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4 446 Hiroki Okubo and Mont Hubbard / Procedia ngineering 112 ( 2015 ) Results and discussion Angular velocity combinations of shoulder, elbow and wrist joints are plotted with each set shown in Fig. 2 to produce the optimal ball release speed, angle and backspin, 4.58 m/s, 47.5, 2π rad/s for the short-, 6.62 m/s, 46.0, 4π rad/s for the medium-, and 9.04 m/s, 46.0, 6π rad/s for the long-range shots [7] in Figs. 3-5 when the shoulder velocity v is zero and there is no slipping between the fingertip and the ball surface at release. The reasonable conditions that Ψ > 0 (the vertical component of the velocity of the elbow is usually positive) and Ψ + Ψ + Ψ < Ψ + Ψ < 0(the angular velocities of the forearm and hand are usually negative and the magnitude of angular velocity of the hand is larger than that of the forearm) are used for calculations. Figure 3 shows the sets of angular velocities of shoulder, elbow and wrist joints for the optimal release conditions in the shoulder-elbow-wrist angular velocity space. Figures 4(a) and (b) show the top and side views of the sets of angular velocities in Fig. 3. ach straight line is a function of Ψ, Ψ + Ψ, and Ψ + Ψ + Ψ W. As the angular velocity of shoulder increases, the magnitude of angular velocity of elbow increases. When the angular velocity of elbow is close to zero, the shoulder angular velocity is small but large wrist joint angular velocity is needed. When the elbow angle is close to180 ( = 59 ), large wrist angular velocity is also required. Ψ Fig. 3. Combinations of angular velocities of upper arm, forearm and hand at release for short-, medium-, and long-range shots. (a) (b) Fig. 4. Relationships between joint angular velocities for short-, medium- and long-range shots. (a) forearm angular velocity vs. wrist angular velocity and (b) wrist angular velocity vs. shoulder angular velocity.

5 Hiroki Okubo and Mont Hubbard / Procedia ngineering 112 ( 2015 ) For Ψ + Ψ = 90 and Ψ W = 0 (both the forearm and the hand are vertical) at release, the vertical component of the fingertip velocity is a function of Ψ, but neither a function of Ψ + Ψ nor of Ψ + Ψ + Ψ W. The shoulder angular velocities can be calculated as Ψ =15. 9, 22.4 and 30.7 rad/s for short-, medium- and long-range shots. houlder rotation contributes to the vertical component of ball velocity and is important to a good arch of the ball trajectory. Fig. 5. Combinations of angular velocities of upper arm, forearm and hand at release for medium-range shots with 1Hz, 2Hz and 3Hz backspin. (a) (b) Fig. 6. Relationships between joint angular velocities for medium-range shots with 1, 2 and 3Hz backspin. (a) forearm angular velocity vs. wrist angular velocity and (b) wrist angular velocity vs. shoulder angular velocity. ome previous studies [3,4,5] measured angular velocities of shooters shoulder, elbow and wrist joints at release. Miller and Bartlett [3] represented the angular velocities of arm joints to produce three different ball release speeds. As release speed of the ball increases, the magnitude of elbow angular velocity increases but the magnitude of wrist angular velocity decreases, and there is almost no difference in the magnitude of shoulder angular velocity for the three ball release speeds. In the results of reference [4], as shot distance increases, the magnitudes of angular velocities of shoulder and elbow increase but there is no correlation between shot distance and wrist angular velocity at release. Miller and Bartlett [9] ignored the wrist action because some previous studies pointed out that there was

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