Asymptotic Properties of Higher Order Cayley Transforms

In this short paper we generalize some previous results on attitude representations using higher order Cayley transforms. We show that the kinematic parameters generated by these higher order Cayley transforms have a a very simple limit, i.e., the well-known Euler vector. Introduction In a recent paper1 we introduced the concept of Higher Order Cayley Transforms (HOCT) as a means of generating three-dimensional parameterizations of the rotation group SO 3 . Since SO 3 is the configuration space of the rotational motion of a freely rotating body, these Cayley transforms can be used to generate new kinematic descriptions of the attitude motion. It is a classical result that the well-known Cayley-Rodrigues parameters (CRP’s) can be generated by a first order Cayley transform2,3. In Ref. 4 and Ref. 1 we showed that a second order Cayley transform can be used to generate the (not so well-known) Modified Rodrigues parameters (MRP’s)5–11. The advantages of the MRP’s over the CRP’s are discussed, for example, in Refs. 8,10,11 and they essentially stem from the fact that the MRP’s are well-defined for all eigen-rotations 2π φ 2π, whereas the CRP’s can be used for describing eigenrotations only in the interval π φ π. Thus, the MRP’s have twice the domain of validity of the CRP’s (measured in terms of the eigenaxis rotation angle φ). Higher order Cayley transforms can be used to increase the domain of φ even further thus, essentially, increasing the region of validity of the corresponding kinematic parameters. In this note we investigate the asymptotic behavior of these higher order Cayley transforms and we show that, as the order of the transformation tends to infinity, the corresponding kinematic parameters tend to the so-called “Euler vector”. This may provide a proof to the conjecture that the Euler vector provides the “best” Assistant Professor, Department of Mechanical, Aerospace and Nuclear Engineering. Senior member AIAA. Copyright c 1998 by Panagiotis Tsiotras. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. three-dimensional attitude description in the sense that it has the smallest kinematic singularity measure . In the process of showing this result, we provide an alternative – more straightforward – proof of the main result in Ref. 1, which reveals the connection between matrix transformations of the form in Eq. (3) and the corresponding parameters given in Eq. (13). Finally, we discuss the implications of these results to attitude kinematics problems. Higher Order Cayley Transforms As usual, let so n denote the space of n n skew symmetric matrices† and let SO n denote the space of all n n proper orthogonal matrices‡. The standard Cayley transform parameterizes a proper orthogonal matrix C SO n as a function of a skew-symmetric matrix Q so n via C I Q I Q 1 I Q 1 I Q (1) The Cayley transform is invertible and its inverse is the transformation itself Q I C I C 1 I C 1 I C (2) More information on the classical Cayley transform for the 3-dimensional case and its use in the description of the attitude motion can be found in Ref. 2. In Ref. 1, drawing on some insightful comments by Halmos13, we interpreted Eq. (1) as a “conformal mapping” in the space of matrices. This allowed a generalization of Eq. (1) to higher order and the introduction of ‘Higher Order Cayley Parameters” (HOCP’s) via the series of “Higher Order Cayley Transforms” (HOCT) defined by C I Q m I Q m I Q m I Q m (3) The kinematic singularity measure of a 3-dimensional parameterization of SO 3 is defined in Ref. 12 as the ratio of all possible configurations over the non-singular configurations. †That is, so n fA IRn n : A ATg. ‡That is, SO n fA IRn n : AAT I det A 1g.