Appendix A Physiological Model of the Elbow in MATLAB/Simulink

Transcription

1 Appendix A Physiological Model of the Elbow in MATLAB/Simulink This section contains a complete description of the implementation of the physiological model of the elbow joint in the MATLAB/Simulink environment. The highest level of the Simulink block model diagram is shown in Fig. A.1. The inputs to the model are from two MATLAB data files: data.mat, which contains the raw EMG signal data from the biceps and triceps muscles, and the motion data from the Polaris Spectra optical tracking system; and tuning.mat, which contains the tuned parameter values for a particular subject. The outputs of the Simulink model include the predicted movement from the physiological model, the actual movement measured by the tracking system and the error between the two values (used to determine RMSE). The submodel EMG Model contains the physiological model and is shown in Fig. A.2. The subsystems of the model have been grouped according to their function (in red), and their structure and purpose will be described in the remainder of this section. A.1 Subsystem A This subsystem performs the LEP of the raw EMG signals from the biceps and triceps muscles. The block model is shown in Fig. A.3 and it outputs the normalised activation level of both muscles. Springer International Publishing Switzerland 2016 S.(S.Q.) Xie, Advanced Robotics for Medical Rehabilitation, Springer Tracts in Advanced Robotics 108, DOI /

2 324 Appendix A: Physiological Model of the Elbow in MATLAB/Simulink Fig. A.1 Top level view of the elbow joint Simulink model Fig. A.2 Physiological model block diagram in Simulink

3 Appendix A: Physiological Model of the Elbow in MATLAB/Simulink 325 Fig. A.3 EMG signal processing subsystem A.2 Subsystems B and C: The Musculotendon Model These subsystems are identical with B for the biceps group and C for the triceps group. The parameters for the model are passed into the embedded MATLAB function from the Simulink model. The code for the function is shown below: 1 % This function is a generic method for calculating the force of a muscle 2 % given its activation level and physical properties. The function should 3 % be able to be used with any muscle given that the relevant properties 4 % have been defined. This includes the input arguments shown below. The 5 % output of the function is total muscle force. 6 7 % A activation level 8 % length current muscle length 9 % lengthmax maximum muscle length 10 % lengthopt optimal muscle length 11 % Fmax maximum muscle force 12 % Cpass tendon stiffness 13 % U_sc muscle size % Author: James Pau 16 % Date: 11 November

4 326 Appendix A: Physiological Model of the Elbow in MATLAB/Simulink 18 % UPDATE: 23 March % Implementing new musculotendon model (one that has been justified from 20 % the literature) function F = calculateforce(length,length_prev,a,fmax,lengthopt, vmax,cpass,sampling_time,u_sc,cdamp) % Find passive force due to the tissue elasticity 26 l_norm = length/lengthopt; 27 Fpass = exp(cpass*10*(l_norm 1))/exp(5)*Fmax; % Find force length relationship 30 F_leng = 1 ((l_norm 1)/0.5)^2; % Find force velocity relationship 33 v_norm = ((length length_prev)/sampling_time)/(0.5*vmax*(a+1)); % From Rosen 36 F_vel = /( exp( 1.409*sinh(3.2*v_norm+1.6))); F_visc = Fmax*Cdamp*v_norm; % Find total force 41 F = U_sc * Fmax * A * F_leng * F_vel + Fpass + F_visc; A.3 Subsystem D: Musculoskeletal Model 1 % This function calculates the moment generated at the elbow joint. The 2 % function is joint specific because it depends on the geometry of the 3 % joint and number of active muscle groups (in this case two). The input 4 % arguments to the function are the forces calculated using calculateforce 5 % and joint geometries. 6 7 % Author: James Pau 8 % Date: 13 November % UPDATE: 6 March % Changed inputs to suit use in Embedded MATLAB function % UPDATE: 8 March 2010

5 Appendix A: Physiological Model of the Elbow in MATLAB/Simulink 327

6 328 Appendix A: Physiological Model of the Elbow in MATLAB/Simulink A.4 Subsystem E: Kinematic Model Implementation of the kinematic model.

