Energy Optimization of Quaternion Spline Curves

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.