apchi technique for rapidly and accurately predicting ...

AAS 12-253

APCHI TECHNIQUE FOR RAPIDLY AND ACCURATELY PREDICTING MULTI-RESTRICTION SATELLITE VISIBILITY Xiucong Sun,* Hongzheng Cui,† Chao Han,‡ and Geshi Tang§ Multi-Restriction Satellite Visibility Prediction (MRSVP) problem is of great significance in space missions such as Earth observation and space surveillance. This paper presents a numerical method to rapidly and accurately compute sitesatellite and satellite-satellite in-view periods, taking multiple restrictions into account. A novel curve fitting method named Adaptive Piecewise Cubic Hermite Interpolation (APCHI) technique is introduced to approximate waveforms of visibility functions derived for corresponding restrictions, featured with autonomous searching for the best interpolation points to guarantee accuracy. Test results obtained from this approach are almost the same with those from conventional trajectory check method. However, this new approach reduces more than 90% of computation time. As this numerical method can apply to all kinds of orbit types and propagators, it proves to be a good choice for satellite constellation design and mission planning.

INTRODUCTION The Multi-Restriction Satellite Visibility Prediction (MRSVP) problem is the study of determining the imaging and communication opportunities for a satellite to observe or communicate with an object either in orbit or on the Earth’s surface. It is closely related to the classical satellite rise-and-set problem, which deals with when a satellite is visible for a given ground station through investigating their geometric relationship.1 However, the MRSVP problem is much more complicated as it takes account of multiple visibility restrictions in order to produce more practical results. For example, optical observation of resident space objects from a space-based platform is subject to a variety of constraints, including the camera’s field of view, sun/moon/Earth light illumination, Earth shadow and so on.2, 3 Under this circumstance, the MRSVP problem’s task is to predict when a bright target will come into and out of the narrow field of view without stray light interference. The MRSVP problem has broad applications. Besides the example of space surveillance mentioned above, there are also many other space missions that have a requirement to solve satellite visibility problem. During the design of satellite constellation for Earth observation, usually the first necessary work to evaluate its coverage performance is to generate the access timetables for

*

Graduate Student, School of Astronautics, Beihang University, Beijing 100191, China. [email protected]. Ph.D., School of Astronautics, Beihang University, Beijing 100191, China. † Post Ph.D., Flight Dynamics Laboratory, Beijing Aerospace Control Center, 100094, China. ‡ Professor, School of Astronautics, Beihang University, Beijing 100191, China. § Professor, Flight Dynamics Laboratory, Beijing Aerospace Control Center, Beijing 100094, China. †

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every member satellite to photograph the ground location we have interest in.4 Analogous work should be done in the first place when scheduling a satellite with an antenna to transmit data with a specific ground station. Moreover, in communication satellite constellation design, the intersatellite link performance can also be assessed by obtaining the time periods when satellites are able to communicate with each other. Given the wide-ranging employment in satellite design and mission planning, the MRSVP problem is of great value to be researched. A conventional approach to solve the MRSVP problem is to let the satellite run through its ephemeris and to check at each instant whether it can “see” the target. In this method, the Kepler equation will be solved hundreds of times per orbit period to yield precise results. One disadvantage of this method is the tremendous computation time, especially when many orbital perturbations and visibility restrictions considered. This is particularly undesirable for real-time onboard operation. Therefore, a fast and accurate algorithm to implement visibility prediction is greatly required, which will be of crucial importance for reducing the cost of both ground operation and onboard autonomy in large amounts of space missions. There have been a few of fast algorithms developed to solve the classical satellite rise-and-set problem.1, 5-7 However, most of them can’t be directly applied to or even adapted for the MRSVP problem, because the finite sensor capacity and other inevitable visibility restrictions will bring in great tricky complications. More recently, Yan Mai and Philip Palmer proposed a new method based on the search for “target closest satellite passage” from nadir tracking. 8 It’s a great progress as they studied the camera field of view for the visibility prediction, but it’s still not available in more complicated conditions where sensors may rotate in the satellite reference system. In addition, the nadir tracking fashion is apparently nullified for satellite-satellite visibility prediction. Among all the publications discussing the satellite rise-and-set problem, Salvatore Alfano’s work provided a numerical method not only valid for all orbit types but also suitable for all perturbed satellite motions.9 In this method, a space curve modeling technique called parabolic blending is used to construct the waveform of the visibility function. This paper improves Alfano’s numerical method and promotes a novel technique named Adaptive Piecewise Cubic Hermite Interpolation (APCHI) to ameliorate the efficiency and accuracy of curve fitting. At the same time, a variety of restrictions affecting satellite visibility are analyzed and modeled to obtain corresponding visibility functions. To examine and validate this new method, two synthetic simulations are carried out and the results are compared with those of the conventional brute method. It proves to be a time-saving, high-accuracy and widely applicable numerical method to solve the MRSVP problem. APCHI TECHNIQUE FOR CURVE FITTING Given a second-order continuous time-varying function and a time interval, we can use a firstorder continuous curve which consists of numerous piecewise third-order polynomials to accomplish approximation. For each subinterval, corresponding cubic polynomial is created by the function values and their first-order time derivatives at the beginning and ending points. This interpolation method is known as Cubic Hermite Interpolation.10 Apparently, the prerequisite for precise global approximation is to ensure every cubic polynomial fits local function waveform exactly. In most cases, the original waveform changes values at a varying frequency. Therefore, if we choose a fixed step to complete interpolation, it will probably cause distortion at the fast-changing parts of the waveform. That indicates the accuracy of approximation has a great relationship with the selection of interpolation points. This paper presents an autonomous searching technique to determine interpolation subintervals by checking second-order derivative consistency and comparing extremum variances. The overall method for curve fitting in this paper is designated the name

