Hypersphere Stereographic Orientation Parameters

Abstract

E ULER parameters (EPs), also referred to as quaternions, are a nonsingular set of four attitude coordinates that are constrained to a unit norm. The first analytical mapping from EPs to modified Rodrigues parameters (MRPs) is performed by Wiener in his 1962 dissertation [1], in which he discovered a singularity at the 360 deg rotation. In [2], Marandi and Modi exploit the nonuniqueness property of the MRPs by formulating a nonsingular minimal attitude description. Shuster also mentions the MRPs in his well-known survey of attitude parameterizations, and he gives the parameters the name modified Rodrigues parameters [3]. Tsiotras and Longuski point out that theMRPs can be viewed as the result of a stereographic projection of the EP constraint unit hypersphere onto a threedimensional (3-D) projection hyperplane [4,5]. He also discovers that the natural logarithm function forms an elegant attitude cost (Lyapunov) function in terms of MRPs, which leads to linear MRP feedback with nonlinear stability. Schaub and Junkins [6] further develop this work by showing that theMRP stereographic projection description discovered by Tsiotras can be expanded to describe general families of attitude parameters called the stereographic orientation parameters (SOPs). In particular, [6] presents the subgroup of symmetric SOPs and shows that the MRPs and classical Rodrigues parameters are a subset of this family. Later on, Southward et. al. [7] develop the full kinematic properties of the symmetric SOPs by allowing the projection point to be placed anywhere on the scalar EP coordinate axis within the EP constraint hypersphere. These symmetric SOPs are expressed algebraically in terms of scalar projection point coordinates and yield minimal set attitude coordinates, in which the singularity can occur at any desired orientation within 0 deg< < 360 deg. In contrast, the asymmetric stereographic attitude parameters (ASOPs) of [6] place the projection point at 1 along one of the vector EP coordinate axes. This leads to an interesting behavior, in which singularities are only encountered if a pure rotation about a particular principle body axis is performed. Further, a 180 deg rotation may lead to a singular attitude description, but a 270 deg rotation (exact same orientation) is nonsingular. Only 630 deg in the negative direction leads to a singular description. The nonsymmetric nature of the singular rotations and their dependency of the path to a particular orientation lead to the name of asymmetric SOPs. Other recent attitude coordinates that relate to the MRPs include the higher-order Rodrigues parameters [8]. Here, higher-order Cayley transforms are used to develop attitude coordinates that grow infinitely large a at multiples of 360 deg. These higher-order Rodrigues parameters are convenient to develop minimal sets of attitude coordinates, for which the differential equation can be made arbitrarily linear through the use of higher-order Cayley transformations. Hurtado uses the MRPs to create inner and outer parameters for attitude representations and presents new Cayley-like transformations [9]. This paper investigates a subfamily of attitude coordinates called the hypersphere SOPs (HSOPs), which contain both the previous MRPs (particular set of symmetric SOPs) and theASOP, allowing for all this work to be combined into a single, minimal attitude parameter description. HSOPs allow the projection point to lie at any point on the EP unit hypersphere constraint. Thus, depending on the choice of the project point, these attitude coordinates can display a singular behavior similar to that of the ASOP. The attitude of a spinning body can be described singularity-free, with a minimal three-parameter coordinate set as long as the body is not spinning about a particular combination of principal body axes. Or, the HSOP coordinates can be chosen, such that their singular behavior matches that of the MRPs, for which a particular 360 deg rotation about any body axis leads to a singular description. When different attitude coordinates are combined into a more general family of parameters, such as the joining of classical Rodrigues parameters and MRPs into symmetric SOPs in [7], the result is often a more complex set of algebraic equations. This paper investigates how a general projection point on the surface of the EP constraint hypersphere complicates the associated HSOP differential kinematic equations and their mapping to the shadow set. This paper is organized as follows. Section II describes the geometry and algebra of a general stereographic projection. Section III provides the analytical mapping between HSOPs from EPs, the DCM, as well as the derivation of the shadow sets, the kinematic differential equation, and the singularity condition. Section IV discusses how the HSOPs can be employed in attitude control strategies.