# Hierarchical Model Updating Method for Vector Electric-Propulsion Satellites
## 4 Results and Discussion
### The Result of Hierarchical Model Updating
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Variable**} & \multirow{2}{*}{**Real**} & \multicolumn{4}{c}{**The First Loop**} & \multicolumn{4}{c}{**The Second Loop**} \\ \cline{2-6}  & & **Estimated** & **Error** & **Relative Error** & **Estimated** & **Error** & **Relative Error** \\ \hline \(\varphi_{T_{\text{N}}}^{Install}\), rad & \(-0.84298\) & \(-0.85\) & 0.01 & 5.50\% & \(-0.80\) & 0.05 & 27.58\% \\ \(\varphi_{T_{\text{N}}}^{Install}\), rad & \(0\) & 0.07 & 0.07 & 35.83\% & \(-0.02\) & 0.02 & 9.01\% \\ \(\varphi_{T_{\text{N}}}^{Install}\), rad & \(0\) & 0.41 & 0.41 & 204.90\% & 0.15 & 0.15 & 73.40\% \\ \(X_{T_{\text{N}}}^{Install}\), m & \(-0.5\) & \(-0.74\) & 0.24 & 238.83\% & \(-0.65\) & 0.15 & 153.32\% \\ \(Y_{T_{\text{N}}}^{Install}\), m & \(-1\) & \(-0.88\) & 0.12 & 62.39\% & \(-0.70\) & 0.30 & 149.22\% \\ \(Z_{T_{\text{N}}}^{Install}\), m & \(-0.891\) & \(-0.66\) & 0.23 & 131.56\% & \(-0.81\) & 0.08 & 45.31\% \\ \(F_{T_{\text{N}}}\), N & 0.0441 & 0.04 & 0.00 & 0.00\% & 0.04 & 0.00 & 0.00\% \\ \hline \hline \multicolumn{2}{c}{RMSE} & \multicolumn{4}{c}{0.053958\%} & \multicolumn{4}{c}{0.009247\%} \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The estimation of propulsion state variables. From the simulation result of hierarchical model updating, the RMSE of telemetry variables is significantly reduced by multi-loops iterative updating, which validates the model updating method. For the state variables, the updating results of the second loop are closer to the true values than that of the first loop. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**The First Loop**} & \multicolumn{2}{c}{**The Second Loop**} \\ \cline{2-6}
**Variable** & **Real** & \multicolumn{1}{c}{**Estimated**} & \multicolumn{1}{c}{**Error**} & \multicolumn{1}{c}{**Estimated**} & \multicolumn{1}{c}{**Error**} \\ \hline Force X in stage 1 \(F_{xT_{\text{N}}}\), N & \(-0.0003164\) & \(-0.0030089\) & \(0.0026925\) & \(-0.0009050\) & \(0.0005887\) \\ Force Y in stage 1 \(F_{yT_{\text{N}}}\), N & \(0.0051050\) & \(0.0034721\) & \(0.0016329\) & \(0.0062679\) & \(0.0011629\) \\ Force Z in stage 1 \(F_{xT_{\text{N}}}\), N & \(-0.0438024\) & \(-0.0438600\) & \(0.0000576\) & \(-0.0435619\) & \(0.0002405\) \\ Moment X in stage 1 \(M_{xT_{\text{N}}}\), Nm & \(0.0546021\) & \(0.0475862\) & \(0.0070159\) & \(0.0472978\) & \(0.0073043\) \\ Moment Y in stage 1 \(M_{yT_{\text{N}}}\), Nm & \(-0.0211241\) & \(-0.0283029\) & \(0.0071788\) & \(-0.0269889\) & \(0.0058648\) \\ Moment Z in stage 1 \(M_{xT_{\text{N}}}\), Nm & \(-0.0028563\) & \(-0.0055050\) & \(0.0026488\) & \(-0.0043405\) & \(0.0014843\) \\ Force X in stage 2 \(F_{xT_{\text{N}}}\), N & \(0.0022124\) & \(0.0006701\) & \(0.0015423\) & \(0.0030239\) & \(0.0008114\) \\ Force Y in stage 2 \(F_{yT_{\text{N}}}\), N & \(0.0013324\) & \(0.0008649\) & \(0.0004675\) & \(0.0020675\) & \(0.0012751\) \\ Force Z in stage 2 \(F_{xT_{\text{N}}}\), N & \(-0.0440243\) & \(-0.0440864\) & \(0.0000621\) & \(-0.0438481\) & \(0.0001763\) \\ Moment X in stage 2 \(M_{xT_{\text{N}}}\), Nm & \(0.0510070\) & \(0.0456808\) & \(0.0053263\) & \(0.0434219\) & \(0.0075851\) \\ Moment Y in stage 2 \(M_{yT_{\text{N}}}\), Nm & \(-0.0237159\) & \(-0.0311270\) & \(0.0074110\) & \(-0.0288623\) & \(0.0051464\) \\ Moment Z in stage 2 \(M_{xT_{\text{N}}}\), Nm & \(0.0018456\) & \(0.0000837\) & \(0.0017619\) & \(0.0003308\) & \(0.0015148\) \\ \hline \hline \end{tabular}
\end{table} 
Table 11: The estimation of thrust force and moment. Figure 4: The variation in state variables estimation relative error and telemetry RMSE without the hierarchical method. (**a**) Estimation of \(r_{0}\); (**b**) Estimation of \(\
u_{0}\); (**c**) Estimation of GPS errors; (**d**) RMSE of all telemetry variables.