Kalman filtresi ve dengeleyen eğri olarak polinomlar ve trigonometrik fonksiyonlar
Başlık çevirisi mevcut değil.
- Tez No: 75436
- Danışmanlar: DOÇ. DR. MUHAMMED ŞAHİN
- Tez Türü: Yüksek Lisans
- Konular: Jeodezi ve Fotogrametri, Geodesy and Photogrammetry
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1998
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Jeodezi ve Fotogrametri Ana Bilim Dalı
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 168
Özet
ÖZET Bilindiği gibi navigasyon olaylarında bir noktanın şu andaki ve ilerideki konumunun ne olacağının tahmin edilmesi büyük bir problemdir. Şu andaki GPS teknolojisi ile hareket halindeki veya duran cisimlerin zaman içerisindeki hareketleri istenilen hassasiyette izlenebilmektedir. Ancak gözlemler sonucu elde edilen verilerin uygun bir biçimde değerlendirilmesi gerekir. Özellikle deformasyonlarla ilgili çalışmalarda yersel noktaların şu andaki hareketlerinin nasıl olduğu ve belli bir tarihte nokta konumlarının nasıl olacağının belirlenmesi deformasyon analizi ve sonuçlandırma işlemleri açısından çok önemlidir. Kalman Filitresi adı verilen ve temeli en küçük kareler yöntemine dayanan bir metodla bütün bu sorulara cevap vermek mümkündür. Kalman Fifitresi sadece Jeodezi biliminde değil hemen hemen bütün alanlarda kullanılan çok geniş bir uygulamadır. BtFada Kalman Filitresinin, 3 sene boyunca SLR (Satellite Laser Ranging=Lazerle uyduya uzaklık ölçme tekniği) yöntemi ile elde edilmiş nokta konumlarının zaman farkları esas alınarak uygulaması yapılmıştır. Buradaki amaç, 3 yıllık SLR verilerini tam bir deformasyon analizine sokmak değildir. Bu verilerin deformasyona tabi tutulabilmesi için başka ön işlemlere gerek vardır. Burada sadece örnek veri olması açısından SLR verileri kullanılmıştır. Nokta koordinatlarının kestirim ve dengeleme işlemleri sonunda elde edilen dengelenmiş koordinatları ile gerçek verilerde oluşturulan izleme eğrileri karşılaştırılmıştır. Kalman Filitreleme yöntemi en küçükkareler yöntemine nazaran çok büyük veri yığınlarının düzenlenmesi açısından daha avantajlıdır. Bu araştırmanın arkasından“acaba bu verileri dengeleyen başka eğriler var mı?”sorusuna cevap bulmak için polinomlar ve trigonometrik fonksiyonlar üzerine bir araştırma yapılmıştır. İkinci derece, dördüncü derece polinomların ve trigonometrik fonksiyonların ölçme verilerine uygunluğu araştırılmıştır. Seçilen fonksiyonların katsayıları, zaman farkları esas alınarak hesaplanmıştır. Elde edilen katsayılar yardımıyla fonksiyon sonuçları hesaplanmış ve yine her bir fonksiyon için çizdirilen eğrilerin ölçme verileriyle uyumu araştırılmıştır. En son olarak hem Kalman Filitresinin hem de polinom ve trigonometrik fonksiyonlardan elde edilen sonuçların karşılaştırması yapılmıştır.
Özet (Çeviri)
SUMMARY KALMAN FILTERING AND POLYNOMIALS.TRIGONOMETRIC FUNCTIONS AS THE FITTING CURVE In recent years, the topic of the motion of the mass location is the biggest interest of the researchers. This motion is called deformation. The deformation is the change of the location or the change of the shape. The deformation must be expressed in a mathematical way. There are some models to express the deformations. These models depends on the measurement period and dependent variance. There are 3 groups of deformation models; a) Static Model: It is estimated that the points was fixed in the same period. The differences of the periods give the deformation. b) Kinematic Model: It is evaluated the point velocity and point coordinates are evaluated simultaneously. c) Dynamic Model: The forces which causes the deformation and the transition functions are examined. There are several ways to obtain deformation. The aim of this study's to analyss the deformation. The purpose is to explain that it is possible to determine the crustal motion, as called deformation, by using filtering methods. Hydrographers, surveyors and geophysicists are among the many groups who would benefit from precise position information either in real time or with very little delay. The most appropriate technology is the GPS (Global Positioning System) to serve the need.The Global Positioning System, introduced in 1973, is a system designed by the Department of Defence (USA) to provide continuous positioning and velocity information for a variety of land, sea and air based military applications. Two levels of accuracy are available to users of GPS. The precise positioning service (PPS), restricted to authorized users, is intent to provide instantaneous position and velocity information at accuracy of 15 meters and 0.1 meters/second, respectively. The standard positioning service (SPS), available to the civilian community, is currently expected to provide instantaneous position and velocity information at accuracy's of 100 meter and 0.3 meters/second, respectively. For most navigation applications, SPS levels of accuracy will be sufficient. GPS was primarily designed as a navigation system with a world wide real time capability. Modern high performance algorithms for ship borne positioning are nearly based on a combination of code and carrier phase measurements. Especially, to obtain the deformation between the continents, there is another satellite technique as known SLR (Satellite Laser Ranging). In laser distance measurements to satellites the time of flight of a laser pulse as it travels between a ground station and a satellite is observed. A short laser pulse is generated in the ground station, and is transmitted through an optical system to the satellite. A part of outgoing laser pulse is used to start an electronic time interval counter. The reflected pulse is collected by the receiving telescope to the time counter. The laser and receiving telescope are mounted such that they can automatically follow the