New approaches to satellite formation-keeping and the inverse problem of the calculus of variations

2012-10-19This dissertation handles two topics that are fundamental in understanding aerospace and mechanical systems. In the first topic, a new controller for maintaining a desired formation of multiple satellites is developed in the presence of attitude tracking requirements under various perturbations and model uncertainties. Both orbital and attitude dynamics are simultaneously considered and continuous thrust propulsion systems are assumed. Unlike most studies, no linearizations and/or approximations are made in dynamics or the controllers. The novel controller developed herein provides a remarkable improvement over the current state-of-theart in that the control force and torque to satisfy the given orbital and/or attitude constraints are obtained in completely closed form. This new, analytical solution can be easily used for on-orbit, real-time control with low computational burden. Furthermore, it is useful in estimating the magnitude of the required control inputs and results in some interesting consequences, including the separation principle, which describes when the orbital control can be completely decoupled from the attitudinal control required to be applied to the follower satellite. Also, an additional additive controller that compensates for model uncertainties is developed. This is done by using the desired trajectory of the nominal system as the tracking signal, and is based on a generalization of the concept of sliding surfaces. The resulting closed-form control causes the desired attitude and orbital requirements of the nominal system to be met in the presence of unknown, but bounded, model uncertainties. ❧ The second topic deals with the inverse problem of the calculus variations. To develop the equations of motion for a mechanical system, it is usual to define a Lagrangian function first and then obtain the equations of motion by applying the Lagrange equation when the forces are derivable from a potential function. However, it is interesting and useful to investigate the types of forces that can engendered when substituting the forces into the Lagrange equation guarantees the proper equations of motion even when the forces do not arise from a potential, that is, when they are nonconservative. By extending the existing Bolza’s approach, a more general and systematic way of handling the problem is found for single-degree-of-freedom systems. Then, new results for linearly damped multi-degree-of-freedom systems are obtained using extensions of results for single-degree-of-freedom systems. Conservation laws for these damped multi-degree-of-freedom systems are also found using the Lagrangians obtained. These new findings can be used for development of approximate solutions of nonlinear differential equations or various numerical techniques