Optimization of Multi-Orbit Transfers between Noncoplanar Elliptic Orbits

On optimization of many-revolution low-thrust orbit transfers: Part 1

“…The parameters of the initial and final orbits and the design parameters of the spacecraft are borrowed from [11]. For the maneuver under consideration, the averaged system was solved analytically in [8].…”

“…The solution is shown in the figures by dashed lines. The transfer duration T is selected equal to that obtained by the averaging method for a minimum-time constant-thrust orbit transfer [11]: To overcome this instability of the optimal solution, it is proposed to refuse to search for the optimal trajectory and to examine the possibility of finding a trajectory with quasioptimal control. Such a control law can be found using formulas (2.1) where the components of the conjugate vector l j t ( )are replaced by those obtained by averaging and transformed by formulas (3.4) and (3.5).…”

“…Later, ways were found to average the equations of optimal constant-thrust motion, which is of more practical interest [18,19,29]. Note that the approach proposed in [19] does not allow a qualitative analysis of the averaged equations of optimal motion because their right-hand side contains quadratures that cannot be expressed in terms of elementary functions.Whether a boundary-value problem is solved successfully is still determined by the quality of the initial approximation [5] or by the method used such as the parameter continuation method [11,12]. The approach proposed in [27] and detailed in [5] made it possible not only to evaluate the quadratures, but also to obtain analytic solutions to the averaged equations for maneuvers of practical interest.…”

“…Whether a boundary-value problem is solved successfully is still determined by the quality of the initial approximation [5] or by the method used such as the parameter continuation method [11,12]. The approach proposed in [27] and detailed in [5] made it possible not only to evaluate the quadratures, but also to obtain analytic solutions to the averaged equations for maneuvers of practical interest.…”

Optimal low-thrust orbital transfers in a central gravity field

“…Note that analysis (i) is necessary because the method proposed earlier in [47] for solving the problem in question is still in use, though with minor modifications [2,27]; and the analytic solutions (iii) have been obtained only with the help of approach (ii). The importance of these analytic solutions is in their describing maneuvers that are of independent practical interest.…”