CHAPTER 3 SATELLITES IN FORMATION FLYING

Transcription

1 38 CHAPTER 3 SATELLITES IN FORMATION FLYING 3.1 INTRODUCTION The concept of formation flight of satellites is different from that of a satellite constellation. As defined by the NASA Goddard Space Flight Center, a constellation is composed of two or more spacecraft in similar orbits with no active control to maintain a relative position. Station keeping and orbit maintenance are performed based on geocentric states, so groups of global positioning system (GPS) satellites or communication satellites are considered constellations. In contrast, formation flight involves the use of an active control scheme to maintain the relative positions of the spacecraft. The difference lies in the active control of the relative states of the formation flying spacecraft. A distinction must also be made between formation keeping (referred to here as formation flying) and formation changes. Formation keeping is the act of maintaining a relative position between spacecraft in the presence of disturbances, while formation changing modifies the formation type, changing the relative satellite dynamics. Using a number of small satellites to fly in a formation than to use a single large satellite is the recent trend in space technology. During experiment and analysis with magnetosphere or aurora, high performance satellite is needed which may require large aperture area in its antenna to provide a required coverage area. In spite of using such a high performance satellite, many satellites having small aperture area can be used to fly in circular formation to produce such a large aperture area and this technology is called synthetic aperture radar

2 39 (SAR). By using small satellites in formation flying better results can be produced than single high performance satellite. As satellites used in formation flying have more or less similar configuration, they can be produced in bulk, thus reducing the manufacturing cost of the satellites. Also there is no need for a greater industrial setup for making these small satellites as these can be easily produced with university collaboration. As many satellites are used instead of a single large satellite, the satellites can be configured according to the need i.e. circular formation or hexagonal formation, etc., and there is greater chance for mission success and flexibility. In a single satellite when any instrument fails then total mission will become a failure. But in satellite formation flying the probability of failure occurring in all the satellites is very low. Even when there is failure in one satellite, other satellites can be configured in such a formation to satisfy the requirement. A brief introduction and the types of formation flying and station keeping are described in this chapter. Solution to the Euler Hill s equation and initial conditions for various relative orbits are discussed. The conversion from relative position to orbital elements and the method for finding the relative distance and velocity between chief and deputy satellites are shown. The various perturbation forces affecting the relative motion are analyzed and their simulations by different approaches are discussed in this chapter. The absolute distance keeping between the satellites achieved by a fuzzy logic based and GA based orbit controller is described. GA based orbit

3 4 controller uses orbital element feedback method in which the gains of the method are optimized using genetic algorithm. 3. TYPES OF SATELLITE FORMATION FLIGHT There are different types of formation flying which are classified according to the following category: a) Based on orbits in which they are flown. b) Presence of control. Based on the presence of control they are classified as given below: 3..1 Independent Satellite Formations In this type of formations, during the time of launch, the satellites are placed at their positions for formation and after that there will be no control present to maintain their positions. So whenever the orbits of the constellation are affected by perturbation, the satellites will not maintain their relative positions. 3.. Master Slave Formation In this type of formation, there will be one master and other satellites will not have any control in them. In this, the master satellite will control the relative position of all other satellite. There will be a need for communication system in satellites to receive the control action from the master satellite and also to send the signal to the slave satellites. Master satellite will receive knowledge about slave satellite by inter-satellite communication.

4 Leader Follower Formation In this type of formation, some control will be given to the slave satellites and the satellite that is in the front will make the satellites to follow the correct orbit. The satellites having some control to maintain their relative position with respect to their leader satellite are called follower satellites Peer-Peer formation In this type of formation, control will be distributed equally between all satellites. Whenever there is a failure in any one satellite only that satellite will fail, whereas the other satellites will remain in formation. Satellites will have control for maintaining their position and also their relative position, based on the orbital position in which they are placed 3..5 In-Plane Formation In this type of formation, the orbit of the satellites will be same but the satellites will differ only in their position i.e. both will be in same plane Out of Plane Formation In this type of formation, the satellites will be in different orbits and their right ascension and inclination may be different Circular Formation In this type of formation, the satellites are placed in positions such that they form a circle Projected Circular Formation In this type of formation, the satellites are placed in positions such that they form a projected circle like ellipse.

5 4 3.3 STATION KEEPING In the formation flying of satellites, it is very important to maintain the relative position of satellites and also it is required to make them to follow a correct orbit. This task of making the satellites to maintain their position with respect to other satellites is called station keeping. There are two types of station keeping namely: Absolute station keeping Relative station keeping Absolute Station Keeping In this method, the absolute positions of the satellites in formation flying are maintained thus relative positions also will be maintained. In this method, the frequency of thruster used is high such that the fuel consumption will be high. Also in this method, the position of the satellites will be known in advance and also can be easily monitored from the ground Relative Station Keeping In this method, the thruster will be fired when the satellites move out of their relative positions. As some satellites will experience same amount of disturbance they will deviate from their orbit although their relative position is maintained. At those times even the fuel consumption is reduced and satellites will deviate from their orbit in course of time. The satellites, which are in formation flying, have to know their position in space and also their relative position with respect to other satellites. This is done to control their position, as there is a chance for collision. Usually the satellites will be controlled only from ground station by using antennas. The position of satellites at any particular time is found by

