MODELLING OF PERTURBATIONS FOR PRECISE ORBIT DETERMINATION

Transcription

1 MODELLING OF PERTURBATIONS FOR PRECISE ORBIT DETERMINATION 1 SHEN YU JUN, 2 TAN YAN QUAN, 3 TAN GUOXIAN 1,2,3 Raffles Science Institute, Raffles Institution, 1 Raffles Institution Lane, Singapore yujunauthor@gmail.com Abstract- An orbit determination and propagation package is developed in Python to predict satellite trajectories accurately with position differences of within 1 km after one orbit when compared to simulated satellite true position. The effect of Earth s non-uniform gravitational fields, caused by equatorial bulge and non-sphericity, is modelled using zonal spherical harmonics from J2 to J6 and the orbit is propagated numerically via Runge-Kutta-4 (RK4) method. The full J2-6 gravitational coefficients model yields improvement in position accuracy over the J2-only model. Furthermore, the use of additional state vector observations via differential correction allow for refinement of initial orbital parameters, hence achieving good agreement with simulated satellite positions. The orbit determination and propagation software package is tested using two real-life applications: satellite collision avoidance and Earth observation. The first application is a simulation of the orbital hypervelocity collision between Iridium 33 and Cosmos 2251 using publicly available NORAD Two Line Elements (TLE) data. The two satellites simulated time of closest approach is 16:55:40 UTC, 10 February 2009, accurate to within one minute. The second application is a geo-registration of the International Space Station (ISS) using photographs taken by ISS Expedition 35 crew and corresponding ISS TLEs, calculatingits position over a rotating Earth coordinate frame and generating a ground-track. The location of ISS over an unknown longitude and latitude location on 15:01:42 UTC 27 March 2013 was interpolated to be 1.00 N, E, matching actual ISS position to within 0.1 degrees. Key words- Orbit Propagation, Orbit Determination, Differential Correction, Conjunction Analysis, Geo-registration. I. INTRODUCTION Accurate knowledge of a satellite s orbit is required for mission planning and geo-registration of ground positions. Hence, orbit determination must return reliable orbital elements data, which can provide more accurate positions of the satellite across time. Orbit determination improves initial orbital elements from a set of observation data, such as state vectors (position, velocity) or angles (azimuth, elevation) [1]. These initial elements uniquely define an orbit and thus a satellite s position in time [2]. Fig. 1 shows the flow from observation or sensor-collected data to refine position and velocity information. For an Earth orbit satellite, dominant perturbations are caused by the non-uniform gravitational field around an oblate Earth [3, 4]. In this paper, we modelled J2 to J6 coefficients using numerical propagation by Runge- Kutta-4 (RK4) method to develop an orbit propagation and determination model in Python. To test the orbit propagation and determination model s effectiveness in actual orbital collision avoidance (conjunction analysis) and photo georegistration applications, we calculated the 2009 satellite collision between Iridium 33 and Cosmos 2251 based on open-source Two Line Elements (TLE) data [5]. This was the first, catastrophic hypervelocity orbital collision [6]. The accuracy of the orbit propagator determines the confidence of two objects closest approach distance, ensuring safe space operations. Additionally, ground identification was performed using actual International Space Station (ISS) photos taken during Expedition 35 in 2013 [7] and corresponding TLE sets. II. PROPAGATION MODEL WITH J2-6 COEFFICIENTS Classical two body interaction results in perfect elliptical motion with only a central force from a point-like mass. In reality, for an Earth-orbiting satellite propagation, dominant perturbation effects are the J2 attraction (due to Earth s equatorial bulge) and zonal spherical harmonics from J3 to J6 (higher order non-sphericity terms) [2]. The effects of higher order J- gravitational terms can be solved numerically as no analytical solution exists [8]. The classic RK4 numerical method is used [2]. A detailed propagator flowchart is presented in Fig. 2. The equation of motion is Newton s Second Law (Eq. 1) with gravitational harmonics attached. Fig.1. Overview of orbit determination steps 198

