Guaranteed Control of the Spacecraft on the Elliptical Orbits

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Guaranteed Control of the Spacecraft on the Elliptical Orbits

Introduction

orbital motion spacecraft

From ancient times to present, human beings never stopped exploring the universe. By the efforts of scientists of many generations, human space industry has made great progress. On 4 October 1957, the Soviet Union launched the first artificial Earth satellite, named Sputnik-1. It was a prologue to development of the world's entire aerospace industry. After one year, the United States launched Explorer-1, the Cold War Space Race began between the two nations. In the end of 20 century only a few countries in the world have the capability of independently launching satellites. So far there are 4635 satellites are orbiting the earth, but only 37.5% of orbiting satellites are active. This means that there are 2898 pieces of space trash hurtling around the Earth at high speed. The main purposes for the operational satellites are: Communications 742 satellites, Earth observation 596 satellites, Technology demonstration 193 satellites, Navigation 108 satellites, Space science 66 satellite, Earth science 24 satellites, Space observation 9 satellites. Those satellites 61.6% are orbiting in low-earth orbits(LEO), 30.6% orbiting in geostationary orbits, 5.6% are orbiting in medium-earth orbits and 2.2% in elliptical orbits.

1. Overview of this problem

The tasks of controlling the spacecraft have been solved many times, but each of solutions have their shortcomings. Dynamic model of circular orbits CW equations were derived from Clohessy and Wiltshire in 1960. Most of control methods of circular orbits are based on CW equations, but if use this dynamic mode on the control of elliptical orbits, there will be a large error. Then dynamic models of elliptical orbits: TH equations, Lawden equations and other equations were proposed. Yamanaka and Andersen obtained the solution matrix of the Lawden equation by means of variable substitution. The accuracy of solutions of the Lawden equation is the same as the CW equation, precision of simulation is not very high. Based on these dynamic models, the scholars have proposed different methods; for instance, the multivariable all-coefficient adaptive control method, but this method requires real-time estimation of the parameters and true anomaly of target spacecraft, process complex, the inhibition effect of this method on coupling is substandard. Based on the above research, Inalhan deduced the periodic closure constraints for the initial position of satellites.

We confine ourselves to a few examples of the use of spacecraft where the motion along a given trajectory is subject to a considerable number of small perturbations and where the trajectory of motion often change. Such as the Cassini-Huygens space program of the United States, in 1997 an unmanned spacecraft Cassini was launched. Although the Cassini is the fourth space probe flied to Saturn, it is the first one to surround Saturn. After arrived at Saturn in 2004, it began to explore Saturn and its natural satellites, as well as Saturn's rings. Cassini-Huygens regularly transmits to the earth high-quality photographs and scientific data of object. In view of the large number of Saturn's natural satellites, it is necessary to change the station's orbit not only for the purpose of approaching the objects under task of exploration, but also for avoiding collisions by fragments that rotate around Saturn. As part of the mission, the Cassini - Huygens space probe has been boosted by gravity many times, flied through Venus two times, once again through the earth, through Jupiter and eventually reaching Saturn. With these movements, the ion thruster will continuously act on the station deflecting it from a predetermined orbit.

Availability and Urgency

A reasonable control framework which brings no additional overhead is very important to space activities. According to Tsiolkovsky rocket equation, based on the principle of conservation of momentum, device through a propulsion system that consumes itself, can generate and obtain acceleration at the speed of the original operation. So any change in the trajectory of motion by controlling the engine of the spacecraft is due to a reduction in the mass of the apparatus. Obviously, monitoring the fuel consumption of the spacecraft is a priority. Also, due to the high cost of any space launch, the control of the apparatus must be guaranteed. Collision with even a small object in space can lead to the loss of a spacecraft or departure from a given orbit can be irreversible because of the limited fuel resources.

The relevance of this work can also be in the commercial application of the developed spacecraft control system in elliptical orbits. For private companies such as television, communications company.

Objective of work

Task process

Article structure

On any spacecraft always exist perturbations.

They can be related to the thermal effect: the spacecraft heats up during the time on the solar side and cools down when it hits the shadow of the Earth, the Moon and other celestial bodies; the attraction of other planets is acting on the satellite; collisions with space debris are possible while moving on orbits. Any perturbation can bring a deviation from a predetermined orbit. This article is devoted to solution of this problem.

