## 4 Experimental Test
With 12 landmark dots equally spaced on the circle as shown in Figure 15b, we have 12 equally spaced landmark sets labeled as (1, 5, 9), (2, 6, 10), (3, 7, 11), (4, 8, 12), (5, 9, 1), (6, 10, 2), (7, 11, 3), (8, 12, 4), (9, 1, 5), (10, 2, 6), (11, 3, 7) and (12, 4, 8). As mentioned above, at each R (and \(\theta\)), we took one camera image of the landmark pattern. By using different 3-landmark sets from the same image, we in fact localize the camera corresponding to different \(\phi\). For example, using landmark set (2, 6, 10) is equivalent to panning the camera from landmark set (1, 5, 9) about the global Z direction by \(\Delta\phi=30^{\circ}\). In this way, instead of physically panning the camera, by using different landmark sets to localize the camera, we equivalently obtained the camera localization results corresponding to different \(\phi\). On the other side, at each R (and \(\theta\)), the reference camera pose estimate corresponding to each landmark set was obtained by rotating the calibrated camera pose, which was calculated based on the image of the checker board pattern at R (and \(\theta\)) using the camera calibration toolbox and corresponds to the landmark set (1,5,9), about the global Z axis with a proper \(\Delta\phi\). The localization results from the proposed single-camera localization algorithm are compared with those from the camera calibration process in Table 2. The calibrated camera pose is used as the reference estimate, and the corresponding calibration error reported by the camera calibration toolbox is presented as the standard deviation of the camera pose calibration. The difference between the camera pose estimated using the proposed algorithm and the calibrated camera pose is recorded as \(\Delta\mathbf{p}_{0}\) (position estimation difference) and \(\Delta\mathbf{\theta}_{c}\) (orientation estimation difference), and the mean and standard deviation of these differences are reported. The results in Table 2 show that on average the proposed algorithm attains a localization estimation very close to that attained with camera calibration. The average difference in position estimation is only about 6 mm at an observation distance of R = 4.3 m (corresponding to \(\mathrm{R_{h}}=4\) m in Table 2), and is smaller with shorter observation distances; the average difference in orientation estimation remains lower than 0.3 degree at an observation distance of R \(\leq\) 4.3 m. While camera calibration cannot be used for real-time localization of a moving camera, the proposed algorithm, which provides a comparable estimation of the camera pose on the go, presents an effective online localization approach for many mobile robot applications. We also notice that the difference in camera position estimation between the two compared methods and the associated standard deviation increase as the observation distance increases. This is mainly because the effective size of the pixels increases as the distance increases. Since the proposed algorithm depends on the segmented landmarks from the same image, the camera positioning error clearly increases as the observation distance and thus the effective pixel size increase; while the camera position estimation error with camera calibration increases much slower, because the calibration process takes advantage of all the images taken at different distances and all the grid corners of the checker board pattern to average out the effect of the increasing effective pixel size. On the other side, the difference in camera orientation estimation between the two compared methods and the associated standard deviation do not appear to be affected as the observation distance increases, due to the cancellation between the effect of the observation distance and that of the effective pixel size.