Problem

Design \(n\) missions able to cumulatively remove all the 123 orbiting debris defined in the file debris.csv. One mission is a multiple-rendezvous spacecraft trajectory where a subset of size \(N\) of the 123 orbiting debris is removed by the delivery and activation of \(N\) de-orbit packages. The following cost function has to be minimized: $$ J = \sum_{i=1}^n C_i = \sum_{i=1}^n \left[c_i + \alpha \left(m_{0_i} - m_{dry}\right)^2\right] $$ where \(C_i\) is the cost charged by the contracted launcher supplier for the \(i\)-th mission and it is composed of a base cost \(c_i\) (increasing linearly during the competition time frame) and a term \(\alpha \left(m_{0_i} - m_{dry}\right)^2\) favouring a lighter spacecraft. At the beginning of the \(i\)-th mission \(m_{0_i}\) is the spacecraft mass and \(m_{dry}\) its dry mass. The \(\alpha\) parameter is set to be \(\alpha = 2.0 \cdot 10^{-6}\) [MEUR / Kg\(^2\)]: each Kg of launch mass saved results in a discount over the mission cost (but also in less \(\Delta V\) capability).

The \(i\)-th mission starting epoch is denoted with \(t_{i}^s\) and its end epoch with \(t_i^f\). A mission starts with a launch delivering, at \(t_i^s\), one spacecraft at a chosen debris and ends when all the \(N\) de-orbit packages on-board have been delivered and activated. An orbiting debris is considered as removed if: a) its position and velocity at some epoch \(t\) coincides with the spacecraft position and velocity vector and b) for the following \(t_w \ge 5\) days the spacecraft stays in proximity of the debris while delivering and activating a de-orbit package of mass \(m_{de} = 30\) [kg].

After, the spacecraft is free to ignite its propulsion system again and leave towards the next debris (note that only in-between debris transfers the spacecraft is subject to the full \(J_2\) perturbation and its dynamics is described by the equations of motion. During the removal operations (i.e. for a time \(t_w\)) the position and velocity of the spacecraft will, instead, be considered to be those of the debris as computed by the ephemerides)

The basic cost \(c_i\) of each mission (i.e. not including the \(\alpha\) term), increases linearly during the competition month and is computed as follows: $$ c_i = c_m + \frac{t_{submission} - t_{start}}{t_{end} - t_{start}} (c_M - c_m) $$ where \(t_{submission}\) is the epoch at which the \(i\)-th mission is validated (via the Kelvins web-site) and \(t_{end}\) and \(t_{start}\) are the end and the beginning epochs of the GTOC9 competition. The exact value of \(c_i\) will be visible at all time on the website. The minimal basic cost \(c_m\) is \(45\) MEUR, while the maximum cost \(c_M\) is \(55\) MEUR.

Orbital Manoeuvres

The only manoeuvres allowed to control the spacecraft trajectory are instantaneous changes of the spacecraft velocity (its magnitude being denoted by \(\Delta V\). After each such manoeuvre, the spacecraft mass is to be updated using Tsiolkovsky equation: $$ m_f = m_i \exp \left(-\frac{\Delta V}{v_e}\right), $$ where \(v_e = I_{sp} g_0\). A maximum of 5 impulsive velocity changes is allowed during each transfer between two successive debris. These do not include the departure and arrival impulse.

The Spacecraft

Each spacecraft initial mass \(m_0\) is the sum of its dry mass, the weights of the \(N \ge 1\) de-orbit packages to be used and the propellant mass: \(m_0 = m_{dry} + Nm_{de} + m_p\). All spacecraft have a dry mass of \(m_{dry} = 2000\) [kg] and a maximum initial propellant mass of \(m_p = 5000\) [kg] (less propellant may be used, in which case the launch costs will decrease). Each de-orbit package has a weight of \(m_{de} = 30\) [kg]

Operational Constraints

The debris removal operations during each of the multiple-rendezvous trajectories are complex and demand some control over the schedule of the debris visits:

• The overall time between two successive debris rendezvous, within the same mission, must not exceed 30 days. So that if the arrival epoch to the debris \(\mathcal D_a\)is \(t_{\mathcal D_a}\) and the arrival to the next debris \(\mathcal D_b\) is \(t_{\mathcal D_b}\), then \(t_{\mathcal D_b}-t_{\mathcal D_a} \le 30\) [days].

• In order to avoid operating different missions in parallel, a time of at least 30 days must be accounted for between any two missions so that if \(t_i^s > t_j^s\) then \(t_j^f + 30 \le t_i^s\) [days], \(\forall i \neq j\).

• All mission events (arrivals, departures, etc..) must take place in an allowed window, so that indicating with \( t_{event} \) the epoch of any event, then \(23467 \le t_{event} \le 26419\) [MJD2000].

• The osculating orbital pericenter \(r_p\) cannot be smaller than \(r_{m} = 6600000\) [m] (this is only checked at each event, not in-between)