This thesis studies the attitude problem of a three-vehicle constrained formation, where the vehicles measure the relative direction to other elements of the formation and to known inertial references. However, the formation is constrained and, consequently, two of the vehicles cannot measure the relative direction between them. The solution for the attitude problem is first computed deterministically by comparing candidates for the same attitude using different measurements. Besides the general case which has a unique solution, degenerate and ambiguous configurations are identified, respectively with infinite and exactly two solutions. Moreover, the set of configurations with multiple solutions is shown to be a zero measure subset of the entire configuration space. Next, an approximation of the deterministic solution uncertainty is associated with each configuration and it is shown that the uncertainty increases near the configurations where information is lost. Afterwards, the attitude kinematics is considered by assuming that the vehicles also measure angular velocities. Subsequently, two observers are designed based on the variational principles of mechanics. The first observer uses the deterministic attitude solution, whereas the second observer uses the measurements directly in the state estimation. The first method has an asymptotically stable zero error for almost all initial conditions. Nonetheless, it cannot be implemented for configurations with multiple solutions. The second method has a locally asymptotically stable zero error, even in many configurations with multiple solutions. Nonetheless, the equilibrium set is hard to characterize, which can hinder the stability for initial conditions with large errors. Such issue can be overcome by initializing the observer with the deterministic solution. The numerical simulations showed that the error converged to zero in the general case. This thesis paves the way for the widespread deployment of autonomous vehicle formations, namely contributing for their reliability and the flexibility of mission design, both useful in space applications.
Attitude Determination in Three-Vehicle Constrained Formations