Hz. Table I shows the specifications and dimensions of the Sarcos humanoid robot. TABLE I SPECIFICATION AND DIMENSIONS OF THE SARCOS HUMANOID ROBOT Eye Nec 3 Mouth Shoulder 3 DOF Elbow Wrist 3 Hand 6 Waist 3 Hip 3 Knee Anle 3 Weight 9 g including hydraulic lines Potentiometers and force sensors at all hydraulic joints, Sensors two 6-axis force/torque sensors on the soles of the feet, two IMUs in the head and body, and a stereo camera Control rate Hz main control rate, 5 Hz joint control rate Hydraulic actuator Max 3 psi, flow control servovalve Dimensions (m) Upper leg.3874 Shoulder to shoulder.3945 Lower leg.3875 Upper arm.577 Foot size.3 x. Lower arm.48 Hip to hip.778 Eye to eye.788 Hip to nec Nec to eye.36 III. JOINT POSITION CONTROL STRATEGY In this paper, our motion control strategy is based on joint position control. That is, all joints must follow desired trajectories accurately. For accurate joint position control, we propose a state feedbac controller and switching gain algorithm as a joint position control strategy. Currently, this strategy only considers the lower body of the robot because the upper body does not experience a large external force such as the ground reaction force and joint position control of the upper body is less important than that of the lower body for waling stability. A. Linear State Space Model of Joints To design a state feedbac controller, we need a linear state space model of each joint. However, it is hard to derive exact joint models because many joints are connected in series, and the motion of one joint affects the others. Therefore, we derived the joint models by using experimentally identified hydraulic actuator models and two inds of simple theoretical models: a one-lin mass-damper system, and a two-lin massdamper system. Fig. shows the two mathematical models: l is the lin length, m is the point mass, c is the damping coefficient, τ is the joint torque, and g is the gravitational acceleration. Fig. Two rigid body models We identified the hydraulic actuator dynamics using frequency response experiments for the joints in the leg. During the experiments, the robot s pelvis was fixed on a stand. A target joint was free and all other joints were fixed using PD control. We sent a sinusoidal valve command to the target joint, and recorded the joint torque, angular position and angular velocity. In addition, we used a valve command offset to avoid nonlinearity when the load is zero. The form of the model is & τ = a & θ + aτ + a3u : & θ is the angular velocity of the joint, τ is the joint torque, and u is the valve command (- 3 ~ +3, D/A units) to the flow control servo valve. The first term represents the rate of change of torque due to piston linear velocity, the second term represents the rate of change of torque due to piston leas, and the third term represents a rate of change of torque due to the valve opening. Furthermore, two of the coefficients of the model should be functions of the joint angle because the moment arm varies slightly with the angle. Without the moment arm, the hydraulic actuator dynamics is & = b & θ + b F b u () F + 3 where, F is the linear force produced by the actuator and b, b and b 3 are constant coefficients. If we multiply the variable moment arm L (θ ), to both sides of equation (), then Because of τ θ c l ( θ ) F & = L( θ ) b & θ + L( θ ) b F + L( θ b u () L ) 3 τ = L (θ ) F m One-lin mass-damper model g (optional) m l θ c Two-lin mass-damper model, the equation becomes & τ = ( θ ) b & θ + b τ + L( θ b u (3) L ) 3 m l Therefore, the first and third coefficients vary with the joint angle. However, in this paper, we assume that the moment arm is constant for simplicity. We computed the following models for the joints of the Sarcos humanoid robot through experiments: Anle joints: & τ = & θ 4.43τ +.44u Other joints: & τ = & θ 4.4τ +.97u (4) Fig. 3 shows the comparison of the Bode plot of the experimental data and the model as an example. τ θ c τ 4 Authorized licensed use limited to: Seoul National University of Technology. Downloaded on June 3, 9 at 6: from IEEE Xplore. Restrictions apply.

