## 2 Definitions of Problem and Symbols  
This paper mainly considers the one-dimensional second-order moving object tracking filter assuming that only position and velocity measurements are considered. For the one-dimensional problem, only tracking along the \(x\)-axis is considered. Note that for the performance evaluation, assuming the realistic situation presented in Section 7, an actual two-dimensional tracking in the \(x\)-\(y\) plane is considered, and the one-dimensional tracking filter being considered is implemented for each axis in this simulation.  
The inputs of the filter are measured target position \(x_{\mathrm{o}}\) and velocity \(v_{\mathrm{o}}\). This assumes that the observed data are both position and velocity. Note that many conventional studies on the tracking system adopt a position-only analysis, whereas assuming position/velocity measurements is one of the features of our study. The errors in \(x_{\mathrm{o}}\) and \(v_{\mathrm{o}}\) conform to white Gaussian noise, and their correlations are not considered for simplicity. We assume that the variance of the position measurement errors \(B_{\mathrm{x}}\) and that of the velocity measurement errors \(B_{\mathrm{v}}\) are known.  
The outputs are predicted target position \(x_{\mathrm{p}}\) and velocity \(v_{\mathrm{p}}\) and smoothed (estimated) target position \(x_{\mathrm{s}}\) and velocity \(v_{\mathrm{s}}\). The focus of this paper is the optimization of the steady state accuracy in \(x_{\mathrm{p}}\). To achieve this, the inputs \(x_{\mathrm{o}}\) and \(v_{\mathrm{o}}\) are filtered by some gains. This study assumes that these gains are unknown parameters that we must design. Thus, the purpose of this paper is the design of the tracking filter gains that minimizes the errors in the predicted target position.  
The detailed definitions and explanations of the tracking filters and their design methodology that we focus on are presented in the rest of this paper. Table 1 lists the symbols used in this paper. Furthermore, each symbol is defined at its first appearance.  
\begin{table}
\begin{tabular}{c l c} \hline \hline
**Variables** & **Description** & **Unit** \\ \hline \(T\) & Sampling interval & (s) \\ \(k\) & Discrete sampling index & Dimensionless \\ \(()_{,k}\) & Parameter at index \(k\) & \\ \(x_{\mathrm{p}}\) & Predicted position & (m) \\ \(v_{\mathrm{p}}\) & Predicted velocity & (m/s) \\ \(x_{\mathrm{s}}\) & Smoothed (estimated) position & (m) \\ \(v_{\mathrm{s}}\) & Smoothed (estimated) velocity & (m/s) \\ \(x_{\mathrm{o}}\) & Observed (measured) position & (m) \\ \(v_{\mathrm{o}}\) & Observed (measured) velocity & (m/s) \\ \(\alpha\) & Filter gain for \(x_{\mathrm{s}}\) with respect to \(x_{\mathrm{o}}\) & Dimensionless \\ \(\beta\) & Filter gain for \(v_{\mathrm{s}}\) with respect to \(x_{\mathrm{o}}\) & Dimensionless \\ \(\eta\) & Filter gain for \(x_{\mathrm{s}}\) with respect to \(v_{\mathrm{o}}\) & Dimensionless \\ \(\dot{\theta}\) & Filter gain for \(v_{\mathrm{s}}\) with respect to \(v_{\mathrm{o}}\) & Dimensionless \\ \(\dot{\mathrm{(})}\) & Forecasts & \\ \(\dot{\mathrm{(})}\) & Estimates & \\ \(()^{\mathrm{T}}\) & Transpose of matrix & \\ \(()^{-1}\) & Inversion of matrix & \\ \(\mathbf{x}\) & State vector of target composed of position and velocity & \\ \(\mathbf{z}\) & Measurement vector & \\ \(\mathbf{F}\) & Transition matrix from \(k\) to \(k+1\) & \\ \(\mathbf{P}\) & Error covariance matrix with respect to \(\mathbf{x}\) & \\ \(\mathbf{Q}\) & Covariance matrix of process noise & \\ \(\mathbf{K}\) & Kalman gain matrix & \\ \(\mathbf{B}\) & Covariance matrix of measurement noise & \\ \hline \hline \end{tabular}
\end{table}
Table 1: List of symbols.