## 2 Materials and Methods
### Moving Oysters Prediction
Similarly, \(\hat{\mathsf{P}}_{\text{t}}^{-}\) is the prior estimated value of the error covariance matrix; the prediction uses the state estimation at the previous time after the Low-Pass filter \(\hat{\mathsf{X}}_{\text{t}-1}^{\text{LPF}}\). The updated equations are defined as follows:  
\[\text{K}_{\text{t}}=\hat{\mathsf{P}}_{\text{t}}^{-}\text{H}^{\text{T}}\Big{(} \text{H}\,\hat{\mathsf{P}}_{\text{t}}^{-}\,\text{H}^{\text{T}}+\text{R}\Big{)} ^{-1} \tag{6}\]  
\[\hat{\mathsf{X}}_{\text{t}}=\hat{\mathsf{X}}_{\text{t}}^{-}+\text{K}_{\text{t} }\Big{(}Z_{\text{t}}-\text{H}\hat{\mathsf{X}}_{\text{t}}^{-}\Big{)} \tag{7}\]  
\[\text{P}_{\text{t}}=(\text{I}-\text{K}_{\text{t}}\text{H})\hat{\mathsf{P}}_{ \text{t}}^{-} \tag{8}\]  
where \(\text{K}_{\text{t}}\), \(\hat{\mathsf{X}}_{\text{t}}\), \(\text{P}_{\text{t}}\), and I are, respectively, the Kalman gain matrix, the optimum filter value, the posterior estimated value of the error covariance matrix, and the unit matrix. The Low-Pass filter processing formula is as follows:  
\[\hat{\mathsf{X}}_{\text{t}}^{\text{LPF}}=\alpha\hat{\mathsf{X}}_{\text{t}}+(1 -\alpha)\hat{\mathsf{X}}_{\text{t}-1}^{\text{LPF}} \tag{9}\]  
where \(\hat{\mathsf{X}}_{\text{t}}^{\text{LPF}}\) is taken as the optimal state estimate at the current time and will be used for the prediction at the next time. During the actual experimental process, the conveyor belt carries oysters at a certain speed along the negative direction of the camera coordinate x-axis, and there is no relative sliding between the oysters and the conveyor belt. The external input is considered to approximate zero. During image acquisition, the time interval between adjacent frames is so short that it is assumed that the speed of the target object remains constant in a very short time, that is, the object moves in a uniform linear motion between adjacent frames. Considering that the direction of the conveyor belt motion aligns with the x-axis of the camera coordinate system, although there may be a slight offset in the y-axis direction due to installation errors, this error can be considered negligible. Therefore, a set of two-dimensional vectors \((\text{x}_{\text{t}},\text{v}_{\text{x},\text{t}})\) is used to describe the displacement and velocity of the oysters in the x directions at time t. The motion equation of the oysters can be represented as follows:  
\[\text{x}_{\text{t}}=\text{x}_{\text{t}-1}+\text{v}_{\text{x},\text{t}}\Delta t \tag{10}\]  
The state variable vector of the system is as follows:  
\[\hat{\mathsf{X}}_{\text{t}}^{\text{LPF}}= \begin{bmatrix}\text{x}_{\text{t}} \\ \text{v}_{\text{x},\text{t}}\end{bmatrix}  \tag{11}\]  
The state matrix of the system is as follows:  
\[\text{A}= \begin{bmatrix}1&\Delta\text{t}\\ 0&1\end{bmatrix}  \tag{12}\]  
The state \(\hat{\mathsf{X}}_{\text{t}}^{\text{LPF}}\) of the system is as follows:  
\[\hat{\mathsf{X}}_{\text{t}}^{\text{LPF}}= \begin{bmatrix}\text{x}_{\text{t}} \\ \text{v}_{\text{x},\text{t}}\end{bmatrix} = \begin{bmatrix}1&\Delta\text{t}\\ 0&1\end{bmatrix}\begin{bmatrix}\text{x}_{\text{t}-1}\\ \text{v}_{\text{x},\text{t}-1}\end{bmatrix}  \tag{13}\]Since only the position is observed, the observation matrix is as follows:  
\[\mathrm{H}= \begin{bmatrix}1&0\\ 0&1\end{bmatrix}  \tag{14}\]  
In order to ensure that the number of iterations is sufficient to obtain the optimal information of the oyster state, the KF + Low-Pass filter method is used. When the grasp point reaches the middle of the x-axis field of view of the camera, the position of the oyster t seconds ahead is predicted by evaluation. This prediction is further refined by considering the average state of the oyster over a certain time horizon. Finally, the robot is controlled to intercept and capture the oyster.