## Materials and Methods
### Handover Prediction Models
#### 2.2.2 Kalman Filter  
The Kalman filter was proposed by [PERSON] in the 1960s and is an optimal recursive data processing algorithm. It can generate optimal solutions with high efficiency for most problems. In addition, it has been widely applied for more than 30 years, particularly in autonomous systems and navigation assistance, including robot navigation, control, and military radar systems and missile tracking. The Kalman filter uses a set of efficient mathematical recursive equations to estimate a system process state by minimizing the squared error under the assumption of Gaussian white noise. The process can be used to predict past, present, and future states and can operate under a system with uncertain system states. For a linear system, the Kalman filter can use a system process model to predict the next state of the system.  
Assume that the current system state is \(k\). According to the system model, \(k\) can be predicted on the basis of the previous state, denoted \(k-1\):  
\[\hat{\kappa}_{k}^{-}=\mathrm{A}\hat{\kappa}_{k-1}+\mathrm{Bu}_{k-1} \tag{1}\]  
\[\mathrm{P}_{k}^{-}=\mathrm{AP}_{k-1}\mathrm{A}^{\mathrm{T}}+\mathrm{Q} \tag{2}\]  
where \(\hat{\kappa}_{k}^{-}\) is the predictive system value of state \(k\), which is the estimated occurrence time of the \(k^{\mathrm{th}}\) handover in this study. Through \(\hat{\kappa}_{k}^{-}\), prediction is performed using the predicted result from the previous state, \(\hat{\kappa}_{k-1}\). Additionally, \(\mathrm{u}_{k-1}\) is the input control amount in the state \(k-1\), and \(\mathrm{A}\) and \(\mathrm{B}\) are system parameter matrices. In this study, \(\mathrm{u}_{k-1}\) is set to 0 and \(\mathrm{A}\) is set to the identity matrix because there is no additional input for adjustment of the system. In Equation (2), \(\mathrm{P}_{k}^{-}\) is the estimated error covariance matrix corresponding to \(\hat{\kappa}_{k}^{-}\). \(\mathrm{A}^{\mathrm{T}}\) is the transpose of matrix \(\mathrm{A}\), and \(\mathrm{Q}\) is the system process noise covariance matrix, which represents the noise of the process and is assumed to be Gaussian white noise that does not change with the system state. In this study, \(\mathrm{Q}\) was set to 1\(\times\)10\({}^{-5}\). Equations (1) and (2) are the first two of the five formulas of the Kalman filter and correspond to the prediction part of the system.  
\[\mathrm{K}_{k}=\mathrm{P}_{k}^{-}\mathrm{H}^{\mathrm{T}}\Big{(}\mathrm{HP}_{ k}^{-}\mathrm{H}^{\mathrm{T}}+\mathrm{R}\Big{)}^{-1} \tag{3}\]  
\[\mathrm{K}_{k}=\mathrm{P}_{k}^{-}\mathrm{H}^{\mathrm{T}}\Big{(}\mathrm{HP}_{ k}^{-}\mathrm{H}^{\mathrm{T}}+\mathrm{R}\Big{)}^{-1} \tag{4}\]  
\[\mathrm{P}_{k}=(\mathrm{I}-\mathrm{K}_{k}\mathrm{H})\mathrm{P}_{k}^{-} \tag{5}\]  
After obtaining the prediction of the current state, the Kalman Gain, denoted \(\mathrm{K}_{k}\), is calculated. \(\mathrm{H}\) is the parameter matrix of the measurement system and is assumed to be a constant that is set to the identity matrix in this study; \(\mathrm{R}\) is the measurement noise covariance matrix of the system, which is also assumed to be Gaussian white noise that does not change with the system state. In this study, \(\mathrm{R}\) is set to 0.004\({}^{2}\), and this number affects sensitivity of the system predictions. In Equation (4), \(\mathrm{K}_{k}\), the predicted value \(\hat{\kappa}_{k}^{-}\) of the previous state \(k-1\), and the measured value \(z_{k}\) of the current state are employed to determine the assembly cycle time.