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四元数样条曲线的能量优化

Energy Optimization of Quaternion Spline Curves

  • 摘要: 含参数的四元数曲线在仿真动画、机器人控制等领域中发挥着关键作用, 其独立于控制顶点的形状调节特性为这些应用提供了有力支持. 鉴于与对应的多项式曲线共享许多微分性质, 给出一种四元数样条曲线参数的选取方法. 首先通过对数映射将四元数曲线投影到欧几里得空间, 基于对多项式曲线能量优化, 给出四元数曲线拉伸能量、弯曲能量和扭曲能量约束下参数的选取准则; 然后将上述结果推广到四元数样条曲线. 应用所提方法对6组数据点进行实验, 比较了迭代方法与所提方法得到的最优参数; 分析了不同能量优化目标的差异; 并与其他方法进行对比. 实验结果表明, 该方法其计算复杂度低、误差较小, 得到的刚体旋转角度变化缓慢, 运动姿态更加自然流畅.

     

    Abstract: Quaternion curves with shape parameters play a crucial role in domains such as simulation animation and robot control. Their shape adjustment feature independent of controlling vertices provides strong support for these applications. Considering their shared differential properties with corresponding polynomial curves, this paper presents a methodology for selecting parameters of quaternion spline curves. Firstly, quaternion curves are projected into Euclidean space by logarithmic mapping. Based on the study of stretching, bending and twisting energy optimization for polynomial curves, criteria are provided for selecting parameters of quaternion curves. Then, the above results are extended to quaternion spline curves. Experiments are conducted on six sets of data points using the proposed method. The optimal parameters obtain by the iterative method are compared with those derived from the proposed approach. The differences in energy optimization objectives are analyzed. A comparison with other methods is also performed. The experimental results demonstrate that the proposed method exhibits lower computational complexity and reduced error. Additionally, the changes in rigid body rotation angles are more gradual, resulting in smoother and more natural motion postures.

     

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