A simplified representation of low thrust trajectories allows for an efficient global optimisation process. Exponential sinusoids are a promising shape
We can define the Lambert's problem also for exponential sinusoids. Given the shape parameter k2 an extraordinary similarity arises between ballistic arcs and exponential sinusoid arcs. See Izzo D., "Lambert's problem for exponential sinusoids", Journal of Guidance Control and Dynamics, Vol.29, No. 5, pp.1242-1245, September 2006
In the case of non multiple revolutions the objective function for a minimum mass transfer between planets does not get significantly modified substituting ballistic arcs with exponential sinusoids. See the completed Ariadna study 05/4106 "Spiral Trajectories in Global Optimisation of Interplanetary and Orbital Transfers" studied in cooperation with the University of Glasgow.
Low-thrust trajectories are becoming increasingly important in the design of space missions. The technological developments that allowed ion propulsion to become a competitive option for missions such as Smart-1 and Deep Space 1 may be soon available for many other advanced propulsion systems that promise to deliver long duration thrust with high specic impulses and high reliability. Methods used to find the optimal trajectories for spacecraft equipped with low-thrust capabilities, are based on direct and indirect approaches highly dependent on the initial guess solution.
These approaches, though successful in the detailed design of trajectories, fail in providing the designers with a tool to effciently explore a large number of options at the early design phase. With this respect, the use of analytical methods able to create solutions to the low-thrust problem could make a dierence. In his Ph.D. thesis Petropoulos introduced for the first time a new analytical solution to the problem of low-thrust trajectories. His solution, based on the use of an exponential sinusoid, is based on an inverse dynamical calculation that leads to analytical expressions both for the thrusting history and for the derivative of the polar anomaly, whenever tangential thrust is assumed. His discovery added up to the small list of analytical solutions available for this increasingly important problem. Moreover, his shape based method, inspired many other researchers.
The ACT started this project on the exponential sinusoids to find out their potentialities in a preliminary optimisation process. Along the way some amazing results have been obtained, suggesting that these shapes have actually "something" more than their pure geometrical interpretation. Together with the researchers at the University of Glasgow, the ACT deepened the understanding on this type of trajectory representation. In the 2nd edition of the Global Trajectory optimisation Competition(GTOC2), organized by JPL, the ACT used the exponential sinusoids to produce the final trajectory.