Author Contributions
Conceptualization, J.Y., W.J., Z.L. and L.Y.; Data curation, J.Y. and W.J.; Methodology, J.Y., W.J., Z.L. and L.Y.; Project administration, J.Y. and Z.L.; Software, J.Y.; Resources Z.L.; Writing—original draft J.Y., W.J. and L.Y.; supervision, Funding acquisition, Z.L., W.J. and L.Y. All authors have read and agreed to the published version of the manuscript.
Figure 1.
Four markers: (a) Checkerboard; (b) ArUco; (c) AprilTag; (d) Circular ring; (e) The proposed Circular-ring marker size design.
Figure 1.
Four markers: (a) Checkerboard; (b) ArUco; (c) AprilTag; (d) Circular ring; (e) The proposed Circular-ring marker size design.
Figure 2.
Cross-ratio invariance.
Figure 2.
Cross-ratio invariance.
Figure 3.
(a) The input image; (b) Gaussian blurred image; (c) Canny edge detection image; (d) Floodfill image.
Figure 3.
(a) The input image; (b) Gaussian blurred image; (c) Canny edge detection image; (d) Floodfill image.
Figure 4.
One of the eight regions.
Figure 4.
One of the eight regions.
Figure 5.
Steps 1, 2, 3, 4, and 6 decode the schematic image.
Figure 5.
Steps 1, 2, 3, 4, and 6 decode the schematic image.
Figure 6.
The relationship between the threshold and the light source intensity.
Figure 6.
The relationship between the threshold and the light source intensity.
Figure 7.
The relationship between this inclination difference and . The parameter is the semi-major axis of the outer ellipse. The parameter is the semi-minor axis of the outer ellipse.
Figure 7.
The relationship between this inclination difference and . The parameter is the semi-major axis of the outer ellipse. The parameter is the semi-minor axis of the outer ellipse.
Figure 8.
The marker identification rate versus distance.
Figure 8.
The marker identification rate versus distance.
Figure 9.
The time of exposure is 100 µs, 1000 µs, 3300 µs, and 6600 µs.
Figure 9.
The time of exposure is 100 µs, 1000 µs, 3300 µs, and 6600 µs.
Figure 10.
The 0° and 45° angular position images at 2 and 4 m.
Figure 10.
The 0° and 45° angular position images at 2 and 4 m.
Figure 11.
Six DoF correspond to repeatability errors if these markers are at 0°: (a) Translation error along X-axis; (b) Translation error along Y-axis; (c) Translation error along Z-axis; (d) Rotation error along X-axis; (e) Rotation error along Y-axis; (f) Rotation error along Z-axis.
Figure 11.
Six DoF correspond to repeatability errors if these markers are at 0°: (a) Translation error along X-axis; (b) Translation error along Y-axis; (c) Translation error along Z-axis; (d) Rotation error along X-axis; (e) Rotation error along Y-axis; (f) Rotation error along Z-axis.
Figure 12.
Six DoF correspond to repeatability errors when these markers at 45°: (a) Translation error along the X-axis; (b) Translation error along the Y-axis; (c) Translation error along the Z-axis; (d) Rotation error along the X-axis; (e) Rotation error along the Y-axis; (f) Rotation error along the Z-axis.
Figure 12.
Six DoF correspond to repeatability errors when these markers at 45°: (a) Translation error along the X-axis; (b) Translation error along the Y-axis; (c) Translation error along the Z-axis; (d) Rotation error along the X-axis; (e) Rotation error along the Y-axis; (f) Rotation error along the Z-axis.
Figure 13.
The translation and rotation accuracy of X and Y-axis at 2 m and 4 m: (a) Translation error along the X and Y-axis at 2 m; (b) Rotation error along the X and Y-axis at 2 m; (c) Translation error along the X and Y-axis at 4 m; (d) Rotation error along the X and Y-axis at 4 m.
Figure 13.
