tailieunhanh - Dual quaternions in spatial kinematics in an algebraic sense

This paper presents the finite spatial displacements and spatial screw motions by using dual quaternions and Hamilton operators. The representations are considered as 4 × 4 matrices and the relative motion for three dual spheres is considered in terms of Hamilton operators for a dual quaternion. | Turk J Math 32 (2008) , 373 – 391. ¨ ITAK ˙ c TUB Dual Quaternions in Spatial Kinematics in an Algebraic Sense Bedia Akyar Abstract This paper presents the finite spatial displacements and spatial screw motions by using dual quaternions and Hamilton operators. The representations are considered as 4 × 4 matrices and the relative motion for three dual spheres is considered in terms of Hamilton operators for a dual quaternion. The relation between Hamilton operators and the transformation matrix has been given in a different way. By considering operations on screw motions, representation of spatial displacements is also given. Key Words: Dual quaternions, Hamilton operators, Lie algebras. 1. Introduction The matrix representation of spatial displacements of rigid bodies has an important role in kinematics and the mathematical description of displacements. Veldkamp and Yang-Freudenstein investigated the use of dual numbers, dual numbers matrix and dual quaternions in instantaneous spatial kinematics in [9] and [10], respectively. In [10], an application of dual quaternion algebra to the analysis of spatial mechanisms was given. A comparison of representations of general spatial motion was given by Rooney in [8]. HillerWoernle worked on a unified representation of spatial displacements. In their paper [7], the representation is based on the screw displacement pair, ., the dual number extension of the rotational displacement pair, and consists of the dual unit vector of the screw axis AMS Mathematics Subject Classification: 53A17, 53A25, 70B10. 373 AKYAR and the associate dual angle of the amplitude. Chevallier gave a unified algebraic approach to mathematical methods in kinematics in [4]. This approach requires screw theory, dual numbers and Lie groups. Agrawal [1] worked on Hamilton operators and dual quaternions in kinematics. In [1], the algebra of dual quaternions is developed by using two Hamilton operators. Properties of these operators are used to find .