A Fast Prediction Algorithm of Satellite Passes
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SSC00-VI-5 A Fast Prediction Algorithm of Satellite Passes P. L. Palmer & Yan Mai Surrey Space Centre University of Surrey,Guildford,GU2 7XH, UK Tel +44 1483 259278, Fax +44 1483 259503 [email protected] Abstract Low cost, fast access and multi-functional small satellites are b eing increasingly used to provide and exchange information for a wide variety of professions. They are particularly useful, for example, as a resource in very remote areas where they can provide useful information such as to rescue teams for changing conditions in a disaster zone and monitoring the sea state to warn approaching shipping. Unlike terrestrial communication systems, the receiver/transmitter in these di erent application areas needs to be powered on and contact to sp ecialised satellites to exchange data at sp eci c time rather than consuming valuable power at all the time. This, therefore, requires accurate knowledge of when these satellites will pass over the horizon of the given lo cation over a timescale of months in some cases. On the other hand, long term orbit estimation with high accuracy is also a key part for mission analysis and Earth observation op eration planning. The same algorithm is also needed onb oard satellites for autonomous on-b oard data management. The principal diculty of predicting satellite passes over such long timescales is to take account of the e ects of atmospheric drag. In this pap er, we present a fast algorithm for the prediction of passes of a LEO satellite over any given lo cation which provides high accuracy over a long p erio d. The metho d exploits sophisticated analytic mo dels of the orbit and provides direct computation of rise-set times and nadir tracking without the need of orbit propagation for hill climbing. This provides for a very small fast algorithm so more suitable for low-end computers and hand-held sets. Since the atmospheric drag is the key factor that a ects the accuracy for long-term estimation for satellite in LEO, this mo del not only includes secular p erturbation and p erio dic p erturbations, on the other hand a drag mo del based on the well acknowledged NASA atmosphere statistics is incorp orated. Di erent from those in other orbit prediction metho ds, for example, the most widely used SGP4, the drag mo del here has a variable parameter which is sub ject to mo dify as time b eing on according to p erio dical atmosphere prop erties changing. Simulation result shows it can provide quite accurate estimation for long lo ok-ahead p erio d. 1 Intro duction cost, fast access and multi-functional small satellites to provide and exchange information for a wide range of applications, which includes communication in a very Small satellites are b ecoming more and more exible remote area for changing conditions, disaster warning and p owerful to enable military and civil applications for approaching ships in the sea and so on. Di er- suchaslow cost store-and-forward communication, re- ent from ordinary communication stations which has mote facility metering, disaster warning for global ship- sucientpower supply, the communication mo dule in ping service and some Earth Observation missions. The these applications, say, a rescue team trekking in a replacement of traditional spacecraft in these applica- south American forest, has only very limited power tions is motivated by the reality of shrinking govern- capacity, therefore requires the receiver/transmitter to mental budgets and commercial interest in deploying be powered on and contact to the spacecraft at sp e- low-cost small satellites for a wide variety of profes- ci c time rather than consuming valuable p ower at all sions. the time. This, therefore, requires accurate estimation In particular, there has b een big trend to use low of when the satellites will start to be visible rise to P.L.Palmer 1 14th Annual AIAA/USU Conference on Small Satellite computers on-b oard spacecraft. Furthermore, in addi- a given lo cation on the Earth and similarly, the time tion to taking account of secular p erturbation and p eri- when the satellite disapp ears from the horizon set, o dic p erturbations, this algorithm includes a straight over a timescale of months in some cases. Meanwhile, forward atmospheric drag mo del derived from the well long term and highly accurate orbit estimation, esp e- acknowledged NASA atmosphere statistics in order to cially rise-and-set time computation, also plays a key overcome the diculties involving long-term prediction part in the pre-request information for mission analysis without incurring complex computational overhead. Dif- and