\r\n\tThe study of populations and plant communities in their different aspects; ecological, structural, functional and dynamic, it is essential to establish a posteriori models of forest and agricultural management.
\r\n\r\n\tFor this, the methodological approaches on the type of sampling are considered essential, since there are differences between the purely ecological and the phytosociological methods, despite the fact that both pursue the same objective.
\r\n\tAlthough the ecological method for the knowledge of the vegetation is widely extended, the phytosociological one is no less so, since in the European Union it has been developed as a consequence of policies on sustainability, through which regulations have been issued, such as the habitats directive.
\r\n\tOn the other hand, research on plant dynamics and knowledge of the landscape in an integral way, have multiplied in the last 30 years, which has favored a deep knowledge of the floristic and phytocenotic wealth, which is fundamental for agricultural management, livestock and forestry.
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He has participated in national and international congresses with about 250 papers. He has held a number of different academic positions, including Dean of the Faculty of Experimental Sciences at the University of Jaen, Spain, and founder and director of the International Seminar on Management and Conservation of Biodiversity.",coeditorOneBiosketch:"Ricardo Jorge Quinto Canas is currently an Invited Assistant Professor in the Faculty of Sciences and Technology at the University of Algarve – Portugal, and a member of the Centre of Marine Sciences (CCMAR), University of Algarve. His current research projects focus on Botany, Vegetation Science (Geobotany), Biogeography, Plant Ecology, and Biology Conservation, aiming to support Nature Conservation.",coeditorTwoBiosketch:"Ana Cano Ortiz's fundamental line of research is related to botanical bioindicators. She has worked in Spain, Italy, Portugal, and Central America. It presents more than one hundred works published in various national and international journals, as well as books and book chapters; and has presented a hundred papers to national and international congresses.",coeditorThreeBiosketch:"Carmelo Maria Musarella is a biologist, specialized in Plant Biology. He is a member of the permanent scientific committee of the International Seminar on “Biodiversity Conservation and Management” guested by several European universities. He has participated in several international and national congresses, seminars, and workshops and presented oral communications and posters.",coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"87846",title:"Dr.",name:"Eusebio",middleName:null,surname:"Cano Carmona",slug:"eusebio-cano-carmona",fullName:"Eusebio Cano Carmona",profilePictureURL:"https://mts.intechopen.com/storage/users/87846/images/system/87846.png",biography:"Eusebio Cano Carmona obtained a PhD in Sciences from the\nUniversity of Granada, Spain. He is Professor of Botany at the\nUniversity of Jaén, Spain. His focus is flora and vegetation and he\nhas conducted research in Spain, Italy, Portugal, Palestine, the\nCaribbean islands and Mexico. As a result of these investigations,\nDr. Cano Carmona and colleagues have directed 12 doctoral theses\nand more than 200 publications among articles, books and book\nchapters. He has participated in national and international congresses with about\n250 papers/communications. He has held a number of different academic positions,\nincluding Dean of the Faculty of Experimental Sciences at the University of Jaen,\nSpain and founder and director of the International Seminar on Management and\nConservation of Biodiversity, a position he has held for 13 years. He is also a member of the Spanish, Portuguese and Italian societies of Geobotany.",institutionString:"University of Jaén",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"5",totalChapterViews:"0",totalEditedBooks:"1",institution:{name:"University of Jaén",institutionURL:null,country:{name:"Spain"}}}],coeditorOne:{id:"216982",title:"Dr.",name:"Ricardo Quinto",middleName:null,surname:"Canas",slug:"ricardo-quinto-canas",fullName:"Ricardo Quinto Canas",profilePictureURL:"https://mts.intechopen.com/storage/users/216982/images/system/216982.JPG",biography:"Ricardo Quinto Canas, Phd in Analysis and Management of Ecosystems, is currently an Invited Assistant Professor in the Faculty\nof Sciences and Technology at the University of Algarve, Portugal, and member of the Centre of Marine Sciences (CCMAR),\nUniversity of Algarve. He is also the Head of Division of Environmental Impact Assessment - Algarve Regional Coordination\nand Development Commission (CCDR - Algarve). 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Kalman filters have been demonstrating its usefulness in various applications. Kalman filters have relatively simple form and require small computational power. However, it is still not easy for people who are not familiar with estimation theory to understand and implement the Kalman filters. Whereas there exist some excellent literatures such as [1] addressing derivation and theory behind the Kalman filter, this chapter focuses on a more practical perspective.
