Estimation of satellite orbits using ground based radar concept Jonas Gabrielsson Master thesis, 30 hp Master of Science programme in Engineering Physics, 300 hp Spring term 2021 Acknowledgements A big thank you to my supervisor Carl Kylin for being enthusiastically helpful and providing invaluable guidance during this project. Thank you to all of ”Teknisk Fysik” at Umea˚ University for being such a great programme. Thank you to my family and friends not least the ones I have gotten to know during my studies. Finally a big thank you to SAAB for giving me this opportunity, it has been an amazing learning experience. June 28, 2021 Master thesis ”Estimation of satellite orbits using ground based radar concept” Jonas Gabrielsson Abstract Today an abundance of objects are circulating in earth captured orbit. Monitoring these objects is of national security interest. One way to map any object in orbit is with their Keplerian elements. A method for estimating the Keplerian elements of a satellite orbit simulating a ground based radar station has been investigated. A frequency modulated continuous wave radar (FMCW) with a central transmitter antenna and a grid of receivers was modeled in MATLAB. The maximum likelihood estimator (MLE) was obtained to estimate the parameters from the received signal. The method takes advantage of the relations between the Cartesian position and velocity and the Keplerian elements to confine the search space. For a signal to noise ratio (SNR) of 10dB, the satellite was followed during a time period of 0.1s where the positions were found within average error of range ±1:4m, azimuth ±2:0·10−6 rad and elevation ±8:4 · 10−7 rad. Using a linear approximation of the velocity the Keplerian elements were found within average error of i : ±0:0050 rad, W : ±0:0050 rad, w : ±0:0058 rad, a : ±2:60 · 105m, e : ±0:0021 and n : ±0:24 rad. A method to obtain more accurate estimates is proposed. 3 Contents 1 Background5 2 Ethical considerations5 3 Theory 5 3.1 Orbital Dynamics.............................................................5 3.2 Radar geometry.............................................................7 3.3 Signal.....................................................................7 3.4 Direction of arrival............................................................8 3.5 Signal matching..............................................................8 3.6 Maximum likelihood estimation..................................................9 3.7 Orbital parameters.......................................................... 10 4 Method 10 4.1 Assumptions............................................................... 10 4.2 Constraints................................................................ 11 4.3 Signal construction.......................................................... 11 Generating orbital parameters• Dividing the information 4.4 Strategy................................................................... 11 4.5 Estimation................................................................. 12 5 Results 12 5.1 Position estimation.......................................................... 13 5.2 Orbital parameter estimation.................................................. 15 6 Discussion 16 6.1 Estimation................................................................. 16 6.2 Signal modulation........................................................... 16 6.3 Simplifications.............................................................. 16 7 Conclusions 17 References 18 Master thesis ”Estimation of satellite orbits using ground based radar concept” — 5/21 1. Background The atmosphere of the Earth is becoming more and more populated by satellites. Keeping track of these satellites is of importance from a national security perspective. For an aerospace and defence company like SAAB, there is understandably an interest in de- veloping a tracking system for satellite traffic. The idea is to use a ground based radar station to track the satellites. One way to map orbiting objects is via their so called ”Keplerian elements” also known as ”orbital parameters”. The aim of this thesis has Figure 1. A satellite in elliptical orbit around been to estimate these orbital parameters from sim- Earth. The closest passage of the satellite is the ulated radar signals. This study is based on work periapsis at r p. The satellite location is defined by done in a previous master thesis project by Kylin, r which is at an angle, true anomaly (n), from r p. Kukic [11] where the satellites where restricted to a specific orbit. Now the task is to extend this model Where h is the specific angular momentum and m allowing any orbit to be found. is the standard gravitational parameter. a and b are the semi-major and minor axis respectively, which q 2. Ethical considerations b2 describe the size of the ellipse and e = 1 − a2 Since the study aims to develop a method which is the eccentricity which essentially describes how would be of use within the military realm, the need ”non-circular” the