POLÍTICAS DE GASTO PÚBLICO

A T 0 2 / Antenna AT03 ' Antenna * AT04 > X O B F S Antenna ATOl

C hapter Five: Investigation into the Perform ance o f GPS A ttitude D eterm ination

The relationship between the GPS antenna array and the body frame is illustrated in Figure 5.9. The phase centre of GPS antenna ATOl was chosen as the origin of the body frame, o. The direction of ATOl to AT02 was chosen as the body frame y-axis. The body frame x-axis lies in the plane defined by the antennas and pointing right of the y-axis. The body frame z-axis then forms a right-handed system with the x-axis and y-axis. The body frame coordinates and the approximate attitude were determined by using the same process as described in 5.4.1. The resulting body frame coordinates are shown in Table 5.13.

Table 5.13: Antenna body frame coordinates using BFS 1234

Antenna x(m ) y (m) z(m )

ATOl 0.0000 0.0000 0.0000

AT02 0.0000 39.9322 0.4589

AT03 -9.8457 18.3205 0.2622

AT04 9.8987 21.4946 0.2805

5.4.2.1

Ambiguity Function Values

The search size chosen for this test is ten (on either side of the initial attitude) which leads to a total number of trial solutions of 9261. Using the GRAPE software, the ambiguity function values (and hence attitude parameters) were determined independently for each epoch. Figure 5.10 shows the ambiguity function values for GPS time 215500 seconds. It is evident that the true attitude, whose ambiguity function value is nearly equal to unity is dominant over the other trial attitudes.

Chapter Five: Investigation into the Perform ance o f GPS A ttitude D eterm ination

mo ' 0 9-J‘S6 m e b e r i e s Plot BPS I 2 3 4 mln —0 5 126

1000 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0 7 0 0 0 8 0 0 0 9 0 0 0

Trial Attitude 2 1 5 5 0 0 AFV GPS t i m e 2 1 5 5 0 0 s

Figure 5.10: Values of ambiguity function for GPS time 215500s derived from BFS 1234

SA,2 2 Pitch, Roll and Yaw Results

Figures 5.11, 5.12 and 5.13 respectively illustrate the pitch and roll in arc minutes, and yaw in degrees, derived using the GRAPE software. The following conclusions can be drawn from these results :

(a) Figure 5.11 shows that the maximum pitch is 1.191 arc minutes and the

minimum pitch is -1.058 arc minutes; Figure 5.12 shows that the maximum roll is 2.423 arc minutes and the minimum roll is -2.081 arc minutes; and Figure 5.13 shows that the maximum yaw is -22.295 degrees and the minimum yaw is -22.310 degrees. The mean value of pitch, roll and yaw are 0.042 arc minutes, 0.250 arc minutes and -22.303 degrees respectively. The figures show that the spread in pitch and roll is about 1 arc minutes which is equivalent to 0.58 cm for a baseline length of 20 metres whereas the spread in yaw is about 0.3 arc minutes (0.005 degrees) which is equivalent to 0.17 cm for the same baseline

Chapter Five: Investigation into the Perform ance o f GPS A ttitude D eterm ination

B F S 1 2 3-4- — 1 . 0 5 > S '

5CJ 0 . 3 3 8 S a I O : 3 :0 0 —

Figure 5.11: GPS-derived pitch in arc minutes (BFS 1234)

Figure 5.12: GPS-derived roll in arc minutes (BFS 1234)

— 22.293-4 c J e g R i o t B F S 1 2 3-^ — 22.31 O 1 dog 1OOO 2 0 0 0 0 .0 0 2 1 deg Epoci- -4000 5 0 0 0 I O : -4. 3 :0 0 — 1 2 : -4 5 : OO ÔOOO "7000 — 22.302*7 Cleg

C hapter Five: Investigation into the P erform an ce o f G PS A ttitu de D eterm ination

length. The standard deviation of the pitch, roll and yaw are 0.339 arc minutes, 0.601 arc minutes and 0.126 arc minutes respectively. The reason the yaw component is more accurate is that unlike yaw, the pitch and roll components are directly affected by the height and the height component remains the least well known in all GPS applications.

(b) As can be seen from section (a), the pitch, roll and yaw values are contained within a small range which suggests that the correct ambiguities have been resolved for every epoch over the whole period. By compiling for each epoch the total number of trial attitudes (pA) leading to ambiguity functions greater than 90% of the maximum value of the ambiguity function and comparing it with the total number that satisfy the F-test (pF), it was found that for every epoch both pA and pF are unity, and the ambiguity function value is greater than 0.95. This suggests that the ambiguities have been resolved but unfortunately, this does not guarantee that the correct ambiguities have been resolved.

