1. INTRODUCTION
Some natural disasters occur on the earth every year, and some areas are still in the emergency situation even now. In order to take countermeasures against disasters and emergencies, we need to obtain the detailed information of these particular areas quickly. If we obtain the information by using a conventional earth observing satellite, it is very difficult to get what we need, because the orbit of the satellite is fixed. Therefore we propose the use of DAS to perform the observing mission.
This paper is structured as follows: We derive the mathematical model of the system and the Steepest Descent Method in section 2. We considered the zonal second coefficient J2 and treat the atmospheric drag as the disturbance force. In section 3, we show the optimal controls for various angle differences of the argument of latitude ( ) between the DAS and the observing point. In section 4, we show the results of extending the time of the DAS to reach the observing point. In section 5, we introduce some constellations about the DAS. Finally we summarize the study of this paper in section 6.
2. MATHEMATICAL MODEL
In this paper, we treat the earth as an ellipsoid of revolution. We show its parameters in table 1.
Table 1 The parameters of the earth
Angular velocity
Equatorial radius
Gravity constant
Zonal second coefficient
Flattening rate
We use the geocentric equatorial inertial coordinate system , the orbit reference coordinate system , and the orbit coordinate system (for details see reference [10]).
Fig.1 Coordinate systems and symbols
The transformation matrix A from to is given by Eq. (2.1).
(2.1)
where the symbols gCh and gSh show the abbreviation of gCosh and gSinh. The symbols , and show the argument of latitude, the right ascension of the ascending node and the inclination of the DASfs orbit, respectively. The transformation matrix B from to is the inverse of A.
We employed the Steepest Descent Method to solve the optimal control problems. The system equations of the problems are shown as follows,
(2.2)
where , ,and is the fuel specific impulse. The state variables of this system are
(2.3)
where x, y and z are the components of the inertial coordinates, and are the components of velocity vector, m is the satellitefs mass. Three thrust components in the direction of three axes of are selected as control variables
(2.4)
If the satellite body attitude is controlled to direct along with these three axes, then , and are considered as thrust forces of thrusters fixed to the satellite body three axes. The following constrains are also imposed on control variables.
(2.5)
The control force components , and in Eq. (2.2) are expressed in relation to control variables
(2.6)
The , and in Eq. (2.2) are disturbance force components. In this study, the DAS dives into very low altitude, and the atmospheric drag is far greater than the other disturbance forces, therefore we only considered the atmospheric drag here. Its components are given by
(2.7)
where A is the reference area of the satellite, is the atmospheric density, which is a function of the altitude, is the velocity of DAS, is the atmospheric drag coefficient, which is a function of the altitude too.
In the Steepest Descent Method some constraint conditions
(2.8)
are imposed, and the terminal time is determined from the following stopping condition.
(2.9)
These equations will be explained in the next section. An optimal observing problem is defined as, finding the optimal control histories under the above conditions to maximize the performance index,
(2.10)
which means to minimize the fuel consumption.
3. OPTIMAL CONTROL FOR VARIOUS BETWEEN THE DAS AND THE OBSERVING POINTS
Our previous studies show that the control of inclination change requires a lot of fuel, and the satellite loses its normal observing function, therefore we only perform coplanar orbit transfers to observe an area. As the earth is rotating, when we use a DAS to observe a particular area, we must consider the next three conditions: The distance between the area and the observing point that is above the area; the angle difference of the argument of latitude ( ) between the DAS and the observing point; the time of the DAS to reach the observing point. Table 2 shows the parameters we have employed as the initial conditions of the calculations.
