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DGPS Kinematic Carrier INTRODUCTION Use of GPS signal simulators for precise Phase Signal Simulation differential GPS positioning and velocity determination testing is not widespread in the Analysis for Precise GPS research community. A recent report [l] addresses the capabilities of satellite signal Velocity and Position generators in areas such as differential GPS. There are numerous advantages in using GPS Determination signal simulators for testing and evaluation. One primary advantage is the provision of cost-effective truth position and velocity information in a well- controlled environment, which is typically difficult, if not impossible, to obtain in a real dynamic M. ELIZABETH CANNON, GERARD LACHAPELLE, environment such as aircraft applications [2,3]. A and MICHAEL C. SZARMES simulator allows for the possibility of rerunning The University of Calgary, Calgary, Alberta exact trajectories/scenarios with the same predefined conditions, which can directly assist in JEFFREY M. HEBERT, JAMES KEITH, and identifying error sources [41. The flexibility in SCOTT JOKERST programming various error sources makes it Holloman Air Force Base, New Mexico possible to identify the impact of individual errors Received January 1997 on system performance. All of these advantages Revised May 1997 favor the use of GPS signal simulators in a test and evaluation environment. The equipment, procedures, and simulated ABSTRACT aircraft trajectories used to analyze carrier phase This paper describes the differential GPS and signal measurements in estimating velocities and 3D simulator equipment, procedures, and simulated aircraft positions are described in this paper. The trajectories used to analyze carrier phase measurements simulations and results presented here focus on in estimating velocities and 30 positions. A differential the impact of high accelerations on precise velocity GPS simulator system was used to generate CIA-code, and position determination. Recent studies in the P-code, and carrier phase signals from segments of the actual GPS constellation. Two pairs of dual-frequency area of airborne GPS and integrated GPS/inertial receivers from different manufacturers were tested. These navigation system (INS) have reported differential units have the capability of tracking the P-code (LlIL2 GPS positioning accuracies at the centimeter level for one of the receivers and L2 for the other). An aircraft and velocity accuracies at the centimeter/second trajectory with accelerations up to 5g was simulated, level for high-performance GPS receivers 12, 3, and analysis is performed in terms of the resulting 5-71. However, verification of results was velocity errors and dynamics. Two methods for accomplished for the most part by comparison with estimating the 30 velocities were tested, namely carrier independent measurement systems (such as phase with fixed-integer and real-number (float) software simulators, INS, or different ambiguities. Both raw Doppler measurements and a manufacturers’ receiver technologies/reference number of carrier-phase-derived Doppler measurements systems) 12, 3, 5-71; by comparison of independent were tested. As a by-product, the 30 positions could also be determined using various carrier phase approaches. processed results (such as foward and reverse The estimated quantities were compared with the processing, and use of code and carrier phase reference quantities known a priori to determine the smoothed code) 12, 6, 71; or through residual performance of the receivers under the various conditions analysis (such as double-difference carrier phase simulated. residuals) [3, 6, 71, as opposed to a comparison against truth data. TEST EQUIPMENT AND SOFlWARE The Central Inertial Guidance Test Facility’s (CIGTF) Northern Telecom STR2760 Differential GPS Simulator System was used for the simulations. It consists of two multichannel, high- dynamic GPS simulators capable of outputting Ll and L2 frequencies (1575.42 and 1227.60 MHz, Journal of The Institute of Navigation NAVIGATION: Vol. 44, No. 2, Summer 1997 respectively), C/A- and P/Y-codes, and the full Printed in U.S.A. navigation message. It is therefore suitable for 231 both pseudorange and carrier phase processing. temperature-compensated crystal oscillator This unit has high signal dynamics that are (VCTCXO). The third-order C/A carrier PLL is capable of simulating + 120 km/s velocities, updated at a rate of 100 Hz, and the raw Doppler & 3600 m/s2 (350 g) accelerations, and + 5000 m/s3 measurement is updated from the PLL at the (500 g/s>jerk, and is appropriate for almost any same rate, but is averaged over 50-100 ms 1131. type of operational performance testing. Signal The receiver can operate under one of four accuracies (RMS maximum over 1 min [81>are dynamic states, with the tracking loop noise given in Table 1. bandwidth varying accordingly, as shown in The trajectories of land, sea, air, and space Table 2 1141.For the test conducted, the default vehicles can be programmed with the vehicle high-dynamics option with fixed 15 Hz bandwidth motion trajectory generator. Trajectory and signal was used. characteristics, such as motion, attitude, Selective Two software packages developed at The Availability @Al/Anti-Spoofing (AS), and signal University of Calgary were used to generate strength can be applied. Various errors can be velocities from the data: FLYKIN’” and simulated, such as clock, multipath, and SEMIKIN’“. FLYKIN” is a DGPS on-the-fly ephemeris errors. Atmospheric conditions can also (OTF) integer ambiguity resolution carrier phase be simulated by applying a tropospheric model and processing package designed to yield centimeter- one of two available ionospheric models. Body level accuracies in position [15, 161. Velocity masking due to aircraft banking can be simulated determination is performed using the raw Doppler as well. measurements when ambiguities are resolved to Two different GPS receivers have been used for their integer values, and velocity accuracies are this analysis-the Ashtech Z-12” and the NovAtel shown to be on the order of a few centimeters/s, MiLLennium’” GPSCard” . !lhvo receivers from depending on the receiver used. During ambiguity each manufacturer were used since differential resolution, however, velocity estimates are based tests were performed. The Ashtech Z-12” unit is a on range and phase measurements. The float 36-channel all-in-view geodetic receiver capable of solution of FLYRIN’” is currently being modified tracking C/A- and P/Y-code signals on Ll and so that velocity estimates will be based on the raw PN-code signals on L2, and provides the full Doppler, as is