6 43 knowing azimuth and elevation angle of the antenna. This information is then propagated to know the position of satellite after one week. Using this data, control signals for firing the thrusters for appropriate number of times, at appropriate positions are sent to the satellites and satellites will fire the thrusters accordingly. In a satellite, which is not in formation with any other satellites, orbital correction will not be done frequently and the satellite s earth viewing angle will be corrected to get the required ground track and to point at the particular area. In case of geostationary satellites orbital correction will be done only when there is huge error in its orbital motion. But in the formation flying, the satellites have to control their position at correct time, so as to maintain the formation; otherwise the satellites will deviate from the formation. So there is a need for an onboard orbit controller, which will take care of orbital corrections needed for the satellite to maintain it in formation. This onboard orbit controller will replace human in the loop and thus human workload will be reduced. This increases precision of the position. In the formation flying of satellites, in order to have the satellites sense their position in the space, Global Positioning System (GPS) can be used. Relative distance between the satellites can be known from the inter satellite communication. GPS will give the position in latitude, longitude and altitude from which satellite position in earth inertial frame can be obtained. This information will be communicated between satellites and deviations from the actual position and velocity can be used to compute the orbital elements and firing of the thrusters can correct the deviations at appropriate time. The satellite will hence be positioned using the control algorithm. In the present work, the initial positions and velocities of the satellites are determined using the Euler-Hill s equations for three types of formation flying configurations viz. leader follower, projected circular orbit

7 44 configuration and general circular orbit configuration. A method for converting orbital elements to relative distance of satellites and the relative distance to orbital elements of the satellites is also given. The variation of relative distance for the satellites with initial conditions determined by using Euler-Hill s equations, due to eccentricity of the master orbit is simulated. The variations in the orbital elements of the satellites due to various perturbing forces are found out by using Gaussian planetary equation, general perturbation effect equation and by using the Newton s gravitational law. The various perturbing forces considered are atmospheric drag, earth s oblateness, earth s triaxiality and solar flux. The amount of variation in the orbital elements are found and thus used for determining the amount of thrust needed to correct the orbital elements. Genetic algorithm is used to determine the optimized gains for the proposed orbital element feedback controller to reduce the deviation in orbital elements and thus to maintain the relative distance. A fuzzy controller using orbital elements for controlling the orbit of the satellite and hence distance between satellites is also designed. 3.4 SOLUTION OF EULER-HILL S EQUATION The solution of the Euler Hill s equation given in equation (.19) can be given in time as, (t) x y n 3x y n x cos n x (t) y x n 3 x y n x n sin cos 3x y n y z(t) z cos z n sin sin (3.1) Taking Laplace transform of equation (.19) and sorting gives equation for x(s), y(s) and z(s). Inverse Laplace of x(s), y(s) and z(s) yields

8 45 equation (3.1). Here is a product of mean motion and time. The Euler-Hill s equations admit bounded periodic solutions, which are suitable for formation flying missions. The solutions to Euler-Hill s solutions are as follows: x = x cos + y sin / y = y cos - x sin z = z cos + z sin / n x nx sin ny cos / (3.) y = -ny sin - nx cos z nz sin z cos The above equations on reduction give following solutions, c1 x sin(nt ) y = c 1 cos (nt + ) + c 3 z = c sin (nt + ) c n cos (nt 1 x y c1n sin(nt ) z cn cos(nt ) ) (3.3) where c 1 1. y x 4 (3.4) c z z n 1. (3.5)

9 46 x Tan y / (3.6) Tan nz (3.7) z The above constants are determined by initial conditions of the satellite position i.e. based on the relative distance between the satellites. The solutions are obtained by satisfying the following constraint on initial conditions. nx (3.8) y The initial conditions, which satisfy the above equation, are referred to as Hill-Clohessy-Wiltshire initial conditions. The initial conditions (Vadali et al 1999) are obtained by substituting t=. x c 1 sin y = c 1 cos + c 3 z = c sin x y z c 1 n cos c1 n sin c n cos (3.9) 3.5 INITIAL CONDITIONS FOR VARIOUS RELATIVE ORBITS Bounded relative orbits of various shapes and sizes can be obtained by choosing arbitrary values for c 1, c, c 3, α, β.

10 Projected Circular Orbit The relative orbit that is obtained by choosing c 1 =c =, c 3 = and α = β is known as Projected Circular Orbit (PCO). The formation is called so because when the motion of slave satellite around the chief satellite is projected on the local horizontal plane (y-z plane), the relative orbit is circular. sin (nt x y cos(nt ) ) (3.1) z sin (nt ) From the above equations, y z (3.11) This represents a circle in the y-z plane with radius. α characterizes the position of the deputies along the circumference of the circle and is a measure of the size of the formation General Circular Orbit There exists another circular orbit of interest for the choice of c 1 =, c = ( 3/), c 3 = and α = β. This results in a circle in three-dimensional space. sin (nt x y cos (nt ) ) (3.1) 3 z sin (nt )

11 48 Equation (3.1) results in x y z (3.13) This is an equation of the circle in three-dimensional space. This relative orbit is known as the General Circular Orbit (GCO). The GCO is used to simulate a large satellite with a circular geometry Leader Follower Formation Choosing c 1 = c = and c 3 =d yields x=, y=d and z=, results in constant along-track separation. This is known as the leader follower configuration because, the deputies either lead or follow the chief by a constant distance in the along track direction. The initial conditions corresponding to the projected circular orbit, general circular orbit and the leader follower configuration will be referred to as PCO initial conditions, GCO initial conditions and the LFC initial conditions, respectively. It should be noted that PCO initial conditions, GCO initial conditions and the LFC initial conditions are three particular cases of Hill-Clohessy-Wiltshire solutions. Of the three formations flying configurations having both the chief and deputy satellites in same inclination and right ascension can form the leader follower configuration. But the projected circular and general circular orbits can be formed only by having the chief and deputy satellites in different right ascension and (or) inclination.