2 Fig.2. RK4 orbit propagation function In Eq. 1, the J x and J y terms are symmetric due to equatorial bulge. All three J terms depend on inclination and are described by Eqs. 2& 3. using STK [10] to simulate satellites position and velocity. Gaussian noise was then added to one set to simulate GPS data ( GPS data), while the other data set was unchanged ( real data). The position error between the propagated result and real set is computed using at every time step by Eq. 4. Fig. 3 shows the position error for the propagator compared to true satellite positions using real simulated data without any noise, as a test of propagator accuracy only. TheJ2-6 model (the entirety of Eqs. 2 & 3) results in 30 km position error over 1 day. The J2onlymodel (up to J 2 in Eqs. 2 &3) gives up to 40 km in position difference, with large periodic fluctuations. In particular, between 500 min to 800 min, the J2 error is larger than J2-6 case by up to 50%. Hence, a thorough numerical solution for J2-6 model yields much better consistency and reliability of propagation. The J2-6 model will be used subsequently throughout this paper. Orbit propagation was developed in numerical and scientific Python libraries, using constants for Earth equatorial radius r E, gravitational coefficients J 2 to J 6 [2].The inputs are initial position and velocity and output is the satellite s full trajectory for a specified mission duration. III. PROPAGATION MODEL ACCURACY WITH SIMULATED GPS DATA To verify the accuracy of the RK4 propagator section, results were compared with simulated satellite position and the error plotted over time. For this paper, two HPOP ephemeris data were generated Fig.3. J2 and J2-6 models position error with real position over 1 day 199

3 Fig.4. Differential correction to improve initial orbital elements IV. ORBIT DETERMINATION WITH DIFFERENTIAL CORRECTION and velocity data, since observations currently come from the same source. (Eq. 7) Based on preliminary elements and subsequent propagation, we can improve our knowledge of the initial orbital elements by considering the rest of the observation data, such as GPS readings [1, 9], to obtain more accurate propagation. The improvement function used in this paper is differential correction as illustrated in Fig. 4. It is coded in numerical Python, incorporating the earlier propagation function. For any propagated quantity in the state vector at time n, the total error comes from initial error in each of the orbital elements used to propagate, in this case classical orbital elements a, e, i, ω,ω and θ. Eq. 5 shows the error in x displacement; the y and z errors are calculated similarly. Each partial derivative at each time step can be linearly approximated with the difference in propagated quantity (x, y, z) when only one initial element is perturbed (by 1% variation) [1]. (Eq. 6) Hence, we can form the residuals matrix B and derivatives matrix A as required in Fig. 4 to solve for the correction matrix C. An identity matrix W is used to weigh the reliability of the observed position To compute A, only the x, y, z components of state vector and their dependence on initial orbital 200