\nFollowing two chapters will devote to introduce algorithms of Kalman filter and extended Kalman filter, respectively, including their applications. With linear models with additive Gaussian noises, the Kalman filter provides optimal estimates. Navigation with a global navigation satellite system (GNSS) will be provided as an implementation example of the Kalman filter. The extended Kalman filter is utilized for nonlinear problems like bearing-angle target tracking and terrain-referenced navigation (TRN). How to implement the filtering algorithms for such applications will be presented in detail.
\nKalman filters are used to estimate states based on linear dynamical systems in state space format. The process model defines the evolution of the state from time \n
where \n
The process model is paired with the measurement model that describes the relationship between the state and the measurement at the current time step \n
where \n
The role of the Kalman filter is to provide estimate of \n
Kalman filter algorithm consists of two stages: prediction and update. Note that the terms “prediction” and “update” are often called “propagation” and “correction,” respectively, in different literature. The Kalman filter algorithm is summarized as follows:
\nPredicted state estimate | \n\n\n | \n
Predicted error covariance | \n\n\n | \n
Measurement residual | \n\n\n | \n
Kalman gain | \n\n\n | \n
Updated state estimate | \n\n\n | \n
Updated error covariance | \n\n\n | \n
In the above equations, the hat operator, \n
The predicted state estimate is evolved from the updated previous updated state estimate. The new term \n
In the update stage, the measurement residual \n
We need an initialization stage to implement the Kalman filter. As initial values, we need the initial guess of state estimate, \n
Note that Kalman filters are derived based on the assumption that the process and measurement models are linear, i.e., they can be expressed with the matrices \n
An example for implementing the Kalman filter is navigation where the vehicle state, position, and velocity are estimated by using sensor output from an inertial measurement unit (IMU) and a global navigation satellite system (GNSS) receiver. In this example, we consider only position and velocity, omitting attitude information. The three-dimensional position and velocity comprise the state vector:
where \n
where \n
where \n
Now, we have the process model as:
where
The GNSS receiver provides position and velocity measurements corrupted by measurement noise \n
It is straightforward to derive the measurement model as:
where
In order to conduct a simulation to see how it works, let us consider \n
We need to generate noise of acceleration output and GNSS measurements for every time step. Suppose the acceleration output, GNSS position, and GNSS velocity are corrupted with noise with variances of 0.32, 32, and 0.032, respectively. For each axis, one can use MATLAB function randn or normrnd for generating the Gaussian noise.
\nThe process noise covariance matrix, \n
Let us start filtering with the initial guesses
where \n
In this simulation, \n
The time history of estimation errors of two Monte-Carlo runs is depicted in Figure 1. We observe that the estimation results of different simulation runs are different even if the initial guess for the state estimate is the same. You can also run the Monte-Carlo simulation with different initial guesses (sampled from a distribution) for the state estimate.
\nTime history of estimation errors.
The standard deviation of the estimation errors and the estimated standard deviation for x-axis position and velocity are drawn in Figure 2. The standard deviation of the estimation error, or the root mean square error (RMSE), can be obtained by computing standard deviation of \n
Actual and estimated standard deviation for x-axis estimate errors.
In real applications, you will be able to acquire only the estimated covariance because you will hardly have a chance to conduct Monte-Carlo runs. Also, getting a good estimate of \n
Source code of MATLAB implementation for this example can be found in [5]. It is recommended for the readers to change the parameters and aircraft trajectory by yourself and see what happens.
\nSuppose you have a nonlinear dynamic system where you are not able to define either the process model or measurement model with multiplication of vectors and matrices as in (1) and (2). The extended Kalman filter provides us a tool for dealing with such nonlinear models in an efficient way. Since it is computationally cheaper than other nonlinear filtering methods such as point-mass filters and particle filters, the extended Kalman filter has been used in various real-time applications like navigation systems.
