以三種數學方法計算標槍繞縱軸旋轉之運動學參數

The calculation of the rotational kinematics about the longitudinal axis of a javelin by three methods

目的：標槍在飛行過程中除了有平移運動外更有三維的旋轉及繞自身縱軸旋轉的運動，其運動型態是頗為複雜的；所以必須透過一些數學及數值運算方法才有辦法正確地計算出繞縱軸旋轉之運動學參數。方法：本研究以兩台同步Redlake高速攝影機（擷取頻率：250Hz，快門：2500Hz）配合標槍上三個不共線之反光標點以尤拉角法、卡當角法及螺旋軸法來計算這些運動學參數。結果：三種方法在角度、角速度及角加速上之平均差距分別為0.048±0.029%、0.03±0.015%及1.34±1.54%。以本研究中受試者而言，在槍體出手時之角速度約為1864deg/s（約為5.1rps），角加速度約為27100deg/s2。結論：三種方法各有其特性及需注意的地方，計算時須特別小心。其中尤拉角及卡當角法需注意旋轉順序的問題，特別是尤拉角法，除了旋轉順序問題外，更要小心Gimbal Lock問題，否則可能誤差極大；而螺旋軸法則較不需考慮這些問題。建議如果想使用尤拉角在類似的標槍實驗中計算，可把標槍縱軸設為第二軸，以減小誤差。

Purposes: The motion of a javelin is complicated after it is released. It is necessary to use some skills to calculate the rotation kinematics about the longitudinal axis of a javelin. In this study, the Euler angle, the Cardan angle and the screw axis methods were used to calculate them. Methods: Two synchronized Redlake high-speed cameras (250 Hz) and a javelin with three fixed reflective markers were used in the experiment. Results: The mean difference among the three methods in angle, angular velocity and acceleration was 0.048±0.029 %, 0.03±0.015 %, 1.34±1.54 %, respectively. Generally the results by the three methods were close, and all of the three methods are suitable for the calculation on these variables. The angular velocity and acceleration about the javelin longitudinal axis were about 1864 deg/s and 27100 deg/s2 at release. Conclusion: In the Euler angle and Cardan angle methods, the order of rotations needs to be taken into account, however no similar problem occurres in the screw axis method. The results suggested that the longitudinal axis should be set as the second axis in the Euler angle method for avoiding singularity or Gimbal Lock problem.