4 elements(classical or Cartesian) are considered. To simplify calculations, we neglect velocity terms. In B, additional observation data is introduced. Thus, A is 3n by 6 and B is 3n by 1, where n is the number of observations. x, y, z components must be calculated individually to obtain a unique trajectory in 3D space [9]. If we have more observation readings than elements, then a Root Mean Square fit is used. Finally, the differential correction process is iterated, using the improved orbital elements. In essence, initial orbital elements are refined. V. DIFFERENTIAL CORRECTION EFFECTIVENESS TEST propagation. Yet, with increasing number of additional readings, the gradient tends towards a minimum, which is bounded by the error between the J2-6 propagator and the commercial HPOP propagator. This minimum occurs when the initial positional error has been fully rectified by orbit determination. Practically, orbit determination is performed using only a limited number of data sets to propagate forward for a much longer epoch (eg. Use 1 hour of readings for 1 day of propagation) [2,4]. Additionally, the oscillatory fluctuations of error has a period of 1 orbit (~90 minutes for Low Earth Orbit), caused by non-symmetric spherical harmonics of Earth s gravitational field [4]. One of the HPOP ephemeris generated has Gaussian error added (Section 3). This is to simulate operational GPS data with it accuracy of about 100m (at high altitudes and velocity) [2, 4].Hence, using a single GPS state vector at t= 0 for forward propagation (using J2-6 propagator) will result in significant deviations from the actual orbit. To refine this state vector at t=0, orbit determination is conducted using subsequent state vectors at t=0, 1, 2 to obtain more accurate propagator data ( prop data). We investigate how the number of subsequent state vectors used affects the overall accuracy of propagated data, by considering the positional error at every time step between prop and GPS data (Eq. 8). To study the error of the overall package, the positional error over time is plotted between propagated data and true HPOP data for various corrected (with orbit determination) and uncorrected propagators (Fig. 5). The number of readings used to orbit determine is varied from 1 to 50, with 1 being a direct uncorrected propagation, at a frequency of 1 position vector per minute. The orbit is propagated for 1 day (1440 min). Fig.5. Total system error with and without orbit determination From the graph, orbit determination results in lower gradient of error, indicating more accurate orbit VI. CASE STUDY - I: CONJUNCTION ANALYSIS FOR COLLISION AVOIDANCE An important application of accurate orbit determination is in conjunction analysis: predicting possible collisions to take evasive action. To analyse satellites (especially foreign or defunct ones) and debris orbits, publicly available Two Line Element (TLE) data sets can be used [2, 6]. These data sets, published by NORAD, provide the 6 classical orbital elements for a satellite at regular intervals (~daily). In this study, TLE sets for both Iridium 33 (satellite 1 ) and Cosmos 2251 (satellite 2 ) were downloaded from CelesTrak [3] and propagated using the Python code developed to find the positions of each satellite in an Earth Centered Inertial (ECI) frame. The collision between Iridium 33 (US commercial) and Cosmos 2251 (Soviet military) on February 10, 2009, 1656 UTC time was catastrophic [6], generating thousands of space debris and highlighting the importance of space situational awareness [9]. The distance between the two satellites ( separation ) can be calculated by Eq. 9: Two satellites have the highest risk of collision at their closest approach (when separation is at a minimum) [6]. Using only one TLE data set results in less accurate propagations; the use of multiple data sets with orbit determination results in more accurate closest approach prediction. Using the J2-6 propagator only, when the last TLEs for each satellite (1 day prior) were used to propagate directly, the time of closest approach was 20:12:03 UTC (Fig. 6) and predicted separation was 12 km inbetween. The satellites are periodically closer and further apart due to their intersecting elliptical orbits around Earth. From the propagated data, there is a global minimum in separation between the two satellites as expected. However, note that the collision timing is inaccurate by 4 hours, suggesting significant error growth from initial TLE inaccuracy. 201

