1 EECS467: Autonomous Robotics Laboratory Prof. Edwin Olson Map Projections and GPS
2 Cartography Several purposes of maps Geographic Information Systems (GIS) - Where is stuff? Measure distances, etc. Navigation - How do I get from point A to point B? - A well-designed map can make it easier to compute routes
3 Navigation Let s ignore obstacles. (We re an airplane at 30,000ft, perhaps) How do I get from Ann Arbor, MI to Pamplona, Follow the rhumb line (or loxodrome) Spain? (42.5, -83.8) to (42.84, -1.68) Travel due east, 6714km
4 Navigation How do I get from Ann Arbor, MI to Pamplona, Spain? (42.5, -83.8) to (42.84, -1.68) Follow great circle, 6423km Initial bearing 60 deg, final bearing 120 deg Follow the great circle
5 Great Circles What is a great circle? Shortest path between two points on a sphere Lies on the intersection of the earth and the plane that goes through p, q, and the earth s center. Efficient, but not necessarily easy Constant course corrections A bit tricky to execute for early sailors p q
6 Simple Maps Our goal is to make navigation easy Draw a line between p and q, read off the bearing. Goal: the line between p and q gives the trajectory for a constant-bearing route How do we draw a map such that it has this property?
7 First Candidate Projection: Platte Carre Uniform spacing of latitude and longitude What are x/y coordinates for a given lat/long? x = long y = lat
8 Are rhumb lines straight on Platte Carre Maps? Let s consider the scaling of the map at various points. First: note this is a cylindrical projection Map is tangent to the globe at the equator Thus scale is perfect there Let s call that scale=1. What is the scale in the y direction? What is the scale in the x direction? (Are areas preserved?)
9 Constant Course on Platte Carre Head north east from Ann Arbor towards (say) south Africa. We become trapped in cartographic singularity at the pole Path on platte carte becomes very long; actual path is very short.
10 Making rhumb lines straight Observation: Given a cylindrical projection, the x scale must increase away from equator. Circumference of lines of latitude getting smaller, map s width staying constant Idea: Adjust y scale so that it is the same as x scale This means different y scales in different parts of the map. What is the xscale factor? and x = longitude We want yscale = 1/cos(theta) too. So what is y(theta)? Must make room for all the y s with smaller thetas width of map perimeter of circle at latitude \theta
11 Mercator projection Defined by: To compute a path: Draw a straight line between start and end points Read of the bearing. Travel in that direction until you get there.
12 The integral The integral turns out to have an interesting history... We could be boring and just give the answer:
13 Mercator- A little history 1569: we... spread on a plane the surface of the sphere in such a way that that positions... shall correspond... in both... true direction and distance... It is for these reasons that we have progressively increased the degrees of latitude towards each pole in proportion to the lengthening of the parallels... - G. Mercator He never actually said how Late 1580s: Thomas Harriot gave a mathematical explanation, but nobody noticed. 1599: Edward Wright showed that the solution involved an integral of the secant function. Gerardus Mercator But no idea how to solve it... we may make a table which shall shew the sections and points of latitude in the meridians of the nautical planisphaere: by which sections, the parallels are to be drawne 1620: Edmund Gunter published a table of logarithms of tangents 1645: Henry Bond notices that Gunter s and Wrights tables appear to similar; conjectures the closed-form answer for y.
14 Mercator- A little history 1665: Nicolaus Mercator (no relation) challenges the Royal Society to prove or disprove Bond s conjecture. Volume 1, Philosophical Transactions 1668: James Gregory presents an ugly proof. the excellent Mr. James Gregory..., not without a long train of Consequences and Complication of Proportions, whereby the evidence of the Demonstration is in great measure lost, and the Reader wearied before he attain it. - Edmund Halley 1670: Isaac Barrow publishes an intelligible proof; an early use of integration by partial fractions. Edmund Halley
15 Views from space Perspective view of earth As distance increases, view approaches orthographic Orthographic projection useful due to simplicity Dates to Hipparchus in 2nd century BC Orthographic projection
16 How much does all this matter? Consider an orthographic projection The center of the map is tangent to the globe Suppose we travel a distance d away from the tangent point. The point on the earth is farther away than the image of that point due to the curvature of the earth. distance error 0 m 0 1 km 4.2 um 10 km 0.42 cm 100 km 4.2 m 500 km 526 m 1000 km 4.2 km 5000 km 775 km Orthographic projection
17 d+e d Orthographic projection distance error 0 m 0 1 km 4.2 um 10 km 0.42 cm 100 km 4.2 m 500 km 526 m 1000 km 4.2 km 5000 km 775 km distance (km) Note: Earth radius ~ 6300km
18 Universal Transverse Mercator (UTM) Standard Mercator projections distort areas away from equator. It s good where the cylinder is tangent to the Earth. Idea: let s use a bunch of maps, arranging the cylinder to be tangent in different places Lines of longitude are natural choices (unlike lines of latitude, they are great circles)
19 Universal Transverse Mercator UTM maps divided into zones Within each zone, using northing and easting (in meters), instead of lat/lon. Coordinates offset to avoid negative numbers Widely used by US military
