The global positioning system

Transcription

1 PHYSICS UPDATE The global positioning system Alan J Walton and Richard J Black University of Cambridge, Department of Physics, Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK University of Glasgow, Computing Science Department, 17 Lilybank Gardens, Glasgow G12 8RZ, UK A hand-held global positioning system receiver displays the operator s latitude, longitude and velocity. Knowledge of GCSE-level physics will allow the basic principles of the system to be understood; knowledge of A-level physics will allow many important aspects of their implementation to be comprehended. A discussion of the system provides many simple numerical calculations relevant to school and first-year undergraduate syllabuses. For a little over 100 it is now possible to buy a hand-held global positioning system (GPS) receiver which displays the user s current latitude, longitude, altitude and velocity (speed and compass bearing). Figure 1 shows one such receiver; its operation relies on receiving information from a set of satellites deployed by the US Department of Defense. Typically, the displayed horizontal position is accurate to 100 m and the altitude to 140 m; the system is inherently capable of greater precision, but the current US policy is to impose selective availability (SA) which degrades its performance to these limits. Most hand-held receivers will have many additional features such as automatically displaying the route being taken and allowing waypoints to be entered. Particular models may be optimized for different applications such as hiking, boating or flying. A GPS receiver and a mobile phone should perhaps be made compulsory safety equipment for every school expedition. Figure 1. A typical hand-held GPS receiver (Garmin model 12). On being shown a GPS receiver, students commonly ask how does it work? The purpose of this article is to show that the basic principles can be understood using only GCSE Phys. Educ. 34(1) January

2 Figure 2. A schematic (two-dimensional) diagram showing that a signal transmitted at time t on the satellite clock is received at a later time t + t on the receiver s clock. physics. To understand how these principles are implemented calls for A-level physics (and beyond). The basics To begin with, we will consider a single satellite in orbit around the Earth. On board is a clock which (for now) we will suppose is identical to and synchronized with a clock in the receiver. By this we mean that if the two clocks are placed next to one another they will forever keep identical time. As the satellite orbits the Earth, it transmits its current time and position. In practice the position is transmitted in the form of the current ephemeris constants ; these specify the equation of motion of the satellite and allow its current position to be deduced from its current time. To put it simply, the satellite is a speaking clock which says My current time is t and my current position is x,y,z. This message is received by a (stationary) GPS set located at some unknown position X, Y, Z not at time t and this is the key point but at a later time t + t. Thus if the satellite transmits a signal at, say, s on its clock and this signal is received 80 ms later at time s on the set s clock, it follows that the receiver is at a distance (a range) ofc t from the satellite, where c is the speed of light, i.e. at a range of ms s = m. To simplify the discussion a little, we will consider the two-dimensional case shown in figure 2. If we draw a circle of radius c t about the satellite s position when it transmitted the signal, then we know that the receiver must be somewhere on this circle. To find out where we must perform a similar timing measurement using a second satellite as shown in figure 3. The receiver is clearly located where the two circles intercept (there are, of course, two such points, but the one up in the sky can be readily discounted). Rather than deducing the receiver position by means of a compass and scale diagram, we can (see figure 3) use Pythagoras theorem to interrelate the signalling positions (x 1,y 1 ) and (x 2,y 2 )of the two satellites, their measured ranges c t 1 and c t 2 and the receiver position (X, Y ). This immediately gives (x 1 X) 2 + (y 1 Y) 2 = (c t 1 ) 2 38 Phys. Educ. 34(1) January 1999

3 Figure 3. A schematic (two-dimensional) diagram showing that the transit times t 1 and t 2 of the signals received from two satellites at known positions (x 1, y 1 ) and (x 2, y 2 ), respectively, will allow the set s location to be determined from the intersection of the circles of radii c t 1 and c t 2, respectively. (x 2 X) 2 + (y 2 Y) 2 = (c t 2 ) 2. These simultaneous equations with their two unknowns X and Y can be solved numerically to find (X, Y ), the set s position. It is not difficult to show that threedimensional navigation demands a minimum of three satellites (again assuming perfect synchronization between the satellite and receiver clocks). If their signalling positions are (x 1,y 1,z 1 ), (x 2,y 2,z 2 ) and (x 3,y 3,z 3 ) and the measured signal transit times are t 1, t 2 and t 3, respectively, then (x 1 X) 2 + (y 1 Y) 2 +(z 1 Z) 2 = (c t 1 ) 2 (1) (x 2 X) 2 + (y 2 Y) 2 +(z 2 Z) 2 = (c t 2 ) 2 (2) (x 3 X) 2 + (y 3 Y) 2 +(z 3 Z) 2 = (c t 3 ) 2 (3) which can be solved numerically to give the receiver s position (X,Y,Z). Clock synchronization problems So far the discussion has assumed that there are identical synchronized clocks on the satellites and on the receiver. In practice the satellites are equipped with highly-stable rubidium and caesium ( atomic ) clocks while the receiver makes do with a less expensive (and lighter!) quartz clock. This means that the clock on the receiver may run fast or slow. If fast by δt (this is called the clock bias error), we must subtract δt from each of the socalled pseudo transit times t 1, t 2 and t 3 to obtain the true transit times required in equations (1), (2) and (3). This gives (x 1 X) 2 +(y 1 Y) 2 +(z 1 Z) 2 = c 2 ( t 1 δt) 2 (4) (x 2 X) 2 +(y 2 Y) 2 +(z 2 Z) 2 = c 2 ( t 2 δt) 2 (5) (x 3 X) 2 +(y 3 Y) 2 +(z 3 Z) 2 = c 2 ( t 3 δt) 2. (6) Since δt is unknown there are now four unknowns and only three equations. The problem is resolved by utilizing a fourth satellite at (x 4,y 4,z 4 ) with a pseudo transit time t 4, giving (x 4 X) 2 +(y 4 Y) 2 +(z 4 Z) 2 = c 2 ( t 4 δt) 2. (7) Phys. Educ. 34(1) January

