Quaternion algebra provides a strong base for the representation of rotation and translation in three-dimensional space and also is useful in the representation of mechanisms and members in spatial plane. This paper addresses the kinematic synthesis of robot manipulator by using quaternion, after a brief introduction to its properties. The tool used for the purpose is based on dual numbers and dual vector. Although a good amount of work has been done previously on the issues of computational efficiency for effecting Three-Dimensional (3D) rotations and their representations using quaternion, none of them addresses parallel implementation of the corresponding algorithms in higher order kinematic chain. A new algorithm has been developed on the basis of D-H algorithm, and the manipulator kinematic synthesis is explained to develop forward kinematic equation of revolute robot manipulator. A comparison is made between the two methods in the solution of the direct kinematic problem. The results are compared, and it is concluded that algorithms based on dual quaternions are computationally more efficient than traditional methods and, in terms of storage requirement, these require less memory space to represent translation and orientation.

Several researchers have proposed alternative methods for representing and solving the kinematic problems of rigid open chain mechanisms. Aspragathos and Dimitros (1998) presented three methods for the formulation of the kinematics equations of robots with rigid links. These three methods were compared for their use in the kinematics analysis of robot arms by using analytic algorithms and recommended for the solution of the direct kinematics problem. Funda and Paul (1990) proposed a computational analysis and comparison of line-oriented representations of general spatial displacements of rigid bodies. They conclude that the dual unit quaternion representation offers the most compact and the most efficient screw transformation formalism. Dai (2006) contributed towards the displacement and transformation of a rigid body, their mathematical formulation and its progress. He studied some contemporary developments of the finite screw displacement and the finite twist representation in the late 20th century. Pernas (1994) defined new operators in differential forms on quaternionic manifold. He shed light on the future holomorphic function theory. This theory led to acceptable results, but failed to contain simple algebraic functions, and the identity is not regular. Horn (1987) developed a relationship between two coordinate frames by using a pair of measurements of the coordinates of a number of points in both systems. He presented a closed form solution of the least square problem by three or more points. The derivation of the problem is simplified by using unit quaternion representing rotation. Clifford (1873) introduced dual numbers in the form of bi-quaternions (called dual quaternions) for studying non-Euclidean geometry. Study (1903) defined dual numbers as dual angles to specify the relation between two lines in Euclidean space.

McAulay (1898) used dual quaternions to describe finite displacement of rigid and deformable bodies. Dimentberg (1950) and Denavit (1958) pioneered kinematic analysis of spatial mechanisms via dual numbers. Using dual numbers, Yang (1963) and Yang and Freudenstein (1964) studied kinematic analysis of spatial mechanisms. Ravani and Roth (1984) studied rigid body displacements via kinematic mapping. Funda et al. (1990) implemented an alternate approach, employing quaternion/ vector pairs as spatial operators and compared with homogeneous transforms in terms of computational efficiency and storage economy.

Quaternion Algebra, D-H Algorithm, 3D Cartesian spaces, Aspragathos, Dual Quaternion Synthesis, Quaternionic Analysis, Theoretical Development, Kinematic Analysis, Homogeneous Algorithms, Quaternion Transformation, Euler Angle Representation.