## 7 Orbit Determination  
Two Line Elements that are provided by the Joint Space Operations Center (JSpoC) have a low precision due to the analytic SGP4-propagation theory, which can amount to several hundred meters [41; 42], whereas precise orbit data is not accessible to publicity. Nevertheless, the need for independent data, which is achieved by telescope and radar sensors, arises for many applications as for the determination of collision probabilities. These sensors, however, can not track every object at every orbit revolution, which is why the objects have to be propagated into the future to get their orbital data. This propagation increases the uncertainty. Thus, the initial uncertainty has to be as low as possible. This is achieved by the field of precise orbit determination (POD). In POD, new measurements of position and velocity of objects have to be achieved--for example, by telescopes, radars or GPS. These measurements are used to correct the predicted orbit using, e.g., least square methods or Kalman filters [43]. The latter refine the state vector without processing previous detections, and thus these are applicable to real-time applications. Figure 10: Comparison of conventional Database Management Systems (**top**) and Data Stream Management Systems (**bottom**). The different filled shapes represent different data, the different empty shapes represent queries. Illustration by German Wikipedia user [PERSON], derivative work by German Wikipedia user [PERSON] and [PERSON], License: CC-BY-SA 3.0. At the Institute, different Kalman filters are in use. These are the Extended Kalman Filter (EKF) [44], Unscented Kalman Filter (UKF) [38; 45], and the Ensemble Kalman Filter (EnKF). The first linear Kalman filter (KF) was proposed in [46]. The basic idea of Kalman filters is to use the detections and additionally a mathematical model, which describes the orbit of the objects [43], for the refinement of the detections. Kalman filters consist of two basic parts: the time update and the measurement update [47]. Within the time update, the state vector and its uncertainty (usually described by covariance matrices) are propagated to the epoch of the next available detection. The uncertainty of the model, the process noise, has to be considered. Within the measurement update, the propagated state vector is updated using the respective detection. Firstly, the Kalman Gain is calculated. This is a matrix (for multivariate distributions) that states whether the state vector or the real detection corresponds with the truth. Then, the propagated state vector and covariance matrix are modified by the Kalman gain to achieve the update values. The KF is applicable to linear applications, and, thus, it is not suitable for orbit determination. The EKF extends the time update by a linearization using Jacobi matrices to propagate the uncertainty [43]. However, Jacobi matrices are complicated to implement and yield to instability [48]. The UKF, however, does not use Jacobi matrices, but propagates the uncertainty using a set of sampling points, which are called sigma points [48]. These sigma points map the uncertainty of the state vector, or, more precisely, the uncertainty of the position and velocity. Figuratively, the sigma points span an ellipsoid around the mean, which can be seen in Figure 11.  
They are threatened as usual state vectors that are propagated within the filtering process. For simplicity, only six sigma points are shown. The propagation yields to a new distribution of the sigma points that describes the uncertainty of the propagated state vector. To generate these sigma points, a Gaussian distribution of the uncertainty is considered, whereas the state vector is its mean and the covariance matrix contains the standard deviations in the respective coordinate directions. Applying the Unscented Transformation (UT), 13 equally distributed sigma points are sampled [48; 49]. In the literature, different statements are given whether the state vector can be regarded as a sigma point or not [48; 50].