Approximate Solution of Required Velocity in Elliptical Earth Model Using Piecewise Linear

In this paper, an approximate solution of required velocity with final position constraint is derived using a piecewise linear gravity assumption for elliptical earth model. In this approach, the total flight time is divided into several time intervals and the gravitational acceleration is assumed to be linear at each interval. The solution gives an explicit relation in terms of the current position vector, desired final position and flight time in three dimensions. The accuracy and computational burden of the method are obtained numerically in terms of the number of time intervals, and compared with linearized solution and Zarchan's iterative algorithm. Numerical solution shows that the present method has better accuracy than the two mentioned approaches with the same computational burden up to a range angle of 18 deg for minimum energy trajectory in an elliptical earth model. The presented method can be extended for two or multi-body problem and also for the computation of sensitivity matrix of required velocity.