The translation and rotation accuracy of X and Y-axis at 2 m and 4 m: (a) Translation error along the X and Y-axis at 2 m; (b) Rotation error along the X and Y-axis at 2 m; (c) Translation error along the X and Y-axis at 4 m; (d) Rotation error along the X and Y-axis at 4 m.
Figure 14.
Translation error along the Z-axis at 2 to 4 m.
Figure 14.
Translation error along the Z-axis at 2 to 4 m.
Figure 15.
Experiment images under different marker dimensions: (a) Markers dimension is mm; (b) Markers dimension is mm; (c) Markers dimension is mm; (d) Markers dimension is mm.
Figure 15.
Experiment images under different marker dimensions: (a) Markers dimension is mm; (b) Markers dimension is mm; (c) Markers dimension is mm; (d) Markers dimension is mm.
Figure 16.
Transform the markers position: (a) Position 1; (b) Position 2; (c) Position 3; (d) Position 4.
Figure 16.
Transform the markers position: (a) Position 1; (b) Position 2; (c) Position 3; (d) Position 4.
Figure 17.
The positioning accuracy experiment of each marker at greater distance: (a) Circular ring; (b) AprilTag; (c) ArUco; (d) Checkerboard.
Figure 17.
The positioning accuracy experiment of each marker at greater distance: (a) Circular ring; (b) AprilTag; (c) ArUco; (d) Checkerboard.
Figure 18.
Rotation for angle estimation test: (a) The yaw of 0°; (b) The yaw of 70°; (c) The roll of 0°; (d) The roll of 80°.
Figure 18.
Rotation for angle estimation test: (a) The yaw of 0°; (b) The yaw of 70°; (c) The roll of 0°; (d) The roll of 80°.
Table 1.
The size of each parameter value.
Table 1.
The size of each parameter value.
| K | L | |
---|
Experimental Value | 0.2 | 2 pixels | 0.5 |
Maximum Value | 0.055 | 1.025 pixels | 0.297 |
Preset Value | 0.06 | 1.13 pixels | 0.33 |
Table 2.
The markers distance range of accurate identification [m].
Table 2.
The markers distance range of accurate identification [m].
| Circular-Ring | AprilTag | ArUco | Checkerboard |
---|
Identification | 1.7–5.8 | 1.3–6.1 | 0.9–5.8 | 1.1–4.3 |
Table 3.
The effective range of the exposure time [µs].
Table 3.
The effective range of the exposure time [µs].
| Circular-Ring | AprilTag | ArUco | Checkerboard |
---|
Low exposure | 400 | 100 | 200 | 100 |
High exposure | 6600 | 3600 | 4200 | 3800 |
Table 4.
Average processing time of single marker image [s].
Table 4.
Average processing time of single marker image [s].
| Circular-Ring | AprilTag | ArUco | Checkerboard |
---|
Time | 0.077 | 1.581 | 0.052 | 1.347 |
Table 5.
The maximum Euclidean distance of translation error and rotation error.
Table 5.
The maximum Euclidean distance of translation error and rotation error.
| Circular-Ring | AprilTag | ArUco | Checkerboard |
---|
(0°)/mm | 3.99 | 13.56 | 19.17 | 33.43 |
(0°)/° | 2.410 | 3.713 | 4.205 | 6.974 |
(45°)/mm | 32.94 | 81.95 | 93.23 | 190.69 |
(45°)/° | 1.915 | 4.798 | 4.108 | 6.651 |
Table 6.
Comparison of positioning accuracy under different resolutions.
Table 6.
Comparison of positioning accuracy under different resolutions.
Relative Error [%] |
---|
| Circular Ring | AprilTag | ArUco | Checkerboard |
---|
| avg | max | avg | max | avg | max | avg | max |
---|
| 0.43 | 0.96 | 1.31 | 4.29 | 1.07 | 4.80 | 1.35 | 4.49 |
| 0.26 | 0.50 | 0.88 | 1.27 | 0.64 | 1.84 | 0.78 | 1.83 |
| 0.18 | 0.32 | 0.18 | 0.41 | 0.39 | 0.60 | 0.19 | 0.39 |
Table 7.
