EXPRESSION OF DUAL EULER PARAMETERS USING THE DUAL RODRIGUES PARAMETERS AND THEIR APPLICATION TO THE SCREW TRANSFORMATION

Dual numbers and dual vectors are widely used in spatial kinematics [3,515,18]. Plücker line coordinates of a straight line can be represented by a dual unit vector located at the dual unit sphere (DUS). By this way, the trajectory of the screw axis of a rigid body in 3 R (the real three space) corresponds to a dual curve on the DUS. This correspondence is done through Study Mapping [8,9]. Conversely a dual curve on DUS obtained from the rotations of the DUS represents a rigid body motion in 3 R [8]. The dual Euler parameters are used in defining the screw transformation in 3 R [8], but originally in this paper these parameters are constructed from the Rodrigues and the dual Rodrigues parameters [15]. Key WordsKinematics, Study Mapping, Dual Euler Parameters, Screw Transformation.


INTRODUCTION
The dual representation of a line is simply the Plücker vector written as a dual unit vector [9].For any operation defined on a real vector space, there is a dual version of it with similar interpretation [5].
Olinde Rodrigues, the French mathematician, wrote a paper on rigid body kinematics in 1840.This paper is well known for its contributions to spherical kinematics [17].Rodrigues revealed that every translation can be represented in an infinite number of ways by composition of two rotations of equal but opposite angle about parallel axes [16].Similarly Euler showed that every displacement can be described by a rotation followed by a translation.
There is a detailed survey ranging from Chasles motion to the Rodrigues parametrization and also from the theoretical developments of the rigid body displacements to the finite twist in Dai [20].
Regarding the historical developments of the rigid body displacement, the studies in this field are associated with the finite twist in the 1990s.The finite twist representation and transformation and its ordered combination for several manipulators which is based on the Lie group operation are investigated by Dai, Holland and Kerr in 1995 [19].
In our paper, the dual Euler parameters are used for defining the transformation of screws in 3  R .The dual Euler parameters are constructed from the Rodrigues and the dual Rodrigues parameters (see [15]) which are obtained from the rotations of the DUS.

When a dual vector
x  ˆ is rotated to the dual vector x   ˆ in DUS, this movement corresponds to a screw transformation in 3  R .This transformation can be given by the dual Euler parameters.In other words, this paper discusses the usage of dual Euler parameters for the transformations of screws in 3  R and these parameters are defined in terms of the Rodrigues and the dual Rodrigues parameters.
Quaternions and dual numbers were combined and generalized to form what is referred as " Clifford Algebra" as first discussed by Clifford in 1882.Application to kinematic analysis is discussed by [1], [13].A comprehensive introduction to dual Quaternions can be found in [8].
Assembling the Euler parameters of a rotation into the quaternion ), rotations in real space can be identified.If a vector , then the rotation from , where the conjugate is defined as is given by the dual Euler parameters In section 1, we introduce the dual numbers and the Study mapping.The theoretical background of the dual Euler parameters is developed in section 2 and the application of dual Euler parameters on the screw transformation is discussed by an example in section 3.

Dual numbers
A dual number is a formal sum * ˆa a a , where a and * a are real numbers.Similar to the complex unit , we have here 0 2   .Addition and multiplication are given by For a given real analytic function f we can extend its definition to dual numbers by letting x e e * ˆ  

Dual Vectors
is called a dual unit vector.Therefore a . The set of dual unit vectors defines the dual unit sphere (DUS), which is also called the Study Sphere (For detailed algebraic properties of dual numbers see also [18]).

Study Mapping
A point l p  ( p can be written as a vector, p  , from origin to l ) and a unit direction vector g  of l determine the equation of the straight line l in 3 R .A unit force with respect to the origin acting to l gives the moment vector .
The norm of the moment vector is the smallest distance from line to the origin [9].
The compenents of ) , , , , the dual vector * ˆg g g defines a point on DUS.The mapping which assigns to an oriented line of Euclidean space the dual vetor * ˆg g g is called the Study mapping.

The Cayley Formula
Performing the Cayley formula [8] for the dual spherical motion with the dual rotation matrix A ˆ (it is clear that A ˆ is orthogonal), we obtain the skew symmetric dual matrix B ˆ and the dual Rodrigues vector (see section 3).In these computations, similar to the real case (that is 2 tan Using the algebra of dual numbers one can simply obtain

THE DUAL EULER PARAMETERS AND THE SCREW TRANSFORMATION
The dual Rodrigues vector b  ˆ is the axis of rotation of DUS.Let us define the dual unit . Using the dual rotation angle  ˆ and the dual unit vector s  ˆ we get the dual parameters which are known as the dual Euler parameters [8].
Reviewing the method of transformation of vectors given in real space, the similar method for the dual case can be proposed.As it is discussed, the rotation from R is given by the quaternion equation .This coordinate transformation can be represented by a dual quaternion The dual quaternion Z ˆ is the sum of real Z and * Z components.Z is the quaternion obtained from the rotation matrix ] , where , where D is the quaternion, , relates to translation vector ) , , ( , (see [8]).On the other hand from a given dual rotation matrix ] [ A , the dual Euler parameters hence the dual quaternion Z ˆ can be obtained.
Originally in this paper the dual Euler parameters and the dual quaternion Z ˆ results from the Rodrigues 2 and the dual Rodrigues parameters ) 2 is the angular velocity and ) , , ( is the translation velocity) is defined by the dual quaternion then the final screw (the transformed screw), Let us expand the dual Euler parameters i c ˆ given by (2) as follows; From (1) we get Substituting the equation ( 5) into (4) yields On the other hand, the expansion of (3) gives,

The Transformed Velocities
Substituting the dual Euler parameters 0 ĉ in (2) and

The relation between the rotation of DUS and the Screw Transformation
).
It is seen from ( 8) and ( 9) that the transformed screw is computed directly from the dual rotation angle  ˆ, the Rodrigues parameters and the dual Rodrigues parameters.

APPLICATION OF DUAL EULER PARAMETERS TO THE SCREW TRANSFORMATION
Theoretically the formulas (8) and ( 9) are obtained from the rotations of the DUS.Let us examine ( 8) and ( 9) on an example by taking a dual rotation matrix, that is an Then the Plücker coordinates of l are , where .On the other hand, by the Cayley mapping, angular and linear velocities of the spatial displacement respectively) then the transformed screw w .Since the transformed screw has the coordinates produced from the dual Rodrigues parameters, it has informative coordinates about the rotations of the DUS.
of Z ˆ are known as the dual Euler parameters of the spatial displacement.Using the dual Euler parameters, the dual orthogonal (dual rotation) matrix ] [

3 RFigure 3 .
Figure 3.The application of a given rotation A ˆ to the Screw Transformation