the case for the fixed ambiguity wavelength integrated carrier phase phase. The user can utilize the widelane measurements on Ll and L2. All tracking loops observable if available for faster fixing of the are fixed at 0.1 Hz and are guided by the wide integer ambiguities because of the longer C/A-code tracking loop bandwidth of 25 Hz 191.The wavelength (86 cm as opposed to 19 cm for Ll raw Doppler is estimated by a three-point only). Once the integer ambiguities have been interpolation using the last three 0.5 s phase determined, however, processing reverts to Ll only estimates. The Z-12’” does not employ a third- to exploit the lower noise of this measurement. order phase lock loop (PLL). The internal oscillator Velocity accuracies are highly dependent upon the is a temperature-compensated crystal oscillator dynamics of the mobile and the quality of the raw (TCXO), which is more stable than a voltage- Doppler measurement, as will be shown in the controlled TCXO 1101. results. The NovAtel MiLLennium” GPSCard’” is a SEMIKIN’” is a suite of processing programs high-performance 24-channel Narrow CorrelatorT’ that can process semikinematic differential GPS GPS receiver that can track the Ll C/A-code and carrier phase and pseudorange measurements 1171. the L2 P/Y-code, and provides the full wavelength Static data are processed in a double-difference integrated carrier phase measurements on Ll and least-squares adjustment, while kinematic data L2 111, 121. The unit can operate under dynamics (phase, pseudorange, and phase rate data) are up to 515 m/s and 6 g. The receiver clock is an on- processed in a Kalman filter in fixed or floating board standard 20 MHz, voltage-controlled, modes. A purely float solution was obtained here for comparison with the FLYKIN” fixed solutions. Table 1-Signal Accuracy Specifications for Northern Telecom STR2760 Simulator System Table P-Dynamics Options Available on the Perel?.leter NovAtel MiLLennium” Range Pseudorange +lOmm Option Description Pseudorange rate *lmm/s Stationary 2.5 Hz bandwidth, jerk < 0.001 g/s Delta-pseudorange +5 mm Low 5 Hz bandwidth, jerk < 0.1 g/s Interchannel bias + 50 mm (code) Medium 10 Hz bandwidth, jerk < 1.0 g/s + 0.265 mm (carrier) High 15 Hz bandwidth, jerk < 4.5 g/s (45 m/s9 232 Navigation Summer 1997 SIMULATION DESCRIPTION The simulation tests are characterized by the path of an aircraft trajectory with increasing levels of acceleration at the turns. From the STR2760 Differential GPS Simulator System, output position and velocity truth data are used for verification and analysis of the results. Two differential GPS simulations were conducted: one using the MiLLennium” and the other using the Z-12” receivers. In each test, one of the receivers acted as the fixed reference, while the other acted 95000 gsooo 97000 98000 09ma I as the mobile. A 1 Hz data rate was chosen, and Height the tests lasted 60 min each. To obtain the best d22wj I possible results without the complication of g 1600 additional individual error sources, the 0 1400 1' = loo0 ,,,~~,~,,I)~~./~,,,I,/~,~,,~/1~~~,~,~~~1 simulations were conducted with no %/AS, no 95000 Q6aoa 97000 osoog sgoocI satellite clock errors, no ephemeris errors, and no GPSTime(aee) multipath. The tropospheric and ionospheric Fig. 2-Simulated Aircraft Trajectory Components models were both employed. Signal masking to simulate realistic banking of the aircraR was not velocity of 100 m/s (in 20 s), the aircraft takes off. applied in either case. In further simulation The acceleration during take-off is constant at analysis, it will be possible to determine the effects 0.5 g. At take-off, there are east and up jerk values of additional individual error sources on velocity of 4.7 m/s9 and 5.7 m/s3, respectively. Within 50 s estimates, but this analysis is beyond the scope of from take-off, the aircraft levels off at its cruising the present investigation. elevation of 2255 m, with east and up jerk values of 1.2 m/s3 and 9.5 m/s3, respectively, during this Aircraft Trajectory period. The aircraft continues west at a constant The simulations were conducted simulating real 100 m/s velocity until the first 2 g turn of the first dynamics of a fighter aircraft experiencing loop. The aircraft then flies 5.5 loops of the circuit upwards of 5 g accelerations. The simulated in a clockwise direction, with increasing trajectories used were the same for both tests (see accelerations during each loop. Table 3 shows Figure 1). The trajectory runs 24.7 km east-west these accelerations and their corresponding times. and 1.0 km north-south, and reaches a maximum Truth Position and Velocity Data altitude of 1 km above the base reference station elevation. The base reference station was held The STR2760 Differential GPS Simulator System fixed at the central point of the trajectory, yielding outputs truth position and velocity data for the a maximum distance between the fixed reference mobile receiver at a specified data rate (in this station and aircraft of less than 15 km. case, 1 Hz). The calculated acceleration and jerk Figure 2 shows the aircraft trajectory latitude, values have been derived from the available truth longitude, and height components. In each velocity data in order to assess the correlation that simulation, prior to aircraft movement, there is a may exist between position/velocity errors and 15 min static initialization with both the fixed dynamics. Plots have been made, and Figure 3 reference and remote aircraft stations occupying shows these calculated acceleration and jerk the same point (elevation of 1255 m), as is possible values for the final 2 g turn (Loop 6). with GPS simulations. After static initialization, As illustrated in Figure 3, there are noticeable the aircraft travels west, commencing with a large spikes of large jerk in both the north and east jerk of 4.5 m/s3. After reaching a maximum components at the beginning and end of the turn. This trend is consistent in all other turns of the trajectory. Absolute acceleration and jerk for the entire trajectory are shown in Figure 4. Maximum values of north and east jerk for each ~~~ ,.,,= r *;, 7 ( of the high-dynamic portions of the trajectory are listed in Table 4. -106.3 -106.3 -106.2 -106.2 -106.1 -106.1 -106 -106 -105.9 METHODOLOGY Longitude(degrees) Velocity estimates are typically obtained by Fig. l-Simulated Aircraft Trajectory using the raw Doppler measurement. Depending Vol. 44, No. 2 Cannon, et al.: Kinematic Carrier Phase Signal Simulation 233 Table 3-Accelerations of the Aircraft Trajectory Loop Acceleration (g) GPS Time (8) 1 2 96423-96440 (17 a) 96681- 96698 (17 s) 2 3 96939-96951(12 s) 97193-97204 (118) 3 4 97445-97455 (10 s) 97696-97707 (10 s) 4 