12 CONVERSION FROM RELATIVE POSITION TO ORBITAL ELEMENTS For finding the initial positions of the deputy satellites with respect to the chief satellite, the position of the chief satellite in orbital frame in terms of orbital elements are assumed. Then they are converted to the position in orbital frame in terms of Cartesian coordinates. From the solution of Hill s equations and according to our requirements like size of the formation and type of formation, the initial conditions (initial position and velocity) of the deputy satellite with respect to chief satellite can be found out (in chief satellite s reference frame). From the conditions in chief satellite s frame of reference, they can be converted to the orbital elements corresponding to the deputy s position and velocity. Thus the deputy satellite s position is obtained in terms of orbital elements. For getting the orbital elements of the deputy satellite from its relative position with respect to chief satellite, first the deputy satellite s orbital frame has to be found out, mainly its inclination and its right ascension in particular. In most cases this can be found out by assuming difference in inclination between chief and deputy as zero i.e. deputy has the same inclination as that of the chief satellite. The governing equation for finding the right ascension is given by, x y z cos( i) sin( )sin( i) cos( )sin( i) cos( ) sin( ) sin( i) x sin( )cos( i) y cos( )cos( i) z c c c (3.14) In the matrix equation (3.14), x c, y c, z c represents the deputy s position in the chief s orbital frame. This equation is obtained from the basic

13 5 equation for converting variables from one frame to another. From the above matrix, the value of z is equal to zero, which leads to following equation, cos ( ) sin( i) x sin( i) y cos( )cos( i) z (3.15) c c c But since (3.16) i Equation (3.15) becomes, y c arc tan (3.17) zc From the above equation, the difference in right ascension between the chief and deputy satellite can be found out. It is known that deputy satellite s position is expressed in chief satellite s coordinate frame of reference, which is rotating with the chief satellite. So, at first the deputy satellite s position is converted to position in its own orbital frame in Cartesian coordinates. This can be done from the anomaly angle of the chief satellite in orbital frame. The direction cosine matrix can be found, which is used for finding the deputy s position in chief s coordinate frame. The direction cosine matrix is, cos sin DCM cd sin cos (3.18) 1

14 51 The position and velocity of the deputy in its orbital frame is given by, c 1 cd z y x ) (DCM z y x c 1 cd z y x ) (DCM z y x By knowing the right ascension, inclination and assuming the argument of perigee as zero (as it is a circular orbit), the position and velocity of deputy satellite in earth centered inertial frame of reference can be found out in Cartesian coordinates. 1 cos sin sin cos ) ( A Z (3.) i cos sin i i sin i cos 1 (i) A x (3.1) 1 cos sin sin cos ) ( A Z (3.) (3.19)

15 5 z y x ) ( A (i) A ) ( A Z Y X 1 z 1 x 1 z z y x ) ( A (i) A ) ( A Z Y X 1 z 1 x 1 z From the position and velocity in earth-centered inertial frame in cartesian coordinates, the orbital elements of the satellite can be found out. 3.7 CONVERSION FROM ORBITAL ELEMENTS TO THE POSITION AND VELOCITY IN CHIEF SATELLITE S COORDINATE FRAME OF REFERENCE The orbital elements of both chief and deputy satellite are converted to their position and velocity in their orbital frame in Cartesian coordinates. The deputy satellite s position and velocity are converted to the chief s orbital frame by rotating by an angle of difference in right ascension and the inclination. The deputy satellite s position and velocity are further rotated to coincide with the direction of revolution of chief satellite (i.e. to convert to chief s coordinate frame). The orbital elements can be converted to their position and velocity in their orbital frame in Cartesian coordinates by set of equations (3.4). (3.3)

16 53 x a cos ae y asin 1 e z = x 3 a x a y sin (3.4) y 3 a x a y 1 e cos z The position and velocity of the deputy in its orbital frame is converted to chief s orbital frame by the following conversions. For (3.5) d c If (3.6) Then cos sin A y ( ) 1 (3.7) sin cos If (3.8) Then For cos sin A y ( ) 1 (3.9) sin cos i i d i (3.3) c If i (3.31) Then 1 A x ( i) cos i sin i (3.3) sin i cos i If i (3.33)

17 54 Then i cos i sin i sin i cos 1 i) ( A x (3.34) d y x c z y x ) ( A i) ( A z y x d y x c z y x ) ( A i) ( A z y x Thus equation (3.35) gives the position and velocity of deputy satellite in chief s orbital frame. The deputy s position and velocity in chief s coordinate frame can be found out by rotating the deputy s position and velocity in chief s orbital frame in the direction of velocity vector of the chief satellite. This is done by the following equations. 1 cos sin sin cos DCM cd (3.36) d cd z y x ) (DCM z y x d cd z y x ) (DCM z y x Thus the orbital elements of the deputy satellite are converted to the position and velocity of the deputy in chief s coordinate frame. The relative distance and velocity vectors can be obtained by subtracting deputy s position and velocity from chief s position and velocity vector. (3.35) (3.37)