A more convincing approach is by taking the mean of the attitude parameters for the whole epoch, substituting these values as the only trial attitude and then re­ processing the data sets again. It was found that the differences between these new attitude values and the old ones are very close to zero, thus confirming that the correct ambiguities have been resolved.

Table 5.14 summarises the ambiguity resolution results. The table shows that in this experiment there is a 100% success rate with which the integer ambiguities could be correctly resolved.

Chapter Five: Investigation into the Perform ance o f GPS Attitude D eterm ination

Table 5.14: Ambiguity resolution results for BFS 1234

Number Percentage

Number of Epochs 7321

Successful Ambiguity Resolution 7321 100%

Wrong Ambiguity Resolution 0

Ambiguity Resolution Failed

(c) Figures 5.11, 5.12 and 5.13 indicate that there are attitude differences caused by the changes in the satellite geometry. This can clearly be seen by comparing Figures 5.11, 5.12 and 5.13 with Figure 5.3; a sudden jump in the attitude represents a change in the satellite geometry. As can be seen from Table 5.4, seven such incidents occurred as a result of the satellite elevation angle cut-off which in this case was 20 degrees. Therefore if the elevation angle of a satellite is less than 20 degrees, it will not be used to compute the attitude. Note that the differencing satellite is that satellite with the highest elevation angle. In this case it was SV30 from the first epoch until epoch 4864 and then replaced by SV6 until the end.

(d) Figures 5.11, 5.12 and 5.13 show low frequency patterns occurring in all three components. Flowever, since the effects of this noise is of such small magnitude, it is certain that this has been caused by multipathing at one, or all of the antennas. The errors are mostly random and can be partly attributed to the receiver noise as the baseline distances between receivers are so small (Barnes et al, 1998).

C hapter Five; Investigation into the P erform an ce o f G PS A ttitu de D eterm ination

5.4.2.3

Double Difference Phase Residuals Results

The LI and L2 double difference phase residuals graphed in Figures A. 13 through to A. 18 are obtained from the GRAPE processing of a direct attitude system whereas Figures A. 19 through to A.24 are obtained from the GASP processing of a single baseline system. By comparing the graphs in Figures A. 13 through to A. 18 with its corresponding graphs in Figures A. 19 through to A.24, it can be seen that all the graphs have similar shape and pattern. The figures show random low frequency patterns of small magnitude, which is most likely to be due to multipathing at one, or all of the antennas along with receiver noise.

Tables 5.15 through to 5.20 summarise the LI and L2 double difference phase residuals for baselines AT01-AT02, AT02-AT03 and AT03-AT04 as derived from a direct attitude system and a single baseline system. Note that SV30 is the differencing satellite from the first epoch through to epoch 4864 and then is replaced by SV6 for the remaining epoch.

The largest maximum value of LI double difference phase residuals derived from a direct attitude system is 0.011 m (at SV24 for baseline AT01-AT02) and for a single baseline system is 0.012 m (at SV24 for baseline AT01-AT02) whereas the largest maximum value of L2 double difference phase residuals derived from a direct attitude system is 0.016 m (at SV24 for baseline AT02-AT03) and for a single baseline system is 0.017 m (at SV24 for baseline AT02-AT03).

The smallest minimum value of LI double difference phase residuals derived from a direct attitude system is -0.012 m (at SV24 for baseline AT02-AT03) and for a single baseline system is -0.012 m (at SV4 for baseline AT02-AT03) whereas the smallest minimum value of L2 double difference phase residuals derived from a direct attitude system is -0.017 m (at SV24 for baseline AT01-AT02) and for a single baseline system is -0.016 m (at SV4 for baseline AT01-AT02).

C hapter Five: Investigation into the Perform ance o f GPS Attitude Determ ination

The maximum standard deviation of LI double difference phase residuals derived from a direct attitude system is 0.004 m (at SV24 for baseline AT01-AT02) and for a single baseline system is 0.004 m (at SV24 for baseline AT01-AT02) whereas the maximum standard deviation of L2 double difference phase residuals derived from a direct attitude system is 0.006 m (at SV4 for baseline AT01-AT02) and for a single baseline system is 0.005 m (at SV4 for baseline AT01-AT02). The maximum difference in standard deviation between a direct attitude and a single baseline results is less than 2 mm.