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Adaptive Piecewise Cubic Hermite Interpolation. Here we give a general description of this technique and Appendix A will provide a complete algorithm. Consider a function f t  with t  a, b . If its function value and first two derivatives at an arbitrary point  in the given interval are known, denoted by f  , f  , f  , we can initiate the search for support abscissas 0 , 1 , , n , with a  0  1     n  b . Once the interval partition   a  0  1     n  b is determined, Cubic Hermite Interpolation will be used to create a cubic polynomial Ci t  to approximate f t  on the ith subinterval i 1 , i  , i  1,2, , n . The expression of the cubic polynomial Ci t  is written as Ci  t  

3hi  t  i 1   2  t  i 1  2

hi3

3

hi3  3hi  t  i 1   2  t  i 1  2

f i  

hi3

 t  i 1   t  i  f    t  i 1  t  i    i 2 2 2

hi

2

hi

3

f i 1 

(1)

f i 1 

where hi  i  i 1

(2)

The search for support abscissas is a sequential optimization process. Each step for a search of  i has an optimization objective to make sure the differences between the second-order derivatives, i.e. f i  vs. Ci i  and f i 1  vs. Ci i 1  are small enough. That is called “second-order derivative consistency check”, given by

 

1  scale f  hi2  0.1  scale  f  48

(3)

where

 



scale f  max f i   Ci i  , f i 1   Ci i 1  scale  f   max  f i  , f i 1  



(4) (5)

 

In Eq.(1), the left term is an integration of scale f within half-length subinterval and the right term is a precision criterion for approximation. The coefficient 0.1 can be adjusted by users to change the balance between speed and accuracy. Eq. (1) itself is not enough to guarantee the cubic curve of Ci  t  coincides with the waveform of f  t  . Thus, a double check called “extremum variance comparison” is advanced, if extremums exist. It aims at ensuring the extremums of Ci  t  are close to the function values of f  t  at the extremum points. Let  represents one of the extremum abscissas of Ci  t  , the criterion is given by

f    Ci    0.01 scale  f 

(6)

In Eq.(6), the right term is also a precision criterion and the coefficient 0.01 can also be adjusted by users. For these two criteria, scale  f  represents the magnitude of function f  t  . Eq.(1) transforms the second-order derivative differences to the level of function values and Eq.(2) is a simple comparison of the extremums. When scale  f  is around 1, these two criteria reach the best effect.

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The APCHI technique is a good choice to exactly approximate function waveforms, as long as we know the function values and first two derivatives at any abscissa. This method changes its step of interpolation according to the frequency of the waveform changes values. It ensures the accuracy of approximation as well as achieves fast solution. SATELLITE VISIBILITY FUNCTIONS Satellite 1

r1 Satellite 2

r2

Earth Center

(a) Satellite communication with antenna Satellite 1

r1 Satellite 2

r2

Earth Center

rs sun

(b) Satellite imaging with camera Figure 1. Viewing Restrictions of Satellite Visibility.