\nThe extended Kalman filter can be viewed as a nonlinear version of the Kalman filter that linearized the models about a current estimate. Suppose we have the following models for state transition and measurement
where \n
All you need is to obtain the Jacobian matrix, first-order partial derivative of a vector function with respect to a vector, of each model in each time step as:
Note the subscripts of \n
Predicted state estimate | \n\n\n | \n
Predicted error covariance | \n\n\n | \n
Measurement residual | \n\n\n | \n
Kalman gain | \n\n\n | \n
Updated state estimate | \n\n\n | \n
Updated error covariance | \n\n\n | \n
As in the Kalman filter algorithm, the hat operator, \n
We are going to estimate a 3-dimensional target state (position and velocity) by using measurements provided by a range sensor and an angle sensor. For example, a radar system can provide range and angle measurement and a combination of a camera and a rangefinder can do the same. We define the target state as:
where \n
The process noise has the covariance of \n
The measurement vector is composed of line-of-sight angles to the target, \n
where \n
where \n
In the simulation, the sensor is initially located at \n
Trajectory of the sensor and the target.
In the filter side, the covariance matrix for the process noise can be set as:
where \n
\n\n
The above equation means that the error of the initial guess for the target state is randomly sampled from a Gaussian distribution with a standard deviation of \n
Time history of an estimation result for x-axis position and velocity is drawn together with the true value in Figure 4. The shape of the line will be different at each run. The statistical result can be shown as Figure 5. Note that the filter worked inconsistently with the estimated error covariance different from the actual value. This is because the process error covariance is set to a very large number. In this example, the large process error covariance is the only choice a user can make because the measurement cannot correct the velocity. One can notice that the measurement Eq. (26) has no term dependent on the velocity, and therefore, matrix \n
Time history of an estimation result for x-axis position and velocity.
Actual and estimated standard deviation for x axis estimate errors.
Source code of MATLAB implementation for this example can be found in [5]. It is recommended for the readers to change the parameters and trajectories by yourself and see what happens.
\nTerrain-referenced navigation (TRN), also known as terrain-aided navigation (TAN), provides positioning data by comparing terrain measurements with a digital elevation model (DEM) stored on an on-board computer of an aircraft. The TRN algorithm blends a navigational solution from an inertial navigation system (INS) with the measured terrain profile underneath the aircraft. Terrain measurements have generally been obtained by using radar altimeters. TRN systems using cameras [7], airborne laser sensors [8], and interferometric radar altimeters [9] have also been addressed. Unlike GNSS’s, TRN systems are resistant to electronic jamming and interference, and are able to operate in a wide range of weather conditions. Thus, TRN systems are expected to be alternative/supplement systems to GNSS’s.
\nThe movement of the aircraft is modeled by the following Markov process:
where \n
Conventional TRN structure.
Typical TRN systems utilize measurements of the terrain elevation underneath an aircraft. The terrain elevation measurement is modeled as:
where \n
Relationship between measurements in TRN.
The process model in (33) and the measurement model in (34) can be linearized as:
where \n
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. Interpolation for two-dimensional gridded data can be realized by interp2 function in MATLAB. Cubic spline interpolation is used in this example.
\nThe DEM we are using in this example has a \n
Contour representation of terrain profile.
An aircraft, initially located at \n
The process noise \n
Let us consider \n
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 \n
The time history of RMSE of the navigation is shown in Figure 9. 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 \n
Time history of RMSE.
Source code of MATLAB implementation for this example can be found in [5]. It is recommended for the readers to change the parameters and aircraft trajectory by yourself and see what happens.
\nIn this chapter, we introduced the Kalman filter and extended Kalman filter algorithms. INS/GNSS navigation, target tracking, and terrain-referenced navigation were provided as examples for reader’s better understanding of practical usage of the Kalman filters. This chapter will become a prerequisite for other contents in the book for those who do not have a strong background in estimation theory.
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