Research On The Bézier Quaternion Spline Curve

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 moment,and in practice,a concise and computationally efficient unit quaternion notation is often used to describe the orientation of a rigid body.In practical problems,we often need to construct a smooth Bézier quaternion spline curve with flexible shape control ability to interpolate the given data points,and then obtain the corresponding smooth and natural rigid body motion.However,most of the existing construction methods have certain limitations.For example,For the Bézier quaternion spline curve with high smoothness,it is necessary to solve the nonlinear equation system to determine its control vertex;the method of evading the complex calculation process by adjusting the basis function cannot meet the demand for higher smoothness.Therefore,the purpose of this thesis is to provide a concise and effective construction method ofC~2 continuous quaternion spline curve,so that the curve can be accurately interpolated to the given data.First,the properties of the Bézier quaternion curve are studied,and the first-order,second-order and third-order derivatives at the endpoints are calculated.Then,apply the above properties.Based on the high-order derivatives at the control vertices at the beginning and end of the curve,the analytical calculation method of the intermediate control vertices is given,and two C~2 continuous Bézier quaternion splines are constructed:Firstly,the relationship between the intermediate control vertex and the beginning and end points of the curve and its velocity and acceleration is clarified,and the construction scheme of the Hermite quaternion spline curve is given;The relationship between the control vertices of two adjacent Bézier quaternion curves is also given,the problem of smooth splicing is solved,and the C~2 construction scheme of uniform quaternion splines is further given.Finally,two numerical experiments are given to apply the above two schemes respectively to prove their feasibility.In this thesis,the higher-order derivatives of the Bézier quaternion curves at the endpoints are calculated,and the constructed quaternion splines have higher-order smoothness.The analytical calculation method of its control vertex is given,which does not need to solve the nonlinear equation system,which greatly improves the operation efficiency.