number of iterations is increased further). When
LS
and
LM
are
used in the
BA
algorithms, both algorithms converge to nearly
the
MSE
of the reprojections. Although
LM
can handle the
singularity of XYZ parametrization by introducing a damp-
ing factor, the number of iterations required in
BA
+
LM
is very
large, and the final
MSE
after 16 iterations is still larger than
that of parallaxBA obtained with only 7 iterations.
From the fifth column and sixth column in Table 8, the
parameters (
Δ
X
,
Δ
Y
,
Δ
Z
,
φ
,
ω
,
κ
) in
F
C
based on
WTLS
are much
closer to the true data than are those based on
TLS
, indicating
that the weight matrix of the tie points with the different dis-
tance is a better choice for the visual localization. In the ninth
row, the errors
σ
0
based on
TLS
and
WTLS
are 3.4 mm and 3.1
mm, respectively, so the accuracy of the rover pose estimation
based on
WTLS
is higher than that for
TLS
.
From the second row, third row and the fourth row, the
absolute error values of the rover pose estimation
∆
∆
∆
∆
∆
∆
X X Y Y Z Z
true
true
true
− ′
(
)
+
− ′
(
)
+
− ′
(
)
2
2
2
are 0.33 m,
0.32 m, 0.33 m, 0.30 m, and 0.25 m. The symbols
Δ
X
′
,
Δ
Y
′
,
Δ
Z
′
, represent the rover pose estimation based on the above
five algorithms. The symbols
Δ
X
true
,
Δ
Y
true
,
Δ
Z
true
, correspond
to the true data of the rover poses. In this experiment, the
result indicates that
WTLS
has better performance than
BA
+
LS
,
parallaxBA,
BA
+
LM
and
TLS
in the rover pose estimation.
The scale change parameter
ρ
can be regarded as one of
the indexes to evaluate the localization accuracy of the rover.
When
WTLS
is applied to the rover’s localization problem,
the higher the localization accuracy is, the closer the scale
will approach 1.000, which indicates that the corresponding
algorithm is more suitable for pose estimation. From the fifth
row in Table 8, the weight matrix of the coordinate observa-
tions of the tie points can improve the rover’s pose estima-
tion precision. However, there are some examples in which
the calculated values of scale change parameter
ρ
deviate
from 1.000 (such as 1.410, 1.500) when false matching occurs
due to brightness, contrast, scale, noise, or rotation differ-
ence between the images from adjacent stations or when the
distribution of the tie points is bad. In this case, we can set
the threshold value (1.050) of the scale change parameter to
distinguish whether the chosen tie points meet the conditions
of high accuracy and good distribution.
In general, a variety comparison test and analysis between
WTLS
and
BA
in terms of accuracy, efficiency and convergence
should be performed. The stereo images of the lunar rover
cameras are obtained in twelve positions, and the yaw angles
of the rover are very large (the eighth row in Table 8). The
results are listed in Table 9.
In Table 9, the “Number” column represents the numbers
of the previous and current stations, the “Relative distance”
column represents the 3D Euclidean distance of the lunar
rover in the two adjacent stations, the “Absolute localiza-
tion accuracy” column represents the 3D Euclidean distance
between the calculated position in the current station and the
measurement positions in
F
C
, and the “Relative localization
accuracy” column represents the ratio of the absolute local-
ization accuracy to the relative distance (Table 10). The five
algorithms use the same data, including the matching image
points and the calibration parameters of the rover’s cameras.
The comparison results are as follows:
1. Accuracy
. The mean localization accuracy of the lunar
rover based on
WTLS
is equal to that of parallaxBA or
TLS
and much higher than
BA
+
LS
and
BA
+
LM
. From the fourth
column, the mean relative localization accuracies based
on
WTLS
,
TLS
,
BA
+
LS
,
BA
+
LM
, and parallaxBA are 4.59%,
4.65%, 5.68%, 5.49%, are 4.56%, respectively. Although
the mean accuracy of
WTLS
is only slightly lower than
that of parallaxBA, the differences in their accuracies are
negligible, and both position parameters can be effectively
supported in the rover’s navigation.
In “C3-C4,” the reason for the low pose estimation accuracy is
that the distribution of the six tie points is relatively central-
ized. In cases “C2-C3,” “C4-C5,” “D1-D2,” “D2-D3,” “D3-D4,”
“D4-D5,” “D5-D6” and “C7-C8,” the planar distribution of the
tie points is more even. In “D5-D6,” the rover’s visual local-
ization methods are applied with different tie points, and the
results are as follows.
The experimental results indicate that the even distribu-
tion of the tie points is one key factor in improving the rover’s
pose estimation accuracy.
In “C1-C2,” when all tie points (Nos. 1, 2, 3, 4, 5, 6, 7, 8,
9, 10) are used, the absolute localization accuracy is lower
than in other numbers. However, the corresponding value of
Table 9. Summary localization results of the lunar rover in the indoor test field.
Number
Relative
distance (m)
Absolute localization accuracy (m)
Relative localization accuracy (%)
BA+LS parallaxBA BA+LM TLS WTLS BA+LS parallaxBA BA+LM TLS WTLS
C1-C2
13.83
0.882
0.854
0.861 0.813 0.812 6.38
6.17
6.23
5.88 5.87
C2-C3
8.633
0.225
0.204
0.224 0.224 0.221 2.61
2.36
2.59
2.59 2.56
C3-C4
4.806
0.412
0.309
0.401 0.393 0.39
8.57
6.43
8.34
8.18 8.11
C4-C5
6.711
0.259
0.21
0.248 0.249 0.24
3.86
3.13
3.70
3.71 3.58
D1-D2
5.377
0.389
0.337
0.35 0.281 0.278 7.23
6.27
6.51
5.23 5.17
D2-D3
6.476
0.393
0.367
0.378 0.244 0.241 6.07
5.67
5.84
3.77 3.72
D3-D4
7.331
0.227
0.102
0.22 0.193 0.189 3.10
1.39
3.00
2.63 2.58
D4-D5
9.581
0.269
0.218
0.267 0.26 0.254 2.81
2.28
2.79
2.71 2.65
D5-C6
7.384
0.257
0.133
0.251 0.235 0.227 3.48
1.8
3.4
3.18 3.07
C6-C7
10.19
0.701
0.396
0.689 0.539 0.536 6.88
3.89
6.76
5.29 5.26
C7-C8
9.427
0.744
0.682
0.71 0.396 0.391 7.89
7.23
7.53
4.20 4.15
C8-C9
9.756
0.908
0.793
0.891 0.821 0.81
9.31
8.13
9.13
8.42
8.3
Table 10. Localization results of the lunar rover in “D5-D6”.
example
Relative
distance (m)
Absolute localization accuracy (m)
Relative localization accuracy (%)
BA+LS parallaxBA BA+LM TLS WTLS BA+LS parallaxBA BA+LM TLS WTLS
a)
7.384
0.257
0.133
0.251 0.235 0.227 3.48
1.80
3.40
3.18 3.07
b)
7.384
0.531
0.463
0.526 0.464 0.458 7.19
6.27
7.12
6.28 6.20
a) The tie points contain Nos. 2, 5, 8, 9, 10, 11, 12; b) The tie points contain Nos. 2, 5, 8, 9, 10 without No. 11 and No. 12.
614
October 2018
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
