OPTIMAL CONTROL OF PROPELLANT CONSUMPTION DURING INSERTION OF ROCKET INTO A CIRCLE ORBIT OF THE EARTH

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.


INTRODUCTION
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. Its solution leads to searching for the local or global extremum of a beforehand defined functional determined on the set of the solutions of the controlled dynamic system satisfying some conditions [1][2][3]. As a rule, the conditions can be both internal and boundary to the control process. Thus, we consider a rocket or a spacecraft to be the controlled dynamic system. Applying some restrictions to it we have some formulation of the optimization problem [4][5][6]. It is well known that its solution is found with the maximum principle by Pontryagin transferring the optimization problem to the boundary problem [7][8][9]. Besides, we have to determine explicitly the performance criteria and restrictions [1,4,10]. There are two models of rocket engine performance [11][12][13]. The first of them matches the non-controlled engine when the reactive force and the relative velocity of exhaust gazes are considered to be constant [14][15][16]. The engine just can be turned on or off. That is the most realistic model. The second of them matches the ideal limited power engine when the power of the engine is constant [17]. Under this restriction, we can vary the reactive force and the exhaust velocity [14]. In this work, we vary both the reactive force and the power of the rocket engine by varying the consumption of propellant and keeping the exhaust gases velocity. The optimal control problem is to find the trajectory corresponding to the minimal consumption of propellant.
A problem of insertion of a rocket into an orbit of the Earth at the height h1 with the first orbital velocity Υ1 during the time T1 supplying minimal propellant consumption is considered. A similar problem has been solved using the regularization method in [18][19][20]. In this work, the regularization method is applied to solve integral equations of the first kind determining components of the velocity and the acceleration. If there are the horizontal component of the velocity υ x (τ) and the vertical one υ y (τ) then the set of the equation of motion of a body with the varying mass m(τ) in atmosphere is [18] (Eq. 1) with the initial conditions (Eqs. [2][3]. Where μ≤m(τ)≤m is the variable mass of a rocket with propellant, kg; μ is the mass of construction of a rocket, kg; υ(τ) is the velocity of a rocket; w(τ) is the control function equal to the consumption of propellant trough one second, kg/s; a=const=2500 m/s is the relative velocity of exhaust gases; 0≤c[h(τ)]≤0.2•10 −7 kg/m is the generalized ballistic coefficient of air; g=9.81 m/s 2 is the free-fall acceleration. (1) The optimal control function w˜(τ) must be positive at a time interval 0≤τ≤T 1 . Gradual decrease in the consumption of the mass of propellant begins at the time instant τ=0 when the velocity is equal to Υ 0 . The optimal control function w˜(τ) and the time instant T 1 when burning of propellant is stopped are desired while (Eq. 4) is the velocity of a rocket equal to the first orbit velocity Υ 1 at the height h 1 reached at the instant T 1 (Eq. 5). Where G= 6.6743• 10 −11 m 3 s −2 kg −1 is the gravitational constant; M=5.97•10 24 kg is the mass of the Earth; R 0 = 6.371• 10 6 m is the radius of the Earth [21][22][23].
If From the set of (Eq. 13) the differential equation for the varying mass m(τ) is gotten which is connected with the velocity υ(τ) with the initial condition (Eq. 14). As far as (Eqs. [15][16][17] there is (Eq. 18).
The consumption of propellant is found from the same set of equations as (Eq. 19).
The procedure of searching for the optimal consumption of propellant using solutions of the integral equations of the first kind is the next. The consequence of couples of the numbers {h (n) 1 ,T (n) 1 } is set, and to each the height h (n) 1 there is the first space velocity Υ (n) 1 . For each couple of the numbers using the regularization method, the integral equations of the first kind (Eq. 6) in the velocity υ y (τ) and (Eq. 9) in the acceleration υ′ x (τ) are solved. The acceleration (Eq. 20) can be calculated and the velocity υ′ x (τ) from the ordinary differential equation (21) with the initial condition (Eq. 22) can be found [24][25][26].
Substituting the functions υ x (τ), υ′ x (τ), υ y (τ), υ′ y (τ) into the (Eq. 18) the mass m(τ) and the consumption w(τ) from the (Eq. 19) are found. Then from the sequence of couples of the numbers {h (n) 1 ,T (n) 1 } such a couple {h (m) 1 ,T (m) 1 } is found on which the propellant consumption (Eq. 19) reaches its minimum (Eq. 23): As a result, the functions w (m) ,h (m) 1 ,T (m) 1 are gotten which are considered to be approximate regularized solution of the problem of optimal control.