Table 2 The parameters of satellite and initial orbit
Satellitefs parameters
Initial mass 350kg
Fuel mass 175
Shape 1~1~1(m3)
Thrusts (}500),(}18),(}500)N
313s
Initial orbital parameters
Altitude 400km
Inclination 90‹
Tangential velocity 7.66856km/s
Period 5553.63s
Initial position (2792.867,4792.867,0)km
In order to obtain detailed information about the area, we need to descend the DAS to low altitude to reduce the observing distance. In this section, we set the observing distance for 130km, because the lowest altitude of many other dive and ascent satellites is 130km, and the atmospheric drag is not so large. Therefore the constraint conditions are
(3.1)
where is the terminal altitude of the calculation, and is the terminal component of velocity of orbit coordinate system. These constraints make the observing point as the perigee or apogee of the controlled orbit. As the DAS must flies to the observing point, so the stopping condition is
(3.2)
Although we need to observe an area with arbitrary , we conducted the next typical 5 cases, that is, the are ƒÎ/2, 5ƒÎ/8, 3ƒÎ/4, 7ƒÎ/8 and ƒÎ. After finishing the observing mission, we need to control the DAS to fly back to its normal orbit, so the final stopping conditions of are 3ƒÎ/2, 9ƒÎ/8, 7ƒÎ/4, 15ƒÎ/8 and 2ƒÎ, and final altitude of all cases are 400km.As the paper space is limited we only show the results about the are ƒÎ/2 and ƒÎ.
First, we show the optimal calculation results about is ƒÎ. We obtained the optimal thrust forcesf pattern of figure 3.1. As the value of is always 0, therefore we the history is abbreviated. The histories of the orbital parameters are shown in figure 3.4, where a is the semi major axis, r is the distance from the DAS to the center of the earth, and p is the semi-latus rectum.
Fig. 3.1 Histories of thrust forces
Fig. 3.2 Controlled orbit and normal orbit
Fig. 3.3 Histories of altitude and velocity
Fig. 3.4 Histories of a, r and p
Fig. 3.5 Histories of mass and
After the one cycle of controlled flying, the remained mass of the DAS is 331.831kg. Therefore the mass of the fuel consumption is 18.169kg, which is 5.191% of the initial mass of the satellite. If we perform the similar control once more, the loss of the fuel is 5.191% (17.225kg) of remained mass too. That is we can use all of the fuel to perform the same observing control 13 times (174.97kg<175kg).
We show the calculation results about is ƒÎ/2 in Fig. 3.6 and Fig. 3.7.
Fig. 3.6: Histories of thrust forces
Fig. 3.7 Histories of altitude and velocity
We summarize all the calculation results in table 3 and Fig. 3.8. By the results we know that the mass of fuel consumption increase when the becomes smaller. In the case of is ƒÎ/2, the mass of fuel consumption is 22.444% of the initial mass, which means that we can only perform the same observing control 2 times.
Table 3 The and fuel consumption
Fuel consumption (kg)
ƒÎ/2(90?) 78.555
5ƒÎ/8(112.5?) 51.678
3ƒÎ/4(135?) 36.215
7ƒÎ/8(157.5?) 25.67
ƒÎ(180?) 18.169
Fig. 3.8 The vs. the fuel consumption
4. EXTEND THE TIME OF THE DAS TO REACH THE OBSERVING POINT
In section 3, we can make the fuel consumption minimum to observe the particular area when the is ƒÎ, and the time of the DAS to reach to the observing point is 2697.406s. As the rotation of the earth is too slow, sometimes it needs more than 2697.406s for the particular area to appear on the orbital plane. Therefore, we may be required to extend the time of the DAS to reach the observing point so that it can observe the area. For this purpose, we must control the DAS ascending to higher altitude at first, and then descending to the observing point. In this paper we calculated the next 4 cases, which obtained the start conditions of the optimal calculations by simulations. The other conditions of the calculations are the same as the is ƒÎ in the former section. In these cases, we get the longest time to reach the observing point in case 4, we show the results in Fig. 4.1~4.3.
Table 4 The cases of extending the time to reach to the observing point
Cases Ascending control time (s) Starting time of Descending control(s)
Case1 50 500
Case2 50 1000
Case3 100 500
Case4 100 1000
Fig. 4.1 Histories of thrust forces
Fig. 4.2 Histories of altitude and velocity
Fig. 4.3 Controlled orbit and normal orbit
We summarized all the calculation results in table 5 and in figure 4.4, where case 0 shows the results of section 3. We extended the time up to 2853.569s. Therefore between 2697.406s and 2853.569s, we can observe anywhere which appears on the orbital plane in detail for the is ƒÎ. However, the mass of fuel consumption is 132.345kg, which is 37.813% of the initial mass. Though we use all of the fuel, we can not perform the observing control more than 1 time.
