# OBU for Accurate Navigation through Sensor Fusion in the Framework of the EMERGE Project
## 3 Sensor Fusion: Loosely Coupled Algorithm Implementation
### The Initialisation Loop
, \tag{11}\]

is given by the difference between the position and velocity solution of the GNSS and the corrected inertial navigation solution, plus a term to account for the displacement between the inertial platform and the GNSS antenna, \(\mathbf{l}_{ba}^{b}\). Measurement matrix:

\[\mathbf{H}^{c}= \begin{pmatrix}[\hat{\mathbf{C}}_{b}^{n}\hat{\mathbf{b}}_{ba}^{ b}]_{x}&0&-\mathbf{I}&0&0\\ ([\hat{\mathbf{C}}_{b}^{n}(\hat{\mathbf{\omega}}_{ib}^{b}\times\mathbf{l}_{ba} ^{b})-[\mathbf{\omega}_{ie}^{n}]_{x}\hat{\mathbf{C}}_{b}^{n}\hat{\mathbf{l}}_{ ba}^{b}]_{x}&-\mathbf{I}&0&0&\hat{\mathbf{C}}_{b}^{n}[\mathbf{l}_{ba}^{b}]_{x} \end{pmatrix} . \tag{12}\]

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