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## Mathematical model

In order to demonstrate the validity of the impedance control method, the swing motions of different-arm-mass golfers should be obtained so that the motions can be compared with those from the proposed golf swing robot using the impedance control method. Therefore, a dynamic model developed in Chapter 3 is used to emulate the swing motions of different-arm-mass golfers under a given input torque of the shoulder joint. It should be noticed that the impedance of the wrist joint (wrist action) is also considered in this model. The wrist impedance function can be simply written as

Where c is the viscous damping constant of the wrist joint; τ(f) is the wrist retarding torque due to the impedance of the wrist joint.

Since professional golfers such as legendary Bobby Jones turned the wrist joint freely and felt that the golf club freewheeled at the latter stage of the downswing, the value of c used here is relatively small as compared to that from Milne and Davis. Here, it is assumed that the value of c is equivalent to that of the golf swing robot. So the equations of motion of golf swing are given by.

Where

c is the wrist viscosity coefficient.

## Impedance control design

The dynamic equation of a mechanical system is always expressed as

Where F is the external force; M, C and K are denoted as the inertia, viscosity and stiffness, respectively. These parameters are called as mechanical impedance in the work of Hogan [23]. In this study, a golfer’s arm as a mechanical system is investigated. Since it has been found that a golfer’s hands move in a circular arc about the shoulder joint during the golf swing, the golfer’s arm is assumed to be a rigid body and the stretch reflex of the arm muscle is neglected. The fact that the swing motions obtained from the numerical simulation using the rigid arm link agree well with those from the swing photographs of professional golfers have assured the above assumption [1-7]. Therefore, the dynamic equation of the golfer’s arm can be given by

Where the viscosity and stiff of the golfer’s arm is neglected and the moment of inertia of the arm about the shoulder joint, M, is defined as the mechanical impedance. The virtual system representing the dynamic model of a golfer’s arm and the robot system expressing the dynamic model of a robot’s arm are shown in Figure 5-1 and Figure 5-2, respectively.

The equations of motion of the virtual and robot systems are written as Eq. (5-5) and Eq. (5-6).

Where the subscripts h and r denote the golfer and robot, respectively; fg( α ) = -Fg l1 cos α ; Fg is the gravitational force of the arm; F is the reaction force from the club to arm, and the direction is perpendicular to the arm; N is the reaction torque from the club to arm; Here, we assume L1h = L1r.

In our control method, the dynamic parameters J1h and J1r in Eq. (5-5) and Eq. (5-6) are defined as the mechanical impedance. With the various arm masses for the golfer and robot, the mechanical impedance J1h and J1r are varied. As a result, the swing motion of the robot is not the same as that of the golfer, even if the input torques of the shoulder  joints are equal. In order to realize the dynamic swing motion of the virtual system, the following control algorithm is proposed for the robot.

According to the Euler method, angular acceleration of the arm can approximate to the following expressions.

Where Δt is the sampling time; M is an integer; n n, αα  are the angular velocity and acceleration in the nth sampling period.

Substituting Eq. (5-7) into Eq. (5-5), and Eq. (5-8) into Eq. (5-6), and after some manipulations, Eq. (5-9) and Eq. (5-10) are obtained.

The whole configuration of the control system is described as Figure 5-3.

## Simulation

A fourth-order-Runge-Kutta integration method at intervals of 1.0×0.0001 s was used to drive the golfer swing model and the swing motions of different-arm-mass golfers were obtained.

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