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The Classical Concept of a Positioning System for a Global Navigation

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The Classical Concept of a Positioning System for a Global Navigation AN AUTONOMOUS RELATIVISTIC POSITIONING SYSTEM UROS KOSTIC, MARTIN HORVAT , JAN BOHINEC, ANDREJA GOMBOC University of Ljubljana Fa culty of Mathematics and Physics Jadmnska 19, 1000 Ljubljana, Slovenia Email: uros.kostic@jmj. uni-lj.si Current GNSS systems rely on reference frames fixedto the Earth (via the ground stations) so their precision and stahility in time are limited by our knowledge of the Earth dynamics. To avoid these drawbacks, the constellation of satellites should have the possibility of constituting by itself a primary and autonomous positioning system, without any a priori realization of a terrestrial reference frame. We constructed such a system, an Autonomous Basis of Co­ ordinates, via emission coordinates. Here we present the idea of the Autonomous Basis of Coordinates and its implementation in the perturbed space-time of Earth, where the motion of satellites, light propagation, and gravitational perturbations are treated in the formalism of general relativity. 1 Introduction The classical concept of a positioning system for a Global navigation satellite system (GNSS) would work ideally if all satellites and a receiver were at rest in an inertial reference frame. But at the level of precision provided by a GNSS, one has to take into account curvature and relativistic inertial effects of spacetime, which are far from being negligible. These effects are most consistently and elegantly dealt with in a relativistic positioning system based on emission coordinates i,2,3,4,5. They depend on the set of four satellites and their dynamics, and can be linked to a terrestrial referenceframe . Consequently, the difficultyno longer lies in the conception of the primary reference frame but in its link with terrestrial referenceframes 4 . This allows to control much more precisely all the perturbations that limit the accuracy and the stability of the primary reference frame, ifthe dynamics of the GNSS satellites, described by their orbital parameters, is known sufficiently well. Our previous work shows that it is possible to construct such a system and do the positioning within it: the orbital parameters of the GNSS satellites can be determined and checked internally by the GNSS system itself through inter-satellite links6. In this way, we can construct a reference system called Autonomous Basis of Coordinates (ABC), which is independent of any Earth based coordinate system. Its reference frame consists of the GNSS satellites. The system is self-assembled from continuous exchange of proper times between the satellites, which enables us to determine the parameters of the ABC system with great accuracy. Here we present the results of our recent work, where we have further developed relativistic positioning and the ABC by including all relevant gravitational perturbations, such as the Earth multipoles (up to the 6th), the Earth solid and ocean tides, the Sun, the Moon, Jupiter, Venus, and the Kerr effect. Furthermore, we show that in addition to precise positioning, ABC also offers a possibility to measure the space-time with unprecedented accuracy. 2 Relativistic Positioning System To model a relativistic positioning system, we simulate a constellation of four satellites moving along their time-like geodesics in perturbed Shwarzschild space-time, where the initial orbital parameters (QJ1(0), Pµ(O)) of the geodesics are known 4,5,7,s. At every time-step of the simulation, each satellite emits a signal and a user on Earth receives signals from all satellites - the signals are the proper times of satellites at their emission events and constitute the emission coordinates of the user. The emission coordinates determine the user's "position" in this particular relativistic reference frame defined by the four satellites and allow him to calculate his position and time in the more customary Schwarzschild coordinates. In order to simulate the positioning system, two main algorithms have to be implemented: (1) determination of the emission coordinates, and (2) calculation of the Schwarzschild coordinates. Determination of the emission coordinates The satellites' trajectories are parametrized by their true anomaly A. The event P0 = (t0, x0, y0 , z0) marks user's Schwarzschild coordinates at the moment of reception of the signals from four satellites. These were emitted at events P; = (t;, Xi, Yi, zi) corresponding to Ai with index i = 1, ..., 4 labeling satellites. The emission coordinates of the user at P0 are, therefore, the proper times Ti (Ai) of the satellites at Pi . Taking into account that the events P0 and P; are connected with a light-like geodesic7 we calculate Ai at the emission point Pi using the equation to ti(Ai Qµ(Ti), P (Ti)) = - l µ (1) µ Tr (il;(AilQ (Ti), Pµ(Ti)), Ro) , where fl; = (xi, Yi, zi) and R0 = (x0, y0, z0) are the spatial vectors