On one problem of quadcopter control with given intermediate values of different parts of coordinates

The work is devoted to the issues of mathematical modeling of the spatial motion of a quadcopter and the construction of program control laws that ensure flight with the values of part of the coordinates of the phase vector specified at intermediate times. A structural diagram of a quadcopter with four propeller engines is used, which allows for movement in space, vertical takeoff and landing. Based on the laws of theoretical mechanics, a system of differential equations is obtained that describes the spatial motion of such a quadcopter. For a linearized mathematical model of quadcopter motion, the problem of constructing program control laws with given initial and final values of the phase vector, as well as the values of part of the coordinates of the phase vector at two intermediate moments of time, has been solved. A necessary and sufficient condition for the existence of program control is obtained and the corresponding movement of the quadcopter is described. Control functions and corresponding phase trajectories of motion are constructed. To illustrate the results obtained, for specific initial, final and intermediate values, explicit expressions of the program control function, program motion are obtained and the corresponding graphs are constructed.