Sammendrag
This paper concerns modelling of relative translation and rotation of formation flying spacecraft in Earth orbits. A nonlinear 6 degree-of-freedom (DOF) mathematical model of relative translation and rotation in a leader-follower formation consisting of two spacecraft is derived, with a formulation similar to general Euler-Lagrange system. The model of relative translation is based on the laws of Newton, specifically Newton's law of gravitational attraction from which the two-body problem can be derived. The relative rotation is based on Euler's momentum equations and the attitude is represented by Euler parameters, or unit quaternions. Based on the models of relative translation and rotation, the total 6DOF model is derived. The model is referenced both in a leader orbit coordinate system and in an Earth-fixed inertial coordinate system. The leader orbit coordinate system is located in the centre of mass of the leader spacecraft, whereas the Earth-fixed inertial coordinate system is located in the centre of the Earth. The rotation matrices between the different coordinate systems are presented. The Earth-fixed model is based on work in the field of marine control systems. The system properties for both the models are presented, where properties like symmetry, skew-symmetry and positive definiteness of matrices can be incorporated into the stability analysis when designing control systems. These system properties represent physical properties of the system. Furthermore, simulations of the model referred to leader orbit coordinate system are presented where the impact of perturbing forces and torques are illustrated. The perturbing forces and torques considered in the simulation are due to atmospheric drag and the oblateness of the Earth.
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