Nonlinear Approach in Primer Vector Based Trajectory Optimisation

Trajectory optimisation during the preliminary design phase is crucial for a mission designer to meet the requirement while achieving the mission objectives. A classical indirect optimisation method such as primer vector method still found its use due to its simplicity and concept clarity. It is used to evaluate the optimality of a trajectory and offers information on how to improve a non-optimal trajectory. In an impulsive and fixed time transfer, primer vector provides information regarding the time and position of the corrective impulse required to reduce the total $____Delta v$. This research aims at studying the application of the primer vector theory in trajectory optimisation in a fixed time impulsive transfer during the preliminary mission design phase. Primer vector representation in polar coordinate allows us to de-couple the in-plane and the out-of-plane components of the analytical solution which has been derived in the previous work. This research is focusing on the in-plane solution derivation to provide a complete analytical solution. The result is useful in the optimality analysis of a three-dimensional transfer without the necessity to integrate the state transition matrix to obtain the primer vector profile. The analytical solution application in space trajectory optimisation in this research chose the Earth-Mars transfer as the study case for both single and multi-revolution transfers. In a multi-revolution case with long transfer duration, a linear approximation of the Lambert segment in gradient base optimisation is not sufficient. A novel approach is proposed, using the information from primer vector analysis as the initial guess for mid-impulse position and time, by implementing a higher-order expansion of the flow using differential algebra (DA) to approximate the perturbed trajectory. It will be demonstrated that the non-linear approach will improve the optimisation of a multi revolution transfer problem