18 RELATIVE MOTION SIMULATION Introduction In this, the initial conditions are determined for leader follower, projected circular and general circular configuration of two satellites. The distance of separation is considered to be km. The initial conditions (relative distance) in x, y and z-axis are given below Initial Conditions of Various Configurations The initial conditions of various configurations are given below: Leader Follower Configuration c 1 = c =, c 3 = n =.594 deg/sec x = y = z = x = y = z = km km km km/s km/s km/s Projected Circular Orbit (PCO) Configuration c 1 = c =, c 3 =, = = 9 n =.594 deg/sec x = y = z = x = y = z = 1 km km km km/s -.7 km/s km/s

19 56 General Circular Orbit (GCO) Configuration c 1 =, c = 1 3, c 3 =, = = 9 n =.594 deg/sec x = y = z = x = y = z = 1 km km km km/s -.7 km/s km/s The above initial conditions for all the three configurations are used in the solution of Euler-Hill s equations to know the relative distance behavior of two satellites at all times Simulation of Various Configurations for Given Initial Conditions The simulations of various configurations for the given initial conditions are given below: The relative distance in leader follower configuration is shown in Figure 3.1. Figure 3. shows the motion of the deputy satellite around master in orbit in PCO configuration. Orbital motion is projected in y-z plane.

20 57 Figure 3.1 Relative distance in leader follower configuration Figure 3. Motion of the deputy satellite around master in orbit

21 58 The relative distance between satellites in projected circular orbit configuration and general circular orbit configuration are given in Figures 3.3 and 3.5 respectively. Figure 3.4 shows the motion of the deputy satellite around master during orbit in GCO configuration. Figure 3.3 Relative distances between satellites Figure 3.4 Motion of the deputy satellite around master during orbit

22 59 Figure 3.5 Relative distances between satellites in general circular orbit configuration From the above plots, it can be seen that the relative distance is maintained for all three configurations, given the initial conditions of the deputy satellite in all three configurations Simulation for Elliptic Chief Orbit The solutions for Euler-Hill s equations and the initial conditions of the deputy satellite are found with the assumption that the chief satellite is in the circular orbit. The above found solutions will not be satisfied, when the orbit of the chief satellite has an eccentricity of. (Vadali ). For the orbit with eccentricity, the position of the deputy satellite with respect to chief, in chief satellite s coordinate frame (Inalhan et al ) is given by equations (3.38):

23 6 x ( ) j sin [d 1j e d j d je e H( )] cos (1 e cos) d 3 j y ( ) j d 1j d 4 j 1 ecos d j eh( ) d3 j sin (1 e cos) d 3 j (3.38) cos [d1je d je H( )] z( ) j d 5 j d6 j sin cos 1 e cos (1 e cos) cos where H( ) d (1 e ) 3 (1 ecos) 5 3eE (1 e )sin E e sin E cos E d E (3.39) e cos cos E (3.4) 1 e cos where E is the eccentric anomaly and θ is true anomaly. The values of d ij are given by following equation, d1 d d 3 d 4 j p p p p 3 p p 3 33 p 44 x() x() y() y() j (3.41)

24 61 The values of matrix elements are, P 11 = 1/e, P = (+e) (1+e) /e, P 3 = (1+e 3 )/e P 3 = -[(1+e)/e] P 33 = -[(1+e)/e] P 41 = -[(1+e) /e] P 44 = (1+e) d 5j = (1 + e) z () d 6j = z() (3.4) Substituting the above found values for an eccentricity of., the position of the deputy satellite has a variation with respect to that of the chief satellite, as given in Figure 3.6. Figure 3.6 Motion of the deputy satellite around master in orbit The relative distance between the satellites has variation in Projected Circular Orbit (PCO) configuration as shown in Figure 3.7.

25 6 Figure 3.7 Relative distances between satellites in elliptic chief orbit 3.9 PERTURBATION FORCES AFFECTING THE MOTION OF SATELLITE Atmospheric Drag For the satellites, which are above thousand kilometers from the earth surface, the density of atmosphere is very low. The effect of atmosphere on the satellite i.e. the drag created by the atmosphere will be insignificant and can be ignored. The important effect of drag on the satellite is to reduce its altitude. The atmospheric drag is given by (Daniel et al 1996), F drag = (1/)ρc d v A/m (3.43) ρ - Atmospheric Density (kg/m 3 ) c d - Coefficient Of drag (.) v - Velocity of satellite ( GM/a km/s) m - Mass of the satellite (kg) a - Position of satellite from earth s centre A - Projected area perpendicular to velocity vector (m )

26 Earth Asphericity The earth is not spherical as it is assumed and its poles are flattened and bulged in the equator. Due to this the gravitational potential exerted by the earth is not uniform throughout its surface and is varying according to its mass distribution. The earth s gravitational potential is given by (Bong and Carlos 1), U(r,λ,Φ)=µ/r[1+n (- ) m (-n) (R/r) n P nm sinφ(c nm cosmλ+s nm sinmλ)] (3.44) where R - Radius of earth r - Altitude of the satellite from the centre of earth λ - Longitude Φ - Latitude The asphericity of the earth is classified into two types namely oblateness and triaxiality of the earth. The oblateness of the earth is due to the fact that the equator of the earth is not circular and it is elongated in one side. The triaxiality of the earth is due to the effect of earth s gravitational potential. The gravitational potential can be found by earth s radius, the latitude and longitude Solar Flux The force will be created due to solar flux by the impingement of photons emitted by the sun on the surface of satellite. F solar = KPA s /m (3.45)

