Localizing in Urban Canyons using Joint Doppler and Ranging and the Law of Cosines Method

Abstract

The performance of Global Navigation Satellite System (GNSS) based navigation can be limited in urban canyons and other regions with narrow satellite visibility. These regions may only allow for less than the minimum of four satellites to be visible, leading to a decay of positional knowledge. A scheme with Joint Doppler and Ranging (JDR) and relative positioning, known as the Law of Cosines (LOC) method, is introduced in this paper that utilizes Doppler and pseudorange measurements from a minimum of two GNSS satellites to obtain a position fix. The user’s GNSS receiver was assumed to output both corrected pseudorange and Doppler shift measurements for each tracked satellite. The velocity vector of each satellite was calculated using broadcast satellite ephemerides. Additionally, the location of the reference station was known and Doppler measurements from the GNSS receiver at the reference station were transmitted to the user. Ephemerides from eight GNSS satellites were simulated with the user and reference station approximately 12 km apart in San Francisco. Gaussian error sources were modelled and randomized in Monte Carlo simulations, adding error to the receiver’s known satellite ephemeris, satellite velocity, Doppler, and pseudorange measurements. Each unique pair of 2 satellites was employed for the positioning of the user using the LOC method for over 10,000 Monte Carlo simulations. With reasonable assumptions on measurement error, the average 2D topocentric Root-Mean-Square-Error (RMSE) performance of all pairs of satellites was 23 meters, reducing to 10 meters by removing specific pairs with poor geometry. However, with a new technique called Terrain Assisted – JDR (TA-JDR), which uses accurate topographic information of the user’s region as a faux pseudorange measurement, the average RSME of the satellite pairs was reduced to approximately 7 meters. The use of the JDR-LOC scheme and its variants may not only be useful in urban canyons, but also in other GPS-denied unfriendly environments. Ultimately, the JDR-LOC scheme was capable of achieving navigational solutions with an RMSE as low as 7 meters for users with limited GNSS satellite visibility, with only the use of a GNSS receiver and a reference station. INTRODUCTION The Global Navigation Satellite System (GNSS), which includes the Global Positioning System (GPS), has been deemed successful through countless use-cases. These cases range from regular commercial and private use to research and military use [1]. However, at least four satellites in view are required for GNSS/GPS range measurements and some locations cannot utilize these positioning systems to their full potential. Urban canyons are locations where a user is surrounded by buildings which block GNSS signals and create a canyon-like environment. Positioning problems in these regions include a smaller quantity of visible satellites, multipath, and interference. Temporary loss of GPS signals is caused by structures blocking line of sight to satellites, multipath and signal interference, thus lowering the number of usable satellites in urban environments. Various solutions have been proposed to resolve these issues, ranging from taking advantage of GLONASS systems [2], weighting models [3], shadow matching [4], and fuzzy logic [5]. However, the use of standard range-based trilateration is still assumed in these approaches. The Joint Doppler and Ranging Law of Cosines (JDR-LOC) scheme is reintroduced in this paper to enable relative positioning with as few as two satellites in view. The Law of Cosines (LOC) scheme is a novel positioning scheme that only relies on Doppler measurements made by a user, a satellite(s), and a reference station [6]. Additional hardware or software is not required to obtain Doppler measurements; most GNSS receivers include options for logging Doppler shifts of locked satellites in real time. Therefore, positioning is enabled with relatively low hardware requirements and without the need for a clock bias calculation. However, improvements in performance have been shown with the addition of range measurements along with Doppler measurements. Range and Doppler measurements are integrated in the Joint Doppler and Ranging (JDR) scheme [7]. Additionally, the knowledge of the user’s altitude was used as a pseudorange measurement from a faux satellite at the center of the planet. This faux measurement was known as the surface constraint [7]. Because ranging and Doppler measurements are based on independent states during an instantaneous timestep (position and velocity, respectively), two measurements can be provided by each satellite towards the calculation of position. Therefore, with the calculation of 3 Cartesian coordinates and a clock bias, only a minimum of two satellites are required for positioning. The JDR-LOC scheme is able to position a user in an urban canyon with only two satellites in view. To test this