# An Algorithm to Assist the Robust Filter for Tightly Coupled RTK/INS Navigation System
## 2 Materials and Methods
### The Fixed Solution  
The lambda algorithm is used to search for integer ambiguity solutions after EKF provides the float solution. This section will introduce how to obtain the corresponding INS solution based on the integer ambiguity solution, as many related articles do not introduce the details. We use \(\chi\) to denote a vector containing the INS states and the DD ambiguity, as Equation (25) shows:  
\[\chi = \begin{bmatrix}\chi_{\text{INS}}\\ \chi_{\text{DO}}\end{bmatrix}  \tag{25}\] \[= \begin{bmatrix}\xi\\ v^{\text{h}}\\ p\\ e^{\text{b}}\\ \mathbf{\
abla}^{\text{b}}\\ D_{\mathbf{\kappa}}\end{bmatrix} ,\]  
where \(\xi\) refers to the INS attitude. Notably, the INS-related components in \(\chi\) are not the same as those in Equation (7). The former reflects the INS states, whereas the latter reflects the states' error. We use \(M\) to denote the covariance matrix of \(\chi\) as shown in Equation (26):  
\[M= \begin{bmatrix}M_{\text{INS}}&M_{\text{INS,DD}}\\ M_{\text{DD,INS}}&M_{\text{DD}}\end{bmatrix} . \tag{26}\]  
We use the notation \(\hat{\chi}_{\text{DD}}\) and \(\tilde{\chi}_{\text{DD}}\) to denote the float solution and the fixed solution, respectively. According to [69], the fixed INS solution \(\tilde{\chi}_{\text{INS}}\)can be calculated based on the float INS solution \(\hat{\chi}_{\text{INS}}\), as Equation (27) shows:  
\[\tilde{\chi}_{\text{INS}} =\hat{\chi}_{\text{INS}}-M_{\text{INS,DD}}M_{\text{DD}}^{-1}(\hat {\chi}_{\text{DD}}-\hat{\chi}_{\text{DD}}), \tag{27}\] \[\tilde{M}_{\text{INS}} =\hat{M}_{\text{INS}}-M_{\text{INS,DD}}M_{\text{DD}}^{-1}M_{\text {INS,DD}}^{\text{T}}.\]Equation (27) tells us we should prepare the five values on the right of the equal sign for the fixed INS solution. It is critical to introduce how to prepare the five values. \(\hat{\mathbf{\chi}}_{\text{INS}}\) denotes the INS states that are corrected by the estimated error states given by EKF. The other four values on the right of the minus form a correction. We substract the correction from \(\hat{\mathbf{\chi}}_{\text{INS}}\), the float solution, to obtain \(\hat{\mathbf{\chi}}_{\text{INS}}\), the fixed solution. Here, we call the correction a \"fixed correction.\" \(\hat{\mathbf{\chi}}_{\text{DD}}\) denotes the DD ambiguity states. We can calculate \(\hat{\mathbf{\chi}}_{\text{DD}}\) based on the transition matrix \(\mathbf{D}\) and the SD ambiguity states estimated by EKF, as Equation (28) shows:  
\[\hat{\mathbf{\chi}}_{\text{DD}}=\mathbf{D}\hat{\mathbf{\kappa}}. \tag{28}\]  
\(\hat{\mathbf{\chi}}_{\text{INS}}\) and \(\hat{\mathbf{\chi}}_{\text{DD}}\) are both related to EKF; they are the float solution for the TC RTK/INS integration. The lambda algorithm will be used following the EKF process. The lambda algorithm searches for the appropriate integer around the float DD ambiguity, which means \(\hat{\mathbf{\chi}}_{\text{DD}}\), the fixed DD ambiguity solution, is provided by the lambda algorithm. The covariance matrix \(\mathbf{M}\) derives from the posterior estimation covariance matrix \(\hat{\mathbf{P}}\) in EKF. Notably, the EKF state vector consists of the INS states' error and the SD ambiguity, whereas \(\mathbf{\chi}\) contains the INS states and the DD ambiguity. \(\hat{\mathbf{P}}\) provides the covariance estimation of the INS states' error, whereas \(\mathbf{M}\) reflects the covariance of the INS states. The inconsistency can result in a difference between the corresponding parts of \(\mathbf{M}\) and \(\hat{\mathbf{P}}\). We should calculate the proper \(\mathbf{M}\) based on \(\hat{\mathbf{P}}\). In contrast, \(\hat{\mathbf{P}}\) provides the covariance estimation of the SD ambiguity states, which means we can focus on the inconsistency caused by INS.