5 97947-97955 (8 9) 98196-98204 (8 s) 5 2 98445-98462 (17 s) 98703-98720 (17 s) 6 2 98961-98978 (17 s) Acceleration Tndh - 2 9 Tun measurement will vary in noise. On power-up, the NovAtel MiLLennium’” defaults to high-dynamics mode with a constant tracking loop bandwidth of 15 Hz. The Ashtech Z-12’” uses a wide guiding C/A carrier phase noise bandwidth of 25 Hz, while all other bandwidths are 0.1 Hz 191. The raw Doppler frequency (also known as delta 98955 98980 98985 98970 98975 98980 98985 range), which results from the relative motion between receiver antenna and satellite, is a JerkTruth-2gTun 15 measurement of the phase rate Cp.Theoretically, 1 North the Doppler measurement in a receiver can be computed from the PLL by counting the number of carrier phase cycles over a small time period divided by that time period. The MiLLennium’” updates the Doppler at a rate of 100 Hz, but the -5im I, (, I, I, I1 s I, s I2 Doppler measurement is obtained by averaging 98955 98980 98985 98970 98975 98980 98986 GPS Xme (set) over a period of 50 to 100 ms. The Doppler measurement of the Z-12” is obtained by a three- Fig. 3-North and East Components of Acceleration and Jerk for point interpolation using the last three 0.5 s phase Final 2 g Turn estimates. The resulting Doppler measurement for a particular epoch is therefore based on the AbsohtteAcceleration averaging time interval. 5- To overcome the inherent noise of the raw Q4- Doppler measurement, a derived Doppler can be s 7 determined, based directly on the integrated e ‘i) 3- 1 carrier phase measurements. The resulting = 2- b carrier-phase-derived Doppler will be smoother d 1 : than the raw Doppler output from the GPS oT,,,,,~,n,,,,,,,~,,~,,,,,,,,,,, receiver since the averaging of the phase 1590 1600 1610 1820 1830 1840 IS50 measurements is now done at the data rate used. GPS Time (min) In the analysis, the accuracy of the velocity Abrobb Jerk estimates was compared in terms of the type of 35 Doppler measurement used. Therefore, in addition 30 to the raw Doppler measurement output from the -2 25 GPS receivers being used, a number of % 20 approximation techniques were applied to the c 15 4 10 integrated carrier phase measurements. In these 5 cases, the carrier-phase-derived Doppler 0 measurement replaces the raw Doppler prior to 1590 1600 1610 1820 1830 1840 1850 processing. These approximation techniques GPS Time (min) include the first-, second- and third-order central difference approximations, and natural cubic Fig. I-Absolute Acceleration and Jerk for Simulated Aircraft Trajectoy spline. These approximation approaches utilizing the carrier phase are similar to that of 1181, wherein static baseline velocity/acceleration upon the receiver tracking loop bandwidth, the accuracies are analyzed based on curve fitting over update rate of the tracking loop from which the high-frequency (50 Hz) phase measurements. Doppler is obtained, and the type of internal The central difference approximations are based oscillator used, the resulting Doppler on a Taylor Series expansion and yield an estimate 234 Navigation Summer 1997 Table 4-Maximum Derived Jerk Values for Each Loop of the Aircraft Trajectory Maximum North Jerk (m/s9 Maximum East Jerk (m/d) LOOD TWlIl Turn 2 TUlll Turn 2 1 (2 g) 12.6 13.0 3.8 3.8 2 (3 g) 18.7 18.7 8.3 8.3 3 (4 g) 25.5 25.6 14.2 14.3 4 (5 g) 29.5 31.2 21.5 21.3 5 (2 g) 11.7 10.4 3.7 3.7 6 (2 g) 10.3 - 3.7 - of the derivative. Derivation of these formulae can depends on data availability. If missing epochs be found in 119, 201. Given a function f(x), the were found during the computation of the carrier- Taylor Series at (x + h) expanded about x is phase-derived Doppler, the central difference defined as: approximations could not apply, given the fixed data rate used (i.e., 1 Hz). In these cases, the raw f(x + h) = f(x) + hf’(x) + h.o.t. (1) Doppler measurement was retained for the By combining with (1) the following Taylor corresponding epoch. This affected the accuracy of Series for the function f(x) at (x - h) expanded the resulting velocity estimates at these epochs, as about x the raw Doppler measurement is generally more noisy than the carrier-phase-derived Doppler. fix - h) = f(x) - hf’(x) + h.o.t. (2) The second consideration relates to the and truncating to first order, the first-order central dynamics of the mobile and the subsequent cycle difference approximation to the derivative is slip detection algorithm applied. Cycle slip obtained: detection in kinematic applications is difficult because of the increased dynamics induced by the f’(x) ~ f(x + h) - fix - h) (3) vehicular motion of the remote antenna. If dual- 2h frequency data is available, a cycle slip can be By including more data points about the central detected by comparing the difference between the point of expansion, one can obtain higher-order two integrated carrier phase measurements over central difference approximations. Neglecting the two epochs [211. If the difference exceeds the higher-order terms, equation (4) is the second- wavelength of Ll (19 cm), a cycle slip (either Ll, order central difference approximation: L2, or both) has occurred. To determine whether the cycle slip occurred on Ll, a phase velocity f(x) ~ 8 f(x + h) - f(x - h) trend method can then be applied 1211. 12h SIMULATION RESULTS f(x + 2h) - f(x - 2h) (4) This section presents the results of the two 12h dynamic simulator test flights using the The third-order central difference approximation MiLLennium’” (Test 1) and Z-12’” (Test 2) used is given by receivers and processed using FLYKIN’” with f’(x) ~ 45 f(x + h) - f(x - h) fixed-integer ambiguities. In both cases, the 60h widelane was used for ambiguity resolution, and the Ll data was used thereafter. The accuracies of _ g f(x + 2h) - f(x - 2h) the velocity estimates were compared with the 60h truth velocity data output from the STR2760 GPS Simulator System for each epoch (1 Hz data rate). f(x + 3h) - f(x - 3h) + (5) For clarity, the results are divided into two 60h categories for each test: static and low dynamics Equations (1) through (5) are based on 1191. (considering sections of the trajectory with zero or A natural cubic spline was also applied to the constant velocities), and high dynamics (for integrated carrier phase measurements, and the sections of the trajectory with accelerations > 2 g). velocity estimates were compared with those of the Next, correlations between aircraft dynamics and central difference approximations. In this case, no velocity/position accuracies are presented and smoothing parameter was applied. discussed. Finally, the results of the two tests are Two important considerations were taken into compared, and a detailed analysis of GPS velocity account in the algorithms applying the above errors during periods of high dynamics (> 2 g) is numerical differentiation techniques. The first presented. Vol. 44, No. 2 Cannon, et al.: Kinematic Carrier Phase Signal Simulation 235 During the analysis of the results, anomalies in It is evident that the first-order central the velocity estimates occurred for two reasons. difference approximation gave the best results for First, the simulator data of Test 1 exhibited huge low dynamics, yielding a 3-D RMS error of 2.1 mm/s. variations (upwards of thousands of Hz) in the raw The first-order central difference approximation Doppler measurement for the initial epochs yielded errors in velocity less by a factor of 20 to (lasting about 1 min) of the observation period. 