# Interval Extended Kalman Filter--Application to Underwater Localization and Control
## 1 Introduction  
The dynamic of a mobile robot is generally described by the following discrete time nonlinear state equation  
\[\begin{cases}\mathbf{x}_{k+1}&=\mathbf{f}_{k}(\mathbf{x}_{k},\mathbf{u}_{k}, \mathbf{n}_{k})\\ \mathbf{y}_{k}&=\mathbf{g}_{k}(\mathbf{x}_{k})+\
u_{k}\end{cases} \tag{1}\]  
where \(k\in\mathbb{N}\) is the time, \(\mathbf{x}_{k}\in\mathbb{R}^{n}\) is the state vector, \(\mathbf{u}_{k}\) is the input vector, \(\mathbf{n}_{k}\) is the state noise, \(\
u_{k}\) is the measurement noise, and \(\mathbf{y}_{k}\) is the measured output vector. Two approaches are generally considered to solve the corresponding state estimation problem:  
* The _probabilistic approach_[1, 2] is over-optimistic and might converge to some fake states. In most of the cases, the implementation relies on an _extended Kalman filter_ (EKF) [3], which is efficient and accurate.
* The _set membership_ approach [4] is over-pessimistic. Its principle is to discard states [5] that are inconsistent with the collected data. Since no consistent state is rejected, these approaches have a high level of integrity [6, 7], even if they are considered are not precise. Usually, the implementation uses interval analysis and leads to what we can call an _interval filter_ (IF) [8].  
The main reason why an EKF may converge toward a fake state is that the linearization point that is chosen can be far from the true state [1]. Moreover, this linearization point, taken from the Kalman filter itself, can be totally inconsistent and would thus have been rejected by the interval filter. The idea we want to develop here is to select a linearization point inside the feasible set generated by an interval filter. Of course, this consistent point could still be far from the true one, but we understand that this choice reduces the possibility of bad behavior of the observer. The resulting controller would not be a hybrid between an IF and an EKF, but rather an EKF helped by the IF for the linearization.  
The idea of combining probabilities with interval-based methods to have a method which is both safe and accurate is not new. It is the objective of the _box particle filter_[9, 10] and other filters such as [11, 12], which combines particle filters with set-membershipmethods. It is also the objective of the _interval Kalman filter_[13], which add interval analysis to bound uncertainties in the Kalman Filter. Using a better Kalman filter such as the unscented Kalman filter (UKF) [2] would give better performances than the EKF, but could still converge toward a fake state because of the linearization point. Moreover, it is possible to use a particle filter (PF) [1] instead of the interval filter since it is also a global method. We decided to use the IF because this method has more integrity and generally requires less computations.  
The paper is organized as follows. Section 2 gives a 2D overview of the different filters and the linearization problem. Section 3 provides the three filters to be used: the Extended Kalman filter (EKF), the Interval Filter (IF), and the Interval Extended Kalman Filter (IEKF). Sections 4 and 5 compare the three approaches (EKF, IF, IEKF) on the localization problem of an underwater robot for different scenarios. Section 6 concludes the paper.