Restrictions for satellite communication and imaging bear a great relationship with the property of spaceborne sensors. In this paper, two representative sensors are considered, that is, antenna and camera. As illustrated schematically in Figure 1, viewing restrictions for antenna communication include conic field of view, range limit and Earth occultation, whereas the counterparts for camera imaging include pyramidal field of view, range limit, sun/moon/Earth light illumination, Earth occultation and Earth shadow. These restrictions are abstracted from actual practice and values of some parameters are given from experience, not real data from satellite operations. Outlined next are common restrictions for the two sensors and corresponding visibility functions and criteria to determine visibility. Viewing Restrictions and Visibility Functions for Antenna Let r1 and r2 be the Earth Centered Inertial (ECI) position vectors of an orbiting satellite S1 and a target S 2 . Be S 2 a ground site or a satellite, r1 , r2 and their first two derivatives r1 , r2 , r1 , r2 can be provided by a chosen propagator or ephemeris.

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n

d

S2



r1,2 S1

Figure 2. Antenna’s Conic Field of View.

As depicted in Figure 2, the antenna’s field of view is a cone topped by a dome at the maximum sensor range. Its orientation and size can be characterized by the unit vector of boresight pointing, half cone angle and range limit, denoted by n ,  and d respectively. Firstly, we consider the angle restriction. Only when the viewing angle between antenna boresight and the target is within the limit  , the target has an opportunity to be visible. Thus, the visibility function of viewing angle restriction is defined as f coneangle  t  

n  r1,2

(7)

r1,2

and the visibility criterion is given by fconeangle  t   cos

(8)

where r1,2  r2  r1 and  means the length of a vector. Note that the value range of f coneangle  t  is  1,1 , which is a good character for polynomial curve fitting. Besides, whether the antenna is fixed or spinning in the satellite’s body coordinates, the vector n and its first two derivatives can always be obtained. The visibility function of range limit is defined as

f range  t   r1,2

(9)

f range  t   d

(10)

and the visibility criterion becomes

S1

r12 r1 L RE

O

r2

Figure 3. Earth Occultation.

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S2

An important issue in satellite-satellite communication is the Earth occultation, which means the line of sight between the two objects is obstructed by the Earth. It occurs when S 2 goes behind the Earth. In this case, the shortest distance from the Earth center O to the line S1S2 is less than RE  h , as shown in Figure 3. RE is Earth radius and h is a bias factor responsible for atmospheric interference. So, if the distance is  d , the target is visible. However, this requirement is too harsh because it overlooks a situation where the two objects are at the same side of the Earth. Then, the visibility functions of Earth occultation are defined as f occultation,1  t  

r1  r1,2 r1,2

(11)

foccultation,2  t   r2  r1,2

(12)

foccultation,1  t    RE  h  or f occultation,2  t   0

(13)

and the visibility criterion is given by

Viewing Restrictions and Visibility Functions for Camera

bx

Camera



by

 b

Figure 4. Camera’s Pyramidal Field of View.

As depicted in Figure 4, the camera’s field of view is a rectangle pyramid topped by a dome at the maximum sensor range. Its orientation is decided by three orthogonal unit vectors, bx , by , b . They form a coordinate system named FOV coordinates used to express the field of view. Its size is decided by two angles and a range limit, denoted by  ,  and d respectively. The visibility functions of viewing angle restrictions are defined as  r1,2   r1,2  by  by   b  f rectangle, x  t    r1,2   r1,2  by  by

(14)

 r1,2   r1,2  bx  bx   b  f rectangle, y  t    r1,2   r1,2  bx  bx

(15)

and the visibility criterion is given by f rectangle, x  t   cos and f rectangle, y  t   cos 

6

(16)

Be similar to the antenna, whether the camera is fixed or spinning in the satellite’s body coordinates, the vectors bx , by , b and their first two derivatives can also be obtained. The visibility function and criterion of range limit are the same with Eq. (9) and Eq. (10).

sun r1,s s

S2

r1,2

S1 Figure 5. Sun Light Illumination.

For optical observation, geometric relationships with the sun, moon and Earth are important. If the camera is observing at a bright background, such as the sun, moon and Earth, the resulting light illumination at the focal plane will intervene in imaging. Let rs and rm be the ECI position vectors of the sun and moon. The minimum angles away from them required for imaging are designed arguments denoted by  s and  m , as shown in Figure 5. The visibility functions of the sun and moon light illumination are defined as f sunlight  t  

f moonlight  t  

r1,2  r1, s r1,2  r1, s

r1,2  r1, m r1,2  r1, m

(17)

(18)

and the visibility criteria are given by f sunlight  t   cos s

(19)

f moonlight  t   cosm

(20) (21)

where r1, s  rs  r1 and r1,m  rm  r1 . In terms of Earth light illumination, its visibility function can be combined with Earth occultation. Actually, when the two satellites are at the same side of the Earth, it turns the Earth light illumination problem. When they are separated by the Earth, it turns the Earth occultation problem. The combined visibility function is written as f earthlight  occu  t  

r1  r1,2 r1,2

(22)

and the visibility criterion is fearthlight occu  t   RE  h

7

(23)

A sun

B

r2

O



RE

rs

C

Figure 6. Earth Shadow.

