## 2 GPS Attitude Baseline Determination
### Integer Ambiguity Resolution  
The results of the float solution are real-valued DD ambiguities and their covariance matrix. However, the true ambiguities are integer valued, and they need to be found within the solution space, which is spanned by the estimated float ambiguities and their covariances. This process is basically a search problem, called integer ambiguity resolution, and it is the key to enable mm-accuracies for the baseline parameters. In our approach, three steps are performed (see also Figure 1) to resolve the ambiguities correctly:  
Step (i): In the first step of the integer ambiguity resolution, the MLAMBDA method [33] is applied. The MLAMBDA method is an ambiguity search technique in the ambiguity domain and an extension of the well-known LAMBDA method [21], leading to faster computation times [33]. The MLAMBDA method is based on an integer least squares problem to obtain the estimates of the DD integer ambiguities solving the minimization problem [21]:  
\[min\left(\hat{\mathbf{N}}-\mathbf{N}\right)^{T}\mathbf{Q}_{\hat{\mathbf{N}}}^ {-1}\left(\hat{\mathbf{N}}-\mathbf{N}\right)\;\;\;\text{with }N\in Z^{n} \tag{14}\]  
where \(\mathbf{N}\) is the integer ambiguity vector, \(\hat{\mathbf{N}}\) is the float ambiguity vector and \(\mathbf{Q}_{\hat{\mathbf{N}}}\) represents its covariance matrix. The inputs for the MLAMBDA method are the DD float ambiguities and their covariance matrix. Due to the aid of the additional information about the attitude, known from the Pos/IMU/Mag solution (Section 3), the baseline float solution is more accurate, compared to a GPS-only float solution. The more accurate the float ambiguities are, the more likely is a fast and correct ambiguity resolution with the MLAMBDA method. Thus, there is a good chance that the ambiguities can already be fixed at this step of the ambiguity resolution process in our approach. As outputs of the MLAMBDA method, we consider ten sets of ambiguity candidates. To make a decision if any of these sets can be fixed, we first apply the simple ratio test [36], where the squared norm of the ambiguity residuals of the best set of ambiguities (\(R_{1}\)) and the second best set of ambiguities (\(R_{2}\)) is compared. Only if the quotient \(R_{2}/R_{1}\) exceeds a threshold of three, the ambiguities of the best set will be fixed to integer values. Step (ii): In case the ambiguities could not be fixed within Step (i), further effort is needed. The ten best solutions from the MLAMBDA method are considered further. In the first instance, we try to perform an elimination of single ambiguity parameters, which probably prevent the fixing of all of the other ambiguities. For example, in the case that all ten sets of ambiguities are equal except the ambiguity of one DD observation, this one varying ambiguity parameter is omitted. In the case that more than one ambiguity parameter is varying in the ten sets of ambiguities, always one ambiguity parameter is eliminated in a random order, and the MLAMBDA method is applied again, until the ratio test is positive or a maximum number of iterations is reached. Due to the elimination of single ambiguity parameters, some of the ten sets of ambiguity candidates can be omitted now, since they probably became equal to other ambiguity sets. Step (iii): In the last step of the ambiguity estimation, a transformation into the coordinate domain is performed. This means that all of the remaining sets of ambiguity candidates are used to determine baseline parameters. Afterwards, these baseline parameters are tested in the ambiguity function method (AFM) [32], which is a search technique based on a trigonometric cost function, the ambiguity resolution function (ARF):  
\[\text{ARF}(b_{x},b_{y},b_{z})=\sum_{j=1}^{n}cos2\pi\left[\phi_{obs}^{jk}-\phi_ {calc}^{jk}\left(b_{x},b_{y},b_{z}\right)\right] \tag{15}\]  
By use of the determined baseline parameters, all observed DD carrier phases \(\phi_{obs}^{jk}\) can also be calculated (\(\phi_{calc}^{jk}\)).