3 The linear state space model of the anle pitch joint is x & = Ax + Bu, & T x = [ θ θ τ ] (6) g c where A =, B = l ml ml In equation (5), the second and fifth rows of A are derived from the linearized rigid body dynamics of the two-lin massdamper model, & θ = M( θ) ( τ C( θ, θ& ) G( θ) ) at the equilibrium point, θ =. The properties of the rigid body models were provided by CAD data from Sarcos. B. Design of State Feedbac Controller We use the following state feedbac controller to perform joint position control for the lower body. For the dual joint sets: K = 3 3 For the single joints : 4 4 u = Kx + K r (7) , K ff ff = θ d, r = θ d u = Kx + ff r (8) K = [ 3 ], r = θ d Fig. 3 Comparison of the Bode plot of the experimental data and the model : Input : valve command, Output : joint torque By using the rigid body and hydraulic actuator dynamics, we can derive linear state space models of all joints of the lower body. The two-lin mass-damper model is applied to the hip pitch/nee pitch joints and the hip yaw/anle yaw joints, and the one-lin mass-damper model is applied to the other joints of the lower body. This is because the masses of the thigh and shan are similar, so it is more accurate to use the two-lin mass-damper models for the hip pitch/nee pitch and hip yaw/anle yaw joints. For example, the linear state space model of the hip/nee pitch joints is where, && θ θ A = && θ θ x & = Ax + Bu, T θ & θ τ θ & T x = θ ], u = u u ] (5) [ τ && θ & θ && θ & θ && θ τ 4.4 && θ τ && θ θ && θ θ && θ & θ && θ & θ [ && θ τ.97, B = && θ τ where, K is the state feedbac gain, K ff and ff are the feedforward gains for the tracing control, and r and r are the reference commands. In the state feedbac gain, there are three inds of gains: position, velocity, and torque. The position gain is for stiffness, the velocity gain is for damping, and the torque gain is for compliance. The torque gain is essential because low position gains cannot mae a joint compliant due to the actuator dynamics. To choose the state feedbac gains, we used a LQR design using simple models and then refined the gains by hand through a full rigid body dynamic simulation to compensate for the unmodeled dynamics. Because the leg of a waling robot alternates between ground contact and swing, we cannot expect good performance of joint position control with constant gains. More specifically, the ground contact changes the moment of inertia about each joint. Therefore, three gain modes for each leg were defined according to the contact state of the feet (Fig. 4). The low gain mode represents the swing leg in the single support phase because there is no external load. The high gain mode represents the supporting leg in single support phase. In this phase, the largest load is applied to the foot and therefore, the position gains of this mode should be the largest. The dynamic load of single support may be more than the weight of the robot because of dynamic effects. The middle gain mode is use for the supporting leg in the double support phase. In this case, the dynamic load is lower than the robot s weight but not zero. The position gains should be between the two other modes. An alternative to this approach would be to use a feedforward command without the gain switching. However, 5 Authorized licensed use limited to: Seoul National University of Technology. Downloaded on June 3, 9 at 6: from IEEE Xplore. Restrictions apply.

4 this approach requires an accurate measurement of the dynamic load and robot s posture in the inertial coordinate frame. Therefore we chose a gain switching approach for joint position control. The gains of all joints of the leg in the low gain mode were determined by defining suitable Q and R matrices for a LQR design, and then tuned by a tracing control of a full rigid body simulation. The Q and R matrices were selected based on the rise time, maximum overshoot, and settling time of a step response curve. For example, the initial values for Q and R were set as follows: For dual joint sets: Q = diag[ ], and R = diag [ ] (9) For single joints: [ 6 4 diag ] Q = and R = - () After the gains of the low gain mode were selected for all joints, we then adjusted them to provide good control in double and single support. While simulating up-down, and side to side motions, we increased the position gain until the tracing error was small. To reduce undesirable vibrations, we increased the velocity gain for low frequency vibrations, and decreased it for high frequency vibrations. If the rate of change of the torque is high, we increased the torque gains. Finally, the feedforward gains were tuned by measuring the steady state tracing error. In this manner, we could use simulated experiments to set the gains of the three modes. Table II shows the gains of the right anle pitch joint. C. Online Gain Switching Algorithm We designed an algorithm to smoothly switch between gain modes. We use ZMP, which is equal to the center of the ground reaction force on the ground, to calculate the dynamic load distribution between the legs. Fig. 5 shows the three gain switching boundaries. D is the lateral distance between the origin of the ZMP coordinate frame and the