# Loosely Coupled GNSS and UWB with INS Integration for Indoor/Outdoor Pedestrian Navigation +
## 3 Material and Methods
### Methods
be computed applying a first-order Taylor series expansion to the equation of motion parameterized in the northeast-down (NED) navigation frame and can be simplified in the following form [30]:  
\[\delta\dot{\mathbf{p}}=\mathbf{T}^{\prime}\delta\mathbf{p}^{n}+T\delta\mathbf{v}^{n} \tag{3}\]  
\[\delta\dot{\mathbf{v}}^{n}=\Big{[}\left(\mathbf{C}_{b}^{n}\dot{\mathbf{p}}^{\prime}\right) \Big{]}\delta\mathbf{v}_{nb}^{n}+\mathbf{C}_{b}^{n}\delta\dot{\mathbf{p}}^{\prime}-\Big{[} 2\left(\mathbf{\omega}_{ie}^{n}+\mathbf{\omega}_{en}^{n}\right)\Big{]}\delta\mathbf{v}^{n}- \Big{[}\left(2\delta\mathbf{\omega}_{ie}^{n}+\delta\mathbf{\omega}_{en}^{n}\right) \Big{]}\mathbf{v}^{n}+\delta\mathbf{g}^{n} \tag{4}\]  
\[\delta\dot{\mathbf{v}}_{nb}^{n}\approx-\Big{[}\left(\mathbf{\omega}_{in}^{n}\right)^{ \ast}\Big{]}\delta\mathbf{v}_{nb}^{n}+\delta\mathbf{\omega}_{in}^{n}-\mathbf{C}_{b}^{n} \delta\mathbf{\omega}_{ib}^{b} \tag{5}\]  
where the matrix \(\mathbf{T}\) relates the relates the position errors to their time derivatives, \(\mathbf{C}_{b}^{n}\) is the rotation matrix, \(\delta\dot{\mathbf{p}}^{\prime}\) and \(\delta\mathbf{\omega}_{ib}^{b}\) are the accelerometer and rate gyro output errors and \(\delta\mathbf{\psi}_{nb}^{n}\) is the attitude errors resolved about the NED frame. Therefore, starting from an initial estimation \(\delta\dot{\mathbf{x}}_{k-1k-1}\)(associated with its var-covariance matrix \(\mathbf{P}_{k-1|k-1}\)), a predicted estimation \(\delta\dot{\mathbf{x}}_{k|k-1}\)(with a computed state var-covariance matrix \(\mathbf{P}_{k|k-1}\)) is obtained by applying the state transition model:  
\[\delta\dot{\mathbf{x}}_{k|k-1}=\mathbf{f}\big{(}\delta\dot{\mathbf{x}}_{k-1|k-1}\big{)}+ \mathbf{w}_{k}. \tag{6}\]  
\[\delta\dot{\mathbf{x}}_{k|k-1}^{T}=\mathbf{F}\big{(}\delta\dot{\mathbf{x}}_{k-1|k-1}\big{)} +\mathbf{Q}_{k}. \tag{7}\]  
with the state vector composed by the INS state errors (position \(\mathbf{p}\), velocity \(\mathbf{v}\), attitude \(\mathbf{\psi}\), acceleration and angular velocity biases \(\mathbf{b}_{a}\) and \(\mathbf{b}_{g}\)). Thus, it can be defined as:  
\[\delta\mathbf{x}^{T}=\Big{[}(\delta\mathbf{p})^{T}\quad(\delta\mathbf{\psi}_{nb})^{T} \quad(\delta\mathbf{\psi}_{nb})^{T}\quad\big{(}\mathbf{b}_{a}^{b}\big{)}^{T}\quad \big{(}\mathbf{b}_{g}^{b}\big{)}^{T}\Big{]}\quad(15\times 15) \tag{8}\]  
and the state noise vector \(\mathbf{w}_{k}\), represented by its var-covariance matrix \(\mathbf{Q}_{k}\), can be defined as:  
\[\mathbf{w}_{k}^{T}=\Big{[}\mathbf{0}_{1\times 3}\quad\Big{(}\mathbf{C}_{b}^{n}\mathbf{w}_{a} \Big{)}^{T}\quad\Big{(}\mathbf{C}_{b}^{n}\mathbf{w}_{g}\Big{)}^{T}\quad\Big{(}\mathbf{C}_{ b}^{n}\mathbf{\mu}_{a}\Big{)}^{T}\quad\Big{(}\mathbf{C}_{b}^{n}\mathbf{\mu}_{g}\Big{)}^{T} \Big{]}^{T}\quad(15\times 15) \tag{9}\]  
with \(\mathbf{C}_{b}^{n}\) rotation matrix and parameter of \(\mathbf{w}_{k}^{T}\) as defined in [31]. Obtaining the state noise covariance matrix \(\mathbf{Q}_{k}\) and the state transition matrix \(\mathbf{F}_{k}\), it is possible to compute the state error covariance prediction \(\mathbf{P}_{k|k-1}\):  
\[\mathbf{P}_{k|k-1}=\mathbf{F}_{k}\mathbf{P}_{k-1|k-1}\mathbf{F}_{k}^{T}+\mathbf{Q}_{k}\quad\quad(1 5\times 15) \tag{10}\]  
In a closed loop configuration, at each time of INS integration, the state vector is null and is not propagated forward in time. At the time of update, thus when anindependent solution from the GNSS or UWB system is available, we have the new measurements which are used to correct the predicted solution \(\delta\dot{\mathbf{x}}_{k|k-1}\).