# Calibration of Magnetometers with GNSS Receivers and Magnetometer-Aided GNSS Ambiguity Fixing
## 3 Calibration of Magnetic Flux Sensors
### Attitude Determination with Three GNSS Receivers
#### 3.2.3 Integer Ambiguity Fixing Using Prior Information on Baseline Coordinates
\right)+\eta_{z_{n}}\right). \tag{26}\]The notation is further simplified by defining \(\bar{A}:=P_{\bar{A}^{1}}^{\perp}\bar{H}\). The least-squares estimate of both partially fixed baselines follows as  
\[\left( \begin{array}{c}\tilde{\bar{b}}_{12}^{\rm n}(t_{n})\\ \tilde{\bar{b}}_{13}^{\rm n}(t_{n})\end{array} \right)=\left(\bar{A}^{\rm T} \Sigma^{-1}\bar{A}\right)^{-1}\bar{A}^{\rm T}\Sigma^{-1}\left(z_{n}-\bar{A}^{2 }\left( \begin{array}{c}\tilde{N}_{12}^{2}\\ \tilde{N}_{13}^{2}\end{array} \right)\right). \tag{27}\]  
The accuracy of the baseline estimates depends on the number of fixed ambiguities: if less than three ambiguities are fixed per baseline, the pseudoranges are needed and the accuracy is limited by the pseudorange multipath. If at least three ambiguities are fixed per baseline, the baselines can be estimated solely with carrier phase measurements resulting in a much higher accuracy. The efficiency of the ambiguity fixing is substantially improved by the integer decorrelation of Teunissen [1]. The integer decorrelation transformation \(Z\) is applied to the double difference ambiguities, i.e.,  
\[Z\left( \begin{array}{c}N_{12}\\ N_{13}\end{array} \right)=Z^{1}\left( \begin{array}{c}N_{12}^{1}\\ N_{13}^{1}\end{array} \right)+Z^{2}\left( \begin{array}{c}N_{12}^{2}\\ N_{13}^{2}\end{array} \right):=\left( \begin{array}{c}\tilde{N}_{12}^{1}\\ \tilde{N}_{13}^{1}\end{array} \right)+\left( \begin{array}{c}\tilde{N}_{12}^{2 }\\ \tilde{N}_{13}^{2}\end{array} \right). \tag{28}\]  
The integer decorrelation transformation is invertible and preserves the integer property, i.e., the uncorrelated double difference ambiguities can be obtained from the decorrelated ones by  
\[\left( \begin{array}{c}N_{12}\\ N_{13}\end{array} \right)=Z^{-1}\left(\left( \begin{array}{c}\tilde{N}_{12}^{ 1}\\ \tilde{N}_{13}^{1}\end{array} \right)+\left( \begin{array}{c}\tilde{N}_{12}^{2 }\\ \tilde{N}_{13}^{2}\end{array} \right)\right). \tag{29}\]  
The decorrelated fixed ambiguities \(\tilde{N}_{1r}^{1}\) and float ambiguities \(\tilde{N}_{1r}^{2}\) are introduced in the measurement model of Equation (25), which is rewritten as  
\[\left(z_{n}-\underbrace{A(\lambda)Z^{-1}}_{\tilde{A}}\left( \begin{array}{c} \tilde{N}_{12}^{2}\\ \tilde{N}_{13}^{2}\end{array} \right)\right)=\bar{H}\left( \begin{array}{c} \tilde{b}_{12}^{\rm n}(t_{n})\\ \tilde{b}_{13}^{\rm n}(t_{n})\end{array} \right)+\underbrace{A(\lambda)Z^{-1}} _{\tilde{A}}\left( \begin{array}{c}\tilde{N}_{12}^{1}\\ \tilde{N}_{13}^{1}\end{array} \right)+\eta_{z_{n}}. \tag{30}\]  
The decorrelated _float_ ambiguities are eliminated by a projection on the space orthogonal of \(\bar{A}\). The least-squares baseline estimates for partially fixed decorrelated ambiguities follows as  
\[\left( \begin{array}{c}\tilde{\bar{b}}_{12}^{\rm n}(t_{n})\\ \tilde{\bar{b}}_{13}^{\rm n}(t_{n})\end{array} \right)=\left(\bar{\bar{A}}^{ \rm T}\Sigma^{-1}\bar{\bar{A}}\right)^{-1}\bar{\bar{A}}^{\rm T}\Sigma^{-1} \left(z_{n}-\bar{A}\left( \begin{array}{c}\tilde{N}_{12}^{2}\\ \tilde{N}_{13}^{2}\end{array} \right)\right), \tag{31}\]  
with \(\bar{\bar{A}}=P_{\bar{A}}^{\perp}\bar{H}\). The baseline estimates are used to constrain the search interval in Equations (22) and (23). The reduction of search space enables a much more reliable ambiguity fixing. **4. Fast Initialization of GNSS Attitude Ambiguity Fixing with Calibrated Magnetometers**  
In this section, we describe the benefit of magnetometer-based attitude information for kinematic GNSS-based attitude determination. In a first step, we derive the attitude from calibrated magnetic flux measurements. It is assumed that precise estimates of the sensor bias \(\hat{b}_{m^{\rm S}}\) and of the magnetic flux strength \(\hat{m}^{\rm S}\) are available, e.g.,using a previous calibration.