## 3 Numerical Simulations
### Orbit Determination Refinement
Measurement update \[\mathbf{C}_{\overline{y}_{k}\overline{y}_{k}}=\sum_{i=0}^{2L}W_{i}^{(c)}\Big{[}\mathbf{ \mathcal{Y}}_{i,k|k-1}-\hat{y}_{k}^{-}\Big{]}\Big{[}\mathbf{\mathcal{Y}}_{i,k|k-1}- \hat{y}_{k}^{-}\Big{]}^{\mathrm{T}}+\mathbf{R}^{n}\] (24) \[\mathbf{C}_{\mu_{k}\mathbf{y}_{k}}=\sum_{i=0}^{2L}W_{i}^{(c)}\Big{[}\mathbf{ \mathcal{X}}_{i,k|k-1}-\hat{\mu}_{k}^{-}\Big{]}\Big{[}\mathbf{\mathcal{Y}}_{i,k|k-1 }-\hat{y}_{k}^{-}\Big{]}^{\mathrm{T}}\] (25) \[\mathbf{\mathcal{K}}_{k}=\mathbf{C}_{\mu_{k}\mathbf{y}_{k}}\mathbf{C}_{\overline{y}_ {k}\overline{y}_{k}}^{-1}\] (26) \[\hat{\mu}_{k}=\hat{\mu}_{k}^{-}+\mathbf{\mathcal{K}}_{k}\Big{(}\mathbf{y}_ {k}-\hat{y}_{k}^{-}\Big{)}\] (27) \[\mathbf{C}_{k}=\mathbf{C}_{k}^{-}-\mathbf{\mathcal{K}}_{k}\mathbf{C}_{\overline{y}_ {k}\overline{y}_{k}}\mathbf{\mathcal{K}}_{k}^{\mathrm{T}}\] (28)  
In the previous formulation, \(\mathbf{R}^{n}\) indicates the measurement noise matrix, \(y_{k}\) is the vector of real observations at time epoch \(t_{k}\), i.e., Doppler shift, whereas \(\mathbf{h}\) is the nonlinear transformation to pass from the state estimate to the observation vector estimate \(\mathbf{\mathcal{Y}}_{k|k-1}\). The algorithm is repeated for all the observation instants \(t_{k}\) of the investigated passage. The result, in the end, is an updated estimate of the state vector of the object at the epoch of the last observation instant \(\mathbf{\hat{X}}_{\mathbf{t}_{obs},\mathbf{N}_{obs}}\sim\mathcal{N}\Big{(}\hat{\mu}_{ \mathbf{t}_{obs},\mathbf{N}_{obs}},\hat{\mathbf{C}}_{\mathbf{t}_{obs},\mathbf{N}_{obs}}\Big{)}\). This new estimate replaces the old available estimate, thus updating the database. The described procedure is repeated for all the passages of the list. Every time a new passage of a given object is planned, its state estimate is retrieved from the database, it is propagated from its time epoch to the epoch of the new passage and eventually updated. If the OD refinement process is successful, the new estimate replaces the old one in the database, and so on. Table 2 shows the evolution of the state estimate accuracy for object NORAD ID 00730 in terms of error in position and velocity (\(\varepsilon_{p}\) and \(\varepsilon_{v}\)) and standard deviation in position and velocity (\(\sigma_{p}\) and \(\sigma_{v}\)) obtained without performing OD refinement (\"_free_\" subscript) and updating the estimate with the UKF (\"_OD_\" subscript). For the case under study, only Doppler shift measurements are considered for the measurement update phase of the UKF. The first line shows the initial accuracy of the available estimate, whereas the second line shows the accuracy of the estimate at the epoch of the first available measurement of the first object passage. As can be seen, the errors in position and velocity increase of four and three orders of magnitude, respectively, in slightly less than two days. The third line shows a comparison between the result obtained at the end of the UKF (left) and what would be obtained without performing OD: the UKF guarantees an improvement in both position and velocity accuracy of around three orders of magnitude. Nevertheless, the uncertainty of the estimate obtained at the end of the UKF in both position and velocity is relatively large with respect to the actual errors (2.86 km and 4.17 m/s, respectively). This aspect has an unavoidable drawback in the accuracy of the results obtained when the estimate is propagated. Lines four and five show the comparison between the refined estimate and the old one when propagated one day and two days after the UKF epoch, respectively. As can be seen, the error of the updated estimate rapidly increases in time, and after one day the accuracy of the updated estimate is already lower than the accuracy of the old estimate. That is, the refinement effect granted by the UKF is already lost.