25 than those derived from the raw Doppler These variations were due to the warm-up period estimated from the MiLLennium’“, which yielded of the simulator. After a settling period, however, relatively large velocity errors (3-D RMS error of the raw Dopplers were observed to be consistent 46.8 mm/s). with the phase measurements over time. Because There is a degradation in accuracy as the order of the instability of the initial Doppler and phase of the approximation is increased. Both the third- measurements and the resulting incorrect velocity order and cubic spline approximations performed estimates, this initial period of the test was not worse than the other central difference included in the analysis. Second, because of approximation techniques, representing about a missing epochs within the dataset, the central 50 percent decrease in accuracy as compared with difference approximations cannot be applied, and the first-order results. This degradation is due to the raw Doppler measurement was preserved for the increased level of noise inherent in applying that epoch. Since in most instances the raw more data points in the approximations, and is Doppler is a much noisier measurement than the consistent with results found in 1191.The results carrier-phase-derived Doppler, correspondingly using the raw Doppler based on the FLYKIN’” noisier velocity estimates resulted at these epochs. fixed and SEMIKIN’” float solution are more or In the above two cases where anomalies in the less equivalent. velocity estimates occurred, they were removed from the computation of the statistics. It should be High Dynamics (> 2 g) noted that there is little or no variation in the velocity estimates given a change in the number of Periods of high dynamics for the same simulated satellites. test flight are now considered. Table 6 shows the maximum errors in velocity (in centimeters/s) Test 1: MiL Lennium’” Receiver during the turns of Loop 4 corresponding to a 5 g acceleration. Note that one gets only a general Static and Low Dynamics (Zero and idea of the relative accuracies of the velocity Constant Velocity) estimates from Table 6. It is important to consider the errors more closely during these high- The statistics shown in Table 5 give the RMS acceleration periods, and this is discussed in a velocity errors using the various approximation later section. techniques for Test 1, involving the NovAtel At high accelerations (in this case 5 g>, the MiLLennium’” receiver during periods of low velocity estimates are degraded by a factor of dynamics (during static initialization and constant about 25 in the case of the raw Doppler as velocity sections of the trajectory). In addition, the compared with those of the low-dynamics case results based on a SEMIKIN” float solution are presented in Table 5. It is evident from Table 6 presented for the case of the raw Doppler. During that the velocity estimates during the 5 g turns this simulation, the number of satellites tracked are best for those corresponding to the raw was between 5 and 6. The PDOP remained below Doppler measurements, yielding maximum 3.5 for the section of the trajectory corresponding absolute velocity errors on the order of 23 to to 5 satellites and below 2.8 for the sections 61 cm/s. Both the third-order central difference corresponding to 6 satellites. and cubic spline approximations performed the Table 5--Summary of Statistics- Dynamic Simulator Test Flight (Test 1) During Periods of Low Dynamics (MiLLennium’” Receiver) Velocity Component FtMS Error (cm/s) 3-D FtMS Method of Velocity Estimation North East UP km/s) Raw Doppler FLYKIN’” tied 1.70 1.58 3.47 4.68 Raw Doppler SEMIKIN” float 1.53 1.47 3.77 4.77 1st order 0.07 0.06 0.16 0.21 2nd order 0.09 0.08 0.21 0.27 3rd order 0.10 0.09 0.24 0.31 Cubic spline 0.11 0.10 0.23 0.31 236 Navigation Summer 1997 Table 6-Summary of Statistics-Dynamic Simulator Test Flight (Test 1) During 5 g Turn-(MiLLennium’” Receiver) Maximum Velocity Component Error (cm/s) Method of Velocity Estimation North East Raw Doppler FLYKIN’” fixed 46 36 Raw Doppler SEMIKIN” float 61 23 1st order 600 400 2nd order 300 120 3rd order 240 80 Cubic spline 220 70 next best, at a level about 2 to 5 times worse than position dilution of precision (PDOP) remained the raw Doppler results. The float and fixed below 3.5 for the portion of the trajectory solutions using the raw Doppler are more or less corresponding to 5 tracked satellites and below 2.8 equivalent. The first-order approximation for the portions of the trajectory corresponding to performed about 10 times worse than the raw 6 tracked satellites. There was a problem at the Doppler. fixed reference station as it did not track PRN 15 In the central difference approximations, the for the duration of the test, whereas the aircraft integrated carrier phase measurement at the remote was able to track this satellite. Thus, the central point of expansion is not used (e.g., per number of satellites processed in FLYKIN’” was equation (3)). For example, for the first-order between 4 and 5. This led to problems in approximation, a straight line is projected through determining the correct integer ambiguities. In the two carrier phase measurements adjacent to addition, at the 4 g and 5 g turns, the Z-12’” was the point of interest. It is not evident as yet how unable to track all visible satellites and lost lock not using the carrier phase measurement at the during these periods of high acceleration. central point of expansion correlates with the It is evident that the first-order central velocity errors based on the central difference difference approximation gave the better results, approximations listed in Table 6. The cubic spline yielding a 3-D RMS velocity error of 2.9 mm/s. The is expected to perform the best as it fits a first-order central difference approximation yielded polynomial through the data points to estimate the errors in