# Performance Characterization of GNSS/IMU/DVL Integration under Real Maritime Jamming Conditions
## 3 Methods  
The proposed hybrid navigation system employs the recursive Bayesian estimation (RBE) framework to combine the outputs of several sensors in order to obtain a single navigation solution. The RBE methods deal with the problem of estimating the changing-in-time state of the system applying a priori information about the underlying system dynamics and the observations from the sensors. The probabilistic paradigm has been widely applied to navigation and tracking problems, given its ability to accommodate inaccurate models as well as imperfect sensors [32]. In general, any RBE cycle is performed in two steps:  
**Prediction**: The a priori probability is calculated from the last a posteriori probability using the available process model. **Correction**: The a posteriori probability is calculated from the a priori probability using the measurement model and the current measurements. Among the multiple RBE methods, the Kalman filter (KF) is probably the most well-known solution for the navigation problem. The original KF provides an optimal solution when the models are linear and the probabilities are Gaussian. However, when dealing with nonlinearities in the models, as is often the case for positioning problems, nonlinear extensions of KF such as the extended KF (EKF) or the unscented KF (UKF) can be used. In the case of the EKF, the nonlinear models are linearized around the most recent state estimate, while in the UKF the probability distribution is approximated using a set of deterministically chosen (non-randomly sampled) points in the state space, where every sample is assigned a particular fixed weight. This set of points conserves the Gaussian properties of the distribution under nonlinear transformations [33]. Although the EKF has been for a long time considered as a _de facto_ standard for tracking applications [34], the UKF has emerged as an alternative, claiming even higher accuracy and robustness for nonlinear models [35]. The UKF employs the statistical linearization techniques and implements the scaled unscented transform (UT), where carefully selected samples of the state are propagated through the actual nonlinear functions. Then, the mean and covariance are recalculated back from the propagated points, yielding more accurate results compared to a conventional linearization routine. Although in the presented work we have adopted UKF as a core estimation framework, similar resultsare expected if the estimation process would have been formulated with an EKF with sufficiently high update rate [36]. In general, the navigation filter is formulated as a nonlinear estimation problem for the system governed by the following stochastic models:  
\[x_{k} = f\left(x_{k-1},u_{k},\
u_{k}\right) \tag{1}\] \[z_{k} = h\left(x_{k},e_{k}\right) \tag{2}\]  
where \(x_{k}\) is the state vector of length \(n\), \(z_{k}\) is the vector of observations of length \(m\), \(u_{k}\) is the control input, \(\
u_{k}\sim\mathcal{N}\left(0,Q_{k}\right)\) is a zero mean process noise vector, and \(e_{k}\sim\mathcal{N}\left(0,R_{k}\right)\) is the observation noise vector. The \(f\left(\cdot\right)\) and \(h\left(\cdot\right)\) are the dynamic models for the state propagation and measurement model, respectively. Here the noises neither have to be additive nor have to be of the same dimensionality as \(x_{k}\) or \(z_{k}\). In our work, the vector \(x_{k}\) describing the state of the vessel in a 3D space can be expressed as  
\[x=\left[q^{T}\,v^{T}\,p^{T}\,b_{a}^{T}\,b_{g}^{T}\left(dt\,dt\right)\right]^{T} \tag{3}\]  
where \(q\) represents the attitude quaternion from the vessel local to the the earth-centered, earth-fixed (ECEF) frames, \(v\) and \(p\) are the 3D velocity and position in the ECEF frame, and \(b_{a}\) and \(b_{g}\) are the accelerometer and gyroscope offsets. Finally, for tightly coupled IMU/GNSS integration schemes, the state has to be augmented with GPS receiver clock offset \(dt\) and clock offset rate \(dt\).