27 64 where K - Dimensionless constant between 1 and (K=1: surface perfectly absorbent, K=: surface reflects all light) P - Momentum flux from the sun (Nm - s -1 ) A s - Surface area of satellite normal to solar flux (m ) m - Mass of the satellite (kg) Solar and Lunar Gravity In addition to the gravity exerted by the earth, there will be perturbation due to the gravitational force exerted by sun and moon. These effects are called third body perturbations (Daniel et al 1996). The acceleration due to third body is given by, a d = (µ d /r ds 3 ) (r s + f (q) r d ) (3.46) where µ d - Gravitational constant due to third body r ds - Distance between satellite and sun (Km) r s - Distance between earth and satellite (Km) r d - Distance between earth and third body (Km) q = r s ( r s - r d )/ r d (3.47) f (q) = q((3+3q+q )/(1+(1+q) 3/ )) (3.48) 3.1 DETERMINATION OF EFFECTS DUE TO PERTURBATION FORCES In this work, the perturbations due to atmospheric drag, earth s asphericity and the solar flux are considered and the effect due to each force on the orbital elements of the satellite are calculated using the following methods.

28 Gaussian Planetary Equation The Gaussian planetary equation is a first order differential equation, which gives the variation of all the orbital elements of the satellite due to the forces are acting on the satellite in the coordinates of orbital frame. The Gaussian planetary equation is given in the matrix equation (3.49) (Sidi 1997). a e i M a esin h psin h pcos he b(p cos re) ahe where a - Semi major axis b - Semi minor axis a p hr (p r)cos re h (p r)sin he b(p r)sin ahe e - Eccentricity of the orbit i - Inclination of the orbit Ω - Right ascension of the orbit ω - Argument of perigee M - Mean anomaly rsin( ) u h u rsin( ) h sin i u rsin( )cosi h sin i r h (3.49) h = a(1+e) µ (3.5) µ = GM = (3.51) p = h / µ (3.5) r = P/(1+ecosθ) (3.53) u r - Force acting on the r axis of orbital frame u θ - Force acting on the θ axis of orbital frame u h - Force acting on the h axis of orbital frame

29 66 Forces due to atmospheric drag are: u r = (1/) )ρc d V x A/M (3.54) V x = Ve sinθ (3.55) u θ = (1/) )ρc d V θ A/M (3.56) V θ = V (1+ecosθ) (3.57) u h = (3.58) Forces due to oblateness of earth are: u r = - (3µJ R /r 4 )(1-3sin i sin (ω+θ)) (3.59) u θ = - (3µJ R /r 4 )( sin i sin(ω+θ)) (3.6) u h = - (3µJ R /r 4 )( sini sin(ω+θ)) (3.61) where J Forces due to triaxiality of earth (Bong and Carlos 1) are: u r = - (9µR /r 4 )( C cosλ + S sin λ)cos Φ (3.6) u θ = - (6µR /r 4 )( C sinλ + S cos λ ) cos Φ (3.63) u h = - (6µR /r 4 )( C cosλ + S sin λ) cosφ sinφ (3.64) where C S λ - Longitude at the particular time Φ - Latitude at the particular time. The latitudes and longitudes are found out by first converting the orbital elements into Cartesian coordinates in ECI reference frame. Then they are converted from the Cartesian coordinates into spherical coordinates, which gives the latitude and longitude at a particular time.

30 67 Solar flux is given by, F x = Fsin θ (3.65) F y = F cos θ (3.66) F z = (3.67) Force, F = KPA s (3.68) where K - dimensionless constant between 1 and (K=l: surface perfectly absorbent, K=: surface reflects all light) (Taken as 1.5) P - Momentum flux from the sun ( kg/m/s ) A s - Surface area of satellite normal to solar flux ( km ) 3.1. General Equations of Orbital Elements Variation The following set of equations (3.69) (Sidi 1997) gives the variation of orbital elements due to all the perturbations. a e i Ac D 3J n cosi (1 e ) e a1 cosi n (3.69) M 3J n(5cos i 1) 4(1 e ) n(1 3J (1 3sin 3 (1 e) isin ))

31 Gravitational Force Equation The variation of the orbital elements due to various perturbing forces can be found out from the basic force equation of Newton s universal law of gravitation and algebraically adding it with the perturbing forces, which affect the satellite motion. This is also called as Cowell s method. a GM 1 cd V V r r (3.7) 3 r m In the present work, the effect of atmospheric drag perturbation is only calculated by using the equation (3.7). By numerically integrating the above equation, the velocity vector can be got, which on further integration yields the position vector at all time. The position and velocity vector obtained in the earth-centered inertial frame will be converted to the orbital elements, from which the deviation in the orbital elements can be found out CONTROLLER DESIGN In the formation flying of satellites, control is needed as the satellites orbital parameters are varied by various disturbances, which alters the position and velocity of the satellites. The effect of various disturbances on the position and velocity of the satellite varies with the corresponding variation in the shape and size of the orbit, in which satellite is orbiting. For making the satellites in the formation to maintain their relative distance, control is applied to fire thrusters in appropriate direction and correction is done on orbital elements. The controllers can be made to control the position and the velocity of the satellites directly or can be used to control the orbital elements. There are different types of controllers as continuous controller or impulsive type

