Development of Backdrivable Hydraulic Joint Mechanism for Knee Joint of Humanoid Robots

Transcription

1 2009 IEEE International Conference on Robotics and Automation Kobe International Conference Center Kobe, Japan, May 12-17, 2009 Development of Backdrivable Hydraulic Joint Mechanism for Knee Joint of Humanoid Robots Hiroshi Kaminaga, Junya Ono, Yusuke Nakashima, and Yoshihiko Nakamura Abstract Robots must have similar mechanical impedance characteristics to humans in order to make safe and efficient contact. This impedance requirement applies not only to the surface but also to the actuation mechanisms. The objective of this research is to develop inherently flexible actuator by realizing backdrivability. A class of hydraulic actuation called electrohydrostatic actuator was applied to knee joint in humanoid robots to satisfy flexibility and large torque output simultaneously. This paper explains the methodology of performance evaluation of actuators and design concept of joint mechanism. Mathematical model of electro-hydrostatic transmission is also presented. Evaluation of backdrivability, inertia modification control, and compliance control of developed mechanism are performed. I. INTRODUCTION In general, robots consist of hard material, such as metal, and often have joints controlled with high gain position servo. In contrast, humans and numerous objects around us are soft, or in other word, have low mechanical impedance. It is desirable for robots to have similar impedance characteristics to interact such soft materials. To design robots with low impedance, just having soft skin is not enough; need to have soft joints as well. The difficulty of humanoid robots is that just being soft is not enough. It must produce large torque when it is needed, such as stance phase of bided locomotion. There are several ways to control joint impedance. The method to control joint impedance by software is called impedance control [1]. Albu-Shäffer et al. [2] is one of the examples of the impedance control to upper torso robot. Advantage of this method is that the impedance can be configured just by modifying software. However, it has major disadvantage of being incapable of emulating impedance for input beyond its control frequency. Another approach for controlling joint impedance is hardware. The examples are SEA (Series Elastic Actuators [3] and pneumatic actuators [4], [5]. Their advantage is that they can produce low impedance in wide bandwidth. However, their control bandwidth is limited by mechanical resonance frequency, which are relatively low due to the compliant components. Backdrivability is yet another approach to the low impedance joint mechanism. Backdrivability would not only This work was supported partially by Grant-in-Aid for Scientific Research (No for Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists and partially by IRT Foundation to Support Man and Aging Society under Special Coordination Funds for Promoting Science and Technology from MEXT. H. Kaminaga, J. Ono, Y. Nakashima, and Y. Nakamura are with Department of Mechano-Informatics, The University of Tokyo, Hongo, Bunkyo-Ku, Tokyo , Japan {kaminaga, junya, nakasima, nakamura}@ynl.t.u-tokyo.ac.jp give solution to absorb impulsive input, but combined with impedance control, gives high controllability. CB [6] used hydraulic actuator with pressure control to give actuator level backdrivability to full size humanoid robot and realized flexibility and large torque output capability. In our previous work [7], we developed an anthropomorphic robot hand with servo motor driven HSTs (hydrostatic transmissions, thus EHAs (electro-hydrostatic actuators, to give backdrivability to whole actuator system of the hand. In this paper, approach toward inherently low impedance actuator via backdrivable EHA is applied for joints with larger output torque capacity, such as lower limbs of humanoid robots. Mechanical design concepts and considerations on hydraulic system are mentioned. Actuator performance evaluation method using robot dynamics (macro model and actuator model (micro model is proposed. A model of EHA that shows the symmetric nature is presented. Discussions on performance evaluation, backdrivability, and controller design are based upon this model. An EHA was designed with a performance necessary for walking existing robot. Actuator performance was evaluated and comparison with existing actuator was done. Using the design parameters decided with above method, a prototype knee joint was developed. Backdrivability of the developed prototype was evaluated. Control strategies for inertia modification and compliance was presented and tested on the prototype. II. DESIGN OF THE ACTUATOR A. Model of EHA and Backdrivability To understand the dynamics of the EHA, the system was simplified as shown in Fig. 1. Here, τ i are torque inputs and θ i are positions. Here, (i,ī {(p,m,(m,p} and p and m denote pump and hydraulic motor respectively. p i {1,2} are the oil pressure in chambers, q {p,m} l are internal leak flow rate, Q is the flow rate in the transmission line, k λ is the flow resistance in the tube. Putting the moment of inertia J i, equations of motion is written as follows: J i θi = ( kt i + kq( i p i 1 p i 2 τf( i θ i,p i 1 p i 2+τ i (1 Here, k i t is a coefficient of conversion from pressure to torque, and k i q is coefficient of conversion from leak flow (q i l to drag torque. Friction torque acting on the system (τi f are categorized into friction by the component proportional to δp i = p i 1 p i 2 and p =(p i 1 + p i 2/2. We call p a charge /09/$ IEEE 1577

2 pressure because this pressure is the initial supply pressure from charge pump. The friction model of [8] is assumed. τfm( i θ i,δp i = sgn( θ { i kc i + ( ks i kc i } e ( θ i/ θ s 2 δp +k i v θ i (2 τfs( i θ i, p = sgn( θ { i kcs i + ( kss i kcs i } e ( θ i/ θ s 2 ( p + ps0 i + kvs i θ i (3 and τ i f = τ i fm + τ i fs (4 Here, kc(s i are the Coulomb friction coefficients, ki s(s are maximum static friction coefficients, and kv(s i are viscous friction coefficients. p i s0 is the offset of the pressure that cause friction at 0 gauge pressure. Parameter θ s determines velocity region which the stiction friction is dominant. sgn( is signature function defined by (5. 1 x>0 sgn(x= 0 x =0 (5 1 x<0 τfm i are mainly caused by mechanical contacts as bearings, and τfs i are mainly caused by oil seals. Putting coefficient of internal leak flow per pressure kl i, relationship between chamber pressure is expressed as follows. p i = p i 1 p i 2 = k1 i θ i k2 i θ ī (6 k i {1,2} has the form of following equation, with the relationship between rotation speed and pumping flow rate kd i (. 1+k λ k1 i kīl kd i = kl i + kīl + k (7 λkl ikīl k i 2 = kīd k i l + kīl + k λk i l kīl Resulted equation of motion is as follows: (8 J i θi = k i 1k i 3 θ i + k i 2k i 3 θ ī τ i f( θ i, θ ī +τ i (9 k i 3 is given as follows. k i 3 = k i t + k i q (10 It is worthwhile to note that the presented model resembles that of flexible manipulators as in [9]. However in the case of flexible manipulators, coupling terms of link side equation and motor side equation are positions, where in the case of EHA, coupling terms are velocities. This is due to the existence of internal leakage of oil that makes the relationship between link position and motor position non one-to-one. Backdriving operation is a operation mode of an actuator that actuator output torque direction and actuator velocity direction are opposite. Putting actuator output torque τ out ane actuator speed θ out backdriving mode is expressed as: sgn (τ out θ out < 0 (11 Fig. 1. Conceptual Diagram of Hydrostatic Transmission Using the notation of the EHA model, this is written as: sgn (τ m θ m > 0 (12 Here, the signature is inverted because τ m = τ out.however, in the case of EHA sgn( θ m =sgn( θ p does not always hold, so there are two possibility of backdriving states: 1 Output backdrive: sgn (τ m θ m > 0 2 Total backdrive: sgn (τ m θ p > 0 Backdrivability is defined as a capability of actuator to be operated under backdriving condition. Backdrivability in EHA are defined as follows. Conditions can be derived from (9. 