Proposal of an Electromagnetic Actuator for Prosthetic Knee Joints

APSAEM1 Journal of the Japan Society of Applied Electromagnetics and Mechanics Vol.1, No.3 (13) Regular Paper Proposal of an Electromagnetic Actuator for Prosthetic Knee Joints Noboru NIGUCHI *1, Katsuhiro HIRATA *1 and Ryota NAKAMURA *1 This paper proposes an electromagnetic actuator for a prosthetic knee joint, which mainly consists of a ball screw and brushless motor. Energy consumption can be reduced by using the regenerated energy in the stance phase during the swing phase. The operational principle is described and the required torque-speed curve (N-T curve) is calculated by using the inverse kinematics. Then, a brushless motor that satisfies the required N-T curve is designed. Finally, the N-T curve of the brushless motor is computed by using a coupled magnetic field - electrical circuit - control analysis. Keywords: electromagnetic actuator, brushless motor, ball screw, N-T curve, prosthetic knee joint. (Received: 31 May 1, Revised: 18 June 13) 1. Introduction Nowadays, leg amputees are increasing due to the increase in diabetics and conflicts all over the world. Therefore, the demand for prosthetic legs is increasing, and its enhancement is required so that leg amputees can walk smoothly. The most important capability required from a prosthetic leg is a smooth gait, which consists of a swing phase and a stance phase as shown in Fig. 1. In particular, the prosthetic knee joint (Fig. ) must not buckle in the stance phase and must have a smooth swing in the swing phase. Various prosthetic knee joints that use air or hydraulic power has been developed. These prosthetic knee joints enable the amputee to walk smoothly by using high-performance control technology [1-4]. However, they have drawbacks such as the battery must be charged every day and be changed every year, and thus the QQL (quality of life) of the amputee will be reduced. This paper proposes an electromagnetic actuator for a prosthetic knee joint, which mainly consists of a ball screw and brushless motor. Energy consumption can be reduced by using the regenerated energy in the stance phase during the swing phase. The operational principle is described and the required torque-speed curve (N-T curve) is calculated by using inverse kinematics. Then, a brushless motor that satisfies the required N-T curve is designed. Finally, the N-T curve of the brushless motor is computed by using a coupled magnetic field - electrical circuit - control analysis.. Structure and Operational Principle The prosthetic knee joint in this study mainly consists of a brushless motor and ball screw as shown in Fig. 3, and the ball screw converts the linear motion into a rotational motion and vice versa. A battery and control Correspondence: N. NIGUCHI, Department of Adaptive Machine Systems, Graduate School of Engineering, Osaka University, -1 Yamadaoka, Suita, Osaka 565-871, Japan email: noboru.niguchi@ams.eng.osaka-u.ac.jp *1 Osaka University circuit are also installed (omitted in Fig. 3). Fig. 4 shows a detailed description of the swing phase. In the swing phase, the brushless motor generates a braking torque from when the toe rises off the ground Swing phase Fig. 1. 1 period of a gait. Stance phase Socket Knee joint Foot Fig.. Conventional prosthetic leg. Brushless Ball screw DC motor Fig. 3. System with a brushless motor and ball screw. Braking torque Auxiliary torque Braking torque Auxiliary torque Fig. 4. Swing phase using the electromagnetic knee joint. 47

日本 AEM 学会誌 Vol. 1, No.3 (13) to when knee bending finishes. After knee bending, the knee extension starts and the brushless motor generates an auxiliary torque in order to assist the knee extension. Then, the brushless motor generates a braking torque until the latter period of the knee extension, where the brushless motor again generates an auxiliary torque. On the other hand, the brushless motor generates a braking torque throughout the stance phase. Therefore, the regenerated energy is stored in the capacitor and used in the initial period of the knee extension of the swing phase. 3. Required Specifications for the Brushless Motor 3.1 Required Specifications for the Electromagnetic Knee Joint Required specifications for the electromagnetic knee joint were calculated by applying the inverse kinematics to the measured data of 17 cm, 55 kg able-bodied person who walks at a relatively high speed. The mass and moment of inertia of the upper leg and lower leg were employed from data of an average young Japanese man and specifications of a conventional prosthetic leg. The relationship between the knee bending angle and knee joint torque required for the electromagnetic knee joint is shown in Fig. 5, and the kinematic model of the prosthetic knee joint in the swing phase is shown in Fig. 6. 3. Specifications of the Ball Screw When the ball screw is driven, the resisting force should be considered. The relationships between the rotational and linear motions of the ball screw are given in Eqs. (1) and (). T Fl (1) 1 dz () l dt where T is the torque, F is the thrust force, l is the lead of the ball screw, z is the linear position, and is the angular velocity. The rotational motion equation of the electromagnetic knee joint is shown in Eq. (3) using moment of inertia J. d J T (3) dt From Eqs. (1), () and (3), Eq. (4) can be obtained. d z F J (4) l dt Eq. (4) represents the inertial force, which is the resisting force to the linear motion. If the lead of the ball screw is short, the thrust due to the brushless motor will increase according to Eq. (1), but according to Eq. (4) the resisting force will also increase at the same time. In this study, after considering the trade-offs between thrust and resistance, a ball screw lead of 1 mm was employed. 3.3 N-T Characteristics of the Brushless Motor In this section, the required specifications shown in Fig. 5 will be converted into the N-T characteristics of the brushless motor. The thrust force of the ball screw can be obtained by dividing the knee joint torque by the moment arm length, and is shown in Eq. (5). F Tk / K sin 3 (5) where T k is the knee joint torque, K is the distance from the center of gravity of the lower leg to the brushless motor and 3 is the angle between the lower leg and brushless motor. The motor torque T m is shown in Eq. (6) using the thrust force of the ball screw. T m F l / (6) where is the efficiency of the ball screw. Knee joint torque (Nm) 6 4 - -4-6 initial period middle period latter period end 4 6 Bending angle (deg) start Fig. 5. Required characteristics for the electromagnetic knee joint. Brushless DC motor + Ball screw 3 K z Lower leg Hip joint 1 Upper leg R Center of gravity Knee joint Fig. 6. Dynamics model of the swing phase. 48

