Inside GNSS Media & Research

JUL-AUG 2019

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58 Inside GNSS J U L Y / A U G U S T 2 0 1 9 www.insidegnss.com is article presents a simultaneous tracking and naviga- tion (STAN) framework that addresses the aforementioned challenges (for more, see 2 papers from Morales, et alia). is framework tracks the states of LEO satellites while simultane- ously using pseudorange and Doppler measurements extracted from their signals to aid the vehicle's INS. e performance of the STAN framework is demonstrated in realistic simulation environments and experimentally on a ground vehicle and on an unmanned aerial vehicle (UAV), showing the potential of achieving meter-level-accurate navigation. Pseudorange, Doppler Measurement Model is section describes the LEO satellite receiver pseudorange and Doppler measurement model and discusses the sources of error in LEO-based positioning: (i) satellite position and velocity errors, (ii) satellite and receiver clock errors, and (iii) ionospheric and tropospheric delay rate errors. A. PSEUDORANGE AND DOPPLER MEASUREMENT MODEL e LEO receiver extracts pseudorange and Doppler frequen- cy measurements from LEO satellite signals. A pseudorange rate measurement can be obtained from (1) where is the speed of light and is the carrier frequency. e pseudorange from the m-th LEO satellite at time-step , which represents discrete-time at for an initial time and sampling time T, is modeled as (2) where , represents discrete-time at with being the true time-of-flight of the signal from the m-th LEO satellite; and are the LEO receiver's and m-th LEO satellite's 3-D position vectors, respectively; and are the LEO receiver and the m-th LEO satellite transmitter clock biases, respectively; and are the ionospheric and tropospheric delays, respectively, affecting the m-th LEO sat- ellite's signal; and is the pseudorange measurement noise, which is modeled as a white Gaussian random sequence with variance . e pseudorange rate measurement from the m-th LEO satellite is given by (3) where and are the LEO receiver's and m-th LEO satel- lite's 3-D velocity vectors, respectively; and are the LEO receiver and the m-th LEO satellite transmitter clock dris, respectively; and are the dris of the ionospheric and tropospheric delays, respectively, affecting the m-th LEO satellite's signal; and is the pseudorange rate measurement noise, which is modeled as a white Gaussian ran- dom sequence with variance . B. POSITION AND VELOCITY ERRORS One source of error that should be considered when navigat- ing with LEO satellite signals arises due to imperfect knowl- edge of the LEO satellites' position and velocity. is is due to time-varying Keplerian elements caused by several perturbing accelerations acting on the satellite. Mean Keplerian elements and perturbing acceleration parameters are contained in pub- licly available two-line element (TLE) file sets. e informa- tion in these files may be used to initialize a simplified general perturbations (SGP) model, which is specifically designed to propagate a LEO satellite's orbit. SGP propagators (e.g., SGP4) are optimized for speed by replacing complicated perturbing acceleration models that require numerical integrations with analytical expressions to propagate a satellite position from an epoch time to a specified future time. e tradeoff is in satellite position accuracy: the SGP4 propagator has around 3 km in position error at epoch and the propagated orbit will continue to deviate from its true one until the TLE files are updated the following day. Figure 6 shows the accumulated position and velocity error for an Orbcomm LEO satellite (FM 112). C. CLOCK ERRORS In contrast to GNSS, LEO satellite clocks are not tightly syn- chronized and the clock errors (bias and dri) are unknown FIGURE 6 SGP4 position and velocity errors. FIGURE 7 Time evolution of 1-σ bounds of (a) clock bias and (b) clock drift for a typical OCXO and a typical TCXO over a 10-minute period. FIGURE 8 (a) Skyplot showing the trajectory of an Orbcomm LEO satellite (FM 109) and a GPS MEO satellite (PRN 32) over a 10-minute period. (b) The elevation angle rate of FM 109 and PRN 32 over the 10-minute trajectory. The elevation angle rate of the Orbcomm LEO satellite reaches as high as 60 times that of the GPS MEO satellite. STAN WITH LEO

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