The article presents a formulation of a problem of trajectory optimization using low-thrust engines for an optical space system based on diffractive membranes. A methodology, where first stage nominal trajectories and control programs are selected and then corrected at the longrange guidance, has been developed for solving the problem of optimizing the trajectories of a flight to a geostationary orbit. At the final stage, algorithms for terminal control are formed, which allows to deliver a cosmic optical system based on diffraction membranes to a given point in the geostationary orbit. The end result is acquisition of Pareto-optimal solutions in the coordinates "characteristic speed-duration of the flight", where each point of the set of solutions has a corresponding a measure of accuracy of payload delivery to a geostationary orbit at a given set of coordinates.
Since spacecraft of this type have significant dimensions and, accordingly, mass-inertial characteristics, the process of motion control is significantly hampered. This calls for a solution of the problem of joint optimization of trajectory and angular movements.
Let us consider a ballistic scheme for a transfer of a spacecraft from initial circular orbit to operational (geostationary) orbit, with insertion into a given station and return of the space tug to the initial orbit.
The optimization problem in a general statement is shaped as the problem of tied choice of design parameters, p P , ballistic parameters, bB and set of functions u(t,x), x(t) out of allowable multitude D, ensuring the transfer of a spacecraft from the initial state x(t0 ) x0 to a finite multitude x(t f ) X f with the maximum value of the optimization criterion.
Relative payload mass (relation of the payload mass Мpl to launch mass of the spacecraft М0) is adopted as the main optimality criterion : М pl
max .
М 0
Another criterion, no less important, is the overall transfer time ТΣ, which is a sum of the powered flight time ТP and the time of unpowered legs ТUP. The latter result from the need to make navigational observation and to "phase" precision targeting of an EP-powered spacecraft.
Let us represent the launch weight of a spacecraft as the sum of the masses of its functional elements: power plant (consisting of the reactor, energy transformer and radiator); thruster unit (consisting of cruising EP thrusters and controlling thrusters with their controls); propellant supply for the direct and return transfer, including additional expenses for control; propellant storage and supply system; payload; the body of the spacecraft:
М 0 М PP М EP М P1 М P2 М PSS М PL М B , where М0 is the launch mass of the spacecraft; МP1, МP2 is the propellant mass for the direct and the return transfer respectively; МPL is payload mass; МPP is the power plant mass; МEP is electric propulsion mass; МPSS is the propellant storage and supply system; МB is the spacecraft body mass.
The criterion µ may be represented as [6]: 1 Az1 z 2 z1 z 2 B , um(ta)xU p, b, u(t), T , М 0 um(ta)xU 1 Az2 pP pP (1) (2) where A 1 PSS с PP c EP ; αPP - is the relative mass of the power plant; γEP - is the relative mass of electric
T 2 P propulsion; μB is the relative mass of the body; γPSS is the relation of the propellant storage and supply system mass to propellant W f 1 ; mass; Р is the thrust of the cruising and controlling thrusters; с is the thruster exhaust velocity; z1 1 exp c z 2 1 exp Wcf 2 ; Wf1, Wf2 – is the final characteristic velocity expense for direct and return transfer.
In this statement, the optimization problem is traditionally divided into two parts: dynamic (selection of optimal trajectories and control laws) and parametrical (selection of optimal design parameters).
Thus, two main optimality criteria are set: the relative payload mass μ or, considering the expression (2), the final characteristic velocity expense Wf, as well as the total transfer time TΣ. Then the optimization of the ballistic schemes, trajectories, and control modes as the ultimate goal is reduced to building a Pareto set in coordinates Wf - TΣ. In addition, every point in the Pareto set must correspond to a measure of meeting the final boundary conditions Φ, for example, in the following expression:
Φ xT f x f .
Here x f is the vector of deviation of the final values of the state vector from required values; Λ is a positively determined weighted coefficient matrix.
The general mathematical model of motion includes the equation of the motion of the center of mass in equinoctial elements, dynamic equations of angular motion, kinematic equations in quaternion form.
