Geometrical method for optimal GEO-inclined station keeping
Laurent Beauregard, Arnaud Boutonnet, Frederic Schoutetens and Simon Gruber
In many instances, geo-synchronous satellites are operated at zero inclination (geo-stationary), and the corresponding optimal station-keeping strategies are well known and have been routinely implemented in existing missions. However, for some missions, the inclination can be relaxed up to a maximum value i ≤ i_max leading to the problem of finding the inclination evolution that minimizes the fuel consumption. To a first order approximation, the inclination vector naturally drifts around the invariant “Laplace plane”, located about 7.4 degree with respect to the Earth’s equator, with a period of roughly 53 years. Since station-keeping costs in GEO are dominated by inclination corrections, the problem can be formulated as a constrained trajectory-optimization problem in the inclination-vector space (i_x, i_y), with the objective of minimizing the inclination change provided by the spacecraft, while ignoring eccentricity and longitude drift. Interestingly, unlike the zero-inclination case, the optimal strategy in the constrained-inclination scenario depends on the desired Orbital Mission Lifetime (OML), thus, the full inclination history must be solved for simultaneously. A notable property of this problem is that dramatically different optimal inclination profiles can exist with identical cost (iso-cost solutions). While this problem can be solved with standard optimal-control routines, in this paper, a novel approach is presented that - under the first-order approximation of the dynamics - results in a fully closed-form, purely geometric solution. Moreover, this approach allows the explicit construction of the iso-cost family of solutions that cover all optimal solutions, including the well-known “pig-tail” geometries.
To test and validate this analytical approach, the theoretically optimal results were implemented in DLR GSOC’s Electric Station-Keeping Optimisation Suite (ESKOS), an operational tool used to generate deltaV-optimised manoeuvre sequences for geostationary station keeping. The theoretical inclination evolution was imported into ESKOS as weekly targets for the inclination vector, enabling an assessment of the operationally achievable OML. All computations were performed using the characteristics of a smallGEO platform. Three inclination strategies were analysed, tested and validated both theoretically and operationally, assuming: (1) an initial inclination of 0°, resulting in a “pigtail” strategy; (2) an initial inclination at the maximum allowed value of 4.3°, requiring RAAN targeting at the end of electric orbit raising and producing trajectories drifting along the inclination circle; (3) the lowest possible inclination enabling a 16-year OML.
As the theoretical approach considered only north-south station-keeping manoeuvres, the theoretical deltaV budget includes a 5% margin to approximate east-west control. In contrast, the operational assessment performed with ESKOS explicitly models both north-south and east-west station keeping, including longitude drift and the evolution of the eccentricity vector. As a result, each strategy could be validated for operational feasibility, with the achievable OML reduced by up to 8% depending on the strategy. For strategy 1, the theoretical maximum lifetime of 19.7 years was validated with ESKOS, yielding 18.4 years operationally. Strategy 2, featuring 50° of RAAN drift, has a theoretical OML of 31.4 years, while the operational value was computed as 29 years. Finally, for strategy 3, featuring 104° of RAAN drift, the theoretical OML of 16 years with a minimum inclination of 1.6° was also obtained operationally.
The close agreement between the theoretical and operational results demonstrates that the proposed geometric method for generating station-keeping manoeuvre sequences for inclined GEO satellites is operationally sound and enables optimised orbital mission lifetimes.