32 69 controller. In the continuous type controller, thruster will be used continuously to correct the small changes in the position. In this type of control, the accuracy obtained will be more with reduced lifetime of mission for the same quantity of fuel as in impulsive type control. In impulsive type controller the thruster is used for orbital corrections only at particular times. Thus the accuracy of the relative distance maintenance will be low, but the lifetime of the mission will be more for impulsive type control. In the present work, the control of orbital elements of the satellites is by Gaussian planetary equation. The Gaussian planetary equation, which gives the dynamics of spacecraft in orbit, is given in equation (3.49). The equation (3.49) is used to find out the variation in orbital elements due to effects of various forces acting on the satellite. For controlling the relative distance between the satellites, it is needed to control the orbital elements of the satellites. This in turn requires the amount of thrust to be applied at various directions in orbital frame to maintain the orbital elements of the satellite. The amount of force, which is to be applied at various directions in the orbital frame, can be found out by equation (3.71) (Naasz ). a esin a p h hr psin (p r)cos re h h k1a rsin( ) k e u r h k 3 i u rsin( ) k d 4 u h h sin i k5 pcos (p r)sin rsin( )cosi k6 M he he h sin i b(p cos re) b(p r)sin ahe ahe (3.71)

33 7 In the above equation the various values of proportional gains viz. k 1, k, k 3, k 4, k 5 and k 6, are found. Using these gains the required amount of forces to be applied to the satellite to have desired position and velocity are calculated. The forces are found and are applied in the direction vectors of orbital frame of the satellite. The values of proportional gains can be found out by using genetic algorithm, as the orbital dynamics of satellite is nonlinear. 3.1 SIMULATION OF PERTURBATION EFFECTS ON ORBITAL ELEMENTS The simulation of the orbital elements variation of the satellites due to various perturbations is done in Matlab. The amount of thrust required to maintain the relative distance between the satellites is done by using orbital element feedback control in which genetic algorithm is used to determine the amount of proportional gains needed to calculate the required amount of thrust. The genetic algorithm is implemented in Matlab Gaussian Planetary Equation The perturbing forces that are taken into consideration for determining their effects on orbital elements are atmospheric drag, earth s oblateness, triaxiality of earth and the solar flux from sun. The effect due to each force is simulated separately with the forces converted to the force vectors acting in the different directions in orbital frame of the satellite. The effect on orbital elements due to atmospheric drag is given in Figure 3.8 and the effect on orbital elements due to earth s oblateness is given in Figure 3.9.

34 Figure 3.8 Variations of orbital elements due to atmospheric drag 71

35 Figure 3.9 Variation of orbital elements due to earth oblateness 7

36 73 Figure 3.1. The effect on orbital elements due to earth s triaxiality is given in Figure 3.1 Variation of orbital elements due to earth s triaxiality

37 74 Figure The effect on orbital elements due to solar flux is given in Figure 3.11 Variation of orbital elements due to solar flux

38 75 In the above simulation, for the LEO considered it is found that there is secular effect on the satellite s motion due to atmospheric drag and oblateness of the earth. The other effects are either periodic or very low. The effect of atmospheric drag on the semi-major axis is that the semi-major axis is reduced. The effect of oblateness is on the right ascension of the orbit and is causing the westward nodal regression of the satellite orbit at the rate of.1 deg/day. Due to atmospheric drag, the semi major axis is decreasing at the rate of 3.8 m/day General Orbital Elements Variation Equation The methods of general perturbations are well studied and are used to calculate the effect of perturbative forces on the orbital parameters (Battin 1987, Danby 196, Kaplan 1976 and Roy 198). The effects of perturbation forces on orbital elements are calculated using general orbital elements variation equation and are given in Figure 3.1. In the above simulation, it is found that there is variation in semi major axis, right ascension and argument of perigee. The variation in semi major axis is reduction of 3.8 m/day and that of right ascension is reducing by nearly.15 deg/day, which is same as that found by using Gaussian variation equation Newton s Law of Gravitation In this only atmospheric drag is taken into account. Using this method the reduction in semi major axis is same as that found by using Gaussian variation equation and general equation. It is found to be reducing by 3.8 m/day. Figure The effect on orbital elements due to Cowell s method is given in

39 76 Figure 3.1 Variation of orbital elements using general variation equation

40 Figure 3.13 Variation of orbital elements in Cowell s method 77

41 GAINS OPTIMIZATION USING GENETIC ALGORITHM Orbit controller can correct the deviations in the orbital elements. The mostly used is the orbital element feedback method. The orbital element feedback method has gains for the six orbital elements, five of which are calculated by formulas. One gain has to be arbitrarily chosen. The gain values need the length the firing time also apart from orbital parameters. By using genetic algorithm, the optimized values of the gains are found for the orbital element feedback controller. These gains will calculate the required amount of thrust to be applied to the satellite, to keep them in the required position and velocity by correcting their orbital elements General Procedure of Finding Optimized Gain Using GA The procedure is in steps as explained below: i) The fitness function design is very important to evaluate each individual in one generation. For these problems, minimization of orbital deviation is taken as the performance criteria and tuning is done. So, E = Desired output - Actual output is taken as error, E. Therefore, a smaller E represents a higher fitness (GA maximizes performance). The E is converted to a fitness value of a GA by using, Fitness = 1/ E (Jinwoo et al 1994). ii) In Genetic operations, the entire individuals are expressed as binary strings, not the parameters themselves. In the coding method, the scaling factors generated randomly, are first coded into binary strings. While calculating the fitness values for each individual, these binary strings are converted into corresponding values in the parameter space by using a