1 Output backdrivability: Applying τ m > 0 generates θ m > 0 τ m >kss m ( p + p m s0 2 Total backdrivability: Applying τ m > 0 generates θ p > 0 θ k m > p ss k2(k p p 3 kp s ( p + pp s0 > 0 B. Steady-State Output Torque and Speed of EHA Output characteristics of EHA at steady-state, or when θ i =0can be solved for actuator output speed and torque in closed form under the condition of θ p, θ m,p p > 0 and p m < 0, thus in forward driving condition. Positive direction ( θ p > 0 forward drive steady state output torque and speed are given as follows: [ ] [ ] 1 ([ ][ ] θm A1 0 B1 1 θp = τ m A 2 1 B 2 0 τ p [ k p + cs (p p s0 + p ] kcs m (p m (13 s0 + p where A 1 = (k p 3 + kp ck p 2 A 2 = ( k m 3 + k m c k m 1 k m v k m vs B 1 = (k p 3 + kp ck p 1 + kp v + k p vs B 2 = (k m 3 + k m c k m 2 Hence (13 gives the mapping function from electric motor characteristics to actuator characteristics. 1578

3 C. Performance Evaluation Method of Actuators Based on Robot Dynamics In most design process, performance requirement of actuator was selected using representing values, such as maximum torque and maximum speed. However, this type of selection does not necessarily reflect torque actually required during desired motion. To avoid shortage of performance, usually some safety factor on specification is selected by designers rule of thumb. However, such design strategy has huge risk because relation between torque and speed is not independent, sometimes even nonlinear; often leads to excessive specification or shortage of performance under certain movements. This problem becomes even more evident when the behavior of an actuator is complicated. This can be avoided by mapping robot dynamics (macro dynamics and detailed actuator dynamics (micro dynamics in same space to enable direct comparison. First, dynamics of robot, typically written as (14 with robot s generalized coordinate q, inertia matrix M, Coriolis force B, joint torque τ, and external forces F i. K i are the matrices that transform external force to joint torques. q includes all joint angles and position and attitude of base link of the robot. m M(q q + B(q, q= [ 0 τ ] + i=1 K i F i (14 By calculating inverse dynamics of the robot, sequence of joint torque is calculated for the desired motion. To have the direct comparison of macro and micro dynamics, a common space to be mapped must be decided. Since in most cases, actuator output performance is measured by speed - torque - power relationship, so it is natural to make comparison of the dynamics on the the plane of torque and speed. We put τ limit ( q for performance limitations of actuators and τ j ( q j (t for performance requirements for executing motion θ j (t. Characteristics of the actuator, in general, can be expressed as follows: f = f(x,ẋ,ẍ (15 Here, f is the output force/torque of the actuator and x is the position of the actuator. By estimating limits of f from system limitation, such as speed, heat, and strength of components, performance limitation line f limit given by (16 can be calculated. f limit (ẋ = inf f(ẋ (16 x,ẍ In reality, since actuator inertia is often sufficiently smaller than that of robot s. So steady state characteristics given by (13 is enough for evaluation. We can compare actuator performance limit and requirement by seeing if τ limit τ j (t for all t. τ limit is collection of torque limit curved calculated by (16 for all joints. It should be noted that there are several limitation lines to be considered for performance optimization; i.e. to avoid excessive performance. Fig. 2. System Schematic of Designed Electro-Hydrostatic Actuator Control D. Mechanical Design Knee joints are typical joints in humanoid that support large loads. For such joints with EHA to backdrive smoothly, they must keep designed clearance between components to avoid collisions between them under large stress. On the other hand, it must be light in order for them to be used on the walking robots. For the pump and hydraulic