Comparison of positioning accuracy under different dimensions.
Table 7.
Comparison of positioning accuracy under different dimensions.
Mean Error [mm] |
---|
| Circular-Ring | AprilTag | ArUco | Checkerboard |
---|
mm | 21.1 | 41.8 | 62.0 | 109.8 |
mm | 6.8 | 19.5 | 30.8 | 32.2 |
mm | 3.4 | 17.8 | 7.0 | 15.3 |
Table 8.
Comparison of positioning accuracy under position transformation.
Table 8.
Comparison of positioning accuracy under position transformation.
Absolute Error [mm] |
---|
| Circular Ring | AprilTag | ArUco | Checkerboard |
---|
| avg | max | avg | max | avg | max | avg | max |
---|
1 | 3.4 | 8.0 | 8.8 | 26.5 | 29.0 | 74.4 | 22.5 | 62.5 |
2 | 2.4 | 6.1 | 14.2 | 46.7 | 17.4 | 58.9 | 23.8 | 79.0 |
3 | 3.8 | 10.7 | 19.5 | 47.2 | 7.9 | 17.1 | 32.2 | 101.4 |
4 | 6.8 | 17.6 | 17.0 | 52.7 | 30.8 | 88.2 | 20.1 | 58.3 |
Table 9.
Comparison of positioning accuracy at greater distances.
Table 9.
Comparison of positioning accuracy at greater distances.
| | Circular-Ring | AprilTag | ArUco | Checkerboard |
---|
| | avg | max | avg | max | avg | max | avg | max |
---|
vertical | Absolute error [mm] | 101.9 | 269.4 | 258.5 | 436.6 | 245.0 | 569.8 | 284.0 | 973.0 |
Relative error [%] | 1.67 | 4.42 | 4.24 | 7.16 | 4.02 | 9.34 | 4.66 | 15.95 |
horizontal | Absolute error [mm] | 48.4 | 125.9 | 90.7 | 243.7 | 111.8 | 259.2 | 134.6 | 415.4 |
Relative error [%] | 0.79 | 2.06 | 1.49 | 4.00 | 1.83 | 4.25 | 2.21 | 6.81 |
Table 10.
Errors of angle estimates.
Table 10.
Errors of angle estimates.
Absolute Error [°] |
---|
| Circular Ring | AprilTag | ArUco | Checkerboard |
---|
| avg | max | avg | max | avg | max | avg | max |
---|
Pitch/Yaw | 2.807 | 4.927 | 4.011 | 9.626 | 3.380 | 5.157 | 6.933 | 9.225 |
Roll | 0.745 | 1.776 | 1.948 | 3.438 | 0.573 | 1.490 | 1.604 | 3.380 |
Table 11.
The ratio of the average error of the different fiducial markers to the Circular-ring marker.
Table 11.
The ratio of the average error of the different fiducial markers to the Circular-ring marker.
| | Circular Ring | AprilTag | ArUco | Checkerboard |
---|
5.5 (translation) | 0° | 1 | 3.4 | 4.8 | 8.4 |
45° | 1 | 2.5 | 2.8 | 5.8 |
5.7.1 (translation) | low | 1 | 3.0 | 2.5 | 3.1 |
medium | 1 | 3.4 | 2.5 | 3.0 |
high | 1 | 1 | 2.2 | 1.1 |
5.7.2 (translation) | little | 1 | 2.0 | 2.9 | 5.2 |
medium | 1 | 2.9 | 4.5 | 4.7 |
big | 1 | 5.2 | 2.1 | 4.5 |
5.7.4 (translation) | horizontal | 1 | 1.9 | 2.3 | 2.8 |
vertical | 1 | 2.5 | 2.4 | 2.8 |
5.5 (rotation) | 0° | 1 | 1.5 | 1.7 | 2.9 |
45° | 1 | 2.5 | 2.1 | 3.5 |
5.7.5 (rotation) | Pitch/Yaw | 1 | 1.4 | 1.2 | 2.5 |
Roll | 1 | 2.6 | 0.8 | 2.2 |