The expected accuracy of the differentially corrected GPS location during skiing was further tested using the residuals from Equation 6.3. It was assumed that because the systematic errors were removed by differential correction the residuals of the pseudo-range data should give an indication of the expected accuracy of the calculated location. Pseudo-range data were collected from a GPS receiver attached to an athlete during an alpine ski run. The pseudo- range data were processed as previously described and a differentially corrected location estimated. The residuals for this solution from Equation 6.3 were then plotted (Figure 6.10). Figure 6.10 shows that the magnitudes of the differentially corrected pseudo-range residuals are generally between 1m and 5m during skiing. Errors in the calculated location are also expected to be of similar magnitudes as the errors in the residuals and should be have a 95% confidence interval of ±3m. There is signal loss (not enough satellites visible to calculate a fix and with no fix there can be no residuals) around seventy five seconds into the measurement. Before and after the signal loss the residuals increase, probably because the shape of the terrain caused the reception to deteriorate and also caused multipath effects.
In Figure 6.10 the skiing takes place between 30 and 80 seconds and apart from when the signal is lost at the end of the run, the residuals are not affected by the athlete‟s dynamics. Therefore during skiing the GPS location accuracy is not likely to deteriorate. Therefore, it was expected that the differentially corrected location would be more accurate than the GPS chip specification of 15m RMS for horizontal location during normal operation. The residuals also provide additional information about signal noise and will be used to optimise the FMC algorithm.
6.2.2.
Carrier-frequency and calculating receiver velocity
Carrier-frequency is used to obtain a measurement of relative velocity between the GPS receiver and each satellite. Relative velocity can then be used to calculate receiver velocity which provides useful information about the athlete‟s trajectory. Alternatively, the carrier- frequency data can be fused with the pseudo-range data, using a Kalman filter, in order to improve receiver location accuracy.The expected carrier-frequency of satellite signals received by the GPS chip was
F=1,575,420,000 Hz (SiRF Technology Inc, 2005). The Doppler Effect was used to convert frequency shifts in the carrier frequencies from the satellites to relative velocities. The carrier- frequency shifts (CF) reported by the GPS chips were scaled internally to give units of velocity. The frequency shifts were multiplied by the speed of light (c=299,792,458ms-1) the constant used by the SiRF2 GPS chip (SiRF Technology Inc, 2005) and divided by the expected frequency (F). The reported carrier-frequency (CF) was not the relative velocity because even though the units were velocity, further processing was required to remove the receiver and satellite clock drifts. The drift was the rate of change of clock bias in hertz.
Figure 6.11: Calculation of velocity from carrier-frequency data
To calculate relative velocity the Doppler Effect is used. Doppler Effect explains why an ambulance when travelling towards you appears to have a higher pitched siren than when travelling away from you. As the ambulances approaches the perceived wavelength of the siren is compressed, making it sound higher. As it passes the perceived wavelength of the siren is stretched and the pitch drops. The actual velocity of the ambulance is unchanged as is the actual siren frequency. To calculate receiver velocity, carrier-frequency data from at least four satellites are required. Also required are the satellite to receiver relative location vectors (S2R, Figure 6.11) and the satellite velocity vectors. Figure 6.11 shows how the relationship between relative location vector and satellite velocity vector affects the carrier-frequency. When the two vectors are perpendicular (satellite 2) there is no changes to the carrier- frequency. When the two vectors are aligned, the carrier-frequency increases (satellite 1). When the two vectors are opposed, the carrier-frequency decreases (satellite 3).
The scaled carrier-frequency shift data (CF, with units of ms-1) was acquired from both the base and rover to calculate a differentially corrected receiver velocity. The first step was to converted the carrier-frequency data into a measured relative speed (Speed, Equation 6.4) by subtracting the estimated scaled receiver clock drift (Drift, with units of Hz, extracted from MID 4). The clock drift was scaled by the speed of light (c=299,792,458ms-1) over the frequency (F=1575420000 Hz). It was not necessary to remove the satellite clock drift because it was negligible.
Equation 6.4
The second step was to calculate the normalised relative location vectors from the satellites to the receivers (|S2R|, Equation 6.5) between each satellite location (SL= [SLX, SLY, SLZ]) and the GPS receivers (RL = [RLX, RLY, RLZ]). These vectors affect the measured frequency shifts (see Figure 6.11).
Equation 6.5
The expected relative speed (Speed_Exp, Equation 6.6) is the calculated relative speed between each satellite and the receiver. It is found by the dot product of the normalised relative location vectors (|S2R|) with the relative speed vectors. The relative speed vectors are found by subtracting the satellites velocities (SV = [SVX, SVY, SVZ]) from the GPS receiver velocity (RV = [RVX, RVY, RVZ]). For the base station this calculation was simplified because the base station was stationary (RV = [0 0 0]). The estimates of satellite velocity (SV) were either extracted from the GPS chip (MID 30) or were modelled for improved accuracy.
Equation 6.6
The required differential correction (Speed_Corr, Equation 6.7) was then calculated using the base station data. The correction for each satellite was the difference between the expected relative speed (Speed_Exp, calculated in Equation 6.6) and the measured relative speed (Speed, calculated in Equation 6.4).
Equation 6.7
Example differential carrier-frequency corrections for a skiing run are presented in Figure 6.12. The carrier-frequency corrections are between 0ms-1 and 0.04ms-1. The units of carrier-frequency are metres per second because as explained at the beginning of this section the carrier-frequency is pre-scaled within the GPS chip. The carrier frequency corrections when scaled by dividing by the athlete‟s velocity through a giant slalom race course of approximately 20ms-1 are much smaller than the previous pseudo-range corrections (between 0m and 150m, Figure 6.8) when scaled by the typical giant slalom gate spacing of 20m. It is therefore expected that measurement of the athlete‟s velocity will be more reliable than measurement of the athlete‟s position.
Figure 6.12: Differential carrier-frequency corrections
The differentially corrected GPS velocity of the rover receiver, attached to the athlete, was then calculated. MATLAB‟s non-linear least squares function was used to minimise the residual carrier-frequency error (, Equation 6.8) by optimising of the rover velocity vector. Receiver velocity (RV) is incorporated into Equation 6.8 by the term (Speed_Exp) that is calculated in Equation 6.6. The term (Speed_R) is calculated from the rover data using Equation 6.4. At least four satellites were required to get a fix, three degrees of freedom for the rover velocity vector and one for a new estimated of the receiver clock drift (Drift).
Equation 6.8