Current Global Positioning System
Background and Theory
Stand-Alone GPS Performance
In the GPS positioning problem, a user measures the distance to a satellite and determines the radius of a ‘range ring’ that describes all the possible positions of the user relative to the satellite. Using a minimum of four satellites the user can calculate its position at the intersection of these range rings. Since each one of these range measurements contain errors, the true user position is contained within a region of position error, which varies in size depending on the relative geometry of the satellite constellation. This concept is demonstrated 2-dimensionally in Figures 1 and 2 below. In Figure 1, the two satellites are nearly at right angles to one another, and so this gives a smaller region of position error. In Figure 2, there is a much smaller angle of separation between the two satellites, and as a result, the region of possible user position error is amplified in the direction transverse to the line between the user and the two satellites. Since there is a lack of information of the user position in the ‘transverse’ direction, there is also less certainty about the user position in that direction. In a 3-D application, as with the GPS constellation, the distribution of amplified user position error is described by a geometric Dilution of Precision factor (DOP). Just as in the 2-dimensional case, DOP is a function of the constellation geometry and has axes of information which are generally better than others.
An analysis of the GPS measurement equations can provide insight into the GPS system performance in terms of DOP and the pseudorange measurement errors. A GPS position solution is obtained by varying a linearized estimate of user position until the difference between the estimate and the measured position is minimal. The GPS measurment is described by the following equation:
HDx = Dr (1)
Dx = the offset between measured user position and linearized position
Dr = the offset in pseudorange from the pseudorange values that correspond to the linearization point
H = is a 4x4 matrix describing the direction from the user to the satellite and the user’s time offset from GPS time.
The process of solving for a GPS position solution involves a minimization of the difference between the linearized user position and ‘true’ user position. The offset in user position, Dx , can be expressed as
Dx = xT – xL + dx (2)
xT = the error free position and time
xL = the linearized position and time
dx = the error in the position and time estimate
Solving equation 1 by a method of least squares yields
dx = [(HTH)-1HT]dr (3)
dr = difference between the error-free pseudorange values and the pseudorange values at the linearization point.
By modifying the linearization point until dx is minimized, the final linearization point can be used to describe the measured user position.
The pseudorange errors are considered to be zero mean Gaussian random variables. As a result, the relationship between the GPS position solution error and the pseudorange measurement error can be obtained from the covariance of
cov(dx) = (HTH)-1cov(dr) (4)
The components of dr are usually assumed identically distributed and independent and have a variance equal to the square of the satellite user equivalent range error (UERE). Under this assumption, the covariance of the errors in the computed position and time bias is a scalar multiple of (HTH)-1.
The dilution of precision parameters are defined in terms of the ratio of the components of cov(dx) and sUERE. It is also assumed that local user coordinates are being used in the cov(dx) where the x, y, and z axes correspond to the east, north, and up directions respectively. The vertical and horizontal DOP parameters, VDOP and HDOP, can be expressed in terms of the diagonals of (HTH)-1 by expanding (HTH)-1 in component form
The vertical and horizontal DOP parameters are expressed in terms of (HTH)-1 as follows:
Ultimately, the error in the GPS position solution can be expressed as the product of a DOP geometry factor and a pseudorange error factor.
sp = the standard deviation of the positioning accuracy
DOP = the geometric dilution of precision factor
sUERE = the standard deviation of the satellite pseudorange error
It follows that:
= standard deviation of vertical position (11)
= standard deviation of horizontal position (12)
Expected values for vertical and horizontal position error can be predicted by analyzing VDOP, HDOP, and sUERE.
Pseudorange Errors: sUERE
The GPS measurement errors are caused by corruptions in the time that it takes a signal traveling at the speed of light to traverse the straight-line distance between the user and a satellite. Delays on the signal speed as it passes through the atmosphere, reflections from objects nearby the receiver, and receiver hardware delays all are possible sources of timing error. The total error in the GPS timing measurement can be expressed as:
dtD = dts + dteph + dtatm + dtnoise + dtmp + dthw + dtsa (13)
dts = satellite clock error
dteph = satellite position error
dtatm = atmospheric delays
dtnoise = receiver tracking noise
dtmp = multipath delay
dthw = receiver hardware bias
dtsa = selective availability (if on)
The errors from each of these components are root sum squared to form the total system UERE as is described by Figure 3 below. The result yields that the user equivalent range error is Gaussian distributed with a 1-sigma value of 8 meters. With a Gaussian distribution, 68% of the possible values fall within the ± 1s of the mean; and approximately 95% fall within ± 2s of the mean.