of the satellites and the user, respectively. The function Tr represents the time-of-flight of photons between P0 and Pi as shown in 9 and 4. The equation ( 1) is actually a system of fourequations forfour unknown Ai - once the values of Ai are determined, it is straightforward to calculate Ti for each satellite and thus obtain user's emission coordinates at P0 = ( T1, T2, T3, T4). Calculation of the Schwarzschild coordinates Here we solve the inverse problem of cal­ culating Schwarzschild coordinates of the event P0 from proper times (T1, T2, T3, T4) sent by the four satellites. We do this in the following way: For each satellite, we numerically solve the equation (2) to obtain Ai, where T(AIQµ(T), Pµ(T)) is a known function forproper time on time-like geodesics 7. The Schwarzschild coordinates of the satellites are then calculated from Ai using the solutions of the orbit equation. With the satellites' coordinates known, we can take the geometrical approach presented in 4 to calculate the Schwarzschild coordinates of the user. The final step in this method requires us again to solve (1), however, this time it is treated as a system of 4 equations for 4 unknown user coordinates, i.e., solving it, gives (t0, x0, y0, z0). The accuracy of this algorithm has been tested forsatellites with the followinginitial orbital parameters: for all satellites = 0°, = 30000 km, = 0.007, ta = 0, for satellites 1 and 2 the n a e inclination is l = 45°, while for satellites 3 and 4 it is l = 135°. The arguments of the apoapsis are w1 = 270, w2 = 315, w3 = 275, W4 = 320 forsatellites 1, 2, 3, and 4, respectively. The user's coordinates r0 = 6371 km, 80 = 43.97°, ¢0 = 14.5° remain constant during the simulation. Our algorithms show, that the relative errors, defined as for j=x,y,z , (3) "The light-like geodesics are calculated in Schwarzschild space-time without perturbations, because the effects of perturbations on light propagation are negligible. are of the order 10-32 - 10-30 for coordinate t, and 10-28 - 10-26 for and here t� and R� are user time and coordinates as calculated from the emission coordinax, y,tes. Usingz; a laptoP' for calculations, the user's position (with such errors) was determined in 0.04 s, where we assumed that (1) in real applications of the positioning the true values of orbital parameters would be transmitted to the user together with the emission coordinates, so to account for this in our simulations, we calculated the evolution of parameters from their initial values before starting the positioning, and (2) the position of the user is completely unknown, i.e., we do not start from the last known position. If we did, the times for calculating the position would be even shorter. 3 Autonomous Basis of Coordinates To construct an autonomous coordinate system, we apply the idea of the Autonomous Basis of Coordinates (ABC) presented in 6 to a perturbed satellite system 8, i.e., we simulate the motion of a pair of satellites along their perturbed orbits. At each time-step of the simulation, both satellites exchange emission coordinates as shown in Fig. 1 left, where, for clarity, only communication from satellite 1 to satellite 2 is plotted. These events of emission at proper time of the first satellite and reception at of the second T 7 2 1017 . ,,� 10' 2 �'t[k+l] "' = " 0 '6 .,, 10 01::1 � <.> 't[k +l] 10-7 't[k] 10-15 10-23 0 8000 't[k] � 2000 4000 6000 10000 space coordinate step Figure 1 - Left: A pair of satellites exchanging their proper times. At every time-step k, the satellite 1 sends the proper time of emission T[k] to the satellite 2, which receives it at the time of reception Y[k]. The emission and reception event pairs are connected with a light-like geodesic. Right: The action S(Qµ(O), Pµ(O)) during the minimization process. The first stage takes 1765 steps and the second one takes 8513 steps. satellite are connected with a light-like geodesic, i.e., the differencebetween the coordinate times of emission ti(T) and reception t2(7) must be equal to the time of flight of a photon between the two satellites (cf. ( 1)) (4) where Tr depends on the positions of both satellites. When constructing the relativistic positioning system, it is reasonable to assume that the initial orbital parameters (Qµ(O), Pµ(O)) are not known very precisely. To improve their val­ ues, we sum the differences between the right-hand side and the left-hand side of (4) for all communication events into an action ) S(Qµ(O), Pµ(O)) L (t2,k - ti,k - Tr (R1,ki R2,k (5) = k r bConfiguration: Intel® Core™ i7-3610QM CPU @ 2.30GHz, SGB RAM. expressed via auxiliary symbols tl k tl(T[k]JQµ(T[k]),P (T[k])) , , = µ µ , , t2,k = t2(r[k]IQ (r[k]) Pµ(r[k])) µ , R1,k = R1(T[kJIQ (T[k]),Pµ(T[k])) R2,k R2(r[k] IQµ(r[k]), Pµ(r[k])) , The action S has a minimum value =close to zero forthe true initial values of orbital parameters, ( ) i.e., the 2 x 6 orbital parameters c The search for the correct initial orbital parameters becomes a minimization problem in 12 dimensions.
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