# Hierarchical Model Updating Method for Vector Electric-Propulsion Satellites
## 2 Problem Formulation
### State Equations and State Variables
As for the gyro, besides \(H_{Gyro,\omega},\sigma_{Gyro,\omega}\), there still exists the drift error \(\varepsilon_{Gyro,\omega}\), which changes as  
\[\frac{d\varepsilon_{Gyro,\omega}}{dt}=-\frac{\varepsilon_{Gyro,\omega}}{T}+ \sigma_{Gyro,\omega} \tag{23}\]  
The actuators include wheels, thrusters, and a rotating component of the solar wing. The output process noise is also represented by a normal distribution:  
\[u_{output}=u_{expect}+N(\mu_{Actuator,\mu},\sigma_{Actuator,\mu}) \tag{24}\]where \(u_{expect}\) is the expected output command, \(u_{output}\) is actual output control, and \(\mu_{Actuator,u},\sigma_{Actuator,u}\) are the error mean and variance of the actuator output process, respectively. The sensor model and actuator model are described in Reference [23]. Four vector thrusters are installed on the backboard of the satellite, their installation positions and serial numbers are upper left \(T_{N1}\), upper right \(T_{N2}\), lower left \(T_{S1}\), and lower right \(T_{S2}\), respectively. Each thruster is connected to the satellite body by a vector adjustment mechanism, which is regarded as a structure of two links, each of which can rotate on its axis. The two-degree-of-freedom kinematic model can be described by eight parameters including connecting rod angle \(\alpha_{i-1}\), connecting rod length \(a_{i-1}\), joint angle \(\theta_{i}\), and connecting rod offset \(d_{i}\) (\(i=1,2\)). The meaning of these parameters and the kinematic process are shown in Reference [26]. According to the model, the thrust direction is only related to \(\theta_{i}\), and the application point of thrust is related to all structural parameters. The thruster number is denoted by \(No(No=N1,N2,S1,S2)\); the structural error includes the length error \(\delta a_{i-1,No}\) and offset error \(\delta d_{i,No}\). Three attitude angles \(\varphi_{T_{No}}^{Install},\theta_{T_{No}}^{Install},\psi_{T_{No}}^{Install}\) are available for mounting attitude. Installation points are represented by position vectors \(\left[X_{T_{No}}^{Install},Y_{T_{No}}^{Install},Z_{T_{No}}^{Install}\right]^{T}\). Every time it fires, the deflection error of the thrust direction at the outlet relative to the axis is represented by \(\delta_{T_{No}},\tau_{T_{No}}\), and the output force action point error \(\delta\mathbf{p}_{T_{No}}=\left[\delta p_{x,T_{No}},\delta p_{y,T_{No}},\delta p_{z,T_{No}}\right]^{T}\). The above variables are constant values during the working period of each firing. Furthermore, the output force of the thruster \(F_{T_{No}}\) is not a constant value but a variable that changes with time. In this paper, the parameters of the thruster \(T_{N1}\) are taken as the updating variables. In summary, \(x\) consists of  
\[\mathbf{x}= \left[r_{x},r_{y},r_{z},v_{x},v_{y},v_{z},\varphi,\theta,\psi, \omega_{x},\omega_{y},\omega_{z},\theta_{wing_{1}},\theta_{wing_{2}},\theta_{ SW_{1}},\theta_{SW_{2}},\right.\] \[\left.\eta_{w},P_{wing_{1}},P_{wing_{2}},U_{wing_{1}},U_{wing_{1}},U_ {wing_{2}},I_{wing_{1}},I_{wing_{2}},Q_{Battery},P_{Battery},I_{Battery},m_{Xe},P_{ Xe},T_{i},\right.\] \[\left.H_{GPS,r},\sigma_{GPS,r},H_{GPS,\sigma},\sigma_{GPS,r},H_{ GP\sigma,\omega},\sigma_{Gry,\omega},\sigma_{Gry,\omega},H_{StarSensor},\sigma_{ StarSensor},\right.\] \[\left.\varphi_{T_{N1}}^{Install},\theta_{T_{N1}}^{Install},X_{T_{N1 }}^{Install},Y_{T_{N1}}^{Install},Y_{T_{N1}}^{Install},Z_{T_{N1}}^{Install}, \delta_{T_{No}},\tau_{T_{No}},\delta p_{x,T_{N1}},\delta p_{y,T_{N1}},\delta p _{z,T_{N1}},F_{T_{N1}}\right]\]