日本 AEM 学会誌 Vol. 1, No.3 (13).5.45.4.35.3.5..15.1.5 middle period 5 1 15 4. Brushless Motor Characteristics 4.1 Specifications of the Brushless Motor Considering ease of mounting, the outer diameter and axial length of the stator is fixed to 4 mm and 5 mm, respectively, and a brushless motor that satisfies the N-T characteristics shown in Figs. 7 and 8 was designed. Furthermore, in this design, the voltage which is stored in the capacitor during the stance phase shown in Fig. 9 has to be over 1 V, which is the battery voltage. The cross section diagram and major specifications of the designed brushless motor are shown in Fig. 1 and Table 1, respectively. Fig. 7. Required N-T characteristics in generating the braking torque...18.16.14.1.1.8.6.4. initial period latter period 4 6 8 1 Fig. 8. Required N-T characteristics in generating the auxiliary torque. Next, the linear velocity of the ball screw is shown in Eq. (7). dz d R (7) dt dt where R is the distance from the rotation axis of the knee to the rotation axis of the brushless motor, and is the knee joint angle. The linear velocity of the ball screw can also be expressed using the rotation speed of the brushless motor N, as is shown in Eq. (8). dz Nl (8) dt From Eqs. (7) and (8), the rotation speed of the brushless motor is shown in Eq. (9). R d N (9) l dt The N-T characteristics of the motor are given from Eqs. (6) and (9), and the driven and driving N-T characteristics are shown in Figs. 7 and 8, respectively. 14 1 1 8 6 4 - -4-6.1..3.4.5 Fig. 9. Rotation speed in the stance phase. Laminated silicon steel sheet (5A4) U W V Carbon steel (S45C) Ring-shaped permanent magnet (Br = 1.5 T) Fig. 1. Cross section diagram of the designed brushless motor. V W Table 1 Brushless motor specification Pole number 8 Slot number 6 Rotor outer diameter Air-gap length Coil Phase resistance U mm.5 mm.37mm 55 Y.594 49

日本 AEM 学会誌 Vol. 1, No.3 (13) 4. Fundamental Characteristics of the Brushless Motor In this section, the N-T characteristics of the designed brushless motor are calculated. First, the driven N-T characteristics are computed by using a coupled magnetic field - electrical circuit analysis as shown in Fig. 11. In this analysis, only switch 3 (SW 3) is closed and the rotation speed is increased up to 14 rpm. The current flows through the feedback diode that is parallelly connected to the FET, and is expended by a.1 resistor. The N-T characteristics are shown in Fig. 1. Little braking torque is generated below 5 rpm due to the diode, but we can observe that the target torque is achieved when the rotation speed is over 1 rpm. Next, the driving N-T characteristics are computed by using a coupled magnetic field - electrical circuit - control analysis as shown in Fig. 13. In this analysis, switches 1 and (SW 1 and SW ) are closed. A target rotation speed of 1 rpm, which is much larger than the unloaded rotation speed, is set, and the load is increased up to.45 Nm. The N-T characteristics are shown in Fig. 14, and it is clear that this curve satisfies the target curve. Finally, the stored voltage in the capacitor during the stance phase is calculated by using a coupled magnetic field - electrical circuit analysis. In this analysis, switch 4 (SW 4) is closed and the brushless motor is rotated according to Fig. 9, which is obtained by converting the measured data using inverse kinematics. The computed voltage is shown in Fig. 15, and it is observed that the maximum voltage stored in the capacitor is 1.5 V. Therefore, it is possible to use the stored voltage in the swing phase. The stored energy of the capacitor J c is shown in Eq. (1). 1 J c CV c (1) SW 1 1 V SW SW 3.1 F.1 SW 4 where C is the capacitance, and V c is the voltage stored in the capacitor. The stored energy is used in the knee extension initial period of the swing phase until it becomes equal to the battery voltage. Therefore,.5-J battery consumption can be reduced per gait cycle according to Eq. (1)..7.6.5.4.3..1 Designed brushless motor 5 1 15 middle period Fig. 1. N-T characteristics in generating the braking torque. PI control dq transformation Target I d and rotation speed Gate signal Inverse dq transformation Rotation speed Phase current FEM model PWM Inverter Fig. 13. Flowchart of the coupled analysis..5.45.4.35.3.5..15.1.5 initial period Designed brushless motor latter period Phase voltage 4 6 8 1 Fig. 14. N-T characteristics in generating the auxiliary torque. 5. N-T Characteristics during the Swing Phase 5.1 N-T Characteristics during Knee Bending FEM model Fig. 11. Coupled analysis model. The electromagnetic knee joint is required to operate on the N-T curve. Therefore, the brushless motor, which has an output that is larger than required, must be 41