orbit is, where e = 0 represents to contemplate ethical aspects arises. The end prod- a perfect circle. n is the true anomaly which is uct considered in this study would be a radar which an angle representing the objects position on the would allow surveillance of satellites passing over. orbit and is the only time dependent parameter. The The main goal is thus to keep track of potential directional vectors P and Q are orthonormal vectors threats to national intelligence, hence a defensive that lie in the orbital plane. Using the fact that purpose. The proposed method could not be directly the specific angular momentum is given by h = p used for e.g shooting down or harming satellites, ma(1 − e2) we may obtain that would require a different type of study. a(1 − e2) 3. Theory r = (Pcosn + Qsinn) (1) 1 + ecosn 3.1 Orbital Dynamics For an object in elliptical orbit around a much heav- taking the time derivative can be shown to be (a ier object, one of the foci can be assumed to be cen- more complete derivation is available at [1] chap. tered in the heavier body as in fig.1. Parametrising 2.10) such an orbit can be done using Keplerian orbital parameters which describes the orbit’s shape, size, r m orientation and the position of the object. Assuming r˙ = (−Psinn +Q(e+cosn)): (2) a spherically symmetric gravitational field, the posi- a(1 − e2) tion vector and its time derivatives can be described in a so called perifocal frame as [1] furthermore from the gravitational force and New- tons II law, h2 −m r = (Pcosn + Qsinn) r¨ = r: (3) m(1 + ecosn) jrj3 Master thesis ”Estimation of satellite orbits using ground based radar concept” — 6/21 and its time derivative longitude of the ascending node and argument of periapsis, these are visualized in fig.3. Together ... 3mr · r˙ mr˙ r = r − : (4) with a, e and n, these are what make up the orbital jrj5 jrj3 parameters. The normal to the orbital plane is given by the vec- tor W. Celestial body An Earth Centered Inertial frame (ECI) is a coor- dinate system with origin at the Earth’s center of mass. Consider a J2000 ECI [2] with basis vectors I;J;K where K is parallel to the Earth’s rotational axis, I to the mean equinox and J 90° East about True anomaly ν Argument of periapsis the celestial equator. I;J constitute the equatorial ω plane as in fig.2. The transformation between these Ω و two coordinate systems is given by the following Longitude of ascending node Reference rotations. direction Plane of reference i ☊ Inclination Ascending node Orbit Figure 3. The figure shows angles that can describe the orientation of an orbital plane relative to a reference ECI frame. Figure 2. The figure shows an ECI frame relative to an orbital plane frame I am throughout this report going to use y = [i W w a e n] as a vector containing all orbital parameters. P = (cosWcosw − sinWcosisinw)I + (sinWcosw + cosWcosisinw)J + sinisinwK Q = (−cosWsinw − sinWcosicosw)I+ (−sinWsinw + cosWcosicosw)J+ sinicoswK W = sinWsiniI + −sinicosWJ + cosiK The orbital plane is oriented relative to the reference I−J plane. The angles i;W;w are called inclination, Master thesis ”Estimation of satellite orbits using ground based radar concept” — 7/21 3.2 Radar geometry Since this is study aims to describe the orbit from a radar signal, we must model the position from the radar’s point of view. Let q be the vector going from the Earth’s core to the radar station and ρ be the vector going from the radar station to the satellite as seen in fig.4. The Cartesian position vector can thus be described as r = q + ρ. Figure 4. The figure shows vectors that describe a satellite positions relative both the Earth’s center Figure 5. The figure shows a way to describe a and a radar station satellite positions from a radars point of view Since the coordinate system is not co-rotating with the Earth, the vector q will be time dependant and ρq is described as d = tan−1( ) (7) q 2 2 ρf + ρq q = q[cos(f)sin(q)I+sin(f)sin(q)J+cos(q)K] where q and f are angles as in standard spherical 3.3 Signal coordinates giving the radar stations position. Since The signal I have chosen to study is a frequency the Earth is not simply rotating about a single axis modulated continuous wave (FMCW) signal with both angles will have a time dependence [4]. triangular modulation, where the phase is the in- The vector ρ can be described using an alternative tegral of the instantaneous frequency. The signal coordinate system as seen in fig.5 with basis vec- time t is modelled as, for some total signal time −T T tors qˆ;fˆ;qˆ all being the spherical coordinates as T, t 2 [ 2 ; 2 ]. Using a passband signal, it’s com- described above with radial direction being in q plex baseband representation [5] of the transmitted direction. Hence we get signal will be ˆ ˆ ρ = ρ[cosasindf + sinasindq + cosdqˆ] (5) 2 i2p( fct±(D ft −D ft f Tm)) st(t;t f ) = Ate f ; where ρ is the distance to the satellite or ”range”, a and d are angles known as ”azimuth” and ”eleva- where At;D f ; fc;Tm;t f is the amplitude, frequency tion”.