4 Solving equations (4) to (7) numerically gives the receiver position (X,Y,Z). Likewise, it takes at least three (rather than two) satellites to give a two-dimensional fix. When the number of satellites in view drops from four to three it is the altitude which is sacrificed in a receiver (of no consequence in marine receivers which usually suppress the altitude anyway). It is implicit in equations (4) to (7) that c is the same for the signals coming from all four satellites. This is not warranted, since the signals reaching the receiver from satellites of different elevation will have to pass through different regions of the ionosphere and troposphere. The resulting variation in c is allowed for in the receiver in a mathematical model of the ionosphere and troposphere. It is perhaps worth mentioning that the principles behind GPS but applied in reverse were in operation in the First World War. By timing the arrival of the shock wave produced as a shell left a gun barrel at three microphones located at known coordinates, it was possible to deduce the gun position (Bragg 1921). The three unknowns here which demand three microphones are, of course, the gun two-dimensional coordinates and its firing time. Using this technique, it was possible to locate a gun position to about 45 m. The satellites The American Navstar (Navigational Satellite Timing and Ranging) GPS system contains 24 satellites. There are four satellites on each of six orbital planes. These planes are inclined at 55 with respect to the equator and are equally spaced in right ascension. The satellite orbits are close to circular (to within 2%) with an orbital period chosen so that a satellite completes exactly two orbits while the Earth rotates 360 (one sidereal day). This ensures that the satellite trajectory on the Earth exactly repeats itself twice daily. Given that one sidereal day is 23 h, 36 min and s of mean solar time (or told to calculate it) an A- level student should be able to prove that the radius of the satellite orbit is km and the orbital speed is ms 1. A GCSE student should be able to show that the signal transit time t from an overhead satellite is 67 ms. Although a satellite s orbital speed is only c,itis still necessary to allow both for the time dilation as described by special relativity ( moving clocks run slow ) and for the (opposing) time dilation attributable to the change in gravitational potential between the Earth s surface and the satellite s orbit as described by general relativity. Their combined effect, which yields a net increase in clock speed, is allowed for by offsetting the satellite s clocks prior to launch. The GPS provides a rare example of special and general relativity at work in the mechanical world. The signal structure As has been emphasized, the measurement of transit times lies at the heart of GPS. To measure a range to an accuracy of, say, 30 m therefore demands a timing accuracy of 30 m/( m s 1 ) = 10 7 s, or better. The way that this is accomplished is explained in figure 4. All satellites transmit at the same frequency, but each has its own unique binary code known as the coarse/acquisition (or C/A) code, consisting of a pseudo-random sequence of 0s and 1s. The C/A code is 1023 bits long and repeats every millisecond. Thus each bit (or chip ) has a duration of 10 3 s/1023 = s, which is some ten times greater than the timing accuracy we would require for 30 m resolution. The receiver generates the same C/A code as that transmitted by the satellite, but because of the clock bias δt it will not be synchronized with that being produced in the satellite (see figure 4). The received signal will be time-shifted with respect to the receiver-generated code by t and it is this which is required in each of equations (4) to (7). (The actual signal transit time is t δt; see figure 4.) To find the value of t the receiver time-shifts ( slews ) the receiver-generated code along the time axis until there is a perfect match between the two codes. The matching process is known as cross-correlating. In this process the two curves are multiplied together and the area under their product is calculated. This area will have a maximum value (shown normalized to unity in the bottom line of figure 4) when the slew time is equal to t. This procedure allows transit times to be measured to better than 1/20th of a chip length, i.e. to better than s/20 = s, giving a potential ranging accuracy of around 15 m. The C/A code transmitted by each of the satellites is chosen so that their cross-correlation is (near) zero. This prevents the code produced in 40 Phys. Educ. 34(1) January 1999