anle joint. The ZMP coordinate frame is placed on the center position between the two projected soles on the ground. The boundaries and 3 are placed.8d to the right and left of the origin, and the boundary is placed at the origin. Fig. 6 shows the schematic of the online gain switching algorithm. If the Y ZMP is to the right of boundary 3, the right leg is in the low gain mode and the left leg is in the high gain mode. If the Y ZMP is to the left of boundary, the right leg is in the high gain mode, and the left leg is in the low gain mode. If the Y ZMP is on boundary, both legs are in the middle gain mode. In the areas near boundaries, the gains are linearly interpolated to create a smooth transition. The gains change according to the Y ZMP not the X ZMP. Boundaries and 3 were determined through simulations with hand tuning. To prevent undesirable rapid gain change at the moment of floor contact, we used a low pass filter whose cutoff frequency is 5 Hz after the ZMP calculation. With this algorithm, the system can change the gains smoothly, easily, and effectively. Middle Gain Mode: Supporting leg in DSP High Gain Mode Middle Gain Low Gain Mode Z Y X SSP : Single support phase High Gain Mode : Supporting leg in SSP X DSP : Double support phase Fig. 4 Three gain modes Boundary Boundary D SSP Right Foot Fig. 5 Gain switching boundaries -.8D.8D Gain Mode Fig. 6 Schematic of online gain switching algorithm TABLE II GAINS OF RIGHT ANKLE JOINT IN EACH MODE High Gain Mode Middle Gain Mode Low Gain Mode K ff O IV. SIMULATION A. Simulator We performed dynamic simulations to test the performance of the proposed gain switching algorithm. The SL simulation and Real-Time Control Software Pacage [] and 3D CAD data were used to create a simulator that matched the physical DSP Z Left Foot OXYZ : ZMP Coord. Frame X ( front) o.8d SSP Y Boundary 3 Left Leg Y(left) Right Leg Low Gain Mode : Swing leg in SSP Y ZMP 6 Authorized licensed use limited to: Seoul National University of Technology. Downloaded on June 3, 9 at 6: from IEEE Xplore. Restrictions apply.

5 humanoid shown in Fig.. The ground contact model is a damped-spring static friction model. The hydraulic actuator model was implemented for all joints of the lower body and the state feedbac controllers designed in Section III were used. For the upper body, PD servo controllers were used. B. Joint Position Control Test without Contact Joint position control without contact was performed to test the low gain mode in simulation. The pelvis center was fixed in 3D space. A prescribed waling gait at sec/step,. m step length,.7 m sway amplitude, and.35 m foot lift were used to generate desired joint trajectories. Fig. 7 and Table III show the desired and actual joint trajectories of the right leg and the maximum and average tracing errors. The errors in the plots were magnified by a factor of for visualization. Average errors were calculated as follows: Des θ LAY θ LAY Error(x) E ave Des θ LHR θ LHR Error(x) Des θ LHP θ LHP Error(x) Des θ LHY θ LHY Error(x) Des θ LKP θ LKP Error(x) Des θ LAP θ LAP Error(x) Des θ LAR θ LAR Error(x) N err = θ () N i= Fig. 7 Simulation result of joint position control in low gain mode i TABLE III MAXIMUM AND AVERAGE TRACKING ERRORS IN THE AIR Joint Max. Tracing Error Avg. Tracing Error Left Hip Roll(LHR).64 rad.3 rad Left Hip Pitch(LHP).57 rad.45 rad Left Hip Yaw(LHY).4 rad.7 rad Left Knee Pitch(LKP).5 rad.4 rad Left Anle Yaw(LAY). rad.4 rad Left Anle Pitch(LAP).4 rad.5 rad Left Anle Roll(LAR).5 rad.8 rad C. Joint Position Control Test with Contact Experiments in double support were used to test the gains and the switching algorithm. Up/down motion was used to test the middle gains and side-to-side motion was used to test all three gain modes and switching between them. The pea to pea amplitudes are.8 m for up/down and. m for sideto-side. The frequency is.5 Hz for both motions. These values were determined by approximating the magnitudes and frequencies seen in human waling. Fig. 8 and Table IV show the desired and actual angles of the several joints and the maximum and average tracing errors during the up/down motion. The largest maximum error was.6 rad at the anle pitch joint. All average errors were less than.38 rad. This result shows that the gains for the middle region produce good performance. Next, we performed side-to-side motions with or without the switching algorithm (Fig. 9 and Table V). Without the switching algorithm, the joint angles cannot follow the desired angle well. Table V shows that with the switching algorithm, all joint angles accurately follow the desired angles. Even though Y ZMP moves quicly by. m, the tracing performance is good. The maximum tracing error and average error were.57 rad and.4 rad respectively. Therefore, all gains seem to be appropriate, and the proposed gain switching algorithm provides accurate position control Des θ LAP θ LAP. Error(x) Des θ LHP θ LHP Error(x) Des θ LKP θ LKP Error(x) Fig. 8 Simulation result of joint position control during up-down motion 7 Authorized licensed use limited to: Seoul National University of Technology. Downloaded on June 3, 9 at 6: from IEEE Xplore. Restrictions apply.

