Bézier四元数曲线相关问题研究

Research on the Bézier Quaternion Curve

摘要: 计算机动画和机器人运动学中，对刚体运动曲线的设计尤为重要。刚体的位置和朝向决定其特定时刻的状态，实际中常用形式简洁、计算高效的单位四元数表示法来表描述刚体的朝向。因此，本文的研究目的是提供一种简洁有效的构造C2连续的四元数插值样条曲线的构造方法，使曲线精确地插值于给定的单位四元数序列。首先，对Bézier四元数曲线的性质进行研究，计算出了曲线在端点处的一阶、二阶导矢；然后，利用Bézier四元数曲线在端点处的性质解决了它的光滑拼接问题，进一步给出了一种构造四元数插值样条曲线的方案，明确了中间控制顶点与已知数据点之间的关系; 最后对上述方案进行应用。本文方法无需求解非线性方程组，提高了运算效率。

Abstract: The design of rigid body motion curve is particularly important in computer animation and robot kinematics. The position and orientation of a rigid body determine its state at a specific time. In practice, the simple and efficient unit quaternion representation is often used to describe the orientation of rigid body. Therefore, the purpose of this paper is to provide a concise and effective construction method of constructing continuous quaternion interpolation spline curves, so that the curves can be accurately interpolated into a given unit quaternion sequence. Firstly, the properties of Bézier quaternion curves are studied, and the first-order and second-order derivatives of the curves at the endpoints are calculated; Then, the problem of smooth splicing of Bézier quaternion curves is solved by using its properties at the end point. A scheme for constructing quaternion interpolation spline curve is further given, and the relationship between intermediate control vertices and known data points is clarified; Finally, the above scheme is applied. The proposed method does not need to solve nonlinear equations and improves the operation efficiency.