The motion of a satellite around the Earth or other celestial bodies obeys the laws of celestial mechanics (Kepler's laws). The equations for elliptic orbits referenced the paper of Inalhan [2002] in details. He proposed a mathematical model of satellite motion, on which the present work will be based. It is necessary to take into account the impossibility of carrying out the experiment in real conditions such as high cost of the satellite, the risks of losing control of the spacecraft, the observations speed a lot of time, etc. Due to the rapid development of the capabilities of computer technology, it is possible to build dynamic model models of spacecraft, within the framework of the chosen mathematical model, to calculate flight parameters. There are shortcomings in this approach such as the reasonableness of the equations of motion, the accuracy of numerical solution. However, we can use MatLab solve numerically systems of differential equations with many unknowns.

In the chapter "Mathematical model", a mathematical model of the motion of a spacecraft in an elliptical orbit will be chosen. In this chapter, we will analyze the parameters of the mathematical model and the constraints imposed by the mathematical model.

In the chapter " Integrated control", the theory of constructing a guaranteed control will be introduced, the control is constructed to solve the problem of stabilizing the spacecraft in a given orbit, and also a guaranteed control for the transition of the spacecraft from one elliptical orbit to a given orbit in the presence of perturbations.

In the chapter "Modeling", a mathematical simulation of the motion of a spacecraft will be carried out under the influence of perturbations or without it. The mathematical modeling program of motion in two situations: with and without control. In the presence of the control, modeling of stabilization in orbit and the transition from one orbit to another.

2.Mathematical model

The solution of any control problem begins with the choice of a mathematical model. The motion of the spacecraft in the gravitational field of the center of gravity such a center can be the Sun, Earth, planets, and other celestial bodies obeys the laws of celestial mechanics (Kepler's laws). The orbit of a planet or a spacecraft is an ellipse with the rotation center at one of the two foci.

In the framework of this paper, we will consider a motion along elliptic orbits. For an elliptical orbit, the eccentricity value is in the interval (0,1). For a circular orbit, the eccentricity value is 0. The parameters and b are called the semi-major and semi-minor axes.

The position of the spacecraft in space is described in the general case by three coordinates. Accordingly, we have a system of three differential equations of the second order (according to Newton's second law). From the system of three equations of the second order, we can transfer to a system of six first-order equations, in order to obtain the control problem in the usual form.

x - states of object; A - matrix of object; B - control matrix; y - observation vector; C - observation matrix

A mathematical model of the movement of a spacecraft on an elliptical orbit was obtained by Inalhan [2002] and will be used in this paper. A dynamic model on elliptical orbits is described by equations:

Where , are states of object; where e is eccentricity; where m is mass of satellite; ; where и is the flight path angle from local vertical, value form to ; where u is control equations, the equations are linear and periodic (T = 2р)

For elliptical orbits, the changing rate of :

The chosen mathematical model describes the motion of a spacecraft in an elliptical orbit. Within the framework of this model, the center of gravity is assumed to be stationary (or moving uniformly rectilinearly). In most of situations, the immobility of the center of gravity is justified, such as the case of a very massive body and a body of significantly less mass in two-body system. In the problem of motion of a spacecraft, mass of a spacecraft great less than the mass of the object around. The validity of the assumption of the immobility of the center of attraction is obvious.

In the chosen mathematical model, both the spacecraft and the object around which it refers are taken as material points.

Also, the chosen mathematical model does not take into account the attraction of the spacecraft by other celestial bodies. Within the framework of the present work, influences will be taken into account as perturbations. The chosen mathematical model can theoretically claim universality, in practice, if the perturbations too strong and the control will not be able to stabilize the spacecraft. And in this mathematical model, the control action is continuity.

3. Analyzing parameters of mathematic model

In this part, we are going to analyze the parameters which have the greatest impact on the control of model. The model will be constructed by these influential parameters. That means since use the worst parameters, system with the highest disturbance, so system can be stable in any cases.

Variable substitution:

relative motion of spacecraft is described by equations:

An elliptical orbit can be specified by semi-major and semi-minor axes, true and eccentric anomaly on orbital plane. In the paper we confine ourselves to considering the change in the orbit without changing the plane of motion of the spacecraft and with the preservation of the eccentricity. Geometrically, this means that the orbit changes to a geometrically similar one.

, , .

