{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Lab10 - Introduction to GPS\n", "\n", "#### Authors:\n", "\n", "##### v1.0 (2014 Fall) Rishi Sharma \\*\\*, Sahaana Suri \\*\\*, Paul Rigge \\*\\*, Kangwook Lee \\*\\*, Kannan Ramchandran \\*\\*\n", "\n", "##### v1.1 (2015 Fall) Kabir Chandrasekher \\*, Max Kanwal \\*, Kangwook Lee \\*\\*, Kannan Ramchandran \\*\\*\n", "\n", "In this lab, you will learn how GPS signals are used to estimate the location of an object.\n", "GPS satellites broadcast several different signals.\n", "These signals contain a very accurate measurement of the satellite's time, as well as the satellite's position, velocity, etc.\n", "\n", "GPS receivers make use of the fact that light propagates at a known speed, so the receiver can compute distances from the satellites by measuring how long it takes the GPS signal to propagate from the satellite to the receiver.\n", "This requires very accurate time measurements — light in free space travels 300m in a microsecond, so small timing errors result in huge distance errors. \n", "\n", "The first section of the lab will be to determine how a GPS chip actually goes about receiving and decoding signals. We will step through a subset of problems that must be combatted to successfully send signals from a satellite to your GPS chip. The second portion of the lab will then explore how a GPS chip can use the data it receives to determine its location, assuming the raw, received signals were acquired and decoded. We end with a little open-ended challenge for you. Hope you have fun!\n", "\n", "### $\\mathcal{SCIENCE\\ on\\ the\\ SIDE}$\n", "\n", "Keeping time to an extreme level of accuracy is the crux of the utility of GPS, but even infinite accuracy isn't enough. If engineers don't take into account both Einstein's theories of special and general Relativity, the whole system goes up in flames. \n", "\n", "Consider the following simple calculations:\n", "\n", "Let's say we want our GPS location to be accurate to within $15m$ on Earth. Since distances are measured from the satellite via the change in time of the satellite's clock and the GPS device's clock, multiplied by the speed of light, this implies that we need our device to keep time accurately at the level of $\\frac{15m}{c} = \\frac{15m}{3 \\times 10^{8} \\frac{m}{s}} \\approx 50ns$. \n", "\n", "Since the satellites with which the GPS device communicates with are orbitting the Earth (twice per day) at speeds of $14,000\\ km/hr$, according to special relativity, the clocks on the satellites actually run *slower* relative to those on the Earth's surface according to $T_{sat} = \\frac{T_{Earth}}{\\sqrt{1 - \\frac{v^c}{c^2}}}$. Plugging in the appropriatae values, we see that the satellites' clocks tick more slowly by about $7 \\mu s$ per day.\n", "\n", "However, according to general relativity, objects under the influence of relatively weaker gravitational fields experience time *faster* than those under stronger gravitational fields, according to $T_{sat} = \\frac{T_{Earth}}{\\sqrt{1 - \\frac{2GM}{Rc^2}}}$. Given that these satellites are $20,000km$ above the Earth and that they experience approximately one-fourth the gravity as we do on Earth, the resulting rate at which time passes faster for the satellites is $45 \\mu s$ per day.\n", "\n", "Thus, we have that the net change in the amount of time satellites experience relative to us on Earth is $45 \\mu s - 7 \\mu s = \\boxed{+38 \\mu s / day}$.\n", "\n", "Given that a $50ns$ error in keeping time corresponded to a distance error of $15m$, a $38\\mu s$ error in keeping time corresponds to an error in location of more than $11km$. Even crazier is that this is the amount of error in location that builds up *per day*! Without taking relativity into consideration, GPS devices would be rendered useless within 2 minutes of being synchronized with the satellites' atomic clocks.\n", "\n", "