velocity about 6 times smaller than those quantity at the point of interest. derived from the raw Doppler estimated from the There is a correlation between the magnitude of Z-12’“. Comparing the results based on the raw acceleration and the magnitude of the velocity Doppler for both the MiLLennium’” and Z-12’” error when considering all turns of the trajectory. under low dynamics, it is evident that the Z-12’” In general, the greater the magnitude of yields velocity estimates that are more accurate acceleration, the less accurate the velocity than those of the MiLLennium’” (3-D RMS estimates become. More discussion of this velocity errors of 16.8 mm/s versus 46.8 mm/s for correlation is given in a subsequent section. the Z-12’” and MiLLennium”, respectively). This better performance is due to the method used in Test 2: Z- 12” Receiver the Z-12” to estimate the Doppler measurement. The raw Doppler of the Z-12’” is estimated by a Static and Low Dynamics (Zero and three-point interpolation using the last three 0.5 s Constant Velocity) phase estimates, thereby averaging out much of The statistics shown in Table 7 give the RMS the noise. errors of the raw Doppler and first-order central The results for the carrier-phase-derived difference approximation techniques for Test 2, Doppler based on the first-order central difference involving the Ashtech Z-12’” receiver. During this approximation are about 40 percent worse than simulation, the number of satellites tracked by the those of the MiLLennium’” (3-D RMS error of mobile was between 4 and 6. The corresponding 2.9 mm/s versus 2.1 mm/s for the Z-12” and Table 7-Summary of Statistics-Dynamic Simulator Test Flight (Test 2) During Periods of Low Dynamics (Z-12TYreceiver) Velocity Component RMS Ewor &n/s) 3-D RMS Method of Velocity Estimation North East UP GXnL?J Raw Doppler FLYKIN” fixed 0.45 0.40 1.44 1.68 1st order 0.07 0.06 0.26 0.29 Vol. 44, No. 2 Cannon, et al.: Kinematic Carrier Phase Signal Simulation 237 MiLLennium’“, respectively). The results of the the MiLLennium’” and Z-12’” receivers, the MiLLennium’” are better because of the larger MiLLennium’” test was reprocessed without PRN number of satellites processed. The following 15. It should be noted, however, that there is very section compares the accuracy of the velocity little noticeable difference in the accuracies of the estimates after processing the MiLLennium” data velocity estimates between the MiLLennium’” without PRN 15 in order to make the comparison results processed with and without PRN 15. compatible. Figure 5 shows the number of satellites processed with FLYKIN’” excluding PRN 15 for High Dynamics (> 2 g) Test 1 and for Test 2. This figure shows that during the 4 and 5 g turns, the Z-12’” lost lock and Periods of high dynamics for the same simulated dropped down to 4 and 3 satellites (note that test flight are now considered. Table 8 shows the FLYKIN’” requires a minimum of 4 satellites to maximum errors in velocity during the turns of process at an epoch), whereas the MiLLennium” Loop 2 corresponding to a 3 g acceleration. It was was able to maintain phase lock on all satellites not possible to show the maximum velocity errors through these high dynamics. for the higher-acceleration turns (4 and 5 g) Table 9 shows a comparison of the statistics for because of the Z-12’” losing lock at these levels of the Z-12’” and MiLLennium’” tests without acceleration. Note again that one gets only a PRN 15 during periods of low dynamics. The low- general idea of the relative accuracies of the dynamics results based on the Z-12” raw Doppler velocity estimates from Table 8. It is important to were about 5 to 6 times better than those of the consider the errors more closely during these high- MiLLennium’“. The first-order central difference acceleration periods, and this is discussed in a approximation yielded results that also favor the subsequent section. Z-12’” by about 30 percent, the 3-D RMS errors At high accelerations (in this case 3 g), the being 2.9 mm/s for the Z-12” and 3.8 mm/s for the velocity estimates are severely degraded in the MiLLennium” . case of the raw Doppler as compared with the low- Table 10 shows that during 3 g accelerations, dynamics case presented in Table 7. It is evident the MiLLennium” outperformed the Z-12” when from Table 8 that the velocity estimates during the using the raw Doppler measurements. The results 3 g turns are best for those corresponding to the were generally equivalent for the carrier-phase- raw Doppler measurements, yielding maximum derived Doppler based on the first-order central north and east velocity errors of 2.3 m/s and difference approximation, the Z-12’” slightly 0.8 m/s, respectively. The first-order central outperforming the MiLLennium’” in the north difference approximation performed about 1.5 to component. 2 times worse than the raw Doppler. There is a correlation between the magnitude of DETAILED ANALYSIS OF GPS VELOCITY acceleration and the magnitude of the velocity ERRORS DURING PERIODS OF HIGH DYNAMICS error when considering all turns of the trajectory. In general, the greater the magnitude of The results presented in Tables 6, 8, and 10 acceleration, the less accurate the velocity indicate that for periods of high dynamics (> 2 g), estimates become. More discussion of this there is no improvement from using the carrier- correlation is given in a subsequent section. phase-derived Doppler based on either of the approximation techniques over using the raw Comparison of Results for Test 7 and Test 2 Doppler measurement. However, it is necessary to During the two tests, the numbers of satellites being tracked by the aircraft were the same. Nunber of Satellites Processed (TEST 1) - MiLLerhun [Ezcrding PFtN 151 However, the signal from PRN 15 was not tracked &, by the fixed reference station of the Z-12’” in -5 Test 2. For this reason and to have a valid q II- 93 “““~~~“““““‘~~““‘~~~‘~~‘,,,/~I comparison of performance and results between - 95000 96000 97000 96000 99OOc Nunber of Satellites Processed CIESTP) - 2-12 Table 8-Summary of Statistics- Dynamic Simulator Test Flight (Test 2) During 3 g Turns (Z-12” receiver) Maximum Velocity Component Error (cm/s) 3 Method of Velocity Estimation North East Raw Doppler FLYKIN’” fixed 230 80 Fig. B-Number of Satellites Processed in FLYKIN’” Test 1 1st order 330 150 Without PRN 15 vs. Test 2 238 Navigation Summer 1997 Table 9-Comparison of Statistics-Test 1 Without PRN 15 vs. Test 2 During Periods of Low Dynamics Velocity Component Error bn/s) MiLLennium’“/Z-12” 3-D RMSk.