7 Appendix A: Physiological Model of the Elbow in MATLAB/Simulink 329

8 Appendix B Optimal 4R Mechanism Configurations Optimal 4R mechanism configurations were obtained for reaching 89 uniformly separated upper arm positions in the workspace during the optimisation process. The 89 optimal configurations for the left shoulder are presented in this appendix (Figs. B.1, B.2 and B.3; Table B.1). B.1 Motor Requirements In the selection of a specific motor actuator for each of the five joints in the upper limb exoskeleton, multiple factors are considered including performance, range of motion, weight, size, back-drivability and cost. The performance requirements are torque and velocity which are approximated by considering slightly extreme scenarios of human upper limb movement. The assumed velocity requirement for the shoulder and elbow joints is half a revolution per second or 30 rpm which is reasonably fast compared to movements Fig. B.1 The 89 uniformly separated points in the workspace of the left shoulder Springer International Publishing Switzerland 2016 S.(S.Q.) Xie, Advanced Robotics for Medical Rehabilitation, Springer Tracts in Advanced Robotics 108, DOI /

9 332 Appendix B: Optimal 4R Mechanism Configurations Fig. B.2 Numbers assigned to the 89 workspace points. The red points are not used in the GUI for operating the exoskeleton as they are considered dangerous for the user Fig. B.3 Default configuration of the 5-DOF exoskeleton

10 Appendix B: Optimal 4R Mechanism Configurations 333 Table B.1 Optimal joint angles for reaching the 89 workspace points Point th1 th2 th3 Point th1 th2 th

11 334 Appendix B: Optimal 4R Mechanism Configurations Table B.2 Actuator requirements Exoskeleton joint Requirements Velocity (rpm) Torque (Nm) in performing common tasks. This velocity requirement of 30 rpm is assumed for Joints 1, 2 and 5 for simplicity. The actual velocity requirements of Joints 1 and 2 in the 4R mechanism will be similar to this value for a shoulder rotation velocity of 30 rpm since the 4R mechanism is optimised to operate far away from singular configurations. Due to the smaller range of motion of Joints 3 and 4, the velocity requirement for these two joints is approximated as 20 rpm. The torque requirement considers the weight which the joint needs to support as well as the potential torque applied by the user s limb. The torques caused by inertia, Coriolis and centrifugal effects are small since the exoskeleton operates at low accelerations and velocities and therefore are not considered in the torque requirement calculations. Firstly, the joint torque caused by the weight of the exoskeleton components and the user s upper limb is considered. Since the weight of the exoskeleton upper limb system shifts during operation, simulation of a 3D CAD model of the exoskeleton in Creo Parametric is used to estimate the maximum torques that occur for each joint. These occur when the upper limb is lifted to a straight horizontal position. Next, the user-generated torque is approximated by applying a 25 N force at the hand position of the horizontally positioned limb. The load torque caused by the weight of motors and gearboxes chosen for distal joints are also considered in the calculation. The approximated requirements of joint velocity and torque are provided in Table B.2. Note that the estimated torque requirement of Joint 1 is smaller than all the other joints even though it is supporting more components. This is because the axis of rotation of Joint 1 is constantly tilted resulting in a reduced component of gravitational torque acting about the joint axis. In addition, the majority of the exoskeleton mass is near the vertical plane intersecting the axis of Joint 1 during operation; therefore, the horizontal perpendicular distance to the centre of mass is small (Fig. B.4). Joint 2 has a large estimated torque requirement because the axis of rotation of Joint 2 can enter a near horizontal position during operation which maximises the load carried by the joint. Furthermore, the exoskeleton structures supported by Joint 2 tend towards one side of the joint axis (Fig. B.5). This also applies to Joint 3 which also has a relatively large torque requirement. The torque and velocity output of a motor are inversely proportional; therefore, the actuator requirements in Table B.1 refer to the exertion of torque while operating at the specified velocity. A motor can exert a larger torque than the nominal

12 Appendix B: Optimal 4R Mechanism Configurations 335 Fig. B.4 The moment acting about Joint 1 axis due to gravitational forces is relatively small due to the large angle between the Joint 1 axis and the horizontal plane and the short perpendicular distance to the weight of the system

13 336 Appendix B: Optimal 4R Mechanism Configurations Fig. B.5 The moment acting about Joint 2 axis due to gravitational forces is relatively large due to the small angle between the Joint 2 axis and the horizontal plane and the large perpendicular distance to the weight of the system

