Description
TfMin is a Fortran and Matlab package designed for the numerical solution of continuous 3D minimum time orbit transfer around the Earth (with free final longitude), especially for low thrust engines (e.g., thrust of 0.14 Newton for a 1500 Kilogramme satellite). The optimal control model given by the French Space Agency is the following (we refer to [3] for details):
tf -> min
dx/dt = f0(x)+1/m (u1f1(x)+u2f2(x)+u3f3(x)), t in [0,tf]
dm/dt = -ß|u|
x(0) = x0, m(0) = m0, h(x(tf)) = 0
|u| <= Tmax
where Tmax is the maximum modulus of the thrust. The state of the satellite is described by the geometry of the ellipse osculating to the trajectory and its position on it, x = (P,ex,ey,hx,hy,L), and by its mass m. The numerical method used is single shooting (see the celebrated code [4] for multiple shooting) which turns to be very efficient here. The shooting code is implemented in Fortran 77 and can be used on general optimal control problems. It automatically generates the associated boundary value problem thanks to Automatic Differentiation [1] (see also [2] for a similar approach based on Maple and Scilab). A specific Matlab interface is provided. It allows the user to set the parameters of the min time transfer problem (initial conditions...) and to draw the orbits computed. Results of [3] can be obtained in this way, in the spirit of reproducible research. The Fortran kernel of the software uses two public libraries: Minpack, for the ODE solver RKF45 by H. A. Watts and L. F. Shampine, and the NLE solver HYBRD by B. S. Garbow, K. E. Hillstrom and J. J. More, together with Adifor (version 2.0, see [1]). We are grateful to the authors for making their software available.