Expression of Dual Euler Parameters Using the Dual Rodrigues Parameters and Their Application to the Screw Transformation

Abstract

Dual numbers and dual vectors are widely used in spatial kinematics [3,5-15,18]. 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].

Dual Attitude Representations and Kinematics for Six-Degree-of-Freedom Spacecraft Dynamics

“…The initial conditions of Table 4 are input into the dual attitude equations for the dual quaternion q and the three newly derived dual attitude formulations (ρ, σ, η) from Equations ( 9), (15), and (20). It is worth noting that building directly to the dual attitude equations or using the conversions from q on the right-hand side of equations ( 9), (15), and (20) yield identical results, which is the first step of validation.…”

“…Referencing other dual rodrigues parameters formulations [19], or "twistors" [20] [21], there is a difference in use and identities here The benefit and contribution here for these identities is that they work in terms of the MRPs themselves, provide a direct identity between the dual quaternion, function as a compact and convenient calculation, and build to the new dual MRP dynamics equation. The previous works differ in these qualities.…”

“…Upon review of the existence of dual attitude models and dual attitude kinematics, one author had already derived equations and kinematics for Dual Cayley-Klein parameters [18], and in a similar vein, other dual attitude kinematics models will be derived. There have also been past works that develop models for Dual Rodrigues Parameters [19], or "twistors" [20] [21]. The existing equations maintain the same purpose of position and orientation transformation, but utilize different inputs.…”

Modeling and Control for Spacecraft Relative Pose Motion by Using Twistor Representation

“…In recent years, the concept of the dual Rodrigues parameters (DRPs) or analogs was not revealed until Karakılıç [26] obtained the RP of the dual spherical motion (i.e., DRP), but the presentation was given in scalar form and without further discussions, such as on kinematics equations. Furthermore, in [27], the dual Euler parameters, which is an analog of the dual quaternion, was constructed on the basis of the DRP and was applied to screw transformation. In [28], a complete parameterization framework was constructed for the first time.…”

“…According to the definition given in Sec. II, the twistor B ⌢ BI of O B X B Y B Z B with respect to O I X I Y I Z I can be expressed as follows: B ⌢ BI p BI εΠ BI r I BI(27) …”