Hiroshi Kaminaga 1, Satoshi Otsuki 1, and Yoshihiko Nakamura 1

Transcription

1 th IEEE-RAS International Conference on Humanoid Robots (Humanoids). October 15-17, Atlanta, GA Design of an Ankle-Knee Joint System of a Humanoid Robot with a Linear Electro-Hydrostatic Actuator Driven Parallel Ankle Mechanism and Redundant Biarticular Actuators* Hiroshi Kaminaga 1, Satoshi Otsuki 1, and Yoshihiko Nakamura 1 Abstract Force sensitivity of robots enhances adaptability and dexterity of manipulation. We can hypothesize that this is also true for legged mechanisms since locomotion fundamentally is a manipulation of the center of mass. Legged mechanisms require high resistance to impacts. Recent studies revealed the efficacy of hydraulic actuator in legged robots, but backdrivability was lost due to high resistance in servo valves. Electro-Hydrostatic Actuators (EHAs) use no servo valves and acquired backdrivability and high efficiency. However, it required high rated power in all actuators when they are used in serial manner. In this paper, we propose to design 3DOF ankle with parallel mechanism, 1DOF knee, and one biarticular actuator connecting ankle and knee to provide redundant actuation in order to enhance total power capacity of the leg. Design concept of the biarticular actuator and the design optimization method of the redundantly actuated legged mechanism is discussed. I. INTRODUCTION It is widely acknowledged that the force sensitivity of manipulator or upper body of a humanoid robot greatly enhances manipulation robustness and physical interaction capability under environment with unknown disturbances. It implies importance of force sensitivity in legged mechanisms to enhance locomotion stability, since locomotion fundamentally is a manipulation of the center of mass. Impact is another issue in legged locomotion. Impact due to leg touch down is inevitable. Also, for biped robots, there are cases that falling down is inevitable. It is mandatory for robots to survive even under this extreme condition if we want to discuss about practicality of these robots. Force sensitivity and impact load on actuator are concerns for robots using gear driven actuators, which usually have low strength and large transmission friction that lead to loss of backdrivability. There are force sensing legged mechanisms developed as [1], [2]. DLR Biped [1] is a typical example of legged mechanism with torque sensors. DLR Biped constitutes of a pair of DLR Light Weight Robot-III. COMAN [2] is a humanoid that constitute of Series elastic actuators (SEAs), which are widely used to make robot systems backdrivable. Recent studies revealed the efficacy of hydraulic actuator in legged robots[3], [4], [5]. The main motivation was to overcome the issues of power capacity and impact resistance *This work was supported by Grant-in-Aid for Young Scientists (B) (No ) of the Japan Society for the Promotion of Science. 1 H. Kaminaga, S. Otsuki, and Y. Nakamura are with Department of Mechano-Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Hongo, Bunkyo-Ku, Tokyo, , Japan kaminaga@ynl.t.u-tokyo.ac.jp that were concerns of gear driven robot actuators. However, conventional hydraulic actuators were not force sensitive from its nature due to the large friction in servo valves. Electro-Hydrostatic Actuator (EHA) gives alternative design choice to SEAs and servo valve type hydraulic actuators for backdrivable application. EHAs are displacement type hydraulic systems that control hydraulic motors with pump rotation[6] or pump displacement[7]. EHAs have advantages of hydraulic actuators that they have high strength. EHAs can also estimate output torque by measuring hydraulic pressure acting on hydraulic motors. On the other hand, EHAs lose load-sharing feature of servo valve controlled hydraulics, which have centralized pump. Our aim is to develop a force sensitive legged mechanism that is capable of handling extreme disturbance. We adopt backdrivable EHA to 3 degree of freedom (DOF) ankle mechanism. It is important to use same actuator as much as possible from implementation point of view since in conventional serial mechanism, ratio between pitch (largest) axis and yaw axis (minimum) power is over 6-times. To enable such design, load distributions over multiple actuators are necessary. We use parallel mechanism with linear hydraulic actuators to share the load between actuators. In order to enhance power to weight ratio, we propose to use redundant actuation with biarticular structure. Concept of a biarticular EHA acting on knee and ankle joint is