Why extended kalman filter

The terrain elevation measurement is modeled as:. The elevation measurement is obtained by subtracting the ground clearance measurement from a radar altimeter, h r , from the barometric altimeter measurement, h b. The ground clearance and the barometric altitude correspond to the above ground level AGL height and the mean sea level MSL height, respectively. The relationship between the measurements is depicted in Figure 7.

Note that the terrain elevation that comprises the measurement model in 34 is highly nonlinear. Relationship between measurements in TRN. The process model in 33 and the measurement model in 34 can be linearized as:.

The DEMs are essentially provided as matrices containing grid-spaced elevation data. For obtaining finer-resolution data, interpolation techniques are often used to estimate the unknown value in between the grid points. One of the simplest methods is linear interpolation. Linear interpolation is quick and easy, but it is not very precise. A generalization of linear interpolation is polynomial interpolation. Polynomial interpolation expresses data points as higher degree polynomial.

Polynomial interpolation overcomes most of the problems of linear interpolation. However, calculating the interpolating polynomial is computationally expensive. Furthermore, the shape of the resulting curve may be different to what is known about the data, especially for very high or low values of the independent variable.

These disadvantages can be resolved by using spline interpolation. Spline interpolation uses low-degree polynomials in each of the data intervals and let the polynomial pieces fit smoothly together. That is, its second derivative is zero at the grid points see [ 11 ] for more details. Classical approach to use polynomials of degree 3 is called cubic spline. Because the elevation data are contained in a two-dimensional array, bilinear or bicubic interpolation are generally used.

Cubic spline interpolation is used in this example. The profile of the DEM can be depicted as Figure 8. The figure represents contours of the terrain where brighter color denotes regions with higher altitude. The point 20, 10 in the grid corresponds to the position T in the navigation frame. Contour representation of terrain profile. The aircraft is equipped with a radar altimter and a barometric altimter, which are used for obtaining the terrain elevation.

The radar altimeter is corrupted with a zero-mean Gaussian noise with the standard deviation of 3. The matrices Q and R are following the real statistics of the noises as:. The above equation means the error of the initial guess for the target state is randomly sampled from a Gaussian distribution with a standard deviation of 50 One can observe the RMSE converges relatively slower than other examples.

Because the TRN estimates 2D position by using the height measurements, it often lacks information on the vehicle state. Moreover, note that the extended Kalman filter linearizes the terrain model and deals with the slope that is effective locally. If the gradient of the terrain is zero, the measurement matrix H has zero-diagonal terms that has zero effect on the state correction.

In this case, the measurement is called ambiguous [ 12 ] and this ambiguous measurement often causes filter degradation and divergence even in nonlinear filtering techniques. With highly nonlinear terrain models, TRN systems have recently been constructed with other nonlinear filtering methods such as point-mass filters and particle filters, rather than extended Kalman filters.

Time history of RMSE. In this chapter, we introduced the Kalman filter and extended Kalman filter algorithms. This chapter will become a prerequisite for other contents in the book for those who do not have a strong background in estimation theory.

Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3. Help us write another book on this subject and reach those readers. Login to your personal dashboard for more detailed statistics on your publications. Edited by Felix Govaers.

Edited by Dumitru Baleanu. We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. Downloaded: Abstract We provide a tutorial-like description of Kalman filter and extended Kalman filter.

Introduction Kalman filtering is an algorithm that provides estimates of some unknown variables given the measurements observed over time. More Print chapter. How to cite and reference Link to this chapter Copy to clipboard. Available from:. Over 21, IntechOpen readers like this topic Help us write another book on this subject and reach those readers Suggest a book topic Books open for submissions.

More statistics for editors and authors Login to your personal dashboard for more detailed statistics on your publications. Access personal reporting. The proposed event handling algorithm consists of two parts: relaxing sample time and restricting sample time ST. Relaxing procedure is used to avoid high computational time when no rapid change exists in system dynamics.

