injection into orbit optimization using orthogonal polynomialsinjection into orbit optimization using orthogonal polynomials

in this study, the problem of determining an optimal trajectory of a nonlinear injection into orbit problem with minimum time was investigated. the method was based on orthogonal polynomial approximation. this method consists of reducing the optimal control problem to a system of algebraic equations by expanding the state and control vector as chebyshev or legendre polynomials with undetermined coefficients. the main characteristic of this technique was that it converted the differential expressions arising from the system dynamics and the performance index into some nonlinear algebraic equations, thereby greatly simplifies the problem solution. our research effort focuses on applying a chebyshev series expansion to optimize the trajectory profile of a point-mass satellite launch vehicle (slv).this paper is divided as follows: first, the chebyshev and legender series expansion to optimization are introduced. then, the flight mechanics model of the point-mass slv is given. next, our optimization problem is described and optimization results are presented and discussed.