DETERMINATION OF VERTICAL COMPONENT OF THE VELOCITY AND THE ACCELERATION
For each couple of the numbers {h (n) 1 ,T (n) 1 } the right-hand side of the (Eq. 6) is put approximately, and h (n) where (Eq. 24). The integral equations (25)(26) has the kernel K(h 1 ,τ)=1 and the function (27): The required approximate (regularized) solution of the (Eq. 25), Aυ y =u δ , is the function υ y (τ) which is the solution of the integrodifferential equation (28)  Minimizing the functional (Eq. 31) is a conditional extremum problem. It is solved by the method of undetermined Lagrange multipliers; the function υ y (τ) is found minimizing the smoothing functional (Eq. 33) where [18] (Eq. 34).
This is an unconditional extremum problem, in which the regularization parameter is determined from the (Eq. 35) with the solution (Eq. 36) depending on the discrepancy δ.
The parameter γ may be determined both by the discrepandy (Eq. 35) and other ways [18,27].
Thus, the problem of searching for approximate (regularized) solution of the (Eq. 26), Aυ′ x =u δ , leads to solving the set of linear algebraic equations for the vector (65). Then the vector (66) is found solving the ordinary differential equation (21).

RESULTS OF CALCULATION
The problem of injection into a circle orbit at the height h 1 =500 km during the time T 1 =600 s of a one-stage rocket with the total mass of its construction and payload μ=1000 kg, and the mass of propellant ∆m=1000 kg is considered. Consequently, the start mass of a rocket is equal to m 0 =2000 kg. The velocity υ y (τ) is an approximate solution of the integral equation (6) (Fig. 1a). The acceleration υ′ y (τ) is found by differentiating υ y (τ) numerically (Fig. 1b): Then the velocity (υ y ) M+1 necessary to solve numerically the ordinary differential equation (18) in the mass m i (i= 0,1,…,M) is:  (Fig. 1b). The velocity υ x (τ) is found from the acceleration υ′ x (τ) by solving numerically the ordinary differential equation (21) on the time net (υ x ) i (i=0,1,…,M) (Fig. 1a). Then the velocity (υ x ) M+1 necessary to solve numerically the ordinary differential equation (18) in the mass m i (i=0,1,…,M) is: A one-stage chemical rocket with the velocity of exhaust gazes a=2.5 •10 3 m/s is able to inject into a circle orbit just its own propellant with minimal mass of construction (Fig. 1c). Therefore, one uses multi-stage chemical rockets. To analyze a one-stage rocket engine demonstrative enough another kind of a rocket engine promising at the present is considered. There are projects of plasmic rocket engines with the velocity of exhaust gazes a=2.5 •10 4 m/s reducing by 10 times the consumption of propellant w and keeping the reactive force aw. The consumption of propellant of one-stage plasmic engine injecting into a circle orbit at the height h 1 during the time T 1 a rocket with the start mass is analyzed (72) (Fig. 1d).
The Euler equation for the integral equation (6) in the velocity υ y (τ) corresponds to the right-hand part The first orbital velocity Υ 1 is the right-hand part of the equation (9) in the acceleration υ′ x (τ). A consequence of the heights  (Table 1) as the first orbital velocity decreases when the height increases according to (5).
To solve a problem of keeping predetermined distribution of the velocity υ x (τ), υ y (τ), and consequently the acceleration υ′ x (τ), υ′ y (τ), corresponding to the trajectory x(τ), y(τ), the regularization parameter γ x (τ), from the (Eq. 62) and γ y (τ), from the (Eq. 48) has to be found. The regularization parameter is able to be found analytically by the method of simple iteration or the iteration-variation method [30,31]. In that case the right-hand side of the integral equation (6) in the form of the height of the orbit and the integral equation (9) in the form of the first orbital velocity Υ 1 will deviate from the predetermined one. Similar problems may be formulated and solved to transport a rocket with a payload into desired point. Polar of spherical coordinate system can be also used.

CONCLUSIONS
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 (when a movement is in the plane x y and two integral equations. 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. The integral equations follow from the laws of mechanics: υ y (τ)=dh(τ)/dh,=>υ y (τ)dτ=dh(τ), and υ′ x (τ)=dυ x (τ)/dτ,=>υ′ x (τ)dτ=dv x (τ). The integral equations are solved using the regularization method and an Euler equation on a time net as the set of linear algebraic equations in the velocity υ y (τ) or the acceleration υ′ x (τ).  The problem of insertion of a multy-stage rocket into desired orbit in the view of minimal consumption of propellant is analogous to the problem for a one-stage rocket. But a one-stage rocket injects just itself without any payload. Therefore, working out a plasmic rocket engine with the velocity of exhaust gases more tenfold than chemical one has is promising. Problems of suborbital and interplanetary flights can be solved using the procedure in the spherical or polar coordinate system. Today there are used low power ion-plasma rocket engines for suborbital flights. Manned flights are reasonable on the basis of high power plasmic rocket engines with the reactive force comparable to chemical ones. To search for a solution of the integral equations closed to known distributions of the velocity and acceleration in the time there is a need to find the regularization parameter in the time according to those functions. In that case the right-hand sides of the integral equations deviate from desired values.