# Marine UAV-USV Marsupial Platform: System and Recovery Technic Verification
## 3 Main Method
### Active Modeling of UAV-USV Heterogeneous Platform Motion Model
#### 3.1.2 Model Difference Online Estimation  
In order to eliminate the influence of the model difference (wind, wave, and water flow) on the control system, it must be estimated online in real time and fed back to the controller for online correction. To this end, we introduce a joint estimation based on the simplified LPV model [37; 38].  
Joint estimation processes the state and the model difference simultaneously, that is, by constructing an augmented state that includes the state and the model difference, and building a model about the augmented state, and then using a single state estimation method to estimate the augmented state. Taking a USV as an example, its LPV dynamics is simplified and written as the equation of state as follows:  
\[\dot{x}=A(u)x+u \tag{3}\]  
where \(x=\left[\begin{array}{c}u\\ v\\ r\end{array}\right]A(u)=\left[\begin{array}{ccc}a_{11}+a_{12}|u|&0&0\\ 0&a_{21}u&a_{22}u\\ 0&a_{31}u&a_{32}u\end{array}\right]u=\left[\begin{array}{c}\widetilde{T}\\ b_{2}/b_{1}\widetilde{T}\sin\delta\\ b_{2}/b_{1}\widetilde{T}\sin\delta\end{array}\right]\).  
Assuming that the model difference, \(\Delta x\), of the quasi-LPV simplified model is driven by noise \(h\), then (3) can be rewritten as follows:  
\[\begin{array}{c}\dot{x}^{e}=\mathbf{f}^{e}(\mathbf{x}^{e},\mathbf{u})+\mathbf{w}\\ z=h(\mathbf{x}^{e})+\mathbf{v}=\mathbf{x}+\mathbf{v}\end{array} \tag{4}\]  
where \(\mathbf{x}^{e}=\left[\begin{array}{c}\mathbf{x}\\ \Delta\mathbf{x}\end{array}\right]\mathbf{f}(\mathbf{x}^{e},\mathbf{u})=\left[\begin{array}{ c}A(u)\mathbf{x}+\mathbf{u}+\Delta\mathbf{x}\\ \Delta\mathbf{x}\end{array}\right]\mathbf{v}\) is process noise and satisfies \(\mathbf{w}=\left[\begin{array}{c}0\\ \mathbf{h}\end{array}\right]\), and \(\mathbf{v}\) is the measurement noise.  
From the above estimation equation form, the joint estimation can obtain the optimal or suboptimal estimation of the joint distribution of the state and the model difference, instead of the respective  
Figure 6: Relative dynamics and kinematics of UAV–USV system in free mode.  
optimal or suboptimal estimation of the state and the model difference. In addition, because the joint distribution requires only a single filter, there is no need to modify the original model, and the structure is simpler. It only needs to consider the estimation of the augmented state, so it is widely used when it is necessary to estimate the state and the model difference.  
Suppose the sampling time, \(T_{s}\), is short, then \(\tilde{x}_{k+1}^{e}\approx\frac{x_{k+1}^{e}-x_{k}^{e}}{T_{s}}\), so that (4) can be discretized.  
Considering that the LPV model (3) is a nonlinear model subject to the speed variable parameter (\(u\)) scheduling, UT (Unscented Transformation) is used to calculate the posterior mean and posterior variance of the system state in the nonlinear optimal Gaussian filter. This method can greatly improve the relative modeling accuracy of heterogeneous platforms and provide a basis for subsequent error-based feedforward controller design.