# Performance Characterization of GNSS/IMU/DVL Integration under Real Maritime Jamming Conditions
## 3 Methods
In the case of a loosely coupled integration, an equivalent position solution error covariance can be calculated by using the geometry matrix from the iterated LS solution and a corresponding weight matrix, where the weights are assigned following the assumptions regarding the quality of each link. A simplification, commonly used in practice (and which we actually assume within the equivalent design of the tightly coupled architecture) is that the measurements from the different satellites are uncorrelated. The measurement weighting matrix can then be constructed as an inverse of the diagonal measurement noise covariance matrix. As the GNSS Doppler measurements are usually of a relatively high quality, a constant noise assumption can be easily adopted with an equivalent circular covariance approximation for Doppler LS solution in a loosely coupled configuration. Finally, the DVL measurement model can be considered as the X-Y velocity measurement in the coordinate frame of the sensor including the lever arm compensation with respect to the IMU. In addition to the actual 2D velocity measurement, a constraint along the body vertical axis of the vessel can be employed (velocity projection in the body frame) as one can assume the vertical velocity to be zero on average. The constraint has been implemented within the UKF framework as so-called \"pseudo-measurement\" by extending the true sensor measurement with a third component, setting the measurement to zero with some associated measurement noise. This vertical velocity measurement significantly decreases the vertical position drift by reducing it from being cubic in time to becoming quasi-linear in time. The measurement model for the DVL speed observation \(v_{DVL}\) is expressed as  
\[v_{DVL}=q^{*}\otimes v\otimes q+\omega\times l_{DVL}+\epsilon_{DVL} \tag{6}\]  
where \(l_{DVL}\) refers to the baseline between the IMU and the DVL positions within the local ship frame. Naturally, for lower-cost IMUs the navigation performance is strongly degraded due to a fast accumulation of the errors caused by sensor noises, biases, scale factor errors, etc. Moreover, for non-augmented IMU/GNSS system (e.g., a system without the magnetometer, a gyrocompass, or multiple GNSS antennas), the attitude and some of the inertial sensor errors become weakly observable, and the ability of the system to effectively estimate them is strongly conditioned on the dynamics of the vessel. For these reasons, the baseline observations (non-collinear vector measurements) were incorporated from three spatially distributed GNSS antennas to ensure that the attitude drift is constrained when baseline measurements are available. Recall that the attitude integration is the first integral of three subsequent integrals, which constitute the strapdown inertial mechanization and the attitude errors start to dominate the other error sources relatively fast. A baseline observation is considered to be valid if the RTK-based baseline determination between two antennas resulted in a fixed integer ambiguity solution; therefore, up to three baseline observations can be incorporated into the measurement model at each epoch depending on the quality of the RTK solutions. Here, baseline estimation is realized in a loosely coupled manner, following the methodology presented in [40]. Nonetheless, heading determination can be also tightly coupled to the positioningproblem for a higher exploitation of redundant information [41]. The advantage of a direct baseline vector observation model is that the heading becomes observable even with a single observation of non-vertical baseline (i.e., the baseline non-collinear with the local Earth gravity vector). Note that both pitch and roll angles are effectively observable via the coupling of the position measurements with the Earth gravity and only the heading information cannot be directly determined with a single GNSS antenna for typically limited vessel dynamics.