6 θ LHR θ LHP θ LHY θ LKP θ LAY θ LAP θ LAR Displacement [m] Desired angle Angle without gain switching Angle with gain switching Y ZMP Fig. 9 Comparison of simulation results of joint position control during side to side motion with or without gain switching TABLE IV MAXIMUM AND AVERAGE TRACKING ERRORS OF UP/DOWN MOTION Joint Max. Tracing Error Avg. Tracing Error Left Hip Pitch(LHP).56 rad.3 rad Left Knee Pitch(LKP).59 rad.38 rad Left Anle Pitch(LAP).6 rad.7 rad TABLE V MAXIMUM AND AVERAGE TRACKING ERRORS OF SIDE TO SIDE MOTION Joint Max. Tracing Error (rad) Avg. Tracing Error (rad) w/o GS with GS w/o GS with GS Left Hip Roll(LHR) Left Hip Pitch(LHP) Left Hip Yaw(LHY) Left Knee Pitch(LKP) Left Anle Yaw(LAY) Left Anle Pitch(LAP) Left Anle Roll(LAR) V. CONCLUSION This paper proposed a gain switching algorithm for joint position control of a hydraulic humanoid robot. Our simulation experiments demonstrated that it is possible to obtain accurate joint position control by switching gains based on the location of the ZMP. We plan to apply the control scheme to the Sarcos hydraulic humanoid robot, and test the robot s performance with similar experiments. ACKNOWLEDGMENT This material is based upon wor supported in part by the National Science Foundation under grants CNS-449, ECS-35383, and EEC REFERENCES [] Y. Saagami, R. Watanabe, C. Aoyama, S. Matsunaga, N. Higai, and K. Fujimura, The intelligent ASIMO: System Overview and Integration, in Proc. IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, pp ,. [] K. Aachi, K. Kaneo, N. Kanehira, S. Ota, G. Miyamori, M. Hirata, S. Kajita, and F. Kanehiro, Development of Humanoid Robot HRP-3P, in Proc. IEEE-RAS Int. Conf. on Humanoid Robotics, pp. 5-55, 5. [3] J. Y. Kim, I. W. Par, J. Lee, M. S. Kim, B. K. Cho, and J. H. Oh, System Design and Dynamic Waling of Humanoid Robot KHR-, in Proc. IEEE Int. Conf. on Robotics and Automation, pp , 5. [4] S. O. Anderson, M. Wisse, C. G. Ateson, J. K. Hodgins, G. J. Zeglin, and B. Moyer, Powered Biped Based on Passive Dynamic Principles, in Proc. IEEE-RAS Int. Conf. on Humanoid Robotics, pp. -6, 5. [5] S. O. Anderson, C. G. Ateson, and J. K. Hodgins, Coordinating Feet in Bipedal Balance, in Proc. IEEE-RAS Int. Conf. on Humanoid Robotics, pp.-5, 6. [6] S. H. Hyon, and Gordon Cheng, Passivity-Based Full-Body Force Control for Humanoids and Application to Dynamic Balancing and Locomotion, in Proc. IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, pp , 6. [7] G. Cheng, S. H. Hyon, J. Morimoto, A. Ude, G. Colvin, and W. Scroggin, CB: A Humanoid Research Platform for Exploring NeuroScience, in Proc. IEEE-RAS Int. Conf. on Humanoid Robotics, pp.8-87, 6. [8] S. H. Hyon, and G. Cheng, Gravity Compensation and Full-Body Balancing for Humanoid Robots, in Proc. IEEE-RAS Int. Conf. on Humanoid Robotics, pp.4-, 6. [9] M. A. Jarrah, and O. M. Al-Jarrah, Position Control of a Robot Manipulation Using Continuous Gain Scheduling, in Proc. IEEE Int. Conf. on Robotics and Automation, pp. 7-75, 999. [] B. Paijmans, W. Symens, H. V. Brussel, and J. Swevers, Gain- Scheduling Control for Mechanics Systems with Position Dependent Dynamics, in Proc. Int. Conf. on Noise and Vibration Engineering, 93-5, 6. [] S. Schaal, The SL Simulation and Real-Time Control Software Pacage, unpublished. 8 Authorized licensed use limited to: Seoul National University of Technology. Downloaded on June 3, 9 at 6: from IEEE Xplore. Restrictions apply.