## 2 GPS Attitude Baseline Determination
### GPS Baseline Float Solution
The choice of the values for \(a\) and \(b\) depends on various effects (antennas, receivers, antenna near-fields, _etc._), which may be different for every measurement setup. In order to find realistic values representing the measurement noise of the GPS observations on our system during kinematic applications, we analyzed different flights. Based on the covariance matrices of the parameters and the residuals of the measurements, we found that _a_\(=2\) mm and _b_\(=2\) mm lead to the best results for the parameter estimation with our setup. Finally, in the complete measurement noise matrix \(\mathbf{R}_{k}\), also the DD correlations are regarded:  
\[\Sigma_{\phi,P}=diag\left(\sigma_{\phi,P}^{\text{ 1 2 }}\ \ldots\ \sigma_{\phi,P}^{\text{ 2 }}\right) \tag{8}\] \[\mathbf{R}_{k}= \begin{bmatrix}\mathbf{D}\Sigma_{\phi}\mathbf{D}^{ T}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{D}\Sigma_{P}\mathbf{D}^{T}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\sigma_{\Delta\bar{x}_{b}}\mathbf{I}_{3\times 3}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\sigma_{s}\end{bmatrix}  \tag{9}\]  
where \(\sigma_{\Delta\bar{x}_{b}}\) is the noise of the prior information and \(\sigma_{s}\) the noise of the baseline length constraint. System dynamics model: The standard total state space EKF time update is the following:  
\[\mathbf{x}_{SD,k}^{-}=\mathbf{\Psi}_{k}\mathbf{x}_{SD,k-1}^{+},\ \ \mathbf{P}_{k}^{-}= \mathbf{\Phi}_{k}\mathbf{P}_{k-1}^{+}\mathbf{\Phi}_{k}^{T}+\mathbf{G}_{k} \mathbf{Q}_{k-1}\mathbf{G}_{k}^{T} \tag{10}\]We use a simple random walk model as the system dynamics model. Thus, the state transition matrix is composed of:  
\[\boldsymbol{\Psi}_{k}= \begin{bmatrix}\mathbf{I}_{3\times 3}&\Delta t&\mathbf{I}_{3 \times 3}&\mathbf{0}_{3\times m}\\ \mathbf{0}_{3\times 3}&\mathbf{I}_{3\times 3}&\mathbf{0}_{3\times m}\\ \mathbf{0}_{m\times 3}&\mathbf{0}_{m\times 3}&\mathbf{I}_{m\times m}\end{bmatrix}  \tag{11}\]  
and \(\mathbf{G}_{k}\) is given by:  
\[\mathbf{G}_{k}= \begin{bmatrix}\frac{1}{2}\Delta t^{2}&\mathbf{I}_{3\times 3}& \mathbf{0}_{3\times m}\\ \Delta t&\mathbf{I}_{3\times 3}&\mathbf{0}_{3\times m}\\ \mathbf{0}_{m\times 3}&\Delta t&\mathbf{I}_{m\times m}\end{bmatrix}  \tag{12}\]  
The process noise matrix \(\mathbf{Q}_{k}\) is a diagonal matrix consisting of two types of process noise. The process noise for the first time derivative of the baseline parameters is set to \(\sigma_{\Delta\dot{x}}\) = 1 m/s. This choice reflects the approximate dynamic capabilities of the UAV, but does not smooth the baseline parameters too much. In contrast, the ambiguity parameters are assumed to be constant. However, to allow for a better reaction of the ambiguity parameters on changing satellite configurations, the noise parameter for the ambiguities is not set to zero, but to a very small value: \(\sigma_{N}=1\cdot 10^{-4}\) cycles. \[\mathbf{Q}_{k}=diag\left(\sigma_{\Delta\dot{x}}^{2}\;\sigma_{\Delta\dot{x}}^ {2}\;\sigma_{\Delta\dot{x}}^{2}\;\sigma_{N}^{1\;2}\;\ldots\;\sigma_{N}^{m2}\right) \tag{13}\]  
Since the initialization of the ambiguity parameters, which is based on the difference between the SD pseudoranges and the SD carrier phases, is very imprecise, the process noise for new initialized ambiguities is set to a very large value of \(\sigma_{N}\) = 100 cycles. In this way, new initialized ambiguities do not have any influence on the estimation of the other parameters, but benefit from the current float solution. In the final step of the float solution, the estimated SD float ambiguities and their covariance matrix have to be transformed to DD values, applying the \(\mathbf{D}\)-matrix again.