A Methodology for Tracking Control for Formation Flight of Satellites

Satellite formation flight (SFF) has great potential with numerous advantages such as reducing mission costs, making missions more efficient and flexible, and enabling advanced applications including space-interferometry and high-resolution imaging. However, precision control of satellite formations is still an open problem due to its nonlinear governing dynamics and severe system uncertainties/disturbances. This paper deals with a two-step satellite formation control methodology in the presence of uncertainties and external disturbances. First, assuming a nominal SFF system (with no uncertainties/disturbances) that provides the best deterministic assessment of the real-life uncertain environment, a nonlinear controller is developed without any linearizations/approximations such that a desired formation configuration is exactly achieved. This feedback control strategy is inspired by results from analytical dynamics called the fundamental equation of constrained motion. In addition, Baumgarte's stabilization technique is employed to asymptotically stabilize the SFF system from any initial conditions that do not satisfy the desired configuration requirements, as usually happens in real-world space applications. In the second-step, an adaptive robust controller is designed to compensate for uncertainties and disturbances to which the SFF system may be subjected. This control law is explored based on a generalization of the notion of sliding mode control. Upon defining an error as the difference between the actual trajectory and the nominal trajectory generated by the first nominal controller and feeding this error back to the second controller, the resulting trajectory accurately tracks the reference one even in the presence of uncertainties and disturbances. This new control approach is continuous because it is linear in the sliding variable, thereby removing the problem of chattering. Also, since it is onerous or costly, and often impossible, to exactly measure the magnitude of the uncertainties or the disturbances, an adaptive law that automatically updates the gain is proposed such that the control law adapts itself to the time-varying uncertain SFF dynamics in order to effectively and quickly mitigate the effects of the uncertainties/disturbances. As a result, the error always lies in a user-provided desired bound without a priori knowledge of the uncertainties/disturbances. The new controller may take different forms depending on practical considerations but special attention is paid to a proportional-derivative form with time-varying gain that yields numerous applications. Furthermore, by combining with the Baumgarte's stabilization method, the sliding variable always starts from zero and robustness is guaranteed from the start. The control thrust function is explicitly obtained and the method is not computationally intensive. This makes the proposed approach ideal for on-orbit real-time SFF control. Numerical simulations demonstrate the effectiveness, accuracy, and robustness of the proposed two-step control methodology, in which a desired formation configuration is to be precisely maintained even in the presence of uncertainties in satellite mass, and unknown or uncertain external disturbances.