Symmetric stereographic orientation parameters applied to constrained spacecraft attitude control

Abstract

The full kinematic properties of a minimal set of rigid body attitude coordinates called Symmetric Stereographic Orientation Parameters (SSOPs) are developed. These coordinates result from a stereographic projection of the Euler parameter constraint hypersphere onto a three-dimensional hyper-plane. As discussed in previous work [5], this family contains the well-known classical and modified Rodrigues parameters. Considering general SSOP projection points, transformations to the Euler parameters and the direction cosine matrix are discussed. The set of three SSOP coordinates have the unique feature that the associated singularity can be placed at a desired principal rotation angle by adjusting the projection point. In contrast to the Rodrigues parameters, the SSOP coordinates do not represent a unique orientation. The impact of this non-uniqueness on the constrained spacecraft attitude control problem is discussed. An attitude feedback control law in terms of SSOPs will inherently avoid reaching this singular attitude description, and thus constrain the attitude error response to be within a well-defined cone. Lyapunov's direct method is used to illustrate how a SSOP-based control law can be derived to drive the spacecraft attitude away from the singularity and towards a desired orientation. This control law generalizes the previously developed classical and modified Rodrigues parameter-based attitude control laws for general stereographic projection points.

On the use of Generalized Rodrigues Parameters for Representing Rigid Body Rotations

“…Internal consistency can be achieved, for example, by always taking the plus sign in (6) and (15) if a < 0 (f ++ = 1) and the negative sign if a > 0 (f −− = 1). This is a first extension of the findings in [10]: ignoring all but these two cases, a simple logic based on the sign of a is sufficient to guarantee internal consistency for any rotation. However, this logic does not guarantee external consistency (see section 4).…”

“…The parameter set with the smaller norm is used to switch between between r and r s [7]: this guarantees a bijective mapping between rotation and attitude parameters. More recently, a generalization of the modified Rodrigues parameters was introduced in [9] and [10], in which a three-element vector is formed as…”

“…Moreover, when inverting the transformation, the same rotation matrix maps to two different sets of GRPs. To resolve this duality, it was proposed in [10] -limited to the direct mapping -to employ one set of GRPs if the actual value of the quaternion scalar component q 0 exceeds a, or their shadow counterpart -obtained by switching the sign of a -if it does not. However, this demands an a-priori knowledge of the actual body orientation, which limits the operational use of GRPs.…”

Optimized Data-Driven Prescribed Performance Attitude Control for Actuator Saturated Spacecraft

“…In addition, using the geometric properties of MRP, it can map the MRP vector σ whenever σ > 1 to its corresponding counterpart σ s through [25]:…”

“…Because the optimal approximation weight is unknown, the vector ŵ is utilized to estimate the value of w. Accordingly, the approximation expression of ( 23) and ( 19) can be expressed by (25) and (26).…”