# Loosely Coupled GNSS and UWB with INS Integration for Indoor/Outdoor Pedestrian Navigation +
## 3 Material and Methods
### Methods  
The navigation state estimation of the proposed algorithm is based on an EKF and is composed by position (latitude, longitude, altitude), velocity vectors along the north, east and down axes, attitude angles (roll, pitch, heading), acceleration and gyroscopic biases. The filter is used in a feedback form so that when a new measurement is available from a sensor, the error is computed using the Kalman filter which is then used to correct the inertial sensor measurements and navigation parameters. The EKF is composed by a set of equations applied in two steps recursively: prediction (time update or propagation) and update (measurement aiding or correction). These two steps are based on two estimation models: the state transition model and the measurement model. The state transition model describes the temporal behavior of the states over time and it is used to predict the state vector starting from a previous estimation, whereas the measurement model describes how observations relate to the states being estimated and is used to correct the state vector. State transition model:  
\[\mathbf{x}_{ijk-1}=\mathbf{f}(\mathbf{x}_{k-1|k-1})+\mathbf{w}_{k} \tag{1}\]  
Measurement model:  
\[\mathbf{y}_{ijk}=\mathbf{h}\big{(}\mathbf{x}_{ijk-1}\big{)}+\mathbf{n}_{k} \tag{2}\]  
where \(\mathbf{w}_{k}\) is the state noise vector which follows a normal distribution N(0,Q) and \(\mathbf{n}_{k}\) is the measurement noise vector follows a normal distribution N(0,R). Finally \(\mathbf{f}(x)\) and \(\mathbf{h}(x)\) are two differentiable non-linear functions. During the prediction phase, the state vector is estimated starting from a previous estimation, through the equations of motion. In the case of an GNSS/UWB-INS integration, the estimate should be represented by the time evolution of position, velocity and attitude (\(\dot{\mathbf{p}}\quad\dot{\mathbf{v}}\quad\dot{\mathbf{\psi}}_{nb}\)) of the inertial platform using the mechanization equations as representation of the dynamic of the system. Since these equations are not linear, a linearization of the mechanization equations is required. In particular, the set of equations which represent the time evolution of the INS error (\(\delta\dot{\mathbf{p}}\quad\delta\dot{\mathbf{v}}\quad\delta\dot{\mathbf{\psi}}_{nb}\)) are linear and, therefore, are used to represent the state transition model for GNSS/UWB-INS integration \(\mathbf{f}\big{(}\delta\dot{\mathbf{x}}_{k-1|k-1}\big{)}\) and consequentially to create the state transition matrix \(\mathbf{F}_{k}\) of the INS errors. The error propagation can \begin{table}
\begin{tabular}{c c c c} \hline  & **UWB** & **IMU** & **Processing Unit** \\ \hline
**NEO-MST Ublox** & **Pozyx** & **Pozyx** & **Raspberry Pi 3 Model b+** \\ \hline Constellation: & & 3 axes Accelerometer, & \\ GPS/GLONASS & 3D position accuracy: 30 cm & Gyroscope and & OS: GNU/Linux \\ Galileo/BeiDou & & Magnetometer & \\ \hline
2D position error: & Antenna: Decawave & Roll 2 deg & \\
2.5 m & DW1000 & Pitch 5 deg & Ports: 4 USB 2.0; 1 Ethernet \\  & & Heading 4 deg & \\ \hline Max navigation update rate: & Max Ranging update rate: & \\
5 Hz & 140 Hz & Max Update Rate: 100 Hz & RAM: 512 MB \\ \hline \multirow{2}{*}{-} & Max Positioning update rate: & \multirow{2}{*}{-} & \multirow{2}{*}{-} \\  & 80 Hz & & \\ \hline \multicolumn{3}{c}{Typical LOS range: 30 m} & & \\ \end{tabular}
\end{table} 
Table 1: Performance specification of multisensor system.