日本 AEM 学会誌 Vol. 1, No.3 (13) 1 14 Capacitor voltage (V) 1 8 6 4.1..3.4.5 1 1 8 6 4 middle period.5.1.15 Fig. 15. Capacitor voltage in the stance phase. controlled so that the required rotation speed and torque can be obtained. First, the brushless motor is driven according to Fig. 16 during the knee bending, which is obtained by converting Fig. 7 into the time sequence values. Then, all of the switches except SW 3 are opened, and the braking torque is controlled by applying an on-off control to SW 3. The computed torque characteristics are shown in Fig. 17. The average torque is similar to the target torque. However, the torque ripple is dominant. The long period ripples are due to the cogging torque, which is shown in Fig. 18. On the other hand, the short period ripples are due to the on-off control. This is because the current does not flow in the coils when the SW 3 is open. 5. N-T Characteristics during Knee Extension Next, the auxiliary torque during the knee extension initial period is controlled. The load torque and target rotation speed are shown in Fig. 19, which is obtained by converting Fig. 8 to the time sequence values. SW 1 and SW are closed and the computed torque characteristics are shown in Fig.. The difference between the required and computed N-T characteristics at rotation speeds above 65 rpm is thought to be because of the errors in the current control. Next, the control method for the braking torque during the knee extension latter period is explained. The brushless motor is driven according to Fig. 16, and the braking torque is controlled in the same way as was done during knee bending. The computed torque characteristics are shown in Fig. 1. The torque ripple due to the cogging torque and on-off control is observed. However, the average torque shows a relatively good agreement with the target curve. Finally, the torque characteristics during the knee extension latter period can also be calculated in the same way that the knee extension initial period is calculated. Therefore, it will not be described in this paper...15.1.5 -.5 -.1 Fig. 16. Input rotation speed. Computed curve 1 3 4 Target curve Fig. 17. Torque characteristics during knee bending. Cogging torque (Nm).8.6.4. -. -.4 -.6 -.8 1 3 Rotation angle (deg) Fig. 18. Computed cogging torque waveform. The input energy during the knee extension initial period can be obtained as a summation of the copper loss and output power, and it is.4 J in this study. If the.5 J stored energy in the capacitor during the stance phase is used during the knee extension initial period, 1.5 % energy can be saved. 411

日本 AEM 学会誌 Vol. 1, No.3 (13) Load torque (Nm)..18.16.14.1.1.8.6.4. 1 Target rotation speed 9 8 7 6 5 4 3 Load torque 1..4.6.8 Fig. 19. Load torque and target rotation speed. Torque(Nm).5..15.1.5 -.5 -.1 Computed curve 4 6 8 1 Target curve characteristics approximately satisfy the required characteristics. However, the torque ripple due to the cogging torque and on-off control was dominant. The energy required during the knee extension initial period could be reduced by 1.5% by using the energy that was stored during the stance phase. In the future, the torque ripple will be reduced by enhancing the control method and reducing the cogging torque. Furthermore, a method to store the energy expended during the knee bending and knee extension middle period will also be investigated. References [1] J. Martin, A. Pollock and J. Hettinger, Microprocessor Lower Limb Prosthetics: Review of Current State of the Art, J. Prosthetics and Orthotics, Vol., pp. 183-193, 1. [] K. Tsukishiro, A New Prosthetic Kneejoint System C- Leg, J. Jpn. Soc. Mechanical Engineers, Vol. 17, No. 133, pp. 18-19, 4, (in Japanese). [3] J. Andrysek and G. Chau, An Electromechanical Swing- Phase-Controlled Prosthetic Knee Joint for Conversion of Physiological Energy to Electrical Energy: Feasibility Study, IEEE Trans. Biomedical Eng., Vol. 54, No. 1, pp. 76-83, 7. [4] J. Andrysek, S. Naumann and W. L. Cleghorn, Design and Quantitative Evaluation of a Stance-Phase Controlled Prosthetic Knee Joint for Children, IEEE Trans. Neural Sys. and Rehabilitation Eng., Vol. 13, No. 4, pp. 437-443, 5. Fig.. Torque curve in the knee extension initial period. Computed curve Target curve.6.5.4.3..1 -.1 5 1 15 Fig. 1. Torque curve in the knee extension middle period. 6. Conclusion An electromagnetic knee joint which mainly consists of a brushless motor and ball screw was proposed. The braking and auxiliary torques during the swing phase were computed by using a coupled magnetic field - electrical circuit - control analysis. The computed torque 41