Another prerequisite for optical observation is the target must reflect lights. When a satellite enters into the Earth’s shadow, it will not be possible for observation. To simplify calculation, the umbra shadow is treated as cylinder projection of the Earth and the penumbral effect is ignored, as seen from Figure 6. If S 2 is a satellite, the visibility function for shadow passage is defined as f earthshadow  t  

r2  rs r2  r2   RE  h 

2

(24)

and the visibility criterion of fearthshadow  t   1

(25)

GENERATION OF IN-VIEW TIMETABLES In most cases, the visibility functions developed above are all second-order continuous. Their first two derivatives can be obtained by differentiation. As we can use orbit propagator or ephemeris to predict the ECI position vectors of satellites, the sun and moon, therefore the function values and first two derivatives of those visibility functions can be known at any time. Note that all of the formulae of visibility criteria have constant limits on the right such that they can be represented by horizontal lines in the graph of visibility functions. The time of a visibility function curve crossing the boundary line is the rise or set time. With APCHI technique, the visibility function waveforms can be approximated by cubic curves. It alleviates the process of root finding. In practice, all of the restrictions are not necessary to be considered. Different missions involve different viewing restrictions. Thus, synthetic methods to determine the final in-view periods with multi-restrictions are needed. This paper advances a parallel and a sequential algorithms to adapt for different circumstances. Suppose there are k restrictions involved. The first approach is to obtain k separate in-view timetables simultaneously and implement intersection operation. This is suitable for a parallel work. The other approach is to obtain in-view timetables one by one. The results of a former visibility function provide new time intervals for interpolation of the latter one. That operates like a k  1 multilayer time filter. SIMULATION RESULTS To demonstrate the efficiency and accuracy of the APCHI technique for the prediction of satellite visibility, two separate synthetic simulations are supposed and carried out in this paper. Re-

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sults are compared with those from conventional methods. In the simulations, four satellites are involved: satellite 1 carries a canted-fixed antenna to transmit data with satellite 2 and satellite 3 carries a spinning camera to observe satellite 4. Classical elements of the four satellites and parameters of the two space-borne sensors are all listed in Table 1 and Table 2, included in Appendix B. The epoch time of satellite orbits and the initial time for prediction are prescribed to be 1 Jan 2011 12:00:00.000 UTCG. The simulation time for antenna communication is one day while the time for camera imaging is 3000 seconds. Note that coordinates in Table 2 are set in the satellite body reference frames, which coincide with the VVLH reference frames. Orbital motion is modeled with J 2 effect in the simulation. Users can choose more accurate propagators according to mission requirements.

VISIBILITY FUNCTION OF CONIC VIEW ANGLE

TRUE CURVE ANGLE LIMIT INTERPOLATION POINTS

1.1

cos(theta) 0.85

0.6

0.35 0

15000

30000

45000

60000

75000

86400

Time/s

Figure 7. Visibility Function Curve of Conic Viewing Angle. VISIBILITY FUNCTION OF RECTANGLE VIEW IN X DIRECTION

VISIBILITY FUNCTION OF CONIC VIEW ANGLE

1.35

TRUE CURVE ANGLE LIMIT INTERPOLATION POINTS 1.05 cos(alpha) Time/s

0.95

0.85

0.75

0.65

0

500

1000

1500

2000

2500

3000

Time/s

Figure 8. Visibility Function Curve of Rectangle Viewing Angle in X Direction.

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VISIBILITY FUNCTION OF RECTANGLE VIEW IN Y DIRECTION

TRUE CURVE ANGLE LIMIT INTERPOLATION POINTS

1.05

cos(beta)

0.95

0.85

0

500

1000

1500

2000

2500

3000

Time/s

Figure 9. Visibility Function Curve of Rectangle Viewing Angle in Y Direction.

