## 4 Experimental Test  
An experiment was also carried out to further verify the effectiveness of the proposed single-camera trilateration algorithm, and check the estimation error under the combined effect of various input errors. A landmark pattern was designed as shown in Figure 15a. It consists of a 679 mm \(\times\) 679 mm square-shaped checker board pattern and a circle co-centered with the checker board and with a diameter of 679 mm. There are 12 landmark dots equally spaced on the circle, as shown in Figure 15b. The pattern was hung under the ceiling of an indoor experimental space with the base plane parallel to the flat floor. We used the base plane as the XY plane of the global frame of reference, with the Z axis pointing downwards and the center of the pattern as the origin of the frame. The exact camera, which was calibrated to provide the realistic values of camera parameters for the above simulations, was installed on a Directed Perception PTU-D46-17 controllable pan-tile unit on the top of a Pioneer 3-DX mobile robot (Figure 16) such that the camera was kept on a plane parallel to the base plane at a distance of 1600 mm. We moved the camera (by moving the mobile robot and adjusting the pan-tilt unit) to a number of locations at different observation distances R (which were defined in Section 3.2), one location at one distance. At each location (each R), we adjusted the tilt angle \(\theta\) and pan angle \(\phi\) (which were defined in Section 3.2) of the camera using the pan-tile unit such that the center of the landmark pattern fell on the image center of the camera, and thus kept the camera oriented towards the landmark pattern. At each location, an image of the landmark pattern was taken by the camera, and stored in the onboard laptop computer. The collected landmark images were processed after the data acquisition process. Focusing on evaluating the proposed single-camera trilateration algorithm itself, we segmented the landmarks from each image manually after correcting the image distortion. Because the camera optical center is not a physical point on the camera, it is highly difficult, if not impossible, to directly measure the position of the camera optical center and the orientation of the camera optical axis. Thus it is highly difficult to evaluate the localization accuracy of the proposed algorithm according to the golden truth of the camera poses in the global frame. Instead, we chose to compare the camera localization results from the proposed single-camera trilateration algorithm with those obtained from camera calibration (using the camera calibration toolbox in [62]). The camera calibration process calibrated both the intrinsic parameters of the camera, which were input to the proposed algorithm for camera pose estimation, and the extrinsic parameters associated with each calibration image of the checker board pattern, which were used as the reference camera pose estimate to reflect the localization accuracy of the proposed algorithm. To evaluate the accuracy of the proposed single-camera localization algorithm at different observation distances, we would like to obtain a collection of camera localization results corresponding to different pan angles \(\phi\) at each observation distance R so that we can derive the statistics of localization  
Figure 16: Onboard experimental system. Figure 15: Landmark pattern used in the experiment: (**a**) landmark pattern; (**b**) landmark dot arrangement. accuracy at each R, where, as we adjusted the pan/tile unit to align the camera optical axis with the center of the landmark pattern at each R, the tilt angle \(\theta\) is fixed at each R. Due to the size limitation on the experimental space, it was difficult for us to pan the camera relative to the Z axis of the global frame all around as we kept increasing R. Instead, we took advantage of the design of our landmark pattern.