The primer vector theory, firstly proposed by Lawden, defines a set of necessary conditions to characterise whether an impulsive thrust trajectory is optimal with respect to propellant usage, within a two-body problem context. If the conditions are not satisfied, one or more potential intermediate impulses are performed along the transfer arc, in order to lower the overall cost. The method is based on the propagation of the state transition matrix and on the solution of a boundary value problem, which leads to a mathematical and computational complexity. In this paper, a different approach is introduced. It is based on a polar coordinates transformation of the primer vector which allows the decoupling between its in-plane and out-of-plane components. The out-of-plane component is solved analytically while for the in-plane ones a Hamiltonian approximation is made. The novel procedure reduces themathematical complexity and the computational cost of Lawden’s problem and gives also a different perspective about the optimisation of a transfer trajectory.
Iorfida, E., Palmer, P., & Roberts, M. (2016). Novel approach on the optimisation of mid-course corrections along interplanetary trajectories. In Astrophysics and Space Science Proceedings (Vol. 44, pp. 121–136). Kluwer Academic Publishers. https://doi.org/10.1007/978-3-319-23986-6_9