Modified inverse-polynomial shaping approach with thrust and radius constraints

The shape-based low-thrust trajectory approximation with modified inverse polynomials is studied by considering thrust and radius constraints, which require that the thrust-acceleration magnitude be less than a maximum allowed value and the trajectory radius be between a lower bound and an upper bound. Compared with the original inverse-polynomial method, the polynomial orders in the modified one are optimized. For the time-free transfer between circular orbits, it is proved that the radius monotonously changes, and the maximum thrust acceleration is obtained by solving a cubic polynomial which is the first-order expansion at half of the transfer angle. For a given maximum thrust acceleration, the minimum revolution number is analytically estimated. For the time-fixed rendezvous between circular orbits, the seventh parameter is solved by the secant method in its feasible range considering the radius constraints. The maximum tangent-thrust-acceleration magnitude is estimated by solving real roots of a polynomial, and then the feasible solutions for a given maximum acceleration are determined. Numerical examples show that the maximum thrust acceleration by the proposed modified inverse-polynomial method is less than that by the original inverse-polynomial method for both the orbit transfer and rendezvous problems.