## 5 Discussion  
The authors introduced the VM algorithm in an earlier study and used this algorithm assuming 2D constant velocity model [11]. In that study, we showed that the VM outperforms the ICP algorithm based on data captured by a single sensor. The analysis with using GPS installed on the airplane and used as reference showed that the RMSEs (root-mean-square errors) are 0.72 m/s with VM and 5.29 m/s with ICP even at short (10 m) object-sensor distances. The main reason that the ICP tends to fail is that it requires corresponding points between two point clouds; point clouds acquired by the Velodyne scanners have relatively low point density or resolution, especially when the object-sensor distance is large. Consequently, capturing \"good\" corresponding points is not possible. While the VM performed well for the CV model in our earlier studies, later it turned out that the VM approach cannot adequately handle more complex motions due to the structure of the objective function at small scale. In our next study, we tried to address this issue of approximating the general motion model by applying the constant velocity model for shorter time periods [12]. This solution performed well, yet it was not robust. This study introduced cube trajectories to improve the motion estimation. Figure 16: Estimated coordinates of a landing Cessna airplane using VM-CV (red) and CT (blue) algorithms for the X, Y, and Z directions; X direction is along the centerline of the runway. Figure 15: Reconstruction from sparse point clouds (total number of points is 4320) from data presented in Figure 14: the shape of the plane including the wings, tails are recognizable. Results have to be compared in terms of accuracy, reliability under various aircraft motions. In the following, two indicators are used for qualitatively assessing and comparing the performance. * The reconstruction quality is one indicator that can be used to compare the various algorithms and to assess the performance in general. If the reconstruction is good, the aircraft and its details are recognizable, indicating that the derived motion pattern is also close to the actual trajectory. A fuzzy, blurry reconstruction indicates that the estimated motion is not adequate. The reconstructions of the COG, VM-CA and CT algorithms in Figure 15 show that VM and CT outperform COG and CT provides a better reconstruction than VM. The CT reconstruction is fuzzy compared to the reconstruction provided by the CT algorithm. The reason of the underperformance of the COG method is that different body points are backscattered at different epochs, and consequently, the COG does not accurately represent the center of gravity of the airplane body. Clearly, COG tends to fail when the scans contain small number of points from the tracked object. This is also the case for Test #2, when the object sensor distance is around 33 m, and thus, not enough points are captured from the Cessna-size airplanes. In addition, the sensor network is installed on one side of the runway, which means that the sensors capture points from one side of the airplane. This clearly pulls the COG to the direction of that of the sensors. In contrast to COG, the CT is able to provide a fairly good quality reconstruction (see Figure 15), indicating that the algorithm performs well even if the scans contain a small number of points. * The shape of the trajectory and the velocity profiles can be also used to assess the performance. For instance, in Figure 13 for Test #1, the trajectory derived from COG is not realistic, it is not possible that the airplane could have followed such a curvy trajectory. Note that large tail at the end of the trajectory, which is definitely not feasible motion, is caused by the lack of a sufficient number of captured points. In contrast, the VM-CA and CT results are feasible trajectories. However, it is unclear whether the small \"waves\" of the CT trajectory is the actual motion of the airplane, or it is the result of overfitting due to fourth-degree motion model.