Figure 7 through Figure 9 show the waveforms of visibility functions corresponding to viewing angle limit for antenna, x-direction angle limit and y-direction angle limit for camera. The solid lines represent the true curves obtained from a 1s step calculation and the dashed lines represent the critical value for visibility. The dots are interpolation points searched by APCHI technique. Other viewing restrictions considered in the simulations include range limit, Earth occultation, Earth light illumination and Earth shadow. Parameters of these additional restrictions are listed in Table 3. The sun and moon light illumination are didn’t treated while users can added them to their own simulation. The bias factor h for Earth radius is an empirical value which makes no difference on the process of calculation. Thus it’s set to zero in the simulations. Time/s

Table 4 and Table 5 show the simulation results. The new approach reduces more than 90% of the computation time while the absolute error is no more than 0.2s and there are no conditions of roots missing, which manifests the good performance of this new method. Another question for the APCHI technique is the determination of initial time step for interpolation points searching. Notice that every visibility function has a kind of periodic characteristic as a result of the orbital motion or spinning of sensors. Large amounts of test results suggest 1/20 of the time period is the best initial step for APCHI. CONCLUSION This paper presents a rapid method for prediction of MRSVP problem with adequate accuracy. Visibility functions are second-order continuous under most circumstances and the function values and first two derivatives can be obtained from selected propagators. Users can choose any type of orbit propagators for desired accuracy. This new method can reduce more than 90% of computation time while provide the exact results for satellite visibility prediction. Extensive test results illustrate its broad application for different kind of orbits and restrictions, as long as suitable visibility functions are provided. The numerical method can be used as a powerful tool for orbit designers to analyze the coverage performance of satellites constellation in further research.

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ACKNOWLEDGMENTS This work is supported by the Open Research Foundation of Science and Technology on Aerospace Flight Dynamics Laboratory (Grant No. SFDLHZ2010001). APPENDIX A: ALGORITHM FOR APCHI TECHNIQUE 1. Given a visibility function f  t  and time interval  a, b , determine the initial time step D : D  1 20 P , where P is the characteristic period. It might be an orbital period or the period of sensor spinning. Then, 0  a, 1  a  D . 2. Obtain f 0  , f 1  , f 0  , f 1  , f 0  , f 1 . Implement cubic hermite interpolation at the subinterval 0 , 1  and create a cubic polynomial C1  t  . 3. Carry on the second-order derivative consistency check. If satisfying the criterion, go to setp 4. If not, shorten D to half, and then go back to step 2. 4. Carry on the extremum variance comparison. If satisfying the criterion, go to step 5. If not, shorten D to half, and then go back to step 2. 5. Repeat the process above for the search of next subinterval until it reaches the end of the interval b . APPENDIX B: TABLES OF SIMULATIONS PARAMETERS AND RESULTS Table 1. Classical Elements of Satellites.

Satellite

Semi-major axis (km)

Eccentricity

RAAN (deg)

Inclination (deg)

Argument of Perigee (deg)

True Anomaly (deg)

Satellite 1

15000

0.3

30

30

20

40

Satellite 2

8000

0.05

10

20

0

70

Satellite 3

18000

0

10

40

-

60

Satellite 4

8000

0

150

10

-

278

Table 2. Parameters of Spaceborne Sensors.

Sensor

Field of View

Angle Limits(deg)

Initial Position of Pointing

Pointing Type

Spinning Axis

Spinning Rate(rev/mi n)

Antenna

Cone

  20

(sin30,0,co s30)

Fixed

-

-

Camera

Rectangle pyramid

 5  5

(sin20,0,co s20)

Spinning

(0,0,1)

0.2

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Table 3. Parameters of Additional Restrictions. Sensor

Restrictions

Relevant Parameters

Range Limit

d  20000km

Earth Occultation

RE  6378.14km, h  0

Earth Light

RE  6378.14km, h  0

Earth Occultation

RE  6378.14km, h  0

Antenna

Camera

cylinder projection,

Earth Shadow

RE  6378.14km, h  0

Table 4. One Day In-view Periods for Satellite 1 vs. Satellite 2. APCHI Technique

5s Trajectory Check

Absolute Difference

Start

Stop

Start

Stop

Start

Stop

1189.626

4185.476

1189.634

4185.449

0.008

0.027

11321.865

13434.179

11321.808

13434.146

0.057

0.033

24185.033

26025.015

24185.044

26025.005

0.011

0.010

34418.351

35874.242

34418.368

35874.201

0.017

0.041

36743.162

39390.962

36743.152

39390.981

0.010

0.019

46526.636

48623.549

46526.593

48623.566

0.043

0.017

59329.485

61359.391

59329.462

61359.386

0.023

0.005

69148.089

73594.295

69148.039

73594.168

0.050

0.127

81743.534

83589.152

81743.520

83589.175

0.014

0.023

Table 5. 3000s In-view Periods for Satellite 3 vs. Satellite 4. APCHI Technique

1s Trajectory Check

Absolute Difference

Start

Stop

Start

Stop

Start

Stop

2405.578

2405.846

2405.655

2405.866

0.077

0.020

2677.774

2697.169

2677.752

2697.175

0.022

0.006

2968.756

2989.028

2968.755

29893.039

0.001

0.011

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