Table 5.15: LI double difference phase residuals for baseline AT01-AT02 derived from a direct attitude system (BFS 1234) and a single baseline system

Satellite Direct Attitude System - Single Baseline System ,

(SV) Max (m) Min (m)^ Std Dev (m) Mak (m)"" Min (m) Std Dev (m)

1 0.0080 -0.0083 0.0026 0.0080 -0.0082 0.0026 4 0.0074 -0.0068 0.0019 0.0063 -0.0059 0.0016 5 0.0084 -0.0071 0.0025 0.0086 -0.0075 0.0024 6 0.0072 -0.0077 0.0025 0.0077 -0.0059 0.0022 9 0.0080 -0.0036 0.0019 0.0085 -0.0042 0.0017 24 0.0106 -0.0105 0.0036 0.0119 -0.0079 0.0040 25 0.0096 -0.0060 0.0020 0.0092 -0.0059 0.0022 29 0.0043 -0.0064 0.0014 0.0037 -0.0070 0.0016 30 0.0071 -0.0039 0.0015 0.0067 -0.0038 0.0014

C hapter Five: Investigation into the P erform ance o f GPS A ttitude D eterm ination

Table 5.16: L2 double difference phase residuals for baseline AT01-AT02 derived from a direct attitude system (BFS 1234) and a single baseline system

Satellite (SV)

Direct Attitude System Single Baseline System

Max (m) Min (m) Std Dev (m) Max (m) Min (m) Std Dev (m)

1 0.0121 -0.0099 0.0039 0.0097 -0.0101 0.0036 4 0.0113 -0.0166 0.0058 0.0094 -0.0155 0.0050 5 0.0072 -0.0125 0.0028 0.0078 -0.0094 0.0027 6 0.0085 -0.0125 0.0039 0.0102 -0.0115 0.0038 9 0.0083 -0.0134 0.0044 0.0083 -0.0121 0.0043 24 0.0142 -0.0167 0.0055 0.0105 -0.0136 0.0047 25 0.0082 -0.0082 0.0024 0.0073 -0.0076 0.0021 29 0.0101 -0.0072 0.0025 0.0087 -0.0049 0.0019 30 0.0078 -0.0068 0.0025 0.0075 -0.0046 0.0019

Table 5.17: LI double difference phase residuals for baseline AT02-AT03 derived from a direct attitude system (BFS 1234) and a single baseline system

Satellite Direct Attitude System Single Baseline Systeiri - f - *

(SV) Max (m) Min (m) Std Dev (m) Max (m) Min (m) Std Dev (m)

1 0.0057 -0.0056 0.0018 0.0056 -0.0066 0.0018 4 0.0056 -0.0110 0.0027 0.0039 -0.0115 0.0022 5 0.0086 -0.0106 0.0024 0.0066 -0.0083 0.0022 6 0.0061 -0.0062 0.0018 0.0063 -0.0055 0.0017 9 0.0055 -0.0095 0.0022 0.0063 -0.0087 0.0025 24 0.0070 -0.0122 0.0028 0.0077 -0.0104 0.0029 25 0.0068 -0.0081 0.0019 0.0058 -0.0071 0.0018 29 0.0087 -0.0068 0.0029 0.0087 -0.0058 0.0028 30 0.0048 -0.0052 0.0015 0.0048 -0.0057 0.0017

Chapter Five: Investigation into the Perform ance o f GPS A ttitude D eterm ination

Table 5.18: L2 double difference phase residuals for baseline AT02-AT03 derived from a direct attitude system (BFS 1234) and a single baseline system

Satellite (SV)

Direct Attitude System Single Baseline System

Max (m) Min (m) Std Dev (m) Max (m) Min (m) Std Dev (m)

1 0.0146 -0.0102 0.0041 0.0115 -0.0077 0.0036 4 0.0142 -0.0015 0.0025 0.0149 -0.0027 0.0033 5 0.0169 -0.0096 0.0033 0.0136 -0.0102 0.0032 6 0.0089 -0.0059 0.0026 0.0087 -0.0060 0.0025 9 0.0110 -0.0095 0.0037 0.0108 -0.0073 0.0033 24 0.0160 -0.0118 0.0051 0.0165 -0.0104 0.0044 25 0.0113 -0.0133 0.0032 0.0096 -0.0108 0.0032 29 0.0102 -0.0122 0.0041 0.0077 -0.0108 0.0035 30 0.0070 -0.0059 0.0024 0.0066 -0.0058 0.0019