on-b oard resources management in more general ferent from those in other orbit prediction metho ds, for communication, Earth observation and scienti c space- example, the most widely used SGP4, the drag mo del craft. here has variable parameter which is sub ject to mo dify One conventional way to solve this problem is to as time b eing on according to p erio dical atmosphere let the satellite run through its ephemeris, and check- prop erties changing. Simulation result shows this drag ing at each instant to see whether it just b ecomes vis- mo del works satisfactorily in prediction over long times- ible/invisible to a sp eci c ground lo cation. An orbital cale. propagation is advanced in time by some small time The pap er will b e organised as follows: in section 2, increment, t, and a p ossibilitycheck is p erformed at we describ e the rst phase of the new metho d, which each step, this kind of scheme is called tra jectory check- is called \Coarse Search", it works in two-body, secular ing. This metho d, however, is fairly computational ex- p erturbations arising from the Earth's oblateness and p ensive and therefore not suitable in the circumstances atmospheric drag p erturbations. In section 3, weintro- where p owerful pro cessing resources are absent. Esco- duce the second phase of the metho d, which is called bal [1] prop osed a faster metho d to solve this problem \Re nement" which improves the accuracy of the new by developing a closed-form solution for the visibility metho d. A metho d to up date the atmospheric drag p erio ds. He intro duced a single transcendental equa- parameter consistently for long-term prediction is ad- tion as a function of the eccentric anomaly of the satel- dressed in section 4. Simulation results are presented lite orbit which he called the control ling equation. Nu- in section 5, as well as the comparison of CPU pro- merical metho ds were then used to nd the rise and cessing time b etween the conventional metho d and this set times. The advantage of this equation is that it is new metho d. Finally, in section 6, we set out our con- solved only once per orbital p erio d, in contrast with clusions. the hundreds of times the Keplerian equation must b e solved with the standard step-by-step technique of hill climbing. The controlling equation, however, is only valid for two-b o dy motion. 2 Coarse Search Besides the control ling equation metho d, Lawton [7] has develop ed another metho d to solve for satellite- 2.1 Fundamental Algorithm - Two-Bo dy satellite and satellite-ground station visibility p erio ds Analysis for vehicles in circular or near circular orbits by ap- proximating the visibility function t,byaFourier series. More recently, Alfano [8] further develop ed the t function to suit all orbital typ es. A signi cant diculty, however, of predicting LEO satellite passes over long p erio d is to take account of the e ects of at- mospheric drag. In this pap er, a fast algorithm for the rise-and- set time prediction for LEO satellite is prop osed. It provides high accuracy over a long p erio d. By some further extension, this algorithm also has the p oten- tial to provide maximum elevation angle time predic- tion or nadir tracking problem solving, whichisvery Figure 1: Satellite orbiting around the Earth showing crossings useful for imaging planning using small satellites. The of the Target Latitude Line TLL. new metho d exploits sophisticated analytic mo dels of the orbit and therefore provides direct computation of We can easily estimate the satellite closest approach rise-set times and nadir tracking. This makes it very time bychecking the satellite ascending and descend- suitable for low-end pro cessors in hand-held sets and ing passage once resp ectively per day. Set T = 2=n P.L.Palmer 2 14th Annual AIAA/USU Conference on Small Satellite to b e the orbital period of the satellite and t the time 0 when the satellite rst crosses over a given latitude line on the ascending pass see gure1. We call the circle of constant latitude that runs through the target lo c- ation the Target Latitude Line TLL. The key p oint of our approach is to use the fact that for two-body motion, a satellite will revisit exactly the same p ointin an inertial co-ordinate system after each orbital p erio d T see gure1 . This means that the satellite will make another ascending-pass over the TLL at time t + T . To simplify the discussion we shall ignore the 0 descending passages over the TLL and include them again only at the end. Note, in this metho d, satellite Figure 2: This gure shows the basic idea of our new metho d p osition is expressed by the redundant epicycle co ordin- for satellite rise-and-set times.