Analysis of the matrix A(и). When formulating the control problem, we should take into account the consistency of the changes in the values of the parameters of this matrix, the changes in the values of the roots of the characteristic equation for the matrix A(и). To construct the majorizing model and to synthesize the guaranteeing control, we will take into account the roots of the characteristic equation with the largest real part.

In figture1-3, the eccentricity is changed from 0 to 0.9, we can see the influence of eccentricity `e' on periodic matrix A.

Fig1. the value of when e from 0 to 0.9

Fig2. the value of when e from 0 to 0.9

Fig3. the value of when e from 0 to 0.9

We define as the coefficient in front of matrix B. Picture4 shows the dependence of on eccentricity `e'.

Fig4. the value of when e from 0 to 0.9

Because is the coefficient before the control matrix, the larger can make the control more effective, the system energy consumption lower. It can be noted that the effectiveness of control significantly decreases when and significantly increases near .

We define as the coefficient in front of matrix . Picture5 shows the dependence of on eccentricity `e'.

Fig5. the value of when e from 0 to 0.9

It is seen, that the efficiency of control for the same values of eccentricity and the angle increases by several tens of times in comparison with the graphs presented above. So the most favorable values of the parameters of the system matrices are and .

We define the initial state of system: , ,

, , .

In the framework of the stabilization problem, should change to

in desired time.

In the control of orbit transition, where is:

Integrated control

We assume that the matrix C in the mathematical model has the form of a unit matrix, such as:

In this case, we know completely the state of the object (the spacecraft) in any time. In reality this assumption does not always take place, in practice it is impossible to measure all values of the state vector, and also any measurement has a certain error. In the framework of this paper, we will consider the state of the spacecraft to be completely known.

We define the least favorable parameters of the control matrices and construct the majorizing model of the system.

Here parameters of matrices and are calculated when and .

In the model include perturbations in the roll angle (moving in its orbit, the spacecraft is periodically subjected to sharply variable thermal effects - heating by the rays of the sun during its stay above the illuminated side of the Earth, cooling during flight in the shadow of the Earth, attraction from other planets collision with small passive objects). All this causes deviations from the given orbit. As discussed above, all effects on the spacecraft are included in , except for the attraction to the object around it.

The function must satisfy the condition , where L is the maximum possible perturbations, and these perturbations must be small in comparison with the effect of the center of attraction. In the work in the simulation, we will consider as a harmonic function.

To guaranteed control, we introduce a quality function that includes the weight matrices Q, R, P.

where is the difference between the initial state and the desired stateЈ¬. The representation of has the same form for the stabilization problem and the orbital transition problem.

Control minimizing is described by the function:

,

where S is a positive definite matrix containing constant parameters. And a solution of the algebraic Riccati equation is:

4. Calculation of the control matrix

In order to contain information about the initial state of the object, the control goal and the specified control period for the matrix S, we assign the weight matrices Q, R as follows:

The matrixes R and P at least are positively semidefinite, so they must meet the conditions:

The matrix S with the given initial conditions after solving the Riccati equation is:

Matrix and the roots of the characteristic matrix equation are:

We can see that all roots of the characteristic equation have a negative real part, so the system is stable.

5. Parameters changing in control process

For these two kinds of situations: stabilization and orbit transition. Control is synthesized with the matrix S which calculated from the initial conditions. Consider a control option with the calculation of the matrix S at several discrete time points within the same movement. Let the movement with the help of synthesized control be performed for a period of time T. We divide this period of time T into several intervals and , .

Fig6.

The matrix S with initial conditions was calculated when , and . From time to , the matrices S and K at constant values. At the time , the spacecraft will change its state from to , the matrices S and K: and . In similar manner at the time , the spacecraft will change state from to , the matrices S and K: and . And same as

During this movement, corrections were made to the control parameters n times at each time instant . In the next chapter, modeling will be performed for the matrices S and K and the control parameters changing.

6.Modeling

The vector shows perturbations of the state of the object, the deviations from the nominal values. Take , the perturbation plot will look like this:

Fig7.

Fig8.

7. Stabilization modeling

Guaranteed control will be determined by the control rate:

The object with perturbations and control rate:

Figures 9-12 show transient processes in the control system:

Fig9.

Fig10.

Fig11.

Fig12.

It can be seen from the graphs that the stabilization of the satellite takes place in three seconds. The influence of perturbations acting on the object is successfully parried. The control rates are showed in figures 13-14:

Fig13.

