Computing the GPS Sky-View Factor in Urban Landscapes for Autonomous Driving Simulation

Computing the GNSS Sky-View Factor in Urban Landscapes for Autonomous Driving Simulation GANESH P KUMAR, SHARNAM SHAH, YONGBO QIAN, NAHID PERVEZ Ford Greenfield Labs • Palo Alto • California • 94043 USA Email: (gkumar29, sshah89, npervez2, yqian17)@ford.com TYLER G R REID Xona Space Systems • Vancouver • British Columbia • V6R 2G6 Canada Email: tyler@xonaspace.com Keywords: GNSS, Sky-View Factor (SVF), Line of Sight (LOS), Multipath, Autonomous Vehicle (AV) Motivation It is often cost- and risk-effective to test the response of an Autonomous Vehicle (AV)’s planning and control module to simulated perception output from its sensors, an exercise called perception simulation [1]. The AV’s GNSS Sensor (called the ego GNSS) computes its position in a world coordinate frame (e.g., WGS-84) and feeds directly into the AV’s localization module, influencing downstream operations such as map-relative localization and sensor fusion. Consequently, predicting or simulating GNSS output is extremely useful to determine roadways wherein AV localization may experience degraded performance. However, GNSS output is generally challenging to simulate [4] due to the multiple time- and location-varying sources of error that impact its operation. City landscapes pose modelling challenges in that their buildings and urban canopies (referred to as topographic elements from now) limit the ego GNSS’ view of satellites in the sky, causing sky-impairment [14] that impacts GNSS availability. Further, these topographic elements also precipitate multipath and non-line of sight (NLOS) effects that impact GNSS accuracy. Sky-View Factor (SVF): Since modeling (or simulating) these effects in their entirety for a given urban landscape is computationally nontrivial - primarily due to the need to capture the interference of radio-frequency waves interacting with each topographic element – we will focus on the more tractable problem of computing the Sky-View Factor (SVF) of the landscape. We define the SVF of the (ego) GNSS with respect to a landscape to be fraction of the sky visible to it, unobscured by topographic elements [13]; the SVF is thus a dimensionless quantity between zero (representing a completely obscured sky) and unity (representing a fully unobscured sky), representing the complement of sky impairment. When its SVF is unity, the GNSS’ sky visibility is blocked only by the earth’s curvature, and the GNSS receiver can view the maximum possible number of satellites in its line of sight (LOS). The motivation behind the choice of the SVF as our metric of interest are: its computation is a tractable geometric problem determined only by the shapes of the topographic elements and the GNSS receiver location; the number of satellites visible to the user in its LOS may be determined from it; the data structures used in its computation may be used as a precursor to more complex models of accuracy and availability; it distills a landscape into a single scalar metric that measures how close the AV is located to a city center (or a location rich in topographic elements) - and it may thus be used to characterize cities; and prior work does not compute it directly except for specific dispositions of topographic elements [13]. Thus, the novelty of this abstract lies in identifying and solving the problem of computing the SVF, suggesting approaches to speed up the computation (at the expense of accuracy) for real time applications and outlining further applications of SVF-related data structures. Terminology We paraphrase the following definitions from [14]. A GNSS constellation consists of a satellite set that provides position, navigation, and timing (PNT) information to a GNSS Receiver that is usually located on the earth’s surface. Traditional GNSS constellations reside in Medium Earth Orbit, for example, GPS at an altitude of approximately 20,200 km. Historical GNSS constellation orbital data is available online for example, at [10], although this framework also allows us to examine future satellites including commercial Low Earth Orbit (LEO) Position, Navigation, and Time (PNT) satellites via simulation. We denote the altitude of satellites in a constellation of interest by Rsat. We will also use GPS to mean GNSS receiver throughout. The pseudo-range equation is used to compute the ego position on the earth’s surface using the satellites visible to the GPS. Multipath refers to the reflection of GPS signals off multiple surfaces (e.g., those of buildings) before reaching the GPS receiver, leading to degraded accuracy. Prior Work The pseudo-range GPS equation, sources of GPS error, and satellite navigation performance metrics including availability and accuracy are detailed in [14]. The significance and challenges of GPS modeling for perception simulation are noted in [4, 6], while [11, 20] specify approaches to computing multipath effects in urban environments. Multipath and NLOS effects are computed using simulators in [15, 21]. Our prior work [17] measures the difference between automotive and RTK GPS receiver accuracy over North American Highways. The Sky-View Factor, a term largely used in building and environmental research, is defined and computed in [13]. Problem Statement Given the following data: 1. a discrete time interval (t0, t0 + δt , t0 + 2δt, ...,tf) sampled every δt seconds (the sampling frequency or GPS Epoch), 2. the map of an urban landscape defined by topographic elements T={Ti: 1 ≤ i ≤ n}, specified in WGS-84 coordinates (taken, e.g., from OpenStreetMap or Google Street View, comprising latitude, longitude and altitude), with each roadway taken to be a polygo

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