proposed. The structural design is formulated as an optimization problem. Analysis using human motion captured data, enhancement on power capacity and load distribution were shown. The optimization was performed for stepping (or walking in place), since locomotion is one of the fundamental task that consumes large amount of power. II. LINEAR ELECTRO-HYDROSTATIC ACTUATORS An Electro-Hydrostatic Actuator (EHA) is a type of electric driven hydraulic actuation system. Unlike conventional hydraulic servo system that use servo valves for resistance control, EHA use control of pump in order to control hydraulic motor. Usually a pump and a hydraulic motor are used in pair. EHA can be seen as a replacement of a gear transmission with a hydrostatic transmission. Previously, we proposed designs of EHAs that use vane motors. In this research, we chose to use cylinder as output for ankle actuator as shown in Fig. 1, with following reasons. 1) Output force and stroke can be selected independently /13/$ IEEE 384

2 Fig. 2. Biomimetic Biarticular Muscle Fig. 1. Configuration of Electro-Hydrostatic Actuators Fig. 3. Hydraulic Biarticular Actuator 2) Low friction contact seal can be used between cylinder and piston to minimize internal leakage 3) Parallel linkage actuation to distribute load over multiple actuators is possible with small space occupation For item 1, this is large advantage against a double vane motor since in double vane motor, moving range was limited due to its structure. It limited the choice of surface area and moving range. With cylinders, we have more design freedom. For item 2, since vane motors have more complicated structure around the vane, with square angle at vane corner, it was difficult to use low friction seals. However, the clearance and surface treatment must be properly chosen in order to minimize friction. For item 3, use of parallel linkage is advantageous in reducing the weight for joint with multiple DOF with significantly different power requirement on axes. Although use of cylinder has advantages above, there are disadvantages as well. Use of cylinder in rotary joint induces joint torque nonlinearity. Parallel linkage mechanism to form multiple DOF joint with cylinders generally have small workspace due to singularities. For this reason, we will use cylinder parallel mechanism for ankle, which has relatively small motion range, and vane motor for the knee joint, which has large motion range. III. HYDRAULIC BIARTICULAR ACTUATOR Biarticular muscles are muscles that act on two joints. Humans have biarticular muscles as gastronemius and rectus femoris. Gastronemius acts on the knee joint and the ankle joint. Rectus femoris acts on the hip joint and the knee joint. There are studies on biarticular muscle in biomechanics and bio-mimetic robotics as in [8]. Main interests of the biarticular muscle were synergy, passive compliance ellipsoid, and motion coordination. In this study, we take biarticular muscles, purely as redundant actuators with differential nature to maximize output power of the legged systems while minimizing the actuator weight. Biarticular muscle connects three links with one muscle. Fig. 2 shows simplified structure of biarticular muscle in animals. Biomimetic approach tries to mimic this structure. Tendon drive [9], Pneumatic actuator [10] or differential gear transmission are used for realizing this structure. Those structures can lead to limitation in controllability due to the elasticity of the tendons, or increase in size and weight due to complicated differential mechanism. Furthermore, in practice, realizing structure in Fig. 2 with rigid mechanism makes it difficult to obtain large range of motion. The principle of biarticular muscle is that it gives torque to two joints while leaving 1 DOF underactuated. This is easily realized with hydraulic actuator as shown in Fig. 3. The Fig. 3 shows the example of using linear actuators on both sides, but it is also possible to realize same effect with arbitrary combination of rotary and linear actuator. This method enhances design flexibility utilizing design methodology of conventional joint-wise actuation method, without introducing additional elasticity and use of complicated mechanisms. Another advantage of this method is that the actuators can be connected in reverse way that is shown in Fig. 3. It means, there is a choice of applying torque to two joints in same direction or opposite direction. It is not impossible to do this with biomimetic structure, but the mechanism will become very complicated. Also, in quasi-static state, the proposed mechanism will behave as ideal torque splitting device. Fig. 4 shows the hydraulic schematic of the 2 DOF joint mechanism with conventional monoarticular EHA and biarticular EHA. In the schematic, hydraulic motors are used as actuators, but cylinders can also be used without loss of generality. In this paper, we propose to use a biarticular actuator that acts on knee and ankle pitch direction. The motivation comes from the result of inverse dynamics of human subject while stepping shown in Fig. 5. From this figure, we can see that Fig. 5. Example of Joint Torque Profile of a Human while Stepping 385