Effectiveness of the proposed approach is verified considering the well-known Van der Pol oscillator as a common example of stiff systems. Step size control normally contains event handling procedure to detect the moment in time of specific event. Event handling and zero crossing detection are well-known and well established approaches to handle numerical simulation Watts, ; Zhang et al. The idea of improving the performance by detecting the moment in time at which events take place is known in the field of ODE solvers.

In Esposito and Kumar an event detection algorithm is introduced for non-smooth differential equations. In this paper an extrapolation polynomial is used for selection of the integration step size to detect potential future events. This approaches has advantages especially in the neighborhood of model singularities where the derivative function is undefined in state space. Singular systems can be modeled as Markovian jump singular systems. Investigation of stability, stabilization, control, and filtering of these systems are more complicated compared to the systems modeled in state space Wang et al.

In Bernal the author addressed calculation of time-step sizes to prevent interstep events in multilinear problems. In this work an approach is introduced to locate the moment in time when the stiffness changes in a non-linear system.

This approach can be considered as an interpolation scheme of the time when events occur by using computed responses at the end of corresponding linear step. In Wright and Pei an explicit numerical method is proposed to solve differential algebraic equations DAE considering low-order discontinuities in the restoring force algebraic equations.

The effectiveness of this approach is verified for a specific class of problems namely non-smooth DAE with discontinuities. The algorithm is able to automatically detect each state event location. In reality the system dynamics is modeled by linear or non-linear ordinary differential equations.

The dynamics of system and observer are augmented and described in one linear ODE form. The aforementioned example can be described by a suitable initial value problem IVP. In this case stiff solvers able to handle stiff problems are used to avoid high computational time by relaxing the integration step size when no rapid change exists in system states and disturbance dynamics, as well as to improve the estimation results by decreasing the integration step size in the case of fast dynamical behavior related to disturbances stiff behavior.

Kalman Filter and its extensions EKF and UKF has the same functionality as observers for discontinuous systems and measurements containing statistical noise and other inaccuracies. For example iterated extended Kalman filter IEKF Bell and Cathey, linearizes the system non-linear equations iteratively to compensate the significant non-linearities.

In this algorithm an iterative procedure is considered for measurements update correction part. The iterative procedure is stopped if the maximum number of iterations is reached or the difference between two iterations results is less than a pre-specified threshold.

In Skoglund et al. In this approach the sample time is considered as a fixed value. The measurement update step is calculated based on the Gauss-Newton algorithm or Line Search optimization procedure with variable step length. Here the step size is considered as step size of optimization procedure to ensure the convergence of a predefined criteria.

Measurements obtained at irregular time instants and detected by an encoder. The uncertainty on transition event occurrence time is considered as zero mean random measurement noise acting on rotary position. This measurement noise affected by the sampling time, is considered as part of the system model to be used for Kalman Filter estimation.

This approach adapts the state space formulation, evaluates the observability within each time step, and selects the suitable sub-space that can be used by UKF.

Consequently the unidentifiable parameters related to non-smooth behaviors are detected and excluded from the problem formulation. An event triggered continuous discrete Kalman Filter is introduced in Niu et al.

The MSE of estimation is controlled by detecting the event and taking a new measurement. This approach decreases the sample time at each step to reach the desired estimation error at the current sample time.

The next estimation starts by reinitializing the sample time to the original one. In all aforementioned approaches the sample time of estimation results is defined constant based on the measurements or is just reduced at the current step and is reinitialized for the next time step. The goal is to achieve a suitable estimation performance in the case of event. None of the mentioned approaches focus of sample time control considering event detection by increasing and reducing the sample time during the estimation procedure like ODE solvers.

In this contribution the sample time is controlled based on the estimation performance. It decreases in the case that estimation error increases to achieve the desired performance. Correspondingly it increases when the performance is suitable enough to reduce the computational effort. Event handling and zero crossing procedures have been used and improved over the previous years for solving stiff ODEs. Procedure of KF starts from an initial set of Kalman parameters and system states.