Fig14.

8. Orbit transition modeling

Guaranteed control will be determined by the control rate:

The object with perturbations and control rate:

Figures 15-20 show transient processes in the control system:

Fig15.

Fig16.

Fig17.

Fig18.

Fig19.

Fig20.

We can see in situation of orbit transition, the value of change to after 3-4 seconds.

9. Control efficiency assessment

In this paper, we have synthesized the control for the stable state and the orbit transition. The control was synthesized in two ways: calculated from initial conditions of matrix S; calculated matrix S at several discrete time points.

In this section, we will evaluate the effectiveness of the synthesized control and compare the effectiveness of controls for two cases. It is logical to assume that the control in the second case will be more effective. This assumption is based on the fact that in the control synthesis variant with the calculated matrix S at several discrete instants, the matrix S contains information on the intermediate states of the object in the process of motion; The control is adjusted taking into account the state of the object.

In the synthesis of control with the calculated matrix S at several discrete time points, the matrix S was calculated every second after the start of the motion.

According to iterative calculation of matrix S, the results of control depends on time interval of calculation, the larger the value of this time interval be taken, the smaller difference between S= constant and calculated S.

Based on the simulation results, we can say that the stabilization of the spacecraft or orbit transition occurs in about three to four seconds. So, the time intervals greater than 3 to 5 seconds correspond practically to the case S = constant. Iterative calculation of matrix S, data transfer of motor takes some time in practice, this time depends on the computational power, including the computational power of the Mission Control Center and spacecraft, the distance of the spacecraft from the station, the current state and position of the spacecraft, the computer-engine communication system. Thus, we have limitations on the time interval: . The problem of determining the optimal value of T will not be considered in this paper.

According to Tsiolkovsky rocket equation, monitoring the fuel consumption of the spacecraft is a priority. We will evaluate the effectiveness of control from the point of view of fuel consumption on the spacecraft. The volume of fuel on the spacecraft is determined by the design and dimensions of the apparatus.

The mass flow of fuel on the spacecraft is proportional to the integral over time from control:

Where M is the mass of fuel consumed in the time period from to the time under the effective control u (t), where R is the proportionality coefficient.

We will perform calculations based on the data obtained by modeling the motion with the help of MatLab.

Defining: is the fuel mass consumed for the case S = constant and is the fuel mass for the discrete (in 1 second) case of the calculated matrix S:

We can see that <, this shows that in the case with the discrete (after 1 second) calculated matrix S, the control more efficiently.

Conclusion

In this paper, we study the problem of controlling a spacecraft in elliptical orbit, a reasonable choice of a mathematical model for the motion of a spacecraft in an elliptical orbit was made. The mathematical model is chosen from the article Inalhan [2002]. The parameters of the mathematical model and the constraints imposed by the mathematical model under consideration are analyzed. Control cases for various parameters have been considered, the most favorable and least favorable parameters for control have been found. The efficiency of control at various points of spacecraft on an elliptical orbit has been studied.

The theory of constructing guarantee control within the framework of the selected mathematical model is considered, guaranteeing control for the chosen majorizing model is constructed to solve the problem of stabilizing the spacecraft in a given orbit and also transition from one elliptical orbit to the another.

Program is written on Matlab, including a numerical solution of the system equations describing the motion of the satellite, mathematical modeling of the motion of a spacecraft under the influence of perturbations or without it, the simulation and construction of the corresponding motion diagrams of the spacecraft for cases with perturbations or without perturbations, control of stabilization in orbit or the transition from one orbit to another.

The results show that the constructed guarantee control allows stabilizing the spacecraft on a predetermined elliptical orbit under the action of perturbations. Also, the guaranteeing control allows solving the problem of transferring the spacecraft from one elliptical orbit to another. Efficiency of the synthesized control for different conditions was analyzed.

The method can be used in the guaranteeing control of a satellite with known parameters (mass, the eccentricity of the orbit). It should be noted that in this version of the mathematical model there is an approximation of the continuity of the control action, but it justifies itself for most applied problems.

In the continuation of solving the problems of spacecraft motion, it is possible to set more complex and general tasks. In this article, restrictions were imposed on the mathematical model, such as the fully known state of the object, the continuity of control, decrease in the mass of the spacecraft, and so on.

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