Attitude Error Kinematics

E XACT analytical attitude error kinematic equations for most of the known attitude representations are derived. Consequently, these attitude error kinematicmodels hold for arbitrarily large relative rotations and rotation rates. These attitude errors represent instantaneous departures from a general, smooth reference attitude motion. Numerical integrations of these kinematic equations are performed to validate the machine error accuracy for each attitude representation. Singularities and constraints are discussed for minimum and nonminimum attitude parameter representations, respectively. Applications are expected in estimation for general rotational dynamics as well as for attitude tracking errors. Many attitude representations are available for modeling problems in science and engineering [1–5]. Nonlinearity of the representation of a given physical motion and location of geometric singularities are dependant on three things: 1) the actual motion, 2) the attitude representation, and 3) the particular choice of a moving reference axis. Selecting the appropriate representation is highly linked with the kind of the problem being considered. Themost popular ones are: a) the direction cosine matrix (DCM), b) the principal axis and angle, c) the Euler parameters (quaternion), a nonsingular four component unit-vector, d) the classical Rodrigues parameters (CRPs), a threecomponent vector (minimal parametrization), e) the modified Rodrigues parameters (MRPs) [6,7], a three-component vector, and f) the Cayley–Klein parameters, a complex unitary 2 × 2 matrix. Definitions, characteristics, and transformations between these representations can be found in many references [1–4,8,9]. For applications requiring large and rapid rotational motions there exists a need for developing attitude error kinematic models that exactly describe arbitrary large rotational motions. The main contribution of this work is to develop exact large motion error kinematic differential equations for attitude error relative to a general smooth reference trajectory in a unified presentation. Several attitude parameterizations are compared by solving a nonlinear spacecraft tracking problem. Markley [8] has considered different attitude error representations for estimating the state of a maneuvering spacecraft. He has clarified the relationship between the four-component quaternion representation of attitude and the multiplicative extendedKalman filter. Crassidis et al. [10] investigated a variable-structure control strategy for maneuvering vehicles. In their work, they used a feedback linearizing technique and added an additional term to the spacecraft maneuvers to deal with model uncertainties, which they demonstrated always provides an optimal response. Ahmed et al. [11] extended previous work to consider adaptive asymptotic tracking during maneuvers while estimating inertia properties. They used a Lyapunov argument to generate an unconditionally robust control law with respect to their assumed parametric uncertainty. Bani Younes et al. [12,13] considered generalized optimal control formulations that handle nonlinear system dynamics and enable the development of control gain sensitivities to handle plant model uncertainties during maneuvers. Sharma and Tewari [14] introduced MRPs for parameterization of the orientation error. Theydefined the attitude error as an additive quantity. Theirwork is extended by retaining a rigorous nonlinear MRP-based error equation. Schaub et al. [15] developed a new penalty function for optimal control formulation of spacecraft attitude control problems. This function returns the same scalar penalty for a given physical attitude regardless of the attitude coordinate choice. This role of various performance indices, in conjunction with various coordinate choices, will be considered in a future paper. Junkins [16] discussed the link between designing a good controller and the choice of coordinates to represent the attitude kinematics. He linearized the attitude error equations by defining the departure motion as an additive error from a nominal trajectory. Normally, the position error is described by the distance between the two vectors, which represents the current position state and a prescribed reference position state. However, the error in orientation cannot be rigorously represented as an addition error because of the nonlinear behavior of underlying kinematical descriptions [17]. This work presents attitude variable representations that account for the coupled nonlinear error kinematics for departure from a general motion. No approximations are introduced in the description of the vehicle attitude motion. The resulting expressions have been optimized to obtain the most compact and computationally efficient forms.