Published in IET Control Theory and Applications Received on 3rd February 2008 Revised on 12th August 2008 doi: 10.1049/iet-cta:20080044 ISSN 1751-8644 Generalised dynamic inversion spacecraft control design methodologies A.H. Bajodah Aeronautical Engineering Department, King Abdulaziz University, Jeddah, P.O. Box 80204, Saudi Arabia 21589 E-mail: abajodah@kau.edu.sa Abstract: Generalised dynamic inversion control design methodologies for realisation of linear spacecraft attitude servo-constraint dynamics is introduced. A prescribed stable linear second-order time-invariant ordinary differential equation in a spacecraft attitude deviation norm measure is evaluated along solution trajectories of the spacecraft equations of motion, yielding a linear relation in the control variables. Generalised inversion of the relation results in a control law that consists of particular and auxiliary parts. The particular part works to drive the spacecraft attitude variables in order to nullify the attitude deviation norm measure, and the auxiliary part provides the necessary spacecraft internal stability by proper design of the involved null-control vector. Two constructions of the null-control vector are made, one by solving a state-dependent Lyapunov equation, yielding global spacecraft internal stability. The other is globally perturbed feedback linearising, but yields local stability of the spacecraft internal dynamics. The control designs utilise a damped generalised inverse to limit the growth of the controls coefficient Moore-Penrose generalised inverse as the steady-state response is approached. Both designs guarantee uniformly ultimately bounded attitude trajectory tracking errors. The null-control vector design freedom furnishes an advantage of the approach over classical inverse dynamics, because it can be used to reduce dynamic inversion control signal magnitude. The design methodologies are illustrated by two spacecraft slew and trajectory tracking manoeuvres. 1 Introduction Generalised inversion is a well-known control system design methodology, with numerous applications in the fields of aerospace engineering and robotics. A fundamental advantage of generalised inversion is that it overcomes the limitations of dimensionality and rank that are related to the notion of inversion, which makes the inversion process capable of solving underdetermined problems where requirements can be satisfied in more than one course of action, as well as solving for best approximating solutions to overdetermined problems, when the requirements cannot be satisfied. Another fundamental property of generalised inversion is the characterisation of solution non-uniqueness. This is depicted by the Greville formula [1], which uses the Moore-Penrose generalised matrix inverse (MPGI) [2, 3] to obtain the general solution of a linear algebraic system equations and parameterises the corresponding linear algebraic system matrix nullspace via a free nullvector that appears explicitly in the solution expression. The generalised inversion feature of nullspace parametrisation has been utilised in engineering analysis and design for the purpose of modelling, control and optimization. At the level of analysis, utilisation of this feature in the field of analytical dynamics was made by deriving the Udwadia-Kalaba equations of motion for constrained dynamical systems [4]. The corresponding free nullvector was chosen in order to optimise acceleration energy of the constrained system, yielding its natural accelerations, that is, those obeying Gauss’ principle of least constraints [5], or equivalently yielding constraint forces that satisfy D’Alembert’s principle of virtual work [6]. At the design level, nullspace parametrisation has been insightful in viewing the active perspective of servo- constrained motion, where the design requirements are IET Control Theory Appl., 2009, Vol. 3, Iss. 2, pp. 239–251 239 doi: 10.1049/iet-cta:20080044 & The Institution of Engineering and Technology 2009 www.ietdl.org