5 Fig.6. Closest approaches with last available TLE only Given this limitation, we conduct orbit determination with five prior TLE sets for both satellites to refine the accuracy of initial propagator input. This results in the precise time of closest approach at 16:55:40, accurate to the minute, albeit at a slightly larger separation of 9.8 km (Fig. 7), which is caused by the relative inaccuracy (about 1.5 km) and infrequency (about once daily) of TLE data sets [2]. The significant improvement in time suggests a much more reliable knowledge of position when orbit determination is conducted, as initial input is refined. This paper uses two photos, and the corresponding spacecraft nadir point (point on Earth directly beneath satellite), from Gateway to Astronaut Photography of Earth, a catalogue of NASA astronaut photography [5, 7]. One photo was the unknown target (Singapore, Table 1) and another as a known calibration (Miami, Table 1), taken 1 day apart during ISS Expedition 35. Using TLE data sets from CelesTrak from March 25-30, the motion of the ISS is propagated with orbit determination in the Python code developed. Using known times of image acquisition, ISS nadir point is calculated and corroborated with NASA data. As photos may be taken with the camera not pointing radially down, camera orientation is required to calculate the displacement of the centre point of the photo from the nadir point. Hence, with an accurate orbit determination and propagation system, operators can then predict the motion of satellites and subsequently geo-register ground positions of photos taken [4]. Results of the TLE-based study for ISS location are shown in Table 1, together with photograph details. Table 1 ISS position prediction over Singapore and actual value Fig.7. Improved closest approach with TLE orbit determination VII. CASE STUDY - II: GROUND IDENTIFICATION FOR SATELLITE TRACKING In addition to the ECI frame, the Earth Centered Earth Fixed (ECEF) frame is used to determine a satellite s position relative to a rotating Earth. Considering the rotation rate of Earth ( rad s [1, 2]), and the position of a known ground calibration point (Latitude: θ, Longitude: φ ), a satellite s position can then be converted from ECEF frame to a more familiar notation: longitude, latitude and altitude. Satellite position in ECI frame (x, y, z coordinates) is calculated at calibration ( cali ) and image acquisition ( targ ), and used to determine relative change in latitude and longitude (Eqs. 10 & 11). Latitude = sin sin + θ (10) Longitude = tan y x tan y x +φ ω (t t ) (11) The position of the ISS has been accurately predicted on 27 March 2013 over Singapore with orbit determination. The latitude and longitude values of ISS match completely. With further knowledge of camera orientation, the coordinates of the photo centre can be calculated towithin an error of 1km. This is more accurate because interpolation is conducted with TLE readings both before and after time of image acquisition. The motion of the ISS across Earth can be plotted in a groundtrack (Fig. 8) over a Mercator projection [2, 4]. 14 orbits are shown, from the time ISS passes over Singapore to Miami. The groundtrack shows the satellite s nadir over time as it orbits Earth. Each orbit (curve) is Eastwards of the previous orbit due to Earth rotation, and so the projections move rightwards [2]. These projections indicate the regions where image acquisition is possible. For the ISS, its orbital plane inclination bounds the latitude regions from 56.5 N to 56.5 S. 202

6 Fig.8. Groundtrack of ISS from 27 th to 28 th March 2013 CONCLUSION This study has fully developed and verified a reliable orbit determination Python code using the method of differential correction and a J2-6 gravitational perturbation propagator. The differential correction and propagation methods are collectively tested, producing a good agreement with simulated GPS data with 20 km positional difference after one day. Practical applications considered are orbital collision avoidance based on TLE data propagated and satellite image acquisition. The developed propagator algorithm is shown to predict accurately the real-life case studies of the 2009 Iridium-Cosmos satellite collision and ISS position relative to a rotating Earth. ACKNOWLEDGEMENTS The authors are grateful for the guidance by Dr Lou Kok Yong and Mr Kenny Chen from DSO National Laboratories during the study, and would like to acknowledge the support of Raffles Science Institute in providing the research attachment opportunity and support for attending the conference. REFERENCES [1] R. Bate, D. Mueller, J. White (1971) Fundamentals of Astrodynamics, Dover Publications Inc [2] D. A. Vallado (2007) Fundamentals of Astrodynamics and Applications, 3rd Edition, Springer [3] H. D. Curtis (2010) Orbital Mechanics for Engineering Students, Elsevier [4] O. Montenbruck (2000) Satellite Orbits: Models, Methods, and Applications, Springer [5] CelesTrak. Retrieved from [6] T. S. Kelso (2009) Analysis of the Iridium 33-Cosmos 2251 collision, Advances in the Astronautical Sciences, AAS [7] Gateway to Astronaut Photography of Earth, Earth Science and Remote Sensing Unit, NASA Johnson Space Center. Retrieved from [8] N. Ayat, M. Mehdipour (2006) Accurate Doppler Prediction Scheme for Satellite Orbits [9] D. A. Danielson, D. Canright, D. N. Perini, P. W. Schumacher, Jr. (1999) The Naval Space Command automatic Differential Correction process [10] Systems Toolkit (2015). Analytical Graphics Inc. Retrieved from 203