42 79 decoding procedure. The mapping equation used to map the binary string to the gain value is given below: P = P MIN + b (P MAX -P MIN ) / ( m -1) (3.7) where P - Gain Value. P MIN - Minimum Value of Gain. P MAX - Maximum Value of Gain. b m - Integer Value Corresponding to the m th bit. - No: of Bits. e.g., P MIN =., P MAX = 1., Binary Value = 1 b = 16, m = 8 Bits P = +16 (1-) / ( 8-1) P =.63. iii) Once the fitness values of all individuals in the population are evaluated, the fittest individuals are selected for survival and reproduction. The selection process is based on proportional selection method, i.e. an individual with a high fitness value has a high probability of being selected, as described below, a) p i = f i / Σ f i (3.73) b) p i = p i suitable multiplying factor (3.74) c) p i >1. is selected for the next generation.

43 8 The selected individuals are randomly mated to perform genetic operations. iv) Two mating parents exchange information through simple crossover and are replaced with the new individuals. For a simple crossover, the cut off position is randomly determined. The crossover operation at the cut off position 6 i.e 6 th bit is shown as follows: X = [ ] becomes [ ] Y = [ ] becomes [ ] v) After completing the simple crossover, mutation operation is performed. Mutation of 5 th bit on X changes it to [1 1 1 ]. The specifications for GA are summarized below: Population size : 1 Each string represents the set of solution to the optimization problem. Selection method: Proportionate selection method Probability of crossover: 1 (All the string will undergo crossover) Probability of mutation:.1 (One out of 1 string will undergo a change in bit) During the process of iteration, the genetic algorithm maintains a constant population of individuals. Each individual will undergo evaluation and selection. The surviving individuals will undergo crossover and mutation

44 81 operations. The iteration process is repeated until the termination conditions are satisfied Simulation Result The result of the simulation gives the values of the gains optimized by genetic algorithm. The gains are used to find out the amount of thrust required for correcting the orbital deviations. The thrust is applied to the satellite dynamics to correct for the orbital perturbations. The amount of corrections obtained is also found out. The gains optimized using genetic algorithm are: K 1 = 3.633e-11 K = 1 K 3 = 1 K 4 = K 5 = K 6 = e e-8 The thrusts to be applied are: U r = µn U θ = mn U z =.1871 N

45 8 The corrections in orbital elements made by applying the calculated amount of thrust in corresponding directions are: Correction in semi major axis = m Correction in Eccentricity = e-7 Correction in inclination = rad Correction in right ascension =.469 rad Correction in mean anomaly = -.91 rad The actual amounts of secular variation in the orbital elements are: For semi major axis = 3.8 m For eccentricity For Inclination For right ascension rad =. rad For argument of perigee rad For mean anomaly rad. The response of the orbital elements to the controller is given in Figure It gives a picture of how the orbital elements deviations are corrected by the proposed controller. The initial values of the orbital elements, their variation due to atmospheric drag and their corrected values by the proposed controller can be visualized.

46 Figure 3.14 Deviation and correction of orbital elements 83

47 84 The control law used is a hybrid continuous feedback control law for a local Cartesian relative orbit frame and is a function of differential orbital elements. So, these control laws are valid for both circular and elliptical orbits. Hence the orbital element feedback controller works well for both circular and elliptic orbits. The GA used is just an optimization tool, so the proposed work is applicable for different orbits. The deviations in the orbital elements are reduced to approximately zero, just like that of a circular orbit. This has been proved through results of simulation and is given in Table 3.1. Table 3.1 Performance for various eccentricities Satellite with e=.1 Satellite with e=.1 Satellite with e=.5 Satellite with e=.1 Semi - major Axis (km) Eccentricity Inclination (rad) Mean Anomaly (rad) Right Ascension (rad) Argument of Perigee (rad) Actual Perturbation Correction.38 Approx Actual Perturbation Correction.39 Approx Actual Perturbation Correction.4 Approx Actual Perturbation Correction.46 Approx The amount of thrust required for the orbital correction is of the order of micro-newton. Hence, advanced chemical and low power electric propulsion offers attractive options for small satellite propulsion. Applications include orbit raising, orbit maintenance, attitude control, repositioning, and

48 85 de-orbit of both earth-space and planetary spacecraft. Potential propulsion technologies for these functions include high pressure Ir/Re bipropellant engines, very low power arcjets, Hall thrusters, and pulsed plasma thrusters, all of which are shown to operate in manners consistent with currently planned small satellites (Roger and Steven 1994 and Yashko and Hastings 1996). Any suitable propulsion system can be designed to achieve the distance keeping function FUZZY CONTROLLER In certain low Earth orbit (LEO) satellite missions, it is required that two or more satellites must operate in a certain special configuration relative to each other. This section introduces a simple concept of utilizing aerodynamic drag to achieve this type of constellation control. The key feature is utilizing aerodynamic drag, a natural phenomenon that is normally considered as an unwanted disturbance, especially for low Earth orbit missions. The perturbation due to aerodynamic drag is designed using Gaussian Planetary equation as explained in section The satellites considered for simulation are micro-satellites with a semi-major axis of 7185 Km and eccentricity of.1 and are 1 Km apart. The simulations of the effect of atmospheric drag on the low-earth orbiting satellite shows deviations in their orbital elements, for a day is as given in Table 3.. Table 3. Deviation in orbital elements a e i M 3.8m e-7 rad

