Research On The Quaternion Spline Curves Of The Rigid Body Orientation Trajectory

Quaternion algebra was founded by Irish mathematician Sir William Rowan Hamilton between 1840s and 1860s. Quaternions form a noncommutative extension of complex numbers in the 4-D real number space. It is finite-dimensional associative divison algebra over the real numbers, and is also a sub-algebra of Clifford algebra. In the late 1960s, quaternions began to acquire a practical application in the classical mechanics. In 1985, Shoemake introduced quaternions to computer graphics for representing rotations in 3D space. Ever since then quaternions have been widely used in the fields such as computer graphics, computer animation, computer vision and robotics, etc.In this thesis, combining the quaternion methods with the specialized knowledge of CAGD, especially, B-splines, based on quaternion spherical linear interpolation, and the compact quaternion representation of 3D rotations, taking B-spline, generalized B-spline, triangular parameters spline, and so on as main mathematical tools, we study and discuss some issues in quaternion spline curves representing rigid body orientation trajectory. The main research work and achievements can be divided into two parts:The first part is based on the theory of B-spline and generalized B-spline curves, we construct a class of generalized B-spline quaternion curves; and analyzed its C2 continuity and local control properties. The generalized B-spline quaternion curves generalize Kim’s B-spline quaternion curves, and achieve higher continuity. At the same time, this class of quaternion spline curves also maintains many important differential properties, apart from gaining more shapes from choosing different core functions..In computer animation, it is a fundamental problem to generate smooth rigid body motion whose trajectory interpolates a given sequence of key-frame positions and orientation. So in the second part we study unit quaternion interpolation spline curves. In order to generate B-spline or generalized B-spline quaternion interpolation curves, we must know the control vertices, which lead to a need to solve multivariable nonlinear system of equations. Due to the complexity of calculation in obtaining the control points for the rigid body orientation interpolation curve, this part presents C2-continuous quartic triangular interpolation spline quaternion curves with shape parameters, which can automatically pass the given key orientations. We analyze the C2 continuity and discuss the local shape modification ability and the influence of shape parameters on the curve shape. The modification on one of the shape parameters will only influence the two adjacent curve segments.We establish 3D models, and carry out the simulation experiments in MATLAB. Simulation results verified the feasibility and effectiveness of the proposed quaternion spline curves.