3 Fig. 4. Schematic of Electro-Hydrostatic Biarticular Actuator the peak of torque of knee and ankle come in anti-phase, which implies the possibility of power sharing between ankle and knee joint. IV. DESIGN OPTIMIZATION OF REDUNDANT DRIVEN KNEE-ANKLE JOINT SYSTEMS The proposed joint drive mechanism is redundantly actuated. To design actuation system such structure, load distribution must be decided. It requires some kind of design optimization. We approached this problem as power consumption minimization by optimizing torque split ratio of biarticular actuator between joints. To start with, we formulated the problem as optimization in ankle pitch joint and knee joint. Let τk and τ a be joint torque at knee and ankle respectively. Let τ mi be i-th monoarticular actuator (acts on single joint). τ bi are torque generated by biarticular actuator. Torque splitting can be express as (1) and (2) with ratio a. τ m1 + τ b1 = τ k (1) τ m2 + τ b2 = τ a (2) τ b2 = aτ b1 (3) The power consumption at each actuator P mi and P bi are given as follows, where θ k and θ a are knee and ankle joint speed respectively. The actuator power have upper bound L mi and L b. P m1 = τ m1 θk L m1 (4) P m2 = τ m2 θa L m2 (5) ( P b = τ b θk + a θ ) a L b (6) The cost function in minimizing power consumption is given as follows. Z 1 = P1 2 + P2 2 + P3 2 (7) We put another cost function to suppress sudden change in output torque by (8) where τ mi is the torque of previous frame and the time between frames. ( ) 2 ( ) 2 τ m1 τ m1 τ m2 τ m2 Z 2 = + (8) τ = ( τ m1 τ m2 τ b ) T The equality constraints in (1) and (2) are written in the form of (9). A eq τ = b eq (9) ( ) ( ) τk A eq =, b 0 1 α eq = Inequality constraints given in (4) to (6) are written in the form of (10). Ax b (10) θ k 0 0 L m1 A = 0 θa 0, b = L m2 0 0 θk + α θ a L b The cost function is given as sum of (7) and (8) with a weight β. Z = Z 1 + βz 2 = 1 2 τ T Hτ + f T τ (11) θ k 2 + β 0 0 H = 2 0 θ2 a + β 0 (12) ( θ k + α θ a ) 2 β τ f = 2 m1 β τ 2 m2 (13) 0 The dynamics of biarticular EHA resembles that of conventional EHA[11]. J i θi = k i3 p i τ if ( θ i,p i, p)+τ i (14) The subscript i can be {1, 21, 22} denoting pump and each hydraulic motor. τ if are the friction torques (or forces) and p i are differential pressure acting on each component. p is the mean pressure of the hydraulic system given by: p = 1 n p i (15) n Differential pressure are calculated in forms of (17) to (18), which are natural expansion of conventional EHA. i p 1 = k 14 θ1 k 15 θ21 k 16 θ22 (16) p 21 = k 214 θ21 + k 215 θ22 k 216 θ1 (17) p 22 = k 224 θ22 k 225 θ1 + k 226 θ21 (18) τ a 386

4 TABLE I REQUIRED OUTPUT FOR WALKING EHA type Monoarticular Biarticular Joint Knee Ankle Knee Ankle Maximum Torque (Nm) Maximum Speed (RPM) Fig. 7. Ankle Structure and Actuator Placement Fig. 6. Torque-Speed characteristics of Monoarticular Ankle EHA: Solid line shows maximum continuous output while dashed line shows ideal maximum output. Parameter x i1, y i1 x i2, y i2 r p p i q i TABLE II OPTIMIZATION PARAMETERS Description Cylinder end position of the connecting rod Ankle end position of the connecting rod Cylinder radius Pressure Flow rate Sequential quadratic programming was used as an optimizer. The optimization result is given in Table I. Fig. 6 illustrates output of ankle monoarticular actuator. From the result, the output requirement exceeds continuous operation range for short time. From this observation, we came to idea of using parallel actuation for ankle joint to increase power output. It is interesting to note that the α became negative as the result of optimization. It means the coupling of the human and the optimized result is opposite. It may be implying that the for the flat foot walking, the optimal structure may be different from animals, where the structure is the result of evolution, in which animals walks on tip toe. V. DESIGN OPTIMIZATION OF PARALLEL DRIVEN 3DOF ANKLE JOINT