Modeling, Identification and Control, Vol. 28, No. 1, 2007, pp. 3–13 A model of relative translation and rotation in leader-follower spacecraft formations Raymond Kristiansen 1 Esten I. Grøtli 2 Per J. Nicklasson 1 Jan T. Gravdahl 2 1 Department of Computer Science, Electrical Engineering and Space Technology, Narvik University College, N-8505 Narvik, Norway. E-mail: {rayk,pjn}@hin.no 2 Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. E- mail: {Esten.Grotli,Tommy.Gravdahl}@itk.ntnu.no Abstract In this paper, a model of a leader-follower spacecraft formation in six degrees of freedom is derived and presented. The nonlinear model describes the relative translational and rotational motion of the spacecraft, and extends previous work by providing a more complete factorization, together with detailed information about the matrices in the model. The resulting model shows many similarities with models for systems such as robot manipulators and marine vehicles. In addition, mathematical models of orbital perturbations due to gravitational variations, atmospheric drag, solar radiation and third-body effects are presented for completeness. Results from simulations are presented to visualize the properties of the model and to show the impact of the different orbital perturbations on the flight path. Keywords: Spacecraft formation, relative motion, 3D general orbits, orbital perturbations 1 Introduction 1.1 Background The concept of flying spacecraft in formation is revolution- izing our way of performing space-based operations, and this new paradigm brings out several advantages in space mission accomplishment and extends the possible applica- tion area for such systems. Spacecraft formation flying is a technology that includes two or more spacecraft in a con- trolled spatial configuration, whose operations are closely synchronized, and Earth and deep space surveillance are ar- eas where spacecraft formations can be useful. These ap- plications often involve data collection and processing over an aperture where the resolution of the observation is in- versely proportional to the baseline lengths. Further explo- ration of neighboring galaxies in space can only be achieved by indirect observation of astronomical objects, and space based interferometers with baselines of up to ten kilometers have been proposed. However, to successfully utilize space- craft formations for this purpose, accurate synchronization of both position and attitude of the cooperating spacecraft is vital, depending on accurate dynamical system models of the formation. 1.2 Previous work The simplest model of relative motion between two space- craft is linear and multi-variable, and known as the Hill or Clohessy-Wiltshire equations (Hill, 1878; Clohessy and Wiltshire, 1960). This model originated from the equations of the two-body problem, based on the laws of Newton and Kepler, and has inherently assumptions that the orbit is circular with no orbital perturbations, and that the dis- tance between spacecraft is small relative to the distance from the formation to the center of the Earth. An exten- sion to elliptic Keplerian orbits, yet still assuming no or- bital perturbations, is what is known as the Lawden equa- tions (Lawden, 1954) or also Tschauner-Hempel equations (Tschauner, 1967). Both models were originally presented ISSN 1890–1328 c  2007 Norwegian Society of Automatic Control doi:10.4173/mic.2007.1.1