## 3 Integrated Navigation
### GNSS/INS Integration
#### 3.1.1 Loosely Coupled Approach  
The loose coupling (LC) strategy is also referred to as \"decentralized\" and includes a Kalman Filter (KF) to combine INS and GNSS parameters. Another KF or a LS estimator is used to compute the GNSS navigation solution. The LC scheme is showed in Figure 1. Although the LC approach is relatively simple to implement, the main drawback of the LC approach is that if there are insufficient satellites to compute a standalone GNSS solution, the inertial system is not updated. This ultimately results in higher positioning errors [27] relative to the tight coupling approach (details to follow).  
Figure 1: Loosely coupled scheme.  
To compute the GNSS fix, a LS estimator is preferred herein to simplify a direct LC/TC comparison. Specifically, by using least-squares for LC, the results will be the same as in the TC case so long as enough satellites are available to compute a solution.  
The inertial solution is obtained by applying the mechanization equations for a strapdown configuration to the accelerations and angular rates from the IMU. For this work, the INS mechanization is implemented in the local East-North-Up (ENU) frame. Details of the mechanization equations are widely available in the literature (e.g., [26]) and are thus excluded here.  
The difference between the INS and weighted least-squares (WLS) GNSS position and velocity are used as input measurements to the KF. The WLS covariance matrix is used as measurements covariance matrix \(R\) for the KF:  
\[R=\mathrm{cov}\left(\underline{\Delta x}\right)=\left(H^{T}\!W\!H\right)^{-1} \tag{4}\]  
Implicit in the above is that only the sub-matrices corresponding to the states being used in the update are extracted from the WLS covariance matrix (details follow shortly).  
The state vector of the combined GNSS/INS KF in LC architecture is:  
\[\underline{\delta x}=\left[\underline{\delta P^{u}}\quad\underline{\delta v ^{u}}\quad\underline{\varepsilon}^{u}\quad\underline{\delta b}_{a}\quad \underline{\delta b}_{s}\quad\underline{\delta S}_{a}\quad\underline{\delta S }_{s}\right]^{\!\!  
In contrast to the LC case, the difference between the measured and INS-predicted pseudorange and Doppler observables is used as input measurements to KF. The associated measurement covariance matrix is defined taking into account the inherent accuracies of GNSS measurements and the elevation-dependent accuracy as in the LC case (see Equation (2)).  
The TC KF state vector has the same 21 states as for the LC case (see Equation (5)), but is also augmented with the GPS receiver clock bias and drift. If GLONASS measurements are included, the GPS-GLONASS inter-systems time offset must also be considered and in this work it is modeled as a random constant.  
Both loose and tight strategies are herein implemented in closed loop configuration meaning the navigation, bias and scale factor error states output from the KF are used to correct INS inputs. The closed loop configuration is necessary when low performance INS is used to reduce the inertial error growth [6], which in turn, satisfies the small angle assumptions used to derive the INS error equations.