motor, combination of trochoid pump and vane motor is used as in [7]. To achieve high rigidity, transmission output is supported by cross roller bearings and preloaded in axial direction with angular bearings to eliminate the bearing gap. This also helps the vane motor to keep the clearance between vanes and the casing to avoid the collision and yet keep the clearance minium to enhance the volume efficiency of the EHA. Differential pressure gauge is located in between the vanes to measure the force acting on the actuator output. In the pressure range we use, silicon oil can be seen as incompressive medium. When the pump produces the differential pressure δp p, pressure of discharge port rises to p δp p while pressure of suction port drops to p 1 2 δp p. When the p 1 2 δp p drops below gas disengagement pressure or vaporizing pressure of the oil, bubbles are formed in the hydraulics and degrade system stiffness and output torque considerably. This phenomenon is known as cavitation. For silicon oil, almost 50% of dissolved air will disengage at the pressure of 0.05(MPa in absolute scale[10]. To avoid cavitation, minimum pressure of the oil must be kept sufficiently above atmospheric pressure. We attached pressure charge circuitry [11] to raise p so the pressure at the suction port stays above the atmospheric pressure even at the maximum output pressure of the pump. Note that this pressure does not directly affect the operation of the actuator. The system is driven by brushless DC motor with motor encoder, differential pressure sensor, and output position encoder. Fig. 3 shows the CAD image and outlook of the designed actuator. E. Performance Evaluation of the Designed Actuator As a design reference, we selected UT-θ2[12]. UT-θ2 have 20 D.O.F. in total, 6 in each leg and 4 in each arms. Its full height is 1500(mm and weighs 45(kg. Actuators consist of 1579

4 Fig. 3. Outlook of Designed Knee Joint Fig. 5. Torque - Rotational Speed Relationship of Knee Joint of UT-θ2 while Walking. Fig. 4. The External View of UT-θ2 DC motors and Harmonic Drives. Fig. 4 shows the outlook of UT-θ2. As a reference motion, forward walk with boundary condition relaxation [13] was used. Torque and speed of the knee joint was calculated for walk with step interval 0.8(sec and step length 0.3(m. For dynamics computation, dynamics library by Nakamura and Yamane [14] was used. Fig. 5 shows the relationship between torque and speed of the knee joint. From Fig. 5, it is obvious that the actuator of the UT-θ2 does not satisfy the performance required for this walking pattern. It happened because the actuator selection was done with ideal stall torque and ideal zero load speed of the motor. In reality, often there is large difference in ideal and actual performance curve. With proposed method, it is easy to see if the actual performance curve satisfies the requirement or not. Using the actuator dynamics described in previous section, the performance limitation of the designed actuator was evaluated. Performance limitation comes from motor specification, such as maximum continuous torque curve and maximum instantaneous torque curve. From the evaluation, it was confirmed that the designed actuator satisfies all operating points of the walking motion within the maximum instantaneous output. Also, majority of the points are within the continuous operating range. Few points are outside the continuous operating region, but since it happens only in short duration of the whole cycle so it would give enough time for the motor to cool down. Furthermore, losses regarding detail parameters like internal leakage and viscosity of oil are also considered, which were not in the case of original actuator design of UT-θ2. The designed actuator weigh 2(kg. The estimated output backdriving torque from the datasheet is 3.7 (Nm. The reduction ratio of the EHA is about 1:100. This actuator is comparable to CSG of Harmonic Drive Inc. for the size(diameter of EHA 130(mm vs. Harmonic Drive 135(mm, allowable torque (EHA 1988(Nm vs. H.D. 2700(Nm and weight (EHA 2.0(kg vs. H.D. 1.7(kg. Specified backdriving torque of CSG is 2.9(Nm for gear only, so assuming motor friction torque to be 45(mNm, backdriving torque including motor mould be around 7.4(Nm. Total backdriving torque of EHA was 14(Nm (explained in section IV, and it is expected to come from oil seal friction. We are expecting to reduce the friction by order of 2 by design improvement. III. CONTROL STRATEGY A. Modification of Pump Rotor Inertia Even if EHA was passively backdrivable, it is still desirable to enhance backdrivability for designing responsive impedance controller. Having small pump rotor inertia is desirable to reduce amount of reflected inertia and enhance backdrivability. In this section, dynamics of the pump rotor is modified by local feedback to reduce pump rotor inertia. The equation of motion of the trochoid pump in EHA is as shown in (9. By applying local feedback in similar manner to [2], dynamics of the pump becomes as follows. J p θp = k p 1 kp 3 θ p + k p 2 kp 3 θ m τ p f + u (17 Here, u is the intermediate control input and J p is the desired pump rotor inertia. The actual control input to the pump is 1580

5 given as follows. τ p = J p J p u + ( 1 J p J p k p 3 (p 1p p 2p (18 τ p f is scaled friction term given by following equation. τ p f = J p J p τ p f (19 B. Joint Level Compliance Control of EHA By eliminating θ p from (9, dynamics of the system is expressed as follows. J m θm + k λk m d km 3 +τ m τ m f + km 3 k p 3 θ m = km 3 1 k p 3 J p θ 1+k λ kl m p (20 ( τ pf + u Fig. 6. Result of Passive Backdriving Test When θ m and θ p is sufficiently small, the system becomes as follows. k λ k m d km 3 θ m = τ m τ m f + k m 3 k p 3 1 ( τ pf + u (21 With the feedback of (22, system has the stiffness of K. u = kp 3 k m 3 (1 + k λ kl m K ( θ m θm ref (22 Damping can be modified with the feedback of: u = kp 3 k m 3 (1 + k λ kl m K ( θ m θm ref ( kλ k m d + km 3 1+k λ kl m D θ m (23 IV. EXPERIMENTS A. Backdrivability and Inertia Modification Control Backdrivability of the system was tested. Torque was manually applied to the joint link of the actuator and the angular position of the joint link and pump motor angle was recorded simultaneously. Joint torque was measured by force gauge. Fig. 6 shows the test result for 9 different trials with different force and speed applied to the joint From this experiment, output backdriving torque was measured as 4 (Nm and total backdriving toque was measured as 14 (Nm. With same experiment setup, pump rotor inertia modification was applied to see the effect on backdrivability of the system. Fig. 8 shows the test result with different inertia scale α = J p / J p. From this figure, increasing α reduces backdriving toque of the pump significantly. Fig. 7. Result of Passive Backdriving Test with Origin Magnified to see output backdriving torque. B. Impedance Control Impedance control in (22 was implemented. The test was done by manually applying torque to the joint through the force gauge. The applied toque was measured with force gauge. Fig. 9 shows the result of the compliance control with different compliance value. By changing the paramter K and D, system behavior changes. It became clear that the designed EHA is an over damping system. A. Conclusions V. CONCLUSIONS AND FUTURE WORKS The objective of this paper was to present a design methodology of EHA actuators utilizing robot and actuator dynamics. Followings are result of this paper. 1 Symmetric dynamics model showing the nature of EHA was presented. Discussions on properties and control strategies on EHA were built upon the model. 1581