To overcome the lack of power in ankle joint, we design parallel driven 3DOF ankle joint with a passive spherical joint shown in Fig. 7. Parallel driving of actuators can distribute load to multiple actuators. This structure also has an advantage that the constraint force acting on the joint is supported by the spherical joint. Since the constraint force does not have to be supported with actuators, we can minimize the size and weight of the actuators. Since we have freedom in choosing the design parameters, we chose to optimize parameters shown in Table II. z i1 and z i2 were fixed to a constant from implementation issue, where i shows the index of the cylinder. We used human motion analysis from inverse dynamics of motion capture data. We used stepping data to optimize actuator placement. The cost function f we used is as follows: f(φ) = N F N a { w 2 i + αqi 2 + βg i k=1 i=1 +γ (i,j)(w i w j ) 2 } (19) w i = p i q i (20) {( ) } p 2 g i = max i p 2 + q2 i 0 q0 2 1, 0 (21) φ is the optimization parameter vector consists of parameters listed in Table II. First term in (19) gives the cost for power. Second term gives cost for flow rate calculated with product of rpπ 2 and actuator speed that leads to viscous friction loss. Third term is the penalty term to restrict actuator operation within the permissible range specified by electric motor torque-speed and heat dissipation characteristics. Fourth term distributes actuator load. α, β, γ are weights. N a is the number of actuators and we used 3 in this paper. However, we can use larger number than the number of degree of freedom to optimize actuator placement for redundant actuation. We gave bounds to φ: actuator mainly work in pulling direction to avoid buckling of the connecting rods; maximum pressure becomes lower than maximum pressure axis can hold. Sequential quadratic programming was used as an optimizer. The dynamics of linear EHA is given in identical form to a rotary EHA [11]. { J1 θ1 = k 11 k 13 θ1 + k 12 k 13 θ2 (22) J 2 θ2 = k 22 k 23 θ2 + k 21 k 23 θ1 The (first) subscript 1 denotes pump side parameter and 2 denotes cylinder side parameter. J i are inertial parameter. In case of pump, J 1 is the moment of inertia. In case of 387

5 Fig. 8. Optimized Torque-Speed Relations of Pumps in Ankle Actuators During Stepping Fig. 9. Sequential Optimization Procedure TABLE III PARAMETERS OF THE CYLINDERS Cylinder Parameter Foot (mm) Shank (mm) Pistonφ (mm) x y x y Cylinder Cylinder Cylinder cylinder, J 2 is the driving mass. θ 1 has the unit of radians and θ 2 has the unit of meters. We assume the gap between piston and cylinder being 55μm, piston thickness being 20 mm and viscosity of the oil being 100cSt driven with 200W motor. Fig. 8 shows the optimized result. Blue line shows the maximum continuous output of the motor. We can see the whole movement is possible within the continuous region. Table III shows optimized parameter. VI. DESIGN OPTIMIZATION OF REDUNDANT DRIVEN KNEE-ANKLE JOINT SYSTEMS WITH 3DOF ANKLE Since there is redundancy in selection of monoarticular and biarticular actuators, optimization procedure needs extension. We chose to optimize biarticular actuator design and monoarticular actuator design sequentially to minimize optimization dimension. The optimization sequence is shown in Fig. 9. We first optimize actuator placement with same technique as in previous section and use it as initial configuration of ankle joint actuator placement. Details of each steps are described below. 1) First, we optimize biarticular actuator placement, size, and amount of torque to be supported at time of peak power of ankle joint (See Fig. 10). Torque distribution ratio between monoarticular actuator and biarticular actuator was decided for knee joint. The reason for this choice are as follows. Peak power of ankle was chosen since power requirement of ankle joint is highest. Torque distribution was performed at knee joint since ankle joint is posture dependent where knee joint use vane motor, which has uniform torque characteristics over the range of motion. Cost function has identical form as (19) to (21) with extension to biarticular Fig. 10. Torque Distribution Optimization at Peak Power structure. Additional constraint of torque as in (1) to (3) and power as in (6). 2) Based on actuator placement and size calculated, torque distribution between monoarticular and biarticular actuators are optimized for all frames(see Fig. 11). In this step, torque of biarticular actuator at knee and ankle joint are calculated. 