## 2 Proposed Position Tracking System Based on Composite Sensors
### EKT-Based MEMS IMU and UWB Integrated Positioning Algorithm  
Integrated positioning algorithms of MEMS IMU and UWB sensors have been studied by using various Kalman filters, such as the extended Kalman filter [11; 19] and complementary Kalman filter [20; 21]. We applied [PERSON]'s integrated positioning algorithm with the extended Kalman filter (EKT) [11]. The Kalman filter of a recursive filter estimates the state of a linear dynamic system containing noises that can recursively process the measurement data containing measurement noises, thus being widely used across various fields, including computer vision, robotics, and radar. As shown in Figure 3, the extended Kalman filter consists of five steps, largely comprising a step for predicting the current state and a step for estimating a more accurate value, including the measurement value [6]. We consider a 2D positioning frame for a platform that moves on an x-y plane with a body-mounted IMU and an UWB tag sensor. This means that two orthogonal accelerometers measure platform body acceleration on the x and y axes on the fixed \(E\)_frame_, and measure body angular velocity on the I frame with one gyro along the z axis. The state variables consist of position errors, velocity errors, and attitude angle errors and additional state variables for IMU sensors consists biases (**b**) of accelerations and gyros and misalignment errors. Among them, 8 state error variables \(\mathbf{\epsilon}_{z}\) can be defined [11], as follows;  
\[\mathbf{\epsilon}_{z}=[e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},e_{7},e_{8}]^{T} \tag{18}\]  
where  
\[e_{1}=x_{p}-\hat{x}_{p},\ \ \ e_{2}=v_{xp}-\hat{v}_{xp},\ \ \ e_{3}=b_{x}-\hat{b}_{x},\ \ \ e_{4}=y_{p}-\hat{y}_{p} \tag{19}\]  
\[e_{5}=v_{yp}-\hat{v}_{p},\ \ \ e_{6}=b_{y}-\hat{b}_{y},\ \ \ e_{7}=\psi-\hat{y}_{x},\ \ \ e_{8}=b_{x}-\hat{b}_{x}\ . \tag{20}\]  
By applying the small perturbations in the IMU positioning system, the error dynamic equation can be derived by the state-space form. \[\overset{\mathbf{\kappa}}{\mathbf{x}}_{z}=F_{s}(t)\mathbf{\epsilon}_{z}+\eta \tag{21}\]  
Here, the 8 \(\times\) 8 sparse state-space matrix \(F_{s}(t)\) on any time can be defined by  
\[F_{s}(t)=\left[ \begin{array}{cccccc}0&1&0&0&0&0&0&0\\ 0&0&-\cos(\psi)&0&-\omega_{z}&\sin(\psi)&f_{1}\big{(}\psi,\mathbf{a}^{B}, \mathbf{b}\big{)}&\hat{v}_{yp}\\ 0&0&0&0&0&0&0\\ 0&0&0&0&1&0&0\\ 0&-\omega_{z}&-\sin(\psi)&0&0&-\cos(\psi)&f_{2}\Big{(}\psi,a^{B}_{x},a^{B}_{ y}\Big{)}&-\phi_{xp}\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&-1\\ 0&0&0&0&0&0&0\\ \end{array} \right], \tag{22}\]  
where \(f_{1}\big{(}\psi,\mathbf{a}^{B},\mathbf{b}\big{)}\) and \(f_{2}\big{(}\psi,\mathbf{a}^{B},\mathbf{b}\big{)}\) are computed by the error of body accelerations \(\mathbf{a}^{B}\) and biases \(\mathbf{b}\) to the yaw \(\psi\) on z axis as the following. \[\left[ \begin{array}{c}f_{1}\big{(}\psi,\mathbf{a}^{B},\mathbf{b}\big{)}\\ f_{2}\big{(}\psi,\mathbf{a}^{B},\mathbf{b}\big{)}\end{array} \right]=\left[ \begin{array}{cc}-\sin(\psi)&\cos(\psi)\\ \cos(\psi)&-\sin(\psi)\end{array} \right]\left[ \begin{array}{c}a^{B}_{x}- \hat{b}_{x}\\ a^{B}_{y}-\hat{b}_{y}\end{array} \right] \tag{23}\]  
Next, the state-space parameter \(\eta\) is a vector for gyro measurement defined by  
\[\eta=\big{[}0,\eta_{1},b_{ax},0,\eta_{2},b_{ay},g_{ang},g_{rate}\big{]}^{T} \tag{24}\]  
where \(\big{(}g_{ang},g_{rate}\big{)}\) are gyro angle random walk (ARW) and gyro rate random walk (RRW) and \(\big{(}b_{ax},b_{ay}\big{)}\) are accelerometer biases stabilities. \((\eta_{1},\eta_{2})\) are computed by the accelerometer noises \((n_{ax},n_{ay})\) to the yaw \(\psi\) on the z axis, as follows. \[\left[ \begin{array}{c}\eta_{1}\\ \eta_{2}\end{array} \right]=-\left[ \begin{array}{cc}\cos(\psi)&-\sin(\psi)\\ \sin(\psi)&\cos(\psi)\end{array} \right]\left[ \begin{array}{c}n_{ax}\\ n_{ay}\end{array} \right]+\left[ \begin{array}{c}\vartheta_{yp}g_{ang}\\ -\vartheta_{xp}g_{ang}\end{array}