Abstract
The problem of launching a rocket into the Earth's orbit has already been solved using the regularization method in previous studies. But the regularization method remains relevant for application to solving integral equations of the first kind, which determine the components of speed and acceleration. The problem of optimal control of propellant consumption during the insertion of a rocket into a circle orbit of the Earth is solved using regularized solutions of integral equations of the first kind which are solutions of corresponding Euler equations on discrete-time net. The influence of the regularization parameter and some additional parameters on precision of discredited problem is investigated. Calculations are carried out for existing chemical rocket engine and promising plasmic one. Considered algorithm is summed up easily to problem of suborbital flights by setting desired coordinate system and modifying motion equations. Conclusions were drawn about the required speed for the lowest fuel consumption, as well as about the problem for a single-stage rocket. Thus, the development of a plasma rocket engine with an exhaust velocity is more than ten times higher than that of a chemical one.
Highlights
A problem of the trajectory optimization of a rocket or a spacecraft with a rocket engine belongs to a class of the dynamic systems optimization problems
The problem of insertion of a rocket into the desired orbit in the view of minimal consumption of propellant leads to solving the set of two ordinary differential equations in the components of the velocity
Summarizing the differential equations, the ordinary differential equation in the mass of a rocket from the time connecting it with the free-fall acceleration, the ballistic coefficient of atmosphere depending on the height, the components of the velocity of exhaust gases, and a rocket are gotten
Summary
A problem of the trajectory optimization of a rocket or a spacecraft with a rocket engine belongs to a class of the dynamic systems optimization problems. For each couple of the numbers using the regularization method, the integral equations of the first kind (Eq 6) in the velocity υy(τ) and (Eq 9) in the acceleration υ′x(τ) are solved.
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