%) Method of Velocity Estimation North East UP Mill’“/Z-12” Raw Doppler 23310.45 2.3810.40 8.0611.44 9.5X.68 1st order O.ll/O.O7 0.09lO.06 0.3210.26 0.3810.29 Table lo--Comparison of Statistics-Test 1 Without PRN Velocity North Emx M. Acabrath North- 15 vs. Test 2 During 3 g Turns Maximum Velocity Component Error (cm/s) ~i,~~~, ,) MiLLennium”/Z-12TY Method of Velocity Estimation North East Raw DopplerFLYKIN’” fixed 401230 70180 1st order 3301330 15Ol150 9S410 98420 99430 9S440 95450 Vebcity East Error M. Accehtion East - analyze closely the dynamics of the situation. 0.4 , MiLLemiun: First 2 g Tun rs Dealing with simulator test data and having truth velocity information at each epoch as output from the STR2760 GPS Simulator System, it is possible to derive both acceleration and jerk similar to those presented earlier in Figure 3. The 0.14, I I I I, I, I1 I * I c-1 accelerations are not constant, and there is up to 96410 9S420 95430 96440 9S450 GPS Time (set) 30 m/s3jerk at the 5 g turns. Given knowledge of when the highest values of Fig. d-comparison of Raw Doppler Velocity Errors and acceleration and jerk occur, it is possible to Acceleration for First 2 g Turn MiLLennium” receiver) determine the correlation with the corresponding errors in velocity for both the MiLLennium” and Z-12’” receivers. A number of high g turns were Velocity Nodh Ermr M. Acceleration Notth - MiLLerdun: First 5 g Tun r6 chosen for this comparison to illustrate the -4 correlations discovered. This discussion is divided in terms of the two dynamic simulated tests conducted (Test 1 and Test 2). The analyses are based on the results obtained using raw Doppler measurements. Test 1 with the MiLLennium” was also analyzed in terms of the carrier-phase-derived 97945 97950 97955 Doppler based on the first-order central difference Vekxily East Enw va. Acceleration East - approximation. 0.4 Q 0.3 Test 1 High Dynamics 8 0.2 i Figure 6 shows the north and east velocity errors of the NovAtel MiLLennium” with respect to the truth accelerations for the first 2 g turn. It 97945 97950 97955 97980 is evident that there is a strong correlation GPS Time (sac) between the velocity errors and the magnitude of Fig. 7--Comparison of Raw Doppler Velocity Errors and acceleration. In fact, the relationship is nearly Acceleration for First 5 g Turn CMiLLennium” receiver) linear. Figure 7 shows the north and east velocity errors with respect to the truth accelerations for the first 5 g turn. As with the 2 g turn, there is a trajectory. No direct correlation was found strong correlation between the velocity errors and between the velocity errors using the the magnitude of acceleration, which is nearly MiLLennium” raw Doppler and the magnitude of linear. jerk, however. This correlation with acceleration, A similar near-linear relationship exists between as opposed to jerk, is likely due to the time delay the magnitudes of the velocity error and in the channel tracking loops. Channel tracking accelerations for each of the other turns in the loops are closed synchronous to the navigation Vol. 44, No. 2 Cannon, et a/.: Kinematic Carrier Phase Signal Simulation 239 bits, which are in turn asynchronous to the position error and the magnitude of jerk. Figure 9 measurement strobes. The resulting time lag can shows this negative correlation, which is evident be as much as 10 ms and will vary across in each component. The accuracy of the position channels. This time lag cannot be compensated for estimates for Test 1 was determined to be about in postprocessing. l-2 mm. Hence, for example, the latitude error of From Figures 6 and 7, one can derive the 8 mm illustrated in Figure 9 is determined to be general rule that a velocity error based on the raw strongly correlated with the magnitude of jerk in Doppler (expressed in meters/s) corresponds to the north component. about 10 percent of the magnitude of acceleration A float solution was obtained from SEMIKIN’” (expressed in g) for each of the north and east for Test 1 based on the raw Doppler. Figures 10 components for the MiLLennium” as follows: and 11 show a comparison of the fixed and float 6v(t) = 0.1 * A(t) (6) Error in LetWe - MiLLemiun: First 5 g Tun where 6v(t) is the velocity error in meters/s, and 0.01 , *North r40 A(t) is the magnitude of acceleration in g. Recall that the MiLLennium” bases the Ll raw Doppler measurement on the third-order C/A carrier PLL, capable of maintaining lock during periods of 97940 97945 97950 97955 97990 constant jerk of up to 4.5 g/s (45 m/s3). Because of Enur in Longitude- MiLLemiwn: First 5 g Tun this design, there is no problem in tracking through the jerk experienced during the simulated test flight. ;;~/,,~*+fii It is interesting to analyze the corresponding errors in the position domain to determine whether a similar trend exists between the I 97940 97945 97950 GPS Time (WC) 97955 97960 dynamics and the position errors. The portion of Fig. 9-Comparison of Errors in Latitude and Longitude to Jerk the trajectory analyzed is based on the Ll integer for First 5g Turn (MiLLennium” receiver) ambiguity carrier phase observables. Figure 8 shows the errors in position based on the raw I Vebcity NorthEmx First 2 g Tun I Doppler for the first 2 g turn illustrated in g:: Figure 6, together with the corresponding jerk. 0 Other than the small (5 mm) systematic bias ga.2 that exists in the latitude component, there is no -0.4 1 I ,I III, / ,, , , , / discernible correlation between the magnitude of 1 96415 99420 99425 99430 99435 96440 9S445 I jerk and the corresponding errors in position. Vebcity East Error:First 2 g Tvn There is also no correlation between position TO.2 - FLYKIN Fixed errors and acceleration (compared with Figure 6). zo.1 However, when analyzing the higher-dynamic g o- w.1. 1 I / ,I *1 17 ,, / , , , portion of the trajectory (5 g), some strong 99415 99420 99425 99430 99435 99440 96445 negative correlation is seen to appear between the GPS Time (set) Fig. IO-Comparison of Float and Fixed Solutions for First 2 g Turn, Test 1 (MiLLennium’” receiver) Error in Latitude-MiLLemiun: First 2 g Tun Vebcity NorthError: First 5 g Tun SEMIKIN Flat yjy:&,,,,,j,B ? I- 60.5 FLYKIN Fixed E 0: w.57 I I 1 1 1 1 1 1 99410 99420 99430 99440 99450 I 97940 97945 97950 97955 97990 Error in Longitude- MiLLemhn:Firet 2 g Turn Vebcity East Enur: First 5 g Twn ;o?