The optimal control problem in this statement is extremely difficult, as the state of the controlled object is characterized by a set of variables, some of which are "slow-changing" (trajectorial motion) and some are relatively "fast-changing" (angular motion); besides, the mathematical model of motion includes a perturbation vector. It is apparently problematic to approach this problem with the classical methods of optimal control (the maximum principle and dynamic programming).
Therefore, we propose an approach to the solution of the dynamic problem, which is based on the principle of the extension – narrowing of the class of admissible states and controls [7].
The control vector u will include only the angles of the thrust orientation vector, and other components (controlling torques МХ, МY, MZ) will be subject to constraints. Let us impose constraints also on the state vector x at the final point of the trajectory. In this manner we obtain the following mathematical model:
x* f x*, u* , x* t0 x*0 (3) where x* h, eX , eY , iX , iY , F T , u* , T , with extended functional and constraints x* t f x* f h f , exf , e yf , ixf , i yf , F f T X f , max М x (t) М max_x ; max М y (t) М max_ y ; max М z (t) М max_z . (5)
t t t Here h, ex , e y , ix , i y , F - are the equinoctial elements; , - are the thrust orientation vector angles; а – is reactive acceleration; МMAX_Х, МMAX_Y, MMAX_Z are the maximum possible controlling torques that can be created by controlling thrusters of the spacecraft.
The conditions of (5) are checked as the result of a numerical modeling of the basic system of equations, with optimal control found by solving the problem for the simplified mathematical model.
Assume that the spacecraft is moving along a near circular orbit, so the eccentricity can be taken as zero. Assume also that the radial component of the acceleration is zero.
Then the thrust direction is determined by the angle ψ between the transversal and the thrust vector, and the projections of the reactive acceleration to the axis of the orbital system of coordinates are:
P P aT cos , aS 0, aW sin (6)
M M Let us also exclude from the differential components the equations that describe the changes in the longitude of the ascending node and the perigee argument.
T L V f adt min 0 (4)
The system (3), taking into consideration (6), is transformed to the equations of a near-circular motion of a small-thrust spacecraft. By averaging the "slow-changing" variables r (average radius of a near-circular orbit, equivalent to semi-major axis) and i (orbit inclination) along the "fast-changing" variable u (polar angle of the argument of latitude) we can obtain the "asymptotic" model of motion [8].
Thrust vector control for transfers between non-coplanar orbits requires the sign of the аW to change twice per revolution. The thrust angle orientation control is set in the following way: (W , u) m (W )signcos u , (7) where ψm is the amplitude of oscillations of the angle ψ, W is current characteristic velocity and u is the argument of latitude.
Analytical solutions, obtained within the "asymptotic" model, describe a "universal" trajectory of a transfer between non coplanar orbits, that does not depend on the spacecraft's design parameters (Figure 3). 4. Method of selecting motion control laws of the space optical system on based diffractive membranes during transfer to given point on the geostationary orbit
The method includes an algorithm of developing nominal programs for thrust vector control, an algorithm of EP thrust magnitude adjustment, algorithm of terminal control development using motion models in discrete setting, numerical algorithm of building a set of Pareto-optimal solutions.
The goal of the control at the stage of lifting to GEO is narrowing the area G to an allowable area Ga.
The problem will be solved consequentially, in two stages: - development of an algorithm for moving the final state deflection vector ΔХf into an area G’, where one or more of the components of the vector ΔХf satisfy the required precision (e.g. deflection by inclination ∆if);
- developing the control laws and algorithms for narrowing the area G’ to area Ga, where all components of the vector ΔХf satisfy the required precision conditions. 4.1 Goal of control at the long-range guidance stage
If the deflections are considerable, or it is impossible to build in advance a precise model of perturbing accelerations, which may change significantly during the transfer, it is advisable to use multistage control algorithms with end state prognosis a nd identification of perturbations.
Since sufficiently precise models of perturbations from the Earth's gravity field, solar and lunar perturbations are presently available, the parameter to be adjusted is the magnitude of reactive acceleration.