theory, the user and reference station were assumed to be in San Francisco, separated by a distance of 12 km. All GPS satellites that were visible to the user at a certain epoch were found and their position and velocity vectors were exported from Systems Tool Kit. The position and velocity of the satellites were assumed to be known by the user (to a certain accuracy) through the orbital information broadcasted by each satellite. The clock correction term for the user’s clock was calculated in all analyses, enabling precise pseudorange measurements. The reference station’s clock was synchronized with the GNSS/GPS. This could be assumed with a reference station on the top of a large hill or building, allowing it to see a large portion of the sky. Other errors, including ionospheric and tropospheric delays, were resolved using dual frequency measurements and the effects of relative navigation. In simulation, the true pseudorange and Doppler measurements were calculated and then corrupted with Gaussian error. Additionally, the user’s knowledge of each satellite’s ephemeris and velocity vector were corrupted with Gaussian error. The reference station, which sends its Doppler measurements to the user, also had its measurements corrupted. Just as in other LOC analyses, the rough altitude of the user was assumed to be known and implemented as the surface constraint. All measurements and satellite knowledge were fed into the JDR-LOC scheme and used to calculate the user’s position. This entire process was repeated, with changes only to the Gaussian error with new, random values set by a given error standard deviation. This Monte Carlo simulation was repeated for 10,000 iterations. For each respective run of the JDR-LOC scheme, only two satellites were measured. This meant that all the unique pair combinations of the total 8 visible satellites were tested in each Monte Carlo iteration, totaling 28 pairs. With the JDR-LOC scheme and reasonable assumptions on measurement error, the average 2D topocentric Root-Mean-SquareError (RMSE) performance of all pairs of satellites was 23 meters, reducing to 10 meters by removing specific pairs with poor geometry. Additionally, the use of a new technique called Terrain Assisted – JDR (TA-JDR) was introduced. A high resolution topographic map of the user’s region was utilized in this new virtual instrument. After an initial position fix was calculated from the JDR-LOC scheme, the altitude of that calculated position was pulled from the topographic map and saved as a new surface constraint. The JDR-LOC scheme was run again with all the same measurements and parameters as last time, except with the new surface constraint. This process was repeated until the position fix converged to less than the resolution of the high-resolution topographic map. With TA-JDR, the vertical component of the position fix was resolved with additional information from a map. Additional measurements were not required and the accuracy of the user’s height knowledge was increased. A caveat to this technique was that it did not account for a user not on the surface of the planet, such as in a highrise building. With this new technique, the average RSME of the satellite pairs was reduced to approximately 7 meters. Although this new approach to positioning with limited satellite resources was applied directly to GNSS in urban canyons, there are a myriad of applications of JDR-LOC and TA-JDR. For instance, JDR-LOC can be used to localize a rover on Mars or a user on the Lunar surface with two or even a single satellite (assuming two-way ranging). Additionally, with pre-existing high-resolution topographic maps of the Moon (e.g. the high-resolution Lunar Orbiter Laser Altimeter (LOLA) digital elevation map [8]), TA-JDR can be used to localize a user with greater accuracy than just the standard JDR-LOC. TA-JDR could also be used on other topics centered around JDR. Coupling an IMU is another area of research, which removes the requirement of a static user. Coupling an IMU with JDR through a Kalman filter is expected to improve performance of the scheme [9]. Positioning accuracy of these new schemes can be further improved with TA-JDR. JOINT DOPPLER AND RANGING LAW OF COSINES SCHEME (JDR-LOC) The JDR-LOC scheme is a modification of the original LOC scheme including ranging measurements in addition to Doppler measurements [7]. A review of the JDR-LOC scheme was provided. The visualization of the user (T), the reference station (R), and one of the satellites (C) is described in Figure 1. LL�⃑ ii ′ = � ll1 ll2 ll3 � uu�vv = � vv1 vv2 vv3 � uu�ii = � uu1 uu2 uu3 � PP�⃑ = � xx yy zz � ?⃑?X = �PP�⃑ cccc � RR = � rr1 rr2 rr3 � Figure 1: Visualization of the LOC Technique uu�vv is the satellite’s velocity vector, R is the reference station, T is the user, and C1 is the current satellite Using the Law of Cosines, a cost function was created (Eqn 1). This was the core cost function of the LOC scheme; the relative position P was calculated from the other input measurements.