5 Figure 4. Showing the processing of the C/A code transmitted by a satellite. Here the clock on the receiver is shown running fast by δt; it thus starts to generate the replica C/A code at a time δt earlier than the clock on the satellite starts to generate its code. The cross-correlation process shown in the bottom line measures t and not the true signal transit time t δt. the receiver for, say, satellite 1 mistakenly locking on to the signal received from any other satellite. Since the satellite code repeats every millisecond, a correlation peak is achieved not only at slew time t but at t plus an integer multiple of 1 ms. Hence t is inherently uncertain by multiples of 1 ms or 300 km. (We have already seen that transit times are at least 67 ms.) The additional timing information required to remove this ambiguity is transmitted from each satellite to the receiver as a telemetry signal. The telemetry data, which also includes the current ephemeris constants, is transmitted at a rate of 50 bits s 1. It is superimposed on the C/A code by modulo-2 addition. Whenever a binary 1 occurs in the 50 bit s 1 data stream, the addition has the effect of inverting adjacent binary bits in the C/A code (turning the 0s to 1s and vice versa). (The is, of course, the product of the chipping frequency ( s 1 ) and the telemetry period (1/50 s).) A binary 0 in the data stream leaves adjacent C/A code bits unaltered. The telemetry code is readily removed in the receiver (it can be done in the set by modulo-2 addition of the received signal and the set-generated C/A code). It is worth commenting on how the binary code is transmitted. This is achieved by modulating a carrier frequency of MHz (derived from the on-board atomic clock frequency). The technique used is called phase-shift keying. In this type of modulation, the phase of the carrier shifts by π rad when there is a shift between binary 0 and 1. Selective availability During the initial testing of the C/A GPS, it was discovered that accuracies of m were achievable rather than the intended 100 m. Presumably to limit the value of the system to potential enemies, the US Department of Defense introduced selective availability (SA) to limit positioning accuracy to 100 m horizontally and to 140 m vertically. These accuracies are the two standard deviation (2σ ) values; as such the user can be confident that 95% of the readings will lie within these limits. Physicists may instinctively Phys. Educ. 34(1) January

6 prefer to quote the 1σ values of 50 m horizontally and 70 m vertically (giving the 68% confidence limits). Selective availability involves dithering the satellite clocks and/or falsifying the ephemeris data broadcast in the satellite telemetry message. Since the dithering is different on each satellite, this is not the equivalent of changing the common clock bias in equations (4) to (7). On 29 March 1996 the US President approved a national policy on the use of the US GPS system. This states that It is our intention to discontinue the use of GPS SA within a decade in a manner that allows adequate times and sources for our military to prepare fully for operations without SA... Beginning in 2000, the President will make an annual determination on continued use of GPS SA. There are two ways of circumventing the effects of SA. The first (which is applicable only when the receiver is at rest) is to timeaverage the calculated receiver positions. Several manufacturers produce receivers in which the time averaging commences at the touch of button (e.g. Garmin) or automatically on standing still (e.g. Magellan). In our experience the error quoted on the Garmin 12 XL usually falls to approximately 15 m in about a minute of averaging. Because of short-term biasing in the dithering, the recorded positions of the receiver would have to be averaged for several hours to obtain a position accurate to a few metres. (This biasing also means that the recorded positions should not be used as a source of data for simple error-analysis exercises.) The second and a better way to defeat the effects of SA is to add differential GPS (DGPS) to the receiver. Most GPS manufacturers offer DGPS receivers as an add-on facility. The principle here is that a base station at an accurately surveyed site continuously records its apparent position as given by a GPS receiver. It then transmits the difference between the true and apparent position which is picked up by the user s DGPS beacon. The resulting corrected display can give positions accurate to 1 5 m. Unlike the GPS, which is provided free to users, the DGPS is in the UK a commercial service with a license fee. (In the UK one provider broadcasts it from the Classic FM radio transmitter aerials.) Although DGPS is frequently used on boats and aircraft, we have yet to meet a rambler using such a set. The degree of accuracy to which OS maps can be read just does not justify the enhanced precision available over time-averaging techniques. Further reading Further details on the GPS may be found in the texts of Ackroyd and Lorimer (1990), Logsdon (1995), Leick (1995) and Parkinson and Spilker (1994). These texts are listed in ascending order of difficulty. There are many web sites devoted to GPS; we have listed some of those of general interest, at rjb/gps. Received 3 August 1998 PII: S (99) References Ackroyd N and Lorimer N 1990 Global Navigation: A GPS User s Guide (London: Lloyds of London Press) Bragg W H 1921 The World of Sound (London: Bell) pp Ferguson M 1997 GPS Land Navigation (Boise, ID: Glassford Publishing) Leick A 1995 GPS Satellite Surveying (New York: Wiley) Logsdon T 1995 Understanding the Navstar GPS, GIS and IVHS 2nd edn (New York: Van Nostrand Reinhold) Parkinson B W and Spilker J J (eds) 1994 Global Positioning System: Theory and Applications Progress in Astronautics and Aeronautics vol 163 (Washington: American Institute of Aeronautics and Astronautics) 42 Phys. Educ. 34(1) January 1999