Thrust magnitude has been chosen as the adjusted parameter. Adjustment of actual thrust magnitude will be carried out according to the expression:
P i Pnom 1 T i , Tcalc where Tcalc is the oscillating orbit time at the measured unpowered leg; Ti is the deviation of oscillating orbit time from calculated value.
An example of modeling the trajectorial motion of an EP-powered spacecraft during a transfer to GEO with set values of initial orbit (А0, i0, e0), exhaust speed с and initial acceleration а0 with adjustment of the thrust magnitude depending on the number of corrections N is represented in Table 1. Here ∆P is the systematic error in thrust magnitude; tup is the length of unpowered legs, where the low thrust propulsion unit is, as a rule, shut down to ensure more precise orbit parameter calculat ion and consequent adjustment of thrust magnitude. 4.2 Development of control laws and algorithms at the final stage of GEO transfer
The goal of the terminal control. The goal of the control is to move the end state deviation vector ΔХК from area G’ to area Ga. Assume that the correction maneuver is performed with the help of transversal low thrust, which creates the transversal acceleration аТ.
This problem is set as a terminal control problem with functional
Φ хTf х f min .
Here Λ is the fixed coefficient matrix; x f T f , f , e f T .
The control is structured as a sequence of powered and unpowered leg lengths u 1... i , tup1...tupi T . (8) (9) 2 E Т S Т 0 Here τ and tup are the lengths of the powered and unpowered flight legs respectively. Therefore, the characteristic velocity budget and time required for execution of the maneuver do not depend on design parameters (transversal acceler ation аТ), and are determined only by the initial deflections ΔТ0 and Δλ0. Near optimal control by orbit time and longitude is performed in one step. Here by step we mean a sequence of a powered and an unpowered legs.
2. The problem of terminal control is solved with the help of multistage control algorithm with control parameters adjustment. Let the control law be set by a sequence of powered legs that is taken as decreasing and is defined by the expression [12]: i а 1 i 1 b , (12)
n where i, n are the number of adjustment and total number of adjustments respectively; а and b are the parameters that describe the law of decreasing the lengths of the powered legs.
Then the problem of determining the optimal control law is reduced to a bi-parametric optimization problem, which is stated in the following way: for set initial values of orbital elements, acceleration аТ, number of adjustments n, lengths of unpowered flight legs tup one should find such parameters а and b that ensure the minimum of the functional (9).
Control parameters а and b are found as the result of minimizing the functional of the view (9) and at that for better precision the dependency of the functional from parameter а is approximated by the least squares methods. When the control is adjusted (during motion modeling accounting for perturbations) at every unpowered leg the number of adjustment steps n is also corrected.
A series of calculations of control laws for a transfer of and EP-powered spacecraft to a given station by orbit time and longitude were carried out [9].
The characteristic velocity W budget, depending on the initial deviation by orbit time (∆Т0 = 300…1000 s) is on the order of 4 to 14 m/s.
3. A discrete model for flat motion of a spacecraft under small transversal acceleration (with changes in orbit time, eccentricity and station longitude) was developed in [11], and an analytical solution for the problem of the search for the o ptimal control (lengths of powered and unpowered legs), minimizing final orbit time, eccentricity and station longitude errors, was obtained.
In [23] the authors propose an approximate method for solving the problem based on the three-step control algorithm for circulation period, eccentricity and longitude of the point of standing. Orbit correction is carried out using low-thrust electric rocket engine that produces acceleration in the transversal direction.
