Attitude control with realization of linear error dynamics

An attitude control law is derived to realize linear unforced error dynamics with the attitude error defined in terms of rotation group algebra (rather than vector algebra). Euler parameters are used in the rotational dynamics model because they are globally nonsingular, but only the minimal three Euler parameters are used in the error dynamics model because they have no nonlinear mathematical constraints to prevent the realization of linear error dynamics. The control law is singular only when the attitude error angle is exactly TT rad about any eigenaxis, and a simple intuitive modification at the singularity allows the control law to be used globally. The forced error dynamics are nonlinear but stable. Numerical simulation tests show that the control law performs robustly for both initial attitude acquisition and attitude control. HE conventional approach to attitude control is based on linear control theory. The nonlinear rotational dynamics model is adapted to linear control theory by approximating it with a set of linear dynamic models that are tangent to it at selected design states. The tangent models are in the form of transfer functions or state-space models, depending on whether the frequency-domain or time-domain version of linear control theory is to be used. Linear control theory is applied to each tangent model separately, and the control parameters are scheduled as required in flight. This approach is used for virtually all currently operational aircraft and some spacecraft. It is also still widely used in current research and development. Linear control theory was perhaps the only practical alternative when onboard computer resources were severely limited, but for the following reasons it may not be the best alternative now. First, the rigid-body dynamics and the moment generation are coupled together in the linear tangent models into a single abstract mathematical model without clear physical meaning. Thus, a mere change in the inertia tensor, for example, necessitates in principle a complete resynthesis of the control parameters. Furthermore, only linear moment generation models can be used. Advanced nonlinear aerodynamic models for aircraft cannot be used directly. Also, the attitude error is defined in terms of vector algebra, which is valid only for small angular displacements. Finally, stability away from the design states is difficult or impossible to mathematically guarantee, especially if the dynamic effects of the parameter scheduling are properly considered. Attitude control theory was first introduced by Meyer 1'2; other approaches were given later by Mortensen, 3 Dwyer,4>5 Wie et al.,6'7 and Slotine and Li.8 In contrast to linear control theory, attitude control theory applies directly to nonlinear rotational dynamics. In each approach, the rigid-body dynamics and the moment generation are decoupled into separate mathematical models, and nonlinear moment generation models of arbitrary complexity and sophistication can be used directly. These approaches can be divided into the two major categories outlined below, and this paper introduces a third,