Quaternion models and algorithms for solving the general problem of optimal reorientation of spacecraft orbit

The problem of optimal reorientation of the spacecraft orbit is considered in quaternion formulation. Control (vector of the acceleration of the jet thrust) is limited in magnitude. To solve the problem it is required to determine the optimal orientation of this vector in space. It is necessary to minimize the duration of the process of reorientation of the spacecraft orbit. To describe the motion of the center of mass of the spacecraft we used quaternion differential equation of the orientation of the spacecraft orbit. The problem was solved using the maximum principle of L. S. Pontryagin. Accounting the known particular solution of the equation for the variable conjugated to a true anomaly, we allowed to simplify the equations of the problem. The problem of optimal reorientation of the spacecraft orbit is reduced to a boundary value problem with a moving right end trajectory described by a system of nonlinear differential equations of the fifteenth order. For the numerical solution of the obtained boundary value problem the transition to dimensionless variables was carried out. At the same time a characteristic dimensionless parameter of the problem appeared in the phase and conjugate equations. The original numerical algorithm for finding unknown initial values of conjugate variables, which is a combination of the 4th order Runge – Kutta method, modified Newton method and gradient descent method is constructed. The use of two methods for solving boundary value problems has improved the accuracy of the boundary value problem solution of optimal control. Examples of numerical solution of the problem are given for the case when the difference between the initial and final orientations of the spacecraft orbit equals to a few (or tens of) degrees in angular measure. Graphs of component changes of the spacecraft orbit orientation quaternion; variables characterizing the shape and dimensions of the spacecraft orbit; optimal control are plotted. The analysis of the obtained solutions is given. The features and regularities of the optimal reorientation of the spacecraft orbit are established.