6 Fig. 8. Result of Active Backdriving Test with Inertia Modification Fig. 9. Result of Impedance Control. K=1:1.3:1.3 for dashed line: solid line: dotted dashed line. Dotted dashed line have damping enabled. 2 Method for integrating robot dynamics model (macro model and actuator model (micro model was presented. With this method, more realistic and precise evaluation of the actuator performance limit became possible. 3 Mechanical design of the knee joint was presented. The joint is supported by preloaded bearings to eliminate bearing gap to satisfy high stiffness, high efficiency, and low friction. Also, pressure charge circuitry was attached to avoid cavitation. 4 Performance evaluation was performed using presented evaluation method and dynamics model. The result show that all of the operating points of the reference motion were satisfied. 5 Control strategies of the actuator were presented. Control law for the pump rotor inertia modification and joint impedance control using pressure and position feedback were presented. 6 Experiments on backdrivability, pump rotor inertia modification, and impedande control were performed. Both pump rotor inertia modification and compliance control were implemented with simple pressure and position feedback. The tests were done under different parameters. All tests showed qualitatively correct results. B. Future Works System parameter identification is to be performed. Now, not all parameters used in impedance control are not connected to physical value. It will enable more quantitative design precess, controller stability analysis, and design of disturbance observer for friction compensation. Further analysis on characteristics of the actuator is necessary. It will make clear the limitation and structural issues of current design, which is necessary for the design enhancement. Fundamental restructuring of EHA to reduce oil seal friction is planned. This enahancement will enhance output backdrivability by factor of 50 to 100. REFERENCES [1] N. Hogan, Impedance Control: An Approach to Manipulation: Part I-III, Trans. of ASME J. Dyn. Sys. Meas. Ctrl, vol. 107, no. 1, pp. 1 23, [2] A. Albu-Shäffer, C. Ott, and G. Hirzinger, A Unified Passivity-based Control Framework for Position, Torque and Impedance Control of Flexible Joint Robots, The Int l J. of Robotics Research, vol. 26, no. 1, pp , [3] G. A. Pratt and M. M. Williamson, Series Elastic Actuators, in Proc. of IEEE/RSJ Int l Conf. on Intelligent Robots and Systems, vol. 1, 1995, pp [4] M. Wisse, Three additions to passive dynamic walking: actuation, and upper body, and 3D stability, Int l J. of Humanoid Robotics, vol. 2, no. 4, pp , [5] B. Verrelst, R. V. Ham, B. Vanderborght, F. Daerden, and D. Lefeber, The Pneumatic Biped LUCY Actuated with Pleated Pneumatic Artificial Muscles, Autonomous Robots, vol. 18, pp , [6] G. Cheng, S. H. Hyon, J. Morimoto, A. Ude, G. Colvin, W. Scroggin, and S. C. Jacobsen, CB: A Humanoid Research Platform for Exploring NeuroScience, in Proc. of 6th IEEE-RAS Int l Conf. on Humanoid Robots, 2006, pp [7] H. Kaminaga, T. Yamamoto, J. Ono, and Y. Nakamura, Anthropomorphic Robot Hand With Hydrostatic Actuators, in Proc. of 7th IEEE-RAS Int l Conf. on Humanoid Robots, 2007, in Print. [8] C. C. D. Wit, H. Olsson, K. J. Åström, and P. Lischinsky, A New Model for Control of Systems With Friction, IEEE Trans. on Automatic Control, vol. 40, no. 3, pp , [9] M. Spong, Modeling and Control of Elastic Joint Robots, Trans. of ASME J. Dyn. Sys. Meas. Ctrl, vol. 109, no. 4, pp , [10] Japan Society of Mechanical Engineering, Ed., Fluid Machinery, ser. JSME Mechanical Engineer s Handbook. Japan Society of Mechanical Engineering, 1986, vol. B.5, in Japanese. [11] J. Ono, H. Kaminaga, and Y. Nakamura, HST with Anti-Cavitation Mechanism for Miniature Robot Actuator, in Proc. of JSME Conf. on Robotics and Mechatronics, vol. DVD-ROM, 2008, pp. 1A1 B23, in Japanese. [12] Y. Yamamoto, M. Okada, and Y. Nakamura, Development of Double spherical shoulder joint for humanoid robots, in Proc. of the 21st Annual Conf. of the Robotics Society of Japan, vol. CD-ROM, 2003, p. 2A25, in Japanese. [13] T. Sugihara and Y. Nakamura, A Fast Online Gait Planning with Boundary Condition Relaxation for Humanoid Robots, in Proc. of IEEE Int l Conf. on Robotics and Automation, 2005, pp [14] Y. Nakamura and K. Yamane, Dynamics Computation of Structure- Varying Kinematic Chains and Its Application to Human Figures, IEEE Trans. on Robotics and Automation, vol. 16, no. 2, pp ,