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Raymond Kristiansen, Esten Ingar Grøtli, Per Johan Nicklasson, Tommy Gravdahl

### A model of relative position and attitude in a leader-follower spacecraft formation

Flying spacecraft in formations is revolutionizing our way of performing space-based operations, and this new paradigm brings on several advantages in space mission accomplishment and extends the possible application area for such systems. Spacecraft formation flying is a technology that includes two or more spacecraft in a tightly controlled spatial configuration, whose operations are closely synchronized. Earth and deep space surveillance with radio interferometry and Synthetic Aperture Radar SAR technology is one area where spacecraft formations can be useful. These systems involve data collection and processing over an aperture where the resolution of the observation is inversely proportional to the baseline lengths. Further exploration 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 spacecraft formations for this purpose, accurate synchronization of both position and attitude of the cooperating spacecraft is vital,which again depends on accurate system models of the formation including external elements that might perturb the flight. This paper presents a detailed nonlinear mathematical model in six degrees of freedom of relative translation and rotation of two spacecraft in a leader-follower formation. The model of relative position is based on the two-body equations derived from Newtons inverse square law of force, and the position and velocity vectors of the follower spacecraft are represented in a reference frame located in the center of mass of the leader spacecraft, known as the Hill frame. In addition, rotation matrices between the Hill frame and an earth centered inertial reference frame are given. The relative attitude model is based on Euler's momentum equations, and the attitude is represented by unit quaternions and angular velocities. The model also includes the mathematical expressions for external disturbances originating from gravitational variations, atmospheric drag, solar radiation, and perturbations due to other celestial bodies, known as third body effects. Results from simulations in Matlab are presented to visualize the properties of the model and to show the impact of the different disturbances on the flight path.