Discrete model of plane motion of geostationary satellites under the influence of small transversal acceleration is presented in as [11]: T (k 1) T (k ) 3aТ T3 T (k )3
T3 T (k ) 2π μ
τ(k ), 2π Δλ(k 1) Δλ(k) T3 ΔT (k)
ω3 tп (k) τ(k ),
e(k 1) Δe(k) 2 aT Т3 ΔT (k)
2π μ 1/3
τ(k) , where k = 0,…, N – 1, tП(k) is determined by formulas tп Т0 (1 2m) τ 0T0 , for аТ > 0,
2 2 2π tп Т0 (1 m) Functional τ
2
Where aT – transversal acceleration, 0 - true anomaly angle before correction; ∆ХК = {∆TК, ∆λК, ∆eК}Т – the final state vector, where ΔТК = ТК – ТЗ, ΔеК = еК – еGEO, ΔλК = λК – λР; ТК, еК, λК – values of the orbital period, eccentricity and longitude of the satellite’s standing point on the orbit at the end of the correction maneuver; ТЗ – circulation period of the spacecraft in geostationary orbit, equal to a star day ТЗ = 86164,09 с; еGEO –eccentricity of the geostationary orbit; λР – longitude of the working point of standing of a satellite; ΔТ0 = Т0 – ТЗ, Δе0 = е0 – еGEO, Δλ0 = λ0 – λР, where Т0, е0, λ0 – values of the orbital period, eccentricity and longitude of the point of standing on the orbit before the spacecraft correction maneuver.
Based on the proposed control structure an analytical solution is obtained for 0 , 1 , 2 , tП1 , t П 2 [11] where TС T0 3aт T3 T0 3 T32π μT0 0; where m Z, S 1 1 TС ;
3TС where λC 2π 2π T3 ΔTС τ0 τ 2 2 aт 3 e0 T3 T0 2π μ 1 aт T3 TС 2π μ , τ1
1 aт T3 TС 2π μ , tп2 1 3S TС 12 m τ1 2(11 S3S ) 2
, for аТ > 0, 2 3S 1 S tп2 1 3S TС 1 m τ1 2(1 3S ) 2
, for аТ < 0; 2 3S 2π tп1 λС λB T3 TС
ω3 ,
2π ω3 τ1 T3 TС 1 3aт 3 T3 TС τ1 2ππ ω3 tп2 τ2 , (13) (14) (15) 2π λ B λ0 ω3 τ0.
T3 T0
The presented algorithm has shown high accuracy in modeling of the correction of orbit for a surveillance satellite, fitted with an electric propultion engine. For example, for ΔТ0 = 1000 s, e0 = 0,005, Δλ0 = 0,087 rad the final orbit parameters deviations were: orbital period ΔТK = 1,3 s, standing point longitude ΔλK = 0,150, eccentricity ΔeK = 1×10-4. Durations of the active and passive sections were τ0 = 7758 s, τ1 = 1997 s, , τ2 = 1998 s, tп1 = 260200 s ≈ 3 days, tп2 = 40170 s ≈ 0,46 days.
Figure 4 and 5 shows an example simulation of the orbit control of geostationary spacecraft by using low thrust of EP. 4.3 Building a set of Pareto-optimal solutions of a dynamic problem
Suggested control algorithms make it possible to build a set of Pareto-optimal solutions of the dynamic optimization problem of a space optical system on based diffractive membranes transfer to GEO using low thrust of EP with insertion into a given station.
The sequence of building the set of Pareto-optimal solutions is represented in Table 2. 1. Control program (7) with EP thrust adjustment algorithm (8) and correction of control program without precision GEO formation stage. 1. Control program (7) with EP thrust adjustment algorithm (8) and correction of control program. 2. A near-optimal control by extended set used on the final stage. 1. Optimal control program (7) with EP thrust adjustment algorithm (8) and correction of control program. 2. Control algorithm (12) used on the final stage. 1. Optimal control program (7) with EP thrust adjustment algorithm (8) and correction of control program. 2. Three-stage control algorithm of terminal control (13) – (15) used on the final stage.
Final error margin Φ1 Φ2 Φ3 Φ4
Figure 6 shows a sample calculation of a set of Pareto-optimal solutions for a transfer to GEO with systematic thrust error ΔP = - 2,5%.
A method of solving the dynamic optimization problem of a transfer to a given station on geostationary orbit was developed, including: an algorithm for obtaining nominal thrust control vector programs; algorithm for EP thrust magnitude adjustment on the basis of actual orbit time measurement at the long-range targeting stage; algorithm for obtaining terminal control, using discrete motion models; algorithm for obtaining a set of Pareto-optimal solutions.