satellite, according to the precalculated orbit. Laser ranging is only possible to satellite equipped with approprite reflectors. The incoming laser light must be sent back in exactly the same direction from which it comes. Such types of reflectors are also called retroreflectors.The two way travel time of the signal is derived from the two reading of the user clock, and is scaled into the distance d with the signal propagation velocity c. The basic observation equation is, d= dt_ c c; speed of light (299 792,458km/s) 2 Due to the very high accuracy of laser range observations to satellites field of application in geodesy and geodynamics has been opened. In this study, there are 10 points located in several countries in the world. The SLR data for 1994, 1995 and 1996 are obtain from the NASA Crustal Dynamics Data Information System (CDDIS). A Kalman Filter is a processing technique that is typified by particular measurements and dynamics models. Measurement may be input to the Kalman filter using the least squares techniques. The Kalman Filter is called the linear quadratic Gaussian problem, which is the problem of estimating the instantaneous state of a linear dynamic system perturbed by Gaussian white noise - by using measurement linearly related to the state, but corrupted by Gaussian white noise. The resulting estimator is statistically optimal with respect to any quadratic function of estimator error. The Kalman Filter is also used for predicting the likely future courses of dynamic systems that people are not likely to control, such as flow of rivers during flood, the trajectories of celestial bodies, or the precise of traded commodities. Originally, a filter solves the problem of separating unwanted components of mixtures. In the are of crystal radios and vacuum tubes, the term wasapplied to analog circuits that 'filter" electronic signals. This signals are mixtures of different frequency components and unwanted frequencies. The term of filter is extended in the 1930s and 1940s the separation of signals from noise. Kolomogrov and Wiener used this statistical characterization of their probability distributions in forming on optimal estimate of the signal. With Kalman Filtering the term assumed a meaning that is well beyond the original idea of an inversion problem. It inverts the functional relationship and estimates the independent variables as inverted functions of the dependent (measurable) variables. The classical Kalman Filter is based on the assumption that the data are uncorrelated in time. In practice, however, are often encounters coloured measurements noise. Because the Kalman is only optimal for the case of measurements noise uncorrelated in time, it is important to detect possible time correlations in the data. The dynamic systems are linear, the performance cost functions are quadratic and random processes are Gaussian. There are three general types of estimators for the LQG problem;. Predictors use observations strictly prior to the time the state of dynamic system is to be estimated.. Filters use observations up to and including the time that the state of the dynamic system is to be estimated.. Smoothers use observations beyond the time that the state of the dynamic system is to be estimated. In this study, for application of Kalman Filter, The measurements must be changed into the measurement differences. First of all the measurement differences are counted by using 10 SLR station coordinates. Station coordinates of the first month were assumed to be the base measurements, and the differeces from the other months are then computed. xivThe basic steps of the computational procedure for Kalman Filter are; 1. Compute apriori weight matrix, the transition matrix and a postriori (»variance matrix. Then the predicted values of second measurement can be computed. 2. Compute K Gain matrix, second measurement's covariance matrix and Qe matrix. Then the updated values of second measurement can be computed. 3. Compute the updated value's priori covariance matrix. 4. This process is continued for the third measurement with taking the second measurement priori covariance matrix as apriori matrix of the second measurement. In this study,, the constant velocity motion equations are taken as the motion equation. Here is, there are 6 parameters as known. The measurements are the 3 dimensional coordinates of a point. The model is described as follows; Zi+i = T Zj + a e T=Transition matrix Zj, zm = Measurements vector of i+1th and i th measurements a = acceleration e= white noise At the end of Kalman Filtering process, the predict and update values of the measurements are computed. The comparison of the measurements, predicted and updated values are given in Section 4. As you can see, on the XVFigures 4.1a, 4.1b and 4.1c the updated curves are quite appropriate to the measurement than the predicted curves. On the other hand, having a general information on navigation, polynomials and trigonometric functions can be used as an equilibrium curve. In this study.it is expected to be appropriate for this application, some polynomials and trigonometric functions are selected. They are is given as follows; f(x!) = a + bx + cx2 f(x2) = a + bx + ex2* dx3* ex4 f(x3) = a + bCos(x) + cSin(x) + dx + ex2 First function is a 2nd degree polynomial and the second function is 4th degree polynomial. The third one is different than the others. This is the trigonometric function but it contains the 2nd degree polynomial. The trigonometric functions have periodic curves and the 2nd degree polynomials are curvilinear. If these two functions are summed the new curve will be either having periodic and curvilinear. This curve is quite appropriate for the measurements. XVI
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