Table 5 optimal calculation results
Case Ascending control time (s) Starting time of Descending control (s) Fuel consumption
(kg) Time of reaching to the observing point (s)
Case0 0 0 18.169 2697.406
Case1 50 500 55.328 2755.242
Case2 50 1000 89.636 2800.259
Case3 100 500 80.887 2785.165
Case4 100 1000 132.345 2853.569
Fig. 4.4 Fuel consumption vs. the time of the DAS to reach to the observing point
We also can control the DAS to higher altitude and make it observe the particular area at the apogee of the controlled orbit. We check the case for apogee altitude is 1400km. Therefore, the terminal constrain for the observing control is 1400km, while for the final staying control the is 400km. The optimal calculation results are shown in Fig. 4.5~4.7.
Fig. 4.5 Histories of thrust forces
Fig. 4.6 Histories of altitude and velocity
Fig. 4.7 Controlled orbit and normal orbit
In this case, the time of the DAS to reach the observing point is 3069.996s. It means that we extended the observing time for 372.59s. To perform this control we only need 66.739kg fuel, which is 19.068% of the initial mass, so we can use all of the fuel to perform the same control 3 times (164.466kg). Obviously we can use this method to extend the observing time, though the resolution of the sensors is inferior to that of at the low altitude.
5. THE CONSTELLATIONS OF THE DAS
To cover more area, we need the use of the observing sensorfs pointing function. Now we consider the pointing angle for }44 degrees here. Obviously the observable rangeƒÀis -3.416?~3.416?at 400km, and 11.968?~11.968?at 1400km (see Fig. 5.1).
Fig. 5.1 The observable range
Fig. 5.2 The constellation with 2 DASes
There are two reasons that we need to set 2 DASes on one orbital plane. First, each DAS just needs to cover half of the orbital plane. Second, if one DAS had any trouble, the other one can perform the observing mission too. Fig. 5.2 shows the simplest constellation, which has one polar orbit with 2 DASes. They are symmetrical with respect to the center of the earth.
As the circumference of the equatorial plane is the longest plane on the earth. If we can observe any points on the equatorial plane, which means we can observe the points on the other planes too. Now we consider about observing at the point P1 (see Fig. 5.2). We can start the observing at the time t0=0s, when the satellite D1 is just passing P1, which altitude is 400km and observable range is 0?~3.416?, while the rotating angle of the earth is 0?. Obviously, > . Then at the time t1=3069.996s, we can control the satellite D2 to reach the observing point P1, which altitude is 1400km, and obtain the observable range for 0?~15.434? (3.416?+11.968?). While in 3069.996s, the rotating angle of the earth is 12.827?, then > . Therefore at any time the observable range is larger than the rotating angle of the earth. Consequently, we can observe any points on the equatorial plane by D1 in 24 hours. That is we can cover all the earth by this constellation in 12 hours.
Hence, we can use 4 DASes on 2 orbital planes, they are orthogonal to each other. To avoid the interference at the polar point, we can use the sun-synchronous orbits. This constellation can cover all the earth in 6 hours. Therefore we can cover all over the earth in 2 hours with 12 DASes on 6 sun-synchronous orbits, which is shown in Fig. 5.3.
Fig. 5.3 The constellation with 12 DASes
6. CONCLUSIONS AND FUTURE WORK
In this paper, we propose to observe the particular area which is in the emergency situation or suffered by a natural disaster with the DAS. We obtained the optimal orbital controls by using the Steepest Descent Method. First, we performed the optimal calculations for various between the DAS and the observing point. We can observe the area with the minimum fuel when is ƒÎ. Observing the area with small requires more fuel. Therefore it is a trade-off of observing an area with a proper and saving the fuel. Next, we tried to extend the time of the DAS to reach the observing point. We could extend the time up to 372.59s. Extending the time longer requires more fuel. Therefore it is also a trade-off of extending the time and saving the fuel. The more useful method to extend the time is that we control the DAS to higher altitude and make it observe the particular area at the apogee. We proposed some constellations of DASes. Our final purpose is building an optimal constellation with DASes (optimal orbit, optimal number of DAS), which can observe anywhere on the earth in a short time with minimum fuel consumption.
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