Optimality and guidance for planar multiple-burn orbit transfers

Neighboring optimal control for fixed-time multi-burn orbital transfers

“…As is well known, it is impossible to construct the NOG unless the JC holds along the nominal extremal [4] since the gain matrices are unbounded if the JC is violated. This result was actually obtained by Kelley [2], Kornhauser et al [29], Chuang et al [5], Pontani et al [33,34], and many others who minimize the AMP to construct the NOG. As a matter of fact, given every infinitesimal deviation from the nominal state, the JC, once satisfied, guarantees that there exists a neighboring extremal trajectory passing through the deviated state.…”

“…is a solution of the canonical differential equations ẋ(t) = ∂h ∂p (x(t), p(t), p 0 , ρ(t), τ (t)), ṗ(t) = − ∂h ∂x (x(t), p(t), p 0 , ρ(t), τ (t))), (5) with the maximum condition h(x(t), p(t), p0 , ρ(t), τ (t)) = max…”

“…Once a deviation from the nominal trajectory is measured by navigational systems, a guidance strategy is usually required to calculate a new (or corrected) control in each guidance cycle such that the spacecraft can be steered by the new control to track the nominal trajectory or to move on a new optimal trajectory [1]. Since the 1960s, various guidance schemes have been developed [2][3][4][5][6][7][8][9][10], among of which there are two main categories: implicit one and explicit one. While the implicit guidance strategy generally compares the measured state with the nominal one to generate control corrections; the explicit guidance strategy recomputes a flight trajectory by onboard computers during its motion.…”

“…As far as the author knows, a few scholars, including Chuang et al [5] and Kornhauser et al [29], have made efforts on developing the NOG for low-thrust multi-burn orbital transfer problems. In the work [5] by Chuang et al, without taking into account the feedback on thrust-on times, the second variation on each burn arc was minimized such that the neighboring optimal feedbacks on thrust direction and thrust off-times were obtained. Considering both endpoints are fixed, Kornhauser and Lion [29] developed an AMP for bounded-thrust optimal orbital transfer problems.…”

Optimality Conditions Applied to Free-Time Multiburn Optimal Orbital Transfers

“…Nevertheless, a challenge arizes when we consider a finite-thrust fuel-optimal problem because the corresponding optimal control function exhibits a bang-bang behavior if the transfer time is greater than the minimum transfer time for the same boundary conditions [14]. To the author's knowledge, through testing conjugate points on each burn arc, Chuang et al [3,4] presented a primary study on the sufficient optimality conditions for planar multi-burn orbital transfer problems.…”