/Cy~I;~,,,,,j$~ 99410 99420 99440 99450 97940 97950 L GPSW”(*eC) / GPS Time (sac) Fig. 8-Comparison of Errors in Latitude and Longitude to Jerk Fig. 11-Comparison of Float and Fixed Solutions for First 5g for First 2 g Turn OMiLLennium” receiver) Turn, Test 1 (MiLLennium” receiver) 240 Navigation Summer 1997 solutions for the first 2 g and 5 g turns, order central difference approximation. It is respectively. In addition to the velocity errors of evident from Figure 13 that there is now a strong the two solutions being of the same order of correlation between the velocity errors and the magnitude (the float solution performing slightly magnitude of jerk, as opposed to acceleration, as better than the fixed solution in the east seen previously. This correlation was present in all component), the correlation between velocity error other turns of the trajectory. There is no longer and acceleration is clearly evident. This is any direct correlation between the velocity errors advantageous since, given high dynamics when and the magnitude of acceleration. This fixing of the integer ambiguities may be very corresponds well with the results of the Z-12’“, as difficult, the float solution performs as well as or will be seen subsequently. better than the fixed solution in terms of velocity Also note the 1 s time offset between the truth estimates. data and the computed results. This is due simply A summary plot comparing the maximum raw to the method of computing the magnitudes of jerk Doppler velocity errors and accelerations with from the available truth velocity data as follows: respect to the high g turns for the MiLLennium” v(t) - v(t - 1) a(t) - a& - 1) (7j has been developed. Figure 12 illustrates the a(t) = j(t) = At ’ At strong linear relationship between the velocity errors and the magnitudes of acceleration, which where v, a, and j are velocity, acceleration, and is especially evident in the east component. jerk, respectively. This is of course a very quick Figure 13 shows a comparison of the same 2 g and simple approach for determining the values of turn illustrated in Figure 6, but now using the acceleration and jerk, and was used simply to see carrier-phase-derived Doppler based on the first- whether a correlation exists between velocity error and acceleration/jerk. Computing acceleration as a r function of velocity from equation (7) results in a Maximun Velocity North Error vs. Acceleration - MiLLenniun FLYKIN [Rmv mpplerj 0.5 s offset given a 1 Hz data rate. Computing jerk as a function of this computed acceleration from equation (7) yields an additional 0.5 s offset from the acceleration truth and a 1.0 s offset from the velocity truth, which is evident in Figure 13. 1 2 Absolute a3cceleration40flwn (g) 5 6 From Figure 13, one can derive the general rule Madmun Vebcity East Error ~8. Acceleration - MiLLerhum that a velocity error using the carrier-phase- FLYKIN [Instpd- RarvDopplsr] derived Doppler (expressed in meters/s) based on the first-order central difference approximation corresponds to about 20 percent of the magnitude of jerk (expressed in meters/s3) for each of the 1 2 5 6 north and east components for the MiLLennium’” Absolute kceleratior?of tun (g) J as follows: Fig. 12-Maximum Raw Doppler Velocity Errors vs. Maximum Wt) = 0.2 * J(t) (8) Acceleration with Respect to Each High g Turn (MiLLennium’” receiver) where 6v(t) is the velocity error in meters/s, and J(t) is the magnitude of jerk in meters/s3. Vebcity North Error vs. Jerk North - MiLLenniun Firstoiderappximdon: Fint 2 g Twn Test 2 High Dynamics Figure 14 shows the Z-12’” north and east raw f $+._._>,/!pi Doppler velocity errors compared with the truth jerk for the final 2 g turn of Test 2. It is evident 99415 99420 99425 99430 99435 99440 99445 that there is now a strong negative correlation between the velocity errors and the magnitude of L Vebcity East Error vs. Jerk East - MiLLenniun First ordg appuxidm: Fhst 2 g Tun jerk. In fact, the relationship is nearly linear. Figure 15 shows the north and east velocity errors g;~r,,T~,f~;~g compared with the truth jerk for the first 3 g turn. As with the 2 g turn, there is a strong nearly linear negative correlation between the velocity 99415 99420 99425 99430 99435 99440 99445 errors and the magnitude of jerk. A similar near- L GPS Time (set) linear relationship exists between the magnitudes Fig. 13-Comparison of Carrier-Phase-Derived Doppler Velocity Errors and Jerk for First 2 g Turn Based on First-Order of the velocity error and jerk for each of the other Approximation (MiLLenniumlM receiver) turns in the trajectory. Note, however, the Vol. 44, No. 2 Cannon, et al.: Kinematic Carrier Phase Signal Simulation VelocityNorthErmrvs. Jerk North-Z-12 turns for the Z-12” results using the raw Doppler has been developed. Figure 16 shows these relationships in both the north and east components. It can be seen from this figure that there is a nearly linear relationship between the S8S55 98980 98965 98S70 98S75 98980 98985 velocity errors and the magnitudes of jerk. Note that for the 4 and 5 g turns, the latest velocity error available prior to losing lock was used, resulting in the deviation from the linear relationship illustrated in the figure. A similar correlation between velocity errors based on the various carrier-phase-derived Doppler 98955 98990 98955 98970 SE975 98980 98985 GPS Time (set) measurements and the magnitude of jerk has been observed. Results similar to those of the Z-12” Fig. 14--Comparison of Raw Doppler Velocity Errors and Jerk have been reported in [51. The authors show a for Final 2 g Turn (Z-12’” receiver) comparison between the magnitude of jerk in the north component and the corresponding error in Vebcity Nolth Error vs. Jerk Notth-2-l 2 velocity north for a rocket sled test conducted at Holloman Air Force Base in February 1990. In FLYKIN [Paw Ck@arj: that case, errors in velocities from the Rockwell Collins 3A GPS receiver were also found to be a linear function of jerk. The authors give a 98935 98940 96945 S8S50 98955 preliminary model of this relationship as follows: Vebcity Enur vs. Jerk East - 2-12 I Sv(t) = -0.431J(t - 1) East (10) where Sv(t) is the velocity error in meters/s, J is the magnitude of jerk expressed in meters/s3, and t is in seconds 151. The relatively poorer results of the Z-12’” I 98935 98940 S8S45 GPS lime (set) SW50 S5s55 receiver as compared with the MiLLennium’” under high dynamics can likely be attributed to Fig. 15-Comparison of Raw Doppler Velocity Errors and Jerk the tracking loops used in the Z-12’“, although for First 3 g Turn (Z-12” receiver) further tests and analysis are required to verify this. The Z-12’” does not utilize a third-order PLL difficulty the Z-12’” had in tracking all satellites and is therefore not suitable for tracking under through the higher g turns (> 3 g). There is no high levels of jerk. The MiLLennium’” uses a direct correlation between the velocity errors using third-order PLL, which performs better in the Z-12” raw Doppler and the magnitude of environments of high levels of jerk. The order of acceleration. the PLL directly affects both tracking loop For the 2 g turn (see Figure 141, north and