Table 5.19: LI double difference phase residuals for baseline AT03-AT04 derived from a direct attitude system (BFS 1234) and a single baseline system

Satellite Direct Attitude System# # , -Single, Baseline System, s

' k A W L - ' * ^

(SV) Max (m) Min (m) Std Dev (m) Max (m) Min (m)

■ :: y. Std Dey- (m) 1 0.0068 -0.0080 0.0021 0.0059 -0.0071 0.0021 4 0.0052 -0.0090 0.0021 0.0061 -0.0066 0.0018 5 0.0104 -0.0082 0.0021 0.0096 -0.0082 0.0019 6 0.0079 -0.0054 0.0018 0.0052 . -0.0067 0.0019 9 0.0074 -0.0059 0.0025 0.0061 -0.0053 0.0020 24 0.0061 -0.0081 0.0021 0.0048 -0.0066 0.0016 25 0.0077 -0.0069 0.0022 0.0069 -0.0058 0.0019 29 0.0064 -0.0065 0.0019 0.0044 -0.0048 0.0013 30 0.0057 -0.0058 0.0021 0.0060 -0.0056 0.0019

C hapter Five: Investigation into the Perform ance o f GPS A ttitu de Determ ination

Table 5.20: L2 double difference phase residuals for baseline AT03-AT04 derived from a direct attitude system (BFS 1234) and a single baseline system

Satellite Direct Attitiide System

■■ .V: ■- . .... . ■

' : Single Baseline System

(SV) Max (m) Min (m)'" Std Dev (m)

V . <0: ' Max^m) A n W j 'S td Dev (m) 1 0.0096 -0.0137 0.0040 0.0081 -0.0117 0.0034 4 0.0079 -0.0120 0.0038 0.0061 -0.0104 0.0030 5 0.0117 -0.0113 0.0029 0.0111 -0.0099 0.0026 6 0.0078 -0.0095 0.0025 0.0077 -0.0079 0.0023 9 0.0059 -0.0094 0.0025 0.0041 -0.0094 0.0021 24 0.0096 -0.0080 0.0025 0.0099 -0.0074 0.0021 25 0.0074 -0.0086 0.0025 0.0062 -0.0079 0.0024 29 0.0089 -0.0059 0.0022 0.0067 -0.0047 0.0019 30 0.0051 -0.0055 0.0018 0.0048 -0.0064 0.0019

5.4.2 4

Summary of Results

The results of the test carried out on the body frame system BFS 1234 confirmed that the correct ambiguities, and hence attitudes, have been resolved for the complete data sets. It is evident that the yaw component is more accurate than the pitch and roll components due to the fact that unlike yaw, the pitch and roll components are directly affected by the height. The double difference phase residuals obtained from the GRAPE processing of a direct attitude system are compared with its corresponding residuals obtained from the GASP processing of a single baseline system and the finding shows that the results are effectively equivalent.

C hapter F ive: Investigation into th e P erform an ce o f GPS A ttitu de D eterm ination

5.5

Concluding Remarks

An investigation has been described to test the performance of the GRAPE software. This investigation involved processing the GPS data twice using different types of body frame systems to show that the direct attitude system works whichever way the body frames are defined. The results of the investigation in terms of the ambiguity function values, ambiguity resolution, attitude values, and LI and L2 double difference phase residuals were analysed and discussed. The results of the investigation have shown that complete success may be expected for GPS attitude determination. The following conclusions can be drawn from the investigation :

(a) The true attitude, whose ambiguity function value is nearly equal to unity is dominant over the trial attitudes.

(b) The results of the investigation show that the correct ambiguities and hence

attitudes have been resolved for every epoch over the whole test period.

(c) It was found that the yaw component is more accurate than the pitch and roll components. This is due to the fact that unlike yaw, the pitch and roll components are directly affected by the height and the height component remains the least well known in all GPS applications.

(d) A comparison of the GRAPE processing of a direct attitude system and the

GASP processing of a single baseline system in terms of the double difference phase residuals show that they are effectively equivalent. These results demonstrate that GPS attitude determination using the GRAPE software has been successful.

CHAPTER SIX

INVESTIGATION INTO THE

PERFORMANCE OF GPS AMBIGUITY RESOLUTION