The next estimation step is realized based on the current step information same concept as IVP. Generally the predictor-corrector approaches are based on a pre-specified tolerance. Extending and detailing the initial work, in this contribution an extended and improved version of step size control is proposed for non-linear dynamical behavior considering EKF and UKF.

Furthermore, adjustment of design parameters is investigated in this contribution to get a suitable convergence rate and estimation error. Unscented Kalman Filter is used here as a non-linear discrete filter which can be replaced by any other discrete filter for linear or non-linear systems.

The main contributions of this paper can be summarized as follows:. In the second part main aspects of Matlab-based ODE solvers are introduced e. The last section finalizes the paper with a summary and conclusions. At each step the solver uses the results of previous step considering a particular algorithm Euler's method, RungeKutta methods, etc. Higher-order differential equations can be reformulated at each step of iterative solving procedure of IVP as a system of first-order equations:.

Generally even if function f is a continuous function there is no guarantee that the IVP provides a unique solution. In Atkinson et al. Definition 1. According to Hairer and Wanner relative separation can define the stiffness level. In other words, the eigenvalue responsible for the slowest rates of change should be compared to those leading to the fastest rates of change. Considering non-linear systems the complexity increases because in this case stiffness becomes a global property and therefore cannot be reduced to a local problem by using a solution in the neighborhood of single points.

The non-linear problem may start with non-stiff behavior and become stiff, or vice versa. It may contain stiff and non-stiff intervals. In Ashino et al. Definition 2. If a numerical method is forced to use, in a certain interval of integration, a step length which is excessively small in relation to the smoothness of the exact solution in that interval, then the problem is said to be stiff in that interval.

To solve the numerical integration of stiff ordinary differential Equation 1 a suitable small step size h n is required in the case of fast variation in the solution and correspondingly the step size has to be relatively large relaxed when the solution is smooth.

Consequently a perfect numerical solution is able to solve the stiff and non-stiff ODEs denoted as step size control Watts, Stiff numerical methods have the ability to change the step size during solving procedure. They take small steps to obtain satisfactory results nearby solutions that vary rapidly. The main advantage of stiff solvers is the low computational time compared to non-stiff solvers. Non-stiff solutions can be used for stiff problems with a proper small step size but it takes more time to achieve the final solution because the step size is constant and can not be adapted according to the actual results.

In this section the main ideas of zero crossing and event handling are repeated to be used later introducing the proposed algorithm. Here they are briefly repeated because exactly these ideas will be transferred to the finite difference scheme of calculation digital filters like Kalman Filter. As discussed in section 1, variable step size solvers increases or decreases the step size to achieve error tolerances and required or given performance.

Selection of fixed or variable step size depends on the dynamical model and implementation issues. The fixed step size solver uses one step size for the whole simulation time and consequently the step size has to be small enough to achieve the accuracy requirements. Implicit variable step size solvers stiff solvers can be used to solve stiff problems. The non-stiff solvers are ineffective on intervals where the solution changes slowly because they use time steps small enough to resolve the fastest possible change for the whole estimation time.

Furthermore, the step size of non-stiff solvers has to be defined at the initialization level and can not be changed during the solving procedure. Connect and share knowledge within a single location that is structured and easy to search.

I am working SLAM based problems in robotics and I want to know whether I can use Kalman filter instead of the Extended kalman filter that is predominantly used? The Kalman filter KF is a method based on recursive Bayesian filtering where the noise in your system is assumed Gaussian. The Extended Kalman Filter EKF is an extension of the classic Kalman Filter for non-linear systems where non-linearity are approximated using the first or second order derivative. As an example, if the states in your system are characterized by multimodal distribution you should use EKF instead of KF.

When the system is non-linear the steps are identical for the simulation of the EKF but the main difference is that in step 2 and step 3 we use linearization at the previous step and at the prior respectively, only for those two steps the other steps remain identical for both methods KF and EFK.

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