14 Appendix B: Optimal 4R Mechanism Configurations 337 Table B.3 Specifications of the motor gearbox units Joint Motor gearbox combination Nominal gearbox output Maxon motor Gearbox (r:1) Speed h _ nom (rpm) Torque s nom (Nm) 1 EC 45 Flat 50 W 24 V 156:1 planetary EC 90 Flat 90 W 24 V 91:1 planetary EC 45 Flat 50 W 24 V 319:1 planetary EC 45 Flat 50 W 24 V 47:1 spur (18.6 a ) 3.0 (18.0 a ) 5 EC 45 Flat 50 W 24 V 156:1 planetary a Values after a further reduction of 6:1 by the Joint 4 rail mechanism torque if it is running slower than the nominal velocity and can run at a higher velocity than the nominal velocity if the exerted torque is lower. The exoskeleton is not expected to operate the motors at both the required torque and velocity simultaneously. Therefore, the torque and velocity requirements are an overestimation and are only used as a guideline during motor selection. The weight and especially size of the motors are also important to ensure the motors can fit in the limited space in the exoskeleton structure. B.2 Motor Selection The motors chosen to actuate each of the five exoskeleton joints are Maxon 24V brushless DC motors in combination with reduction gearboxes (Maxon Motor, Switzerland). These motors are back-drivable and have a special flat design which allows them to be mounted directly at the exoskeleton joints without causing interference problems. Maxon ESCON controllers are used to drive each motor. The motor gearbox combinations are carefully selected for each joint (see Table B.3) to ensure the required output torque and velocity specified in Table B.2 can be achieved. Due to the high torque and low speed requirement of the exoskeleton joints, a high reduction ratio is required from the gearboxes. The gearboxes used for Joints 1, 2, 3 and 5 are planetary gearboxes with three stages of reduction for Joints 1, 2 and 5 and four stages of reduction for Joint 3. A spur gearbox with three stage reduction is used for Joint 4. In addition, the weight and size of each motor gearbox unit are also considered in conjunction with the structural design of the exoskeleton to ensure they can be mounted directly at the exoskeleton joints without interfering with the movements of the exoskeleton. Interference analysis is done by simulating a 3D CAD model of the exoskeleton system in Creo Parametric. In particular, extra attention was made in selecting the motor gearbox units and designing the exoskeleton structure for Joints 3 and 4 since attaching the motor gearbox unit onto these joints most likely causes interference. A motor gearbox unit on Joint 3 can interfere with Link 1 (the link between Joint 1 and Joint 2) when it overlaps with Link 2 (the link between Joint 2 and Joint 3) as shown in Fig. B.6a. The structure of Links 1 and 2 is

15 338 Appendix B: Optimal 4R Mechanism Configurations Fig. B.6 Potential occurrences of mechanical interference a between Joint 3 motor and Link 1 and b between Joint 4 motor and Joint 5 carefully designed to ensure this interference does not occur. In the case for Joint 4, there is very limited space available for the Joint 4 motor gearbox unit since the motor gearbox unit of Joint 5 is mounted nearby (Fig. B.6b). Therefore, a small-sized spur gearbox is selected to combine with the motor of Joint 4 instead of the larger planetary gearbox used for the other motors. The nominal output torques and velocities of the chosen motor gearbox units in Table B.3 are all similar or better than the approximated requirements in Table B.2. The motor gearbox unit for Joint 3 has the largest underperformance when compared to the proposed requirements. Although a larger motor gearbox unit would improve the performance for Joint 3, the weight of the motor gearbox unit will increase which means the motors of Joints 1 and 2 will also need to be more powerful. Furthermore, since the motor gearbox unit of Joint 3 can interfere with Link 1 as discussed earlier, a larger motor will require the link designs to be modified in a way which will further increase the load on Joints 1 and 2. Also, the nominal output torque and velocity are the maximum output when the joint is generating both the torque and velocity output simultaneously. This will rarely occur in the exoskeleton since its usage will often involve low velocities when a high load is applied or vice versa. The chosen motor gearbox combination for Joint 3 is the best compromised solution after considering all the trade-offs.

16 Appendix B: Optimal 4R Mechanism Configurations 339 B.3 Sensors Angular position encoders pre-installed onto the motors are available, and this is a convenient option, however this cannot measure the angular displacement of the gearbox without knowing its position when the system starts up. Instead, absolute encoders are installed onto the exoskeleton joint to measure the angular displacement of the joint directly. This is achieved using AMS magnetic rotary encoders (AMS, Austria). These are high-resolution absolute encoders which measure a displacement angle relative to a pre-programmed zero position. Force measurement is achieved using custom-made force sensors comprising of RS strain gauges attached to a specially designed aluminium load structure. This is a low-cost alternative to the otherwise very expensive commercial force sensors. Each axis of force measurement uses a pair of strain gauges, one on each side of the load structure (Fig. B.7). A force applied to the end of the load structure causes it to bend resulting in compression of one strain gauge and extension of the other. This results in an increase in electrical resistance of the strain gauge under extension and a decrease in resistance for the strain gauge under compression. The use of a pair of Fig. B.7 Force sensing method. a Load specimen with no force applied. The Wheatstone bridge is balanced and the output voltage is zero. b Load specimen with force applied. The Wheatstone bridge is unbalanced and the output voltage is a function of the applied force

17 340 Appendix B: Optimal 4R Mechanism Configurations R Strain Gauge #1 V R Strain Gauge #2 Fig. B.8 A pair of strain gauges in a Wheatstone bridge in half-bridge configuration used in the force sensor strain gauges instead of one increases the sensitivity and compensates for changes in resistance due to temperature. The load structure is designed so that the strain gauges deform only when force is applied in the axis it is measuring and experience negligible deformation from force applied orthogonal to this axis. The pair of strain gauges is incorporated in a Wheatstone bridge circuit in half-bridge configuration (Fig. B.8). This circuit uses the change in resistance to generate an output voltage difference. This voltage difference is then amplified using an instrumentation amplifier and is converted to a noise-resistant digital signal using an I2C analog-to-digital converter (ADC).