49 86 The deviations in the orbital elements are corrected by producing the required amount of thrusts. Fuzzy controller determines the amount of thrust required for the satellite. For formation flying and constellation station keeping in near-circular orbits, the deployment and maintenance of the formation or constellation can be done by closely controlling two mean orbital elements (Bainum and Duan 4). This concept has been implemented in this paper by the use of fuzzy control. The inputs for fuzzy controller are semi-major axis a, and eccentricity e. The outputs of the controller are the thrusts in the three directions. The fuzzy controller is represented as a block diagram in Figure a e Fuzzy Controller u h u r u Figure 3.15 Block diagram of fuzzy controller The change in the semi-major axis and eccentricity after a period of time (e.g. one day) is calculated using the Gaussian planetary equation. These two elements are given as inputs to the fuzzy controller. The input variables are then mapped into fuzzy sets. The fuzzy set values are obtained from the triangular membership function. The membership functions are shown in Figures 3.16 and 3.17.

50 87 Figure.3.16 Membership function, a Figure.3.17 Membership function, e The amount of overlap between the different fuzzy sets is optimized through simulation. The saturation point of each input variable is set using an engineering knowledge of the system and optimized using simulation trails. Sample of output membership function for the thrusts is given in Figures The rules of the controller are given in Table 3.3. Figure 3.18 Membership function for output, thrust

51 88 Table 3.3 Rules Da de u r u h u R1 Z Z Z Z Z R Z P Z Z Z R3 Z MP P P P R4 P Z Z Z Z R5 P P P P P R6 P MP P P P R7 MP Z P P P R8 MP P P P P R9 MP MP MP MP MP Rule evaluation is performed using correlation-product encoding, i.e. the conjunctive (AND) combination of the antecedent fuzzy sets. When the result of all the rules is known, the final value is obtained by disjunctively (OR) combining the rule values: y (y ) sgn y min 1, y i1 i1 N N i i i (3.75) The disjunction method can be described as a kind of signed Lukasiewicz OR logic. It is chosen to maximally negatively correlate the rule outputs. For example, opposing rule outputs (different in sign) cancel each other to deliver a small rule base output. Defuzzification is done by centroid method. The crisp output value x is the abscissa under the centre of gravity of the fuzzy set, u i i (x ) x i (x i) i (3.76)

52 89 Here x i is a running point in a discrete universe, and µ(x i ) is its membership value in the membership function. The expression can be interpreted as the weighted average of the elements in the support set. The controller corrects the deviation in the orbital elements due to atmospheric drag. The value of orbital elements at the initial position, their deviation after a day and the corrected values of orbital elements are given in Table 3.4. Table 3.4 Simulation results During Launch Orbit of Sat 1 Perturbed orbit after a day, Sat 1 Orbit Corrected by Fuzzy controller,sat1 During Launch Orbit of Sat Perturbed orbit after a day, Sat Orbit Corrected by Fuzzy controller,sat Semimajor Axis Eccentricity Inclination Right Ascension Argument of perigee Mean Anomaly The result of the simulation for the proposed fuzzy controller is given in Figure The initial orbital elements their perturbation due to atmospheric drag and the correction are indicated in the result.

53 9 Figure 3.19 Orbital elements deviation and correction 3.15 CONCLUSION In this work, a method of finding the initial conditions of the satellites in formation flying in various orbital configurations like leader-

54 91 follower, projected circular orbit and general circular orbit configuration were presented using the solutions of Euler-Hill s equations of relative motion. Simulation of relative distance between the satellites with the given initial conditions was also shown for various configurations. The movement of the satellites away from the formation was simulated for the chief satellite in elliptic orbit. The variations in orbital elements of the satellites due to various perturbation forces were calculated and simulated using various methods of orbital propagation and they were found to agree with each other. A fuzzy and Genetic Algorithm approach has been made to control the orbit and hence to achieve distance keeping for formation flying. The solution proposed in this thesis can be used for formation flying and constellation station keeping and maintaining the relative distance between spacecrafts. Moreover, the deployment or maintenance of the formation or constellation can be done by closely controlling two mean orbital elements, semi-major axis and eccentricity. The method works under the influence of the atmospheric drag. Therefore, it can be effectively used on the low and mid-altitude orbits where this represents the main perturbation effect. A new method to find out the gains of the orbital element feedback controller, to develop a required amount of thrust for absolute station keeping of satellites was done using genetic algorithm. The gains of the classical orbital element controller are generally found by using the orbital elements, length of thruster firing time and other orbital parameters. There is no formula for finding the mean motion gain, and this is chosen depending on how aggressively we want to correct the argument of latitude error. So instead of using the formulas for finding five gains and assuming the sixth gain, the proposed GA based gain optimization technique finds the optimal value for all the gains. The deviations were reduced to almost zero by the proposed control algorithm. The fuzzy controller and the genetic controller are effective