The Optimal Orbital Control and Constellation of Dive and Ascent Earth Observing Satellites

Tianshuang Fu and Fumiaki Imado (Shinshu University)

Abstract: We proposed the use of a Dive and Ascent earth observing Satellite (DAS) to observe the particular area which is in the emergency situation or suffered by a natural disaster. In this paper, we selected the forces of the thrusters fixed to the satellite body three axes as control variables, and obtained the optimal thrust patterns by using the Steepest Descent Method. Some of new ideas and study results are presented. We also proposed the ideas of constellations about the DAS here.

降下・上昇地球観測衛星の最適軌道制御およびコンステレーション

付　天爽，今度　史昭（信州大学）摘要： われわれは降下・上昇地球観測衛星を用いて有事状態の地域と自然災害の被災地等を迅速的に観測することを提案してきた．本稿において，衛星機体軸に固定されたスラスタの推力を制御変数とし，最急降下法を用いて最適な推力パターンを求めた．ここで新しいアイデアおよび研究結果を示し，またDASのコンステレーションを提案する．

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 earthAngular 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]).

Z

X

Y

RX

oY

RY

iη

Ω

equinoxVernal 　

planeEquatorial

RZ

oZ

oX

eωZ

X

Y

RX

oY

RY

iη

Ω

equinoxVernal 　

planeEquatorial

RZ

oZ

oX

eω

Fig.1 Coordinate systems and symbols

The transformation matrix A from to is given by Eq. (2.1).

(2.1)where the symbols “C” and “S” show the abbreviation of “Cos” and “Sin”. The symbols , and show the argument of latitude, the right ascension of the ascending node and the inclination of the DAS’s 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 satellite’s 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

Satellite’s parametersInitial mass 350kgFuel mass 175

Shape 1×1×1(m3)Thrusts (±500),(±18),(±500)N

313s

Initial orbital parametersAltitude 400km

Inclination 90°Tangential velocity 7.66856km/s

Period 5553.63sInitial 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 π/2, 5π/8, 3π/4, 7π/8 and π. 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 3π/2, 9π/8, 7π/4, 15π/8 and 2π, and final altitude of all cases are 400km.As the paper space is limited we only show the results about the are π/2 and π. First, we show the optimal calculation results about

is π. We obtained the optimal thrust forces’ 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

- 50000

5000 - 5000 0 5000

- 8000

- 6000

- 4000

- 2000

0

2000

4000

6000

8000

Y(km)X(km)

Z(km

)

Normal orbit

Controlled orbit

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 π/2 in Fig. 3.6 and Fig. 3.7.

Fig. 3.6: Histories of thrust forces

Fig. 3.7 Histories of altitude and velocityWe 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 π/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 consumptionFuel consumption

(kg)π/2(90￮) 78.555

5π/8(112.5￮) 51.6783π/4(135￮) 36.215

7π/8(157.5￮) 25.67π(180￮) 18.169

0102030405060708090

90 112.5 135 157.5 180The Δ η (degree)

Fuel

cons

umpti

on(kg

)

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 π, 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 π 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 500Case2 50 1000Case3 100 500Case4 100 1000

Fig. 4.1 Histories of thrust forces

Fig. 4.2 Histories of altitude and velocity

- 50000

5000 - 5000 0 5000

- 8000

- 6000

- 4000

- 2000

0

2000

4000

6000

8000

Y(km)X(km)

Z(km

)

Normal orbit

Controlled orbit

Fig. 4.3 Controlled orbit and normal orbitWe 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 π. 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 resultsCase 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 2697.406Case1 50 500 55.328 2755.242Case2 50 1000 89.636 2800.259Case3 100 500 80.887 2785.165Case4 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.

- 500- 400- 300- 200- 100

0100200300400500

0 1000 2000 3000 4000 5000 6000Time (s)

Thru

st (N

)

fxofzo

Fig. 4.5 Histories of thrust forces

0200400600800

1000120014001600

0 1000 2000 3000 4000 5000 6000Time (s)

Altit

ude

(km)

6.8

7

7.2

7.4

7.6

7.8

8

Veloc

ity (k

m/s)

hV

Fig. 4.6 Histories of altitude and velocity

- 50000

5000 - 5000 0 5000

- 8000

- 6000

- 4000

- 2000

0

2000

4000

6000

8000

Y(km)X(km)

Z(km

)

Normal orbit

Controlled orbit

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 sensor’s pointing function. Now we consider the pointing angle for ±44 degrees here. Obviously the observable rangeβis -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

44

D1 or D2P1

h

D1 D2

Nominal orbit

Controlled orbit h1=400k

mh2=1400km

eω

planeEquatorial

Observing point P1 Observing point

P2

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 π.

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.

REFERENCES[1] DAS Study Group., “Preliminary Study on Dive and Ascent Satellite, DAS”, Technical Report of National Aerospace Laboratory, TR-528, 1982.[2] Otsubo, K., Gaodai, T., and Nagasu, H., “Preliminary Trajectory Analysis for the Lowest-Flying Earth Satellite, DAS”, Technical Report of National Aerospace Laboratory, TR-507, 1977.[3] Fu, T. and Imado, F., “A Basic Study on the Optimal Trajectory Control of a Dive and Ascent Satellite,” Proceedings of SICE Annual Conference 2005, (2005) TA2-12-1, PP. 2255-2260.[4] Fu, T. and Imado, F , “A Basic Study about the Optimal Orbital Control of a DAS”, Proceedings of the 15th Workshop on JAXA Astrodynamics and Flight Mechanics, (2005) PP. 395-400.[5] Bryson Jr, A.E., and Denham, W. F., “A Steepest Ascent Method for Solving Optimum Programming problems”, Journal of Applied Mechanics, Vol. 29, June 1962, PP. 247-257.[6] Imado, F., “A Study on the Spacecraft Nonlinear Optimal Orbit Control with Low-Thrusts” Non-linear Problems in Aviation & Aerospace,1,287-296,1999.[7] Takeuchi, S., “Theory of the Orbital Motion of an Artificial Earth Satellite”, Technical Report of National Aerospace Laboratory, TR-807, 1984.[8] Takeuchi, S., “Atmospheric Drag Effects on the Motion of an Artificial Earth Satellite”, Technical Report of National Aerospace Laboratory, TR-748, 1982.[9] Fu, T. and Imado, F., “OPTIMAL TRAJECTORY CONTROL OF DAS,” The 25th International Symposium on Space Technology and Science (ISTS), (2006) ISTS-d-95p 1-6.[10] Fu, T. and Imado, F., “A Study about Optimal Orbit Control of Dive and Ascent Satellite,” The 2006 IEEE Internation Conference on Mechatronics and Automation, PP. 1008-1013.

planeEquatorial