A solution to dynamic errors-in-variables within system equations

Abstract

We noticed that if INS data is used as system equations of a Kalman filter algorithm for integrated direct geo-referencing, one encounters with a dynamic errors-invariables (DEIV) model. Although DEIV model has been already considered for observation equations of the Kalman filter algorithm and a solution namely total Kalman filter (TKF) has been given to it, this model has not been considered for system equations (dynamic model) of the Kalman filter algorithm. Thus, in this contribution, for the first time we consider DEIV model for both observation equations and system equations of the Kalman filter algorithm and propose a least square prediction namely integrated total Kalman filter in contrast to the TKF solution of the previous approach. The variance matrix of the unknown parameters are obtained. Moreover, the residuals for all variables are predicted. In a numerical example, integrated direct geo-referencing problem is solved for a GPS-INS system. Keywords Dynamic errors-in-variables Á System equations Á Integrated total Kalman filter Á Direct geo-referencing

On a New Family of Kalman Filter Algorithms for Integrated Navigation

“…Nevertheless, we can argue that the real dynamic model of the problem should be given by the following equations since some random observed quantities may still exist in which may be neglected such noisy control points. Moreover, if INS is used to produce system equations, one encounters with a dynamic errors-in-variables (DEIV) model (Mahboub et al (2017a). Furthermore, one may need to impose the following quadratic constraint since it is well known that the Quaternion approach is sometimes used to solve the singularity problems of the Euler angles at the 90 degrees angle.…”

“…Therefore Mahboub et al (2016) presented an applicable TKF algorithm with general weight matrixes and named it weight total Kalman filter (WTKF). Then Mahboub et al (2017a) solved the problem which both of the coefficient matrix of the observation equations and system equations are corrupted by random noise and named it integrated Kalman filter (IKF). Eventually Mahboub et al (2017b) developed a constrained integrated total Kalman filter (CITKF) as a solution to DEIV model since a quadratic constraint may appear the navigation problem.…”

“…WTKF algorithmLater onMahboub et al (2017a) proposed integrated Kalman filter (IKF) algorithm in order to solve the problem given by equations (5) to (7) which both of the coefficient matrix of the observation equations and system equations are corrupted by random noise. The algorithm us given by figure 2:Figure 2: IKF algorithm Eventually the constrained integrated Kalman filter (CIKF) proposed by Mahboub et al (2017b) also considers equation (8).…”

A Constrained Total Extended Kalman Filter for Integrated Navigation

“…Nevertheless, motivated by the new advances in remote sensor solutions in combination with traditional navigation sensors, some new systems have been proposed based on fusing remote sensing measurements with Global Positioning System (GPS)-INS measurements to achieve comprehensive, fast and real-time systems (Sheta, 2012). The linearized observation equations and/or system equations of these newly developed systems are referred to as Dynamic Errors-In-Variables (DEIV) models (Mahboub et al, 2017a; 2017b; Schaffrin and Iz, 2008). In such a case the classic EKF algorithm or its alternatives such as the Unscented Kalman Filter (UKF) (Julier and Uhlmann, 1997) may not be useful since they try to approximate the functional and stochastic model rather than properly modelling the dynamic environment.…”

“…These algorithms and the techniques utilised are not ideally suited to the problem which is discussed in this paper. Therefore Mahboub et al (2017a; 2017b) presented a TKF algorithm with general weight matrices and named it the Weight Total Kalman Filter (WTKF).…”