18 Appendix C Supplementary Material on Robot Design Analysis This appendix contains additional information on the design analysis of the ankle rehabilitation robot. This includes the test conditions used to establish the maximum actuator force requirements. Additionally, it also provides a summary of additional results on workspace, singularity and force analysis for three linked designs with different parameters. C.1 Test Conditions for Force Analysis The actuator force requirement was computed for different designs based on application of different moments in different end effector configurations. Table C.1 provides a summary of configuration moment pairing considered in force analysis. Table C.1 Configuration moment pairings used in force analysis End effector orientation in XYZ Euler angles ( ) Applied moments along the Euler angle axes (Nm) h x h y h z s x s y s z ±100 ± ± ± ± ±100 ± ±100 ± ± ± ± ± ± ±40 Springer International Publishing Switzerland 2016 S.(S.Q.) Xie, Advanced Robotics for Medical Rehabilitation, Springer Tracts in Advanced Robotics 108, DOI /

19 342 Appendix C: Supplementary Material on Robot Design Analysis C.2 Simulation Parameters for MIMO Actuator Force Control Table C.2 provides a summary of the parameters used in the analysis and simulation of the MIMO actuator force controller. Table C.2 Parameters used in simulation and analysis of the MIMO actuator force controller Description Symbol Value/expression Motor rotational inertia J m 4: kg m 2 Motor viscous damping b m 2: Nm s/rad Motor torque constant K t 0:0365 Nm/A Belt stiffness k b N/m Belt viscous damping b b 5 N s/m Motor pulley radius r m 0:004 m Ball screw rotational inertia J s 1: kg m 2 Ball screw viscous damping b s Nm s/rad Ball screw pulley radius r s 0:02 m Ball screw transmission ratio G s 2p 0: rad=m Actuator rod mass m r 0:2kg Actuator rod viscous damping b r 0:05 N s/m Force sensor mass m f 0:2kg Force sensor stiffness k f N/m Force sensor damping b f 2500 N s/m Derivative gain K d 0:002 s Sampling time T 0:00075 s 2 Environmental inertia matrix at M e0 0: neutral position 4 0 0: kg m :04 Environmental damping matrix B e 0:1I 3 3 Nm s/rad 0:1I 3 3 þ 0:005K ankle Nm s/rad Environmental stiffness matrix K e 10I 3 3 Nm/rad K ankle Nm/rad Gain matrix in simulation K ½Um 0 v 0 diagð Þ½Um 0 v 0 ½Um 0 v 0 diagð Þ½Um 0 v 0 T T

20 Appendix C: Supplementary Material on Robot Design Analysis 343 C.3 Simulation Parameters for Assistance Adaptation Scheme This appendix provides the parameters used in the simulated case study to evaluate the efficacy of the different assistance adaptation scheme. Additionally, the simulation results and performance measures obtained from the passive-unconstrained, passive-constrained, active-unconstrained and active-constrained motion trials are also presented (Table C.3). Table C.3 Parameters used in the simulation case study for the proposed assistance adaptation scheme Description Symbol Value Rotational offset of environmental coordinates from robot/global coordinates U p 4 rad Stiffness along negative x-axis of environment coordinates k N/m Stiffness along positive x-axis of environment coordinates k N/m Stiffness along negative y-axis of environment coordinates k 3 25 N/m Stiffness along positive y-axis of environment coordinates k 4 25 N/m Zero potential boundary in negative X-direction of the u 01 0:1m environment coordinates Zero potential boundary in positive X-direction of the environment u 02 0:1m coordinates Zero potential boundary in negative Y-direction of the environment u 03 0:1m coordinates Zero potential boundary in positive Y-direction of the environment u 04 0:1m coordinates End effector mass m 1 kg Environmental viscous damping b env 5 N s/m Radius of circular reference trajectory r ref 0:7m Minor radius of elliptic kinematic constraint l a 0:2m Major radius of elliptic kinematic constraint l b 0:8m Rotational offset of semi-minor axis from positive x-axis of global frame u p 4 rad Stiffness of constraint boundary k con N/m Damping of constraint boundary b con 1000 N s/m