east stability and the resulting velocity/position velocity errors of up to 1.6 and 0.35 m/s, accuracy. This error in the Z-12’” tracking loop respectively, result at the periods of maximum jerk. During the interim period of constantly changing acceleration, a maximum north and east Wmun Velocity Nor(hError vs. Jerk Noti -Z-l2 FLYKiN [IIN- WDopplrl velocity error of about 35 cm/s results. Similar results occur for the other 2 and 3 g turns. From Figures 14 and 15, one can derive the general rule that a velocity error based on the raw Doppler (expressed in meters/s) corresponds to about 15 percent of the magnitude of jerk (expressed MscdmunV&city East Etmrw. Jerk East-Z-12 in meters/s31for each of the north and east FLYKIN (Im- RUVDDPM components for the Z-12’” as follows: Sv(t) = -0.15 * J(t) (9) where Sv(t) is the velocity error in meters/s, and J(t) is the magnitude of jerk in meters/s3. A summary plot comparing the maximum Fig. 16-Maximum Raw Doppler Velocity Errors vs. Maximum velocity errors and jerk with respect to the high g Jerk with Respect to Each High g Turn (Z-12” receiver) 242 Navigation Summer 1997 was recently reported by 1221.The authors note a NovAtel GPSCard” static survey over a 7 km significant tracking loop lag for accelerations and baseline yielded 3-D RMS velocity accuracies of jerk of 1.6 g and 0.8 g/s (8 m/s3), respectively, 0.28 mm/s based on carrier phase measurements experienced during a rocket sled test conducted at sampled at 50 Hz 1181. Holloman Air Force Base. Although the Z-12’” was Carrier phase lock stability is a function of the able to maintain lock during these levels of bandwidth and order of the phase tracking loops, dynamics, the double-difference phase residuals as well as dynamics. In the case of the were significantly large (as much as 17 cm) during MiLLennium”, the tracking status bits of the raw the most dynamic portion of the test. measurement log identify carrier phase loss of lock. Analysis showed that there was no loss of DISCUSSION lock for the Ll carrier phase for any portion of the trajectory, including the 5 g turns. There was only The simulated tests were conducted without the minimal loss of lock for the L2 carrier phase for effects of SA/AS, satellite clock errors, multipath, certain satellites, but these occurrences did not or ephemeris errors. Only the simulator correspond to any of the high dynamic portions of tropospheric and ionospheric models were the trajectory. However, this was not the case with employed. This was done to determine the velocity the Z-12’“. The Z-12” was able to maintain lock and position accuracy given the best-case scenario, on all satellites for dynamics up to and including without the complication of additional error the 2 g turns. It was able to maintain lock through sources. It is worth noting, however, how the first 3 g turn, but was unable to maintain lock individual error sources are expected to influence on all visible satellites during the second 3 g turn. the velocity results. FLYKIN’” uses the receiver- It was unable to maintain lock on all satellites for satellite double-difference observable for precise each of the 4 and 5 g turns of the trajectory. It is differential GPS positioning with the carrier not known whether this loss of lock is directly phase. Since double differences are used, a number attributable to the magnitude of acceleration or to of error sources are completely eliminated or jerk. substantially reduced (provided baseline distances are less than about 15 km). By taking double CONCLUSIONS AND FUTURE PROSPECTS differences, receiver and satellite clock errors are eliminated. Orbital and atmospheric errors are It is evident from the analysis presented here reduced. In this investigation, therefore, there that velocity errors are a function of the dynamics should theoretically have been no variation in the of the mobile antenna. In the case of the velocity estimates had SA been employed. This MiLLennium’” based on the raw Doppler, velocity would also be the case for the satellite clock and errors were found to be a near-linear function of ephemeris errors, as long as baselines remained acceleration. Applying a carrier-phase-derived shorter than about 15 km. AS should not affect Doppler based on a first-order central difference velocity accuracies at all since the L2 carrier phase approximation, the velocity error became a linear observable is used only for faster ambiguity function of jerk. During high dynamics (5 g>, resolution within FLYKIN’“. Once ambiguities velocity errors using the raw Doppler were about have been fixed, position and velocity estimates 20 to 60 cm/s. In the case of the Z-12’“, velocity are based on the less noisy Ll carrier phase errors were found to be a linear function of jerk. observable only. AS would only increase the This is consistent with findings reported elsewhere amount of time to integer ambiguity resolution. 151.During high dynamics (3 g), velocity errors Multipath, however, would affect velocity using the raw Doppler were on the order of 35 cm/s accuracies as these errors do not cancel in to 1.6 m/s. Comparison of the MiLLennium’” fixed differential processing. Therefore, there should be and float solutions shows little difference between a degradation in the velocity accuracy given a real these two approaches in terms of the accuracy of multipath environment. the velocity estimates given both low and high Although not reported in this paper, static dynamics. baseline tests have been conducted with both sets It is evident that the relationship between of GPS receivers (2 Hz data rate). In general, the velocity error and dynamics is a function of accuracies of the velocity estimates based on the receiver design. Velocity errors based on the carrier-phase-derived Doppler were found to be of Doppler measurements, in addition to being a the same order of magnitude as those of the function of dynamics, will vary based on the order simulations during low dynamics (i.e., 3-D RMS and noise bandwidth of the tracking loop, the velocity error between 2.5 and 6.0 mm/s>, but were update rate at which the raw Doppler is slightly worse because of the effects of the high estimated, and the internal oscillator used. Since multipath environment. It has been reported that there are strong correlations between the Vol. 44, No. 2 Cannon, et al.: Kinematic Carrier Phase Signal Simulation 243 magnitudes of acceleration and jerk and the errors REFERENCES in velocity during periods of high dynamics, it should be possible to reduce significantly the 1. May, M., Weiss, J., and Haiges, J., Testing velocity errors in postmission processing. 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