P L Palmer
In this paper we present a general analytic formulation for optimal transfer paths
for satellites flying in formation based upon the circular Hill’s problem. Optimisation
is performed to minimise the transfer energy required from the thruster. We consider
the optimisation problem as the choice of trajectory the satellite should follow during
the manoeuvre, while the time taken is fixed as are the boundary conditions. We
show that this optimal control problem has simple analytic solutions which provide a
powerful basis from which to develop formation control strategies.
In formation flying, manoeuvres require low levels of thrust and we assume that the
thruster will be firing throughout the manoeuvre and the optimisation scheme solves
for the magnitude and direction of thrust as functions of time.
We illustrate how we may exploit these analytic results and present examples of
manoeuvres using them. We also present a discussion of a docking problem where we
reverse the analysis and solve for initial conditions fixing the thruster characteristics in
the optimal solutions found. Finally we present a discussion of how we might exploit
the natural dynamics to gain further propellant savings by adjusting the boundary
conditions. This is illustrated by considering a cross track manoeuvre for plane change.
Ref: Palmer, P.L., AIAA J Guidance, Control & Dynamics, 29, No. 3 , pp. 519 – 526, 2006.