3) To take in the effect of biarticular actuator in to account of monoarticular actuators in ankle joint, actuator placement and size are optimized for monoarticular actuators using cost function in (19) to (21). We used stepping motion as in previous section for reference motion. However, there is no guarantee that the steps above will converge. To see the convergence of optimization, torque-speed relations of ankle joint for different iteration are shown in Fig. 12. From this figure, the reduction in average power consumption compared to 1DOF ankle case is observed, which shows the efficacy of use of 3DOF parallel mechanism for the ankle in load distribution among Fig. 11. Torque Distribution Optimization in All Frames 388

6 TABLE V AVERAGE PUMP POWER Average (W) Iteration Pump Pump Pump Pump Pump Total TABLE VI COMPARISON OF ANKLE PEAK POWER (Biarticular) Serial Mechanism Proposed Mechanism sequential optimization of the proposed mechanism, average power consumption over all actuators were reduced by 6%. Difference in power consumption between actuators were reduced by 50%. Compared to serial mechanism, difference in power between actuators was reduced by 91%. From these results efficacy of the proposed mechanism and optimization formulation was confirmed. Fig. 12. The line shows maximum continuous output while the dots show outputs of EHAs on ankle joint during stepping. Top shows 2nd iteration and bottom shows 7th iteration TABLE IV PARAMETERS OF THE CYLINDERS Cylinder Parameter Foot (mm)] Shank (mm) Pistonφ (mm) x y x y Cylinder Cylinder Cylinder Cylinder actuators. Optimized parameters are shown in Table IV. Cylinder 4 is the cylinder of biarticular actuator. Average and standard deviation of the power consumption in different steps are shown in Table V. From the result, we can see power reduction with iteration. The efficacy of proposed mechanism can be seen in Table VI. The ratio between motor with maximum power and minimum power was reduced by 91%. VII. CONCLUSIONS The objective of this paper was to propose a design methodology of redundantly driven ankle-knee joint system with biarticular actuator and 3DOF parallel ankle mechanism. We proposed concept of biarticular actuator using EHA. The power distribution between monoarticular actuator and biarticular actuator was formulated as optimization problem that consider properties of EHA. We selected walking as the design reference, Human motion was analyzed to calculate output speed and torque of the ankle. From the REFERENCES [1] C. Ott, O. Eiberger, J. Englsberger, M. A. Roa, and A. Albu-Schäffer, Hardware and Control Concept for an Experimental Bipedal Robot with Joint Torque Sensors, Journal of Robotics Society of Japan, vol. 30, no. 4, pp , [2] F. Moro, N. Tsagarakis, and D. Caldwell, A Human-like Walking for the Compliant Humanoid COMAN based on CoM Trajectory Reconstruction from Kinematic Motion Primitives, in Proc. of 12th IEEE-RAS Int l Conf. on Humanoid Robots, 2011, pp [3] 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 [4] C. Semini, N. G. Tsagarakis, B. Vanderborght, Y. S. Yang, and D. G. Caldwell, HyQ - Hydraulically Actuated Quadruped Robot: Hopping Leg Prototype, in IEEE/RAS-EMBS Int l. Conf. on Biomedical Robotics and Biomechatronics, 2008, pp [5] G. Nelson, A. Saunders, N. Neville, B. Swilling, J. Boundaryk, D. Billings, C. Lee, R. Playter, and M. Raibert, PETMAN : A Humanoid Robot for Testing Chemical Protective Clothing, J. of Robotics Society of Japan, vol. 40, no. 4, pp , [6] 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, pp [7] S. Alfayad, F. B. Ouezdou, F. Namoun, and G. Cheng, High performance integrated electro-hydraulic actuator for robotics Part I: Principle, prototype design and first experiments, Sensors and Actuators A: Physical, vol. 169, no. 10, pp , [8] M. Kumamoto, T. Oshima, and T. Yamamoto, Control properties induced by the existence of antagonistic pairs of bi-articular muscles - mechanical engineering model analyses, Human Movement Science, vol. 13, pp , [9] K. Radkhah, C. Maufroy, M. Maus, D. Scholz, A. Seyfarth, and O. von Stryk, Concept and design of the biobiped1 robot for human-like walking and running, Int l J. of Humanoid Robotics, vol. 8, no. 3, pp , [10] K. Hosoda, Y. Sakaguchi, H. Takayama, and T. Takuma, Pneumaticdriven jumping robot with anthropomorphic muscular skeleton structure, Autonomous Robots, vol. 28, no. 3, pp , [11] H. Kaminaga, J. Ono, Y. Nakashima, and Y. Nakamura, Development of Backdrivable Hydraulic Joint Mechanism for Knee Joint of Humanoid Robots, in Proc. of IEEE Int l Conf. on Robotics and Automations, 2009, pp