Analytical Impulsive-to-Continuous Thrust Conversion in Linearized Relative Dynamics

The control law proposed in this study has a mathematical analytic bound used to reliably estimate the required transfer time to guarantee that the maximum thrust constraint is not violated. Favored by its analytic and iteration-free nature, the transformation can be embedded into onboard guidance planning algorithms without affecting numerical efficiency and enabling an online assessment of the consequences of a bounded, continuous thrust. The resulting control law is compared against the solution of minimum-energy, minimum-time, and minimum-fuel optimal control problems in terms of propellant consumption and transfer time. Two alternative impulsive-to-continuous thrust conversion approaches are derived, and are referred to as forward and backward conversions. The approach proposed in this study makes use of third-order polynomials to shape the trajectory without iterative procedures.