DGPS Kinematic Carrier Phase Signal Simulation Analysis for by nikeborome

VIEWS: 16 PAGES: 15

									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. This                  Differential GPS Using Satellite Signal Generators,
                                                                 Proceedingsof ION GPS-94, Salt Lake City, UT,
would require additional simulations in order to
                                                                 September 1994.
determine a statistically significant relationship
                                                              2. Owen, T. E. and Wardlaw, R., Evaluating the
between dynamics and velocity error as a function                Velocity Accuracy of an Integrated GPS/INS System:
of each type of receiver tested.                                 Flight Test Results, Proceedings of The Institute of
   To analyze the effect of acceleration and jerk                Navigation National Technical Meeting, San Diego,
and the performance of the tracking loops, a series              CA, January 1992.
of straight-line flight trajectory accelerations could        3. Sun, H., Cannon, M. E., Owen, T., and Meindl, M.,
be performed. In this way, the acceleration and                  An Investigation of Airborne GPSI INS for High
jerk could be increased incrementally, each period               Accuracy Position and Velocity Determination,
being long enough (5 to 10 s) to analyze the                     Proceedings of The Institute of Navigation National
subsequent effect on the velocity determination,                 Technical Meeting, San Diego, CA, January 1994.
                                                              4. Crouch, D., Mosle, C., and Novy, M., The 746th Test
then decreased incrementally. The length of the
                                                                 Squadron: An All-Inclusive GPS Test and
dataset should be sufficient to derive a statistically
                                                                 Evaluation Facility, Proceedings of ION GPS-95,
correct and predictable relationship between                     Palm Springs, CA, September 1995.
velocity error and acceleration/jerk.                         5. May, M., Nguyen, K., and Tanju, B., On GPS
   Measurement domain errors could also be                       Velocity, Proceedings of The Institute of Navigation
 considered as a function of dynamics. Since the                 46th Annual Meeting, Atlantic City, NJ, June 1990.
 algorithms presented in this paper apply to the              6. Cannon, M. E. and Lachapelle, G., Analysis of a
 carrier phase, analysis should be performed                     High-Performance CIA-Code GPS Receiver in
 considering the true rate of change as governed by              Kinematic Mode, NAVIGATION, Journal of The
 satellite/receiver position and velocity truth data,            Institute of Navigation, Vol. 39, No. 3, Fall 1992,
 and the carrier phase measurement and its                       pp. 285-299.
                                                              7. Lachapelle, G., Cannon, M. E., and Lu, G., A
 subsequent estimated rate of change using the
                                                                 Comparison of P Code and High Performance CIA
 carrier-phase-derived Doppler. For this analysis,
                                                                 Code GPS Receivers for on the Fly Ambiguity
 position and velocity truth data for the individual             Resolution, Bulletin Geodesique, 67:185-192, 1993.
 satellites of the simulation would be required. The          8. Northern Telecom, STR2760 Specifications Sheet,
 ephemeris, ionospheric, tropospheric, and                       Northern Telecom Inc., Schaummberg, IL, 1996.
 multipath model error source parameters need not             9. Gourevitch, S., Implications of 3” Technology for
 be incorporated. In this way, attention could be                Civilian Positioning, Proceedings of ION GPS-94,
 focused solely on the performance of the tracking               Salt Lake City, UT, September 1994.
 loops as a function of dynamics, with no other              10. Ashtech, Private communication, Ashtech Inc.,
 effects entering into the analysis. Each error                   Sunnyvale, CA, 1996.
 source could then be applied incrementally to               11. NovAtel, MiLLenniumm GPSCard’” Guide to
                                                                  installation & Operation, OM-20000016, Rev. 1,
 determine the effect of each on the velocity
                                                                  NovAtel Communications Ltd., 1996.
 estimates.
                                                             12. Newby, S., Fenton, P., Neumann, J., and
                                                                  Townsend, B., Core Technology Developments and
                                                                  End-User Products at NouAtel, Proceedings of ION
ACKNOWLEDGMENTS
                                                                  GPS-96, Kansas City, MO, September 1996.
                                                             13. Townsend, B., Private communication, NovAtel
  The authors wish to acknowledge the advice
                                                                  Communications Ltd., Calgary, AB, 1996.
provided by Capt. J. Raquet and Mr. H. Sun,
                                                             14. NovAtel, GPSCard” Command Descriptions
Ph.D. candidates in the Department of Geomatics                   Manual, OM-20000008, Rev. 2.0, NovAtel
Engineering, The University of Calgary. In                        Communications Ltd., 1995.
addition, Mr. B. Townsend of NovAtel                         15. Lachapelle, G., Cannon, M. E., and Lu, G., High-
Communications Ltd., Calgary, and the technical                   Precision Navigation with Emphasis on Carrier-
services staff of Ashtech Inc., Sunnyvale, are                    Phase Ambiguity Resolution, Marine Geodesy,
acknowledged for their interpretations of the                     Vol. 15, 1992, pp. 235-269.
specifications of the NovAtel MiLLennium”                    16. Chen, D., Fast Ambiguity Search Filter (FASF):
GPSCard” and Ashtech Z-12’” GPS receivers,                        A Novel Concept for GPS Ambiguity Resolution,
respectively.                                                     Proceedings of ION GPS-93, Salt Lake City, UT,
                                                                  September 1993.
                                                             17. Cannon, M. E., High Accuracy GPS Semikinematic
Based on a paper presentedat The Institute of                     Positioning: Modeling and Results, NAVIGATION,
Navigation National Technical Meeting, Santa Monica,              Journal of The Institute of Navigation, Vol. 37,
California, January 1997.                                         No. 1, Spring 1990, pp. 53-64.

244                                             Navigation                                             Summer 1997
18. Fenton, P. C. and Townsend, B., NouAteZ                  21. Cannon, M. E., GPS Theory and Applications,
    Communications Ltd.-     What’s New?, Proceedings of         ENGO 625 Lecture Notes, Department of Geomatics
    KIS94, Calgary, AB, 1994.                                    Engineering, The University of Calgary, Fall 1996.
19. Hebert, J. M., Velocity Determination for an Znverted    22. Evans, A. G., Hermann, B. R., Remondi, B. W.,
    Pseudolite Navigation Reference System, MS thesis,           Simpson, P. B., Feist, J. L., and Wiles, G. C., An
    AFIT/GE/ENG/95D-06, School of Engineering, Air               Evaluation of Precise Kinematic on-the-Fly Relative
    Force Institute of Technology (AU), WPAFB, OH,               Positioning for a Rocket Sled Test, Proceedings of
    December 1995.                                               The Institute of Navigation 52nd Annual Meeting,
20. Cheney, W. and Rincaid, D., Numerical                        Cambridge, MA, June 1996.
    Mathematics and Computing, 2nd Edition, Brooks/
    Cole Publishing: Pacific Grove, CA, 1985.




Vol. 44, No. 2                     Cannon, et al.: Kinematic Carrier Phase Signal Simulation                     245

								
To top