The Construction And Application Of Unit Quaternion Curve Based On Polynomial Interpolation Splines

Quaternions form a noncommutative extension of complex numbers in the 4-D real number space. It is 4-dimensional associative division algebra over the real numbers , and is also a sub-algebra of Clifford algebra. Because quaternion multiplication doesn’t meet commutative law, quaternions had not received enough attentions. In the late 1960s, quaternions began to acquire practical applications in the classical mechanics. In1985, Shoemake introduced quaternion to represent rotations in 3D space in computer graphics. With the development of computing technology and space technology, quaternions have been widely used in the realms such as computer graphics, computer animation, computer vision and robotics, and etc.In the thesis, combining the quaternion methods with the spline theories, based on compact quaternion representation of 3D rotations and Kim’s B-spline quaternion curves theory, assisted with the construction method of polynomial blending functions for interpolation spline curve, and the cumulative basis function method for B-spline quaternion curves, we have researched the unit quaternion curves’ roles and applications on the attitude control for the rigid motion. The main research work and contributions are as follows:(1) Based on Kim’s B-spline quaternion curve theory, with the help of the construction method of blending functions for interpolation spline curves, we construct unit quaternion interpolation spline curves. Compared with unit quaternion uniform B-spline interpolation curves, our scheme avoids solving the nonlinear system of equations over quaternion to acquire the control points for B-spline curve, and can produce a curve which not only passes through a given sequence of orientations precisely, but also can achieve desired continuity. This greatly improves the computational efficiency. Simulation results demonstrate the effectiveness of the proposed method.(2) The unit quaternion curve generated by the method in (1) is fixed in shape, so next we construct unit quaternion curve with shape parameters. Based on the properties of the endpoint interpolation and endpoint derivatives of Bezier curve, we construct two groups of interpolation spline basis functions with shape parameters. The proposed unit quaternion curves not only interpolates the given data points, but also can be modified in shape locally. The change in the values of local shape parameters only affects two adjacent segments of curve. Compared with unit quaternion B-spline interpolation curves, our method does not need to solve the nonlinear system of equations, which improves the computational efficiency. Simulation results are also presented to demonstrate the effectiveness of the proposed method.