Predicted Horizontal and Vertical Position Error Using VDOP and HDOP
VDOP and HDOP can be directly calculated from GPS observation measurements using equations 8 and 9. Figure 4 shows variation in HDOP and VDOP over a 24-hour period. Note that the average values for HDOP and VDOP are 1.6 and 2.0 respectively. This shows that for a given sUERE, vertical errors will be generally worse than horizontal position errors.
Since the errors in the computed position are assumed to be zero mean Gaussian errors, it follows that 95% of the measurement error will fall within two standard deviations of the position error predicted by equations 11 and 12. Assuming average values of HDOP and VDOP, estimates of the expected vertical and horizontal position error are given by:
Vertical Position Error:
= 32 meters = ± 105 feet (14)
Horizontal Position Error:
= 2*1.5*8 = 24 meters = ± 79 feet (15)
GPS Onboard the Boeing 777
The 777 is equipped with dual (left and right receivers) which the flight management computer (FMC) uses to update the inertial navigation sensors. The benefit of having two receivers is that the FMC can either use the receiver that has a better view of the GPS constellation, or it can average the output of the two receivers and remove some of the uncommon random errors. If one receiver, for example, incurred significant multipath errors during approach, then the averaged output would have a reduced multipath error. The overall effect would be that unpredictable errors due to multipath and receiver tracking noise would be reduced while the common mode errors would remain unchanged. However, the details of how the dual GPS measurements are processed remain unknown.
The processed GPS position solutions were sampled once every second and recorded in latitude, longitude, and height format. Neither the raw pseudorange measurements, nor the time of signal transmission were recorded by the FMC. For this reason, it was impossible to analyze the processing algorithm or to download the corresponding broadcast ephemeris for the time that the measurements were made.
A major difficulty in analyzing the accuracy of GPS measurements made on an aircraft is determining a reliable estimate the truth. In the case of GPS altitude, the aircraft also measures pressure altitude and inertial vertical speed. Even though the pressure altitude is corrected for local variations in mean sea-level pressure, that parameter is known to have decreased accuracy near the ground due to ground effect. So, when the airplane is near the ground, the baro-corrected pressure gives values approximately 50 to 100 feet lower than expected, and the overall accuracy of the baro-corrected pressure is quoted as ± 100 feet. The vertical inertial sensors integrate measured accelerations on the aircraft and output inertial vertical speed as a function of time. This inertial vertical speed can be integrated to obtain estimates of altitude change over short periods of time; however, the inertial vertical sensors are also known to have drifting tendencies and cannot accurately resolve changes in vertical speed below 10 ft/min2. The changes in altitude measured by the inertial vertical sensors were referenced to the transition time when the landing gear sensors indicated the airplane was on the ground. For this reason, all altitude measurements were referenced to this point where the wheels contacted the ground. The associated altitude measurements are termed as: height above takeoff (HATO) and height above touchdown (HAT) respectively.
‘Truth’ values for latitude and longitude were a little more difficult to obtain. In the case of horizontal position, the only time when the location of an airplane is known is when it is in the vicinity of an airport or a DME. It was determined that the simplest way to approximate the airplane’s position was to use the recorded localizer frequency and the landing location during approach. After determining the exact runway that the airplane was using, surveyed values for each end of the runway could be used to estimate the airplane’s approximate position. Although a problem arose due to the fact that there was no way to precisely determine the airplane’s position along the runway at any point in time.
For this reason, GPS position errors were restricted to the direction transverse to the centerline of the runway. When the localizer deviation is near zero, the aircraft is known to be aligned with the center of the runway; and using surveyed positions for each end of the runway, it was possible to determine the GPS horizontal position error transverse to the runway as a function of time. It was assumed that the along-track position error was similar to the transverse position error.