Spacecraft Generalized Dynamic Inversion Attitude Control Abdulrahman H. Bajodah Abstract— This paper introduces a generalized dynamic in- version control methodology for asymptotic spacecraft attitude trajectory tracking. An asymptotically stable second-order servo-constraint attitude deviation dynamics is evaluated along spacecraft equations of motion, resulting in a linear relation in the control vector. A control law that enforces the servo- constraint is derived by generalized inversion of the relation using the Greville formula. The generalized inverse in the particular part of the control law is scaled by a decaying dynamic factor that depends on desired attitude trajectories and body angular velocity components. The scaled generalized inverse uniformly converges to the standard Moore-Penrose generalized inverse, causing the particular part to converge uniformly to its projection on the range space of the controls coefficient generalized inverse, and driving spacecraft attitude variables to nullify attitude deviation. The auxiliary part of the control law acts on the controls coefficient nullspace, and it provides the spacecraft internal stability with the aid of the null-control vector. The null-control vector construction is made by means of novel semidefinite nullprojection control Lyapunov function and state dependent null-projected Lyapunov equation. The generalized dynamic inversion control signal is multiplied by an exponential factor during transient closed loop response to enhance the control signal in terms of magnitude and rate of change. An illustrating example shows efficacy of the methodology. I. INTRODUCTION Nonlinear dynamic inversion (NDI) is a transformation from a nonlinear system to an equivalent linear system, performed by means of a change of variables and through feedback. The theory of NDI was initially formalized by Su [1] and Hunt et al. [2], and its first reported application to spacecraft attitude control problem is due to Dwyer [3]. Classical NDI is based on constructing inverse mapping of the controlled plant and augmenting it within the feedback control system. Therefore the linearizing transformation de- pends heavily on nature of the plant, and it becomes difficult or impossible as complexity of the plant increases. A paradigm shift was made to NDI by Paielli and Bach in Ref. [4] in the context of spacecraft attitude control. Their approach aims to impose a prescribed dynamics on the errors of spacecraft attitude variables from their desired trajectory values. Rather than inverting the mathematical model of the spacecraft, the desired attitude error dynamics is inverted for the control variables that realize the dynamics. The transformation is global and does not involve deriving inverse equations of motion. Nevertheless, a common feature between the above men- tioned NDI approaches is that the linearizing transformation A. Bajodah is with Faculty of Aeronautical Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia abajodah@kau.edu.sa eliminates nonlinearities from the transformed closed loop system dynamics without distinguishing between types of nonlinearities. For instance, a nonlinearity may cause the spacecraft at a particular time instant to accelerate in a manner that is in favor of the control objective, e.g., in performing a desired maneuver. Yet a needless control effort is made to eliminate that nonlinearity, and an additional control effort is made to satisfy the control objective. This can be extremely disadvantageous as large control signals may cause actuator saturation and control system’s failure. It is therefore desirable to come up with a dynamic inver- sion control design methodology that provides a global lin- earizing transformation, gets around the difficulty of plant’s mathematical model inversion, and requires less control effort to perform the inversion by avoiding blind cancela- tion of dynamical system’s nonlinearity. These features are offered by generalized nonlinear dynamic inversion (GNDI) control [5], [6]. The GNDI methodologies add the flexibility of non-square inversion to the simplicity of NDI by observing that the inverse dynamics problem is underdetermined, i.e., there exist infinite generalized inversion control laws that realize a constrained dynamics. A GNDI spacecraft control design begins by defining a norm measure function of attitude error from desired attitude trajectory. An asymptotically stable second-order linear differential equation in the norm function is prescribed, resembling the desired servo-constraint dynamics. The differ- ential equation is then transformed to a relation that is linear in the control vector by differentiating the norm measure function along the trajectories defined by solution of the spacecraft’s state space mathematical model. The Greville formula [7] is utilized thereafter to invert this relation for the control law required to realize desired stable linear servo- constraint dynamics. The Greville generalized inversion formula exhibits useful geometrical features of generalized inversion. It consists of auxiliary and particular parts, residing in the nullspace of the inverted matrix and the complementary orthogonal range space of its transpose, respectively. The particular part involves the standard Moore-Penrose generalized inverse (MPGI) [8], [9], and the auxiliary part involves a free null-vector that is projected onto nullspace of the inverted matrix by means of a nullprojection matrix. The Greville formula is capable of modeling solution nonuniqueness to problems where requirements can be satisfied in more than one course of action. For that reason, the formula had remarkable contributions towards advancements in science and engineering. 18th Mediterranean Conference on Control & Automation Congress Palace Hotel, Marrakech, Morocco June 23-25, 2010 978-1-4244-8092-0/10/$26.00 ©2010 IEEE 1447