20 North vs. Magnetic North Magnetic north pole at 81.3N, 110.8W Pretty far from true north (90N) It s moving about 35 km/year. Adjustment factor: magnetic declination Offset could be measured at night by comparing to the north star But which was right? - It seems that the star [Polaris] moves like the other stars, and the compasses always seek the truth - Chistopher Columbus Typical 16th century navigation Know the latitude of destination Get on the right latitude, then travel east/west Use magnetic course if you have a good map Use occasional celestial observations (latitude) to confirm position. (Longitude really hard)
21 xkcd 977
22 Gnomonic Projection Great circles are straight lines (doesn t make them any easier to follow, though)
23 Finding our position on a sphere Suppose I tell you the distance between you and N known 3D locations N=1 - You re on a sphere N=2 - You re on a circle N>=3 - Solve for a point
24 GPS Let s do this with satellites They ll tell us their orbit (so we know where they are) We just need to measure distance Idea: Use time of flight c= m/s If our clocks are synchronized with the GPS satellites, this works great. GPS IIR-15 (M), launched September 25, 2006
25 GPS Synchronization Requiring GPS receivers to have an atomic clock isn t sane. Yet. Suppose we make the local receiver s error be an unknown We know have one more unknown variable We can solve for it given one extra satellite observation Thus, 4 GPS readings give us 4 equations, for 4 unknowns (XYZ + time)
26 GPS: Communications L1: bps Each satellite continuously transmits groups of five 6-second, 300 bit subframes Every sub-frame contains the time - New position fixes every 6 seconds Satellite health and GPS time info (1 of 5 frames) Ephemeris data (2 of 5 frames) - Orbital position data (valid for 4 hours) Almanac data (2 of 5 frames) - Long-term predictions of which satellites will be in view - Almanac consists of 50 frames total; takes 12.5 minutes to acquire - Not critical for modern receivers Question: How long to acquire a new satellite (worst-case)?
27 Global Positioning How do you define a location? Easy latitude and longitude How are latitude and longitude defined for point p? Longitude - Angle from Greenwich, to center of earth, to p. Latitude - Angle from equator, to center of earth, to p.
28 A silly question? What is the location of your house? It s not clear Your house moves... tectonic plates.
29 And what about the Earth? It ain t round. It bulges around the equator Many finer-scale deviations We re not even quite sure where the center is. So how do we compute latitude and longitude? So what does this mean for latitude and longitude? They don t uniquely describe a point on the earth. We need additional information.
30 Geoids, Ellipsoids, and Datums: Oh my Geoid: The best estimate of the gravity field of the earth. If there wasn t any land in the way, what would the shape of the earth be? e.g. That thing from the movie before Ellipsoid: An approximation of the Geoid with a major and minor axis. e.g. GRS80 [ m, m] Defines coordinate system for latitude/longitude Datum: The specific alignment of an Ellipsoid with a set of reference points. WGS84, NAD83
31 Latitude/Longitude on Oblate Spheroids ( ellipsoids ) (or geodetic) GPS coordinates are in geodetic coordinates. parameters of our flattened ellipse change your latitude
32 Different models of Earth lead to different Latitude and Longitudes Greenwich Ann Arbor Red and blue datums lead to different angle measurements NAD83/WGS84 close enough (< 3 meters) to be confusing NAD27/WGS84 can be 10s of meters off (150m in Montana)
33 Where do datums come from? Least-squares fits of carefully measured measurements between surveyed points Range Angle Azimuth (angle to landmark like a star projected onto plane)
34 Datums Does it try to reasonably approximate the whole world, or just one region? WGS84 Reasonable fit around whole world Center is at (our best estimate of) the center of mass of the earth. Your house drifts at several cm per year in WGS84 coordinates due to continental drift. NAD83 Same ellipsoid (well, almost) as WGS84. Except, tracks motion of North American plate. Motion of NAD83 with respect to WGS84 modeled as a linear function of time, Dx, Dy, Dz, Rx, Ry, Rz. NAD27 Try to fit North America... never mind elsewhere. Different ellipsoid. Significantly wrong origin. Defined relative to Meads Ranch, Kansas. Param Per Year Dx.9956m.0007m Dy m m Dz m m Rx Ry Rz WGS84 to NAD83 Kansas Earth NAD27 Ellipsoid
35 GPS: Key points When interchanging data, you must agree not only on latitude and longitude, but also the datum Changes to datum can affect the center and shape of the ellipsoid - Especially with NAD27 - Lat/long of a point depends on datum GPS, UTM both based on WGS84 That s a pretty good argument to use WGS84....Unless you want your house to stay put. Then use NAD83.
36 Differential GPS and RTK Errors in GPS receivers are spatially correlated Two nearby receivers will have similar errors (nearby: < 300km) The relative position of the receivers is highly accurate. DGPS- one GPS station remains stationary, and broadcasts pseudo-range correction data to receivers. Sub-meter accuracy possible. You have to provide your own comms channel for this If position of one receiver is known, the position of other receivers can be estimated accurately. - Idea: survey a fixed GPS base station. Real-time kinematic (RTK) Similar to above, but corrections measured in terms of precise phase of the MHz carrier. Sub centimeter accuracy possible.
37 GPS Performance Several contributing factors Noise in underlying observations Geometry of the satellites Multi-path - e.g. urban canyons bad geometry good geometry If we could accurately estimate quality of GPS data, life would be good But quality estimates are often very poor E.g. Geometry is good (based on ephemeris data), but range measurements are bad
38 Some References samplechaps/chapter3_sample.pdf math1132s10/secantintegral.pdf
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