Determination of kinematic state of an orbiting multibody using GNSS signals

Precise attitude determination of the members of a free-flying multi-body system is a not so immediate task, due essentially to the large motion of its appendages coupled with their relevant flexibility effects. In fact, sensors used to this aim in current projects, such as optical encoders usually positioned near the joints of each arm, are almost blind to these effects, and clusters of specific redundant sensors should therefore be required in order to reconstruct both elastic deformations and rigid motion. Satellite navigation systems (GNSS) offer a suitable and reliable solution to this problem. To Exploit the phase of the signal, instead of the traditional pseudo random code, ensures a very high accuracy of the order of magnitude of centimeter. Such a process requires the solution of an initial ambiguity problem, related to the number of integer wavelength included in the length of the member. The aim of the paper is to investigate the capability of this GNSS based technique to reconstruct the kinematics of a flexible multi-body system orbiting around the Earth. This analysis requires a simulation including both the multi-body dynamics and the navigation system constellation to define the satellites lines of sight at each time step. Concerning multi-body equations of motion, a Newtonian formulation is adopted in this work. A special attention is required about the choice of the state variables. As the internal forces are associated to the relative displacements between the bodies, which are small fractions of the distance of the multi-body spacecraft from the center of the Earth, the task of obtaining these forces from inertial coordinates could be impossible from a numerical point of view. So, the problem is reformulated in such a way that the equation of motion of the system contains global equations, with no internal forces, and local equations, with internal forces. In the latter only quantities of the same order of the spacecraft dimensions are present. Accuracies achievable in LEO orbit with current GPS and upcoming Galileo systems are evaluated to show the interest of the proposed technique.