Security information portal. Using a Kalman Filter to Filter Sensor Values Defining a Process Model
In the process of automating technological processes to control mechanisms and units, one has to deal with measurements of various physical quantities. This can be the pressure and flow of liquid or gas, rotation speed, temperature and much more. Measurement of physical quantities is carried out using analog sensors. An analog signal is a data signal in which each of the representing parameters is described by a function of time and a continuous set of possible values. From the continuity of the space of values it follows that any noise introduced into the signal is indistinguishable from the useful signal. Therefore, the incorrect value of the required physical quantity will be received at the analog input of the control device. Therefore, it is necessary to filter the signal coming from the sensor.
One of the effective filtering algorithms is the Kalman filter. Kalman filter is a recursive filter that estimates the state vector of a dynamical system using a series of incomplete and noisy measurements. The Kalman filter uses a dynamic model of the system (for example, a physical law of motion), control inputs, and multiple sequential measurements to form an optimal state estimate. The algorithm consists of two repeated phases: prediction and adjustment. At the first stage, a prediction of the state at a subsequent point in time is calculated (taking into account the inaccuracy of their measurement). On the second, new information from the sensor corrects the predicted value (also taking into account the inaccuracy and noise of this information).
At the prediction stage:
- System State Prediction:
where is the prediction of the state of the system at the current time; – transition matrix between states (dynamic model of the system); – prediction of the state of the system at the previous point in time; – matrix of application of control action; – control action at the previous point in time.
- Covariance error prediction:
where is the error prediction; – error at the previous point in time; – covariance of process noise.
During the adjustment stage:
- Kalman gain calculation:
where is the Kalman gain; – measurement matrix displaying the relationship between measurements and states; – covariance of measurement noise.
where is the measurement at the current time.
- Covariance error update:
where is the identity matrix.
If the state of the system is described by one variable, then = 1, and the matrices degenerate into ordinary equations.
To clearly demonstrate the effectiveness of the Kalman filter, an experiment was conducted with an AVR PIC KY-037 volume sensor, which is connected to an Arduino Uno microcontroller. Figure 1 shows a graph of sensor readings without using a filter (line 1). Chaotic fluctuations in the value at the sensor output indicate the presence of noise.
Figure 1. Graph of sensor readings without applying a filter
To apply the filter, it is necessary to determine the values of the variables , and , which determine the dynamics of the system and measurements. Let us take and equal to 1, and equal to 0, since there are no control actions in the system. To determine the smoothing properties of the filter, it is necessary to calculate the value of the variable, as well as select the value of the parameter.
We will calculate the variable in Microsoft Excel 2010. To do this, it is necessary to calculate the standard deviation for a sample of sensor readings. = 0.62. selected depending on the required level of filtration, take = 0.001. In Figure 2, the second line shows a graph of sensor readings using a filter.
Figure 2. Graph of sensor readings using a Kalman filter.
From the graph we can conclude that the filter coped with the task of filtering out interference, since in steady state the fluctuations in the filtered sensor readings are insignificant.
However, the Kalman filter has a significant drawback. If the quantity being measured by a sensor can change rapidly, the filtered sensor readings will not change as quickly as the quantity being measured. Figure 3 shows the response of the Kalman filter to a jump in the measured value.
Figure 3. Kalman filter response to a jump in the measured value
The filter's response to a jump in the measured value turned out to be insignificant. If the measured value changes significantly and then does not return to the previous value, then the filtered sensor readings will correspond to the real value of the measured value only after a significant period of time, which is unacceptable for automatic control systems that require high performance.
From the experiment we can conclude that it is advisable to use the Kalman filter to filter sensor readings in systems with low performance.
Bibliography:
- GOST 17657-79. Data transfer. Terms and Definitions. – Moscow: Standards Publishing House, 2005. – 2 p.
- Kalman filter // Wikipedia. . Update date: 04/26/2017. URL: http://ru.wikipedia.org/?oldid=85061599 (access date: 05/21/2017).
This filter is used in various fields - from radio engineering to economics. Here we will discuss the main idea, meaning, essence of this filter. It will be presented in the simplest language possible.
Let's assume that we need to measure some quantities of a certain object. In radio engineering, they most often deal with measuring voltages at the output of a certain device (sensor, antenna, etc.). In the example with an electrocardiograph (see), we are dealing with measurements of biopotentials on the human body. In economics, for example, the measured value may be exchange rates. Every day the exchange rate is different, i.e. every day “his measurements” give us a different value. And if we generalize, we can say that most of human activity (if not all) comes down to constant measurements and comparisons of certain quantities (see book).
So, let's assume that we are constantly measuring something. We also assume that our measurements always come with some error - this is understandable, because there are no ideal measuring instruments, and each one produces results with an error. In the simplest case, what is described can be reduced to the following expression: z=x+y, where x is the true value that we want to measure and which would be measured if we had an ideal measuring device, y is the measurement error introduced by the measuring device, and z is the quantity we measured. So the task of the Kalman filter is to guess (determine) from the z we measured, what the true value of x was when we received our z (which contains the true value and measurement error). It is necessary to filter (weed out) the true value of x from z—to remove the distorting noise y from z. That is, having only a sum in hand, we need to guess which terms gave this sum.
In light of the above, let us now formulate everything as follows. Let there be only two random numbers. We are given only their sum and we are required to use this sum to determine what the terms are. For example, we were given the number 12 and they say: 12 is the sum of the numbers x and y, the question is what x and y are equal to. To answer this question, we create an equation: x+y=12. We received one equation with two unknowns, therefore, strictly speaking, it is not possible to find two numbers that gave this sum. But we can still say something about these numbers. We can say that these were either the numbers 1 and 11, or 2 and 10, or 3 and 9, or 4 and 8, etc., also it was either 13 and -1, or 14 and -2, or 15 and -3, etc. That is, we can use the sum (in our example 12) to determine the many possible options that add up to exactly 12. One of these options is the pair we are looking for, which actually gave 12 right now. It is also worth noting that all variants of pairs of numbers giving a total of 12 form a straight line, shown in Fig. 1, which is given by the equation x+y=12 (y=-x+12).
Fig.1
Thus, the pair we are looking for lies somewhere on this straight line. I repeat, it is impossible to choose from all these options the pair that actually existed - which gave the number 12, without knowing any additional clues. However, in the situation for which the Kalman filter was invented, such clues exist. There is something known in advance about random numbers. In particular, the so-called distribution histogram for each pair of numbers is known there. It is usually obtained after sufficiently long observation of the occurrence of these very random numbers. That is, for example, it is known from experience that in 5% of cases the pair x=1, y=8 usually appears (we denote this pair as follows: (1,8)), in 2% of cases the pair x=2, y=3 ( 2.3), in 1% of cases a pair (3.1), in 0.024% of cases a pair (11.1), etc. I repeat, this histogram is given for all couples numbers, including those that add up to 12. Thus, for each pair that adds up to 12, we can say that, for example, the pair (1, 11) appears 0.8% of the time, the pair ( 2, 10) – in 1% of cases, pair (3, 9) – in 1.5% of cases, etc. Thus, we can use the histogram to determine in what percentage of cases the sum of the terms of a pair is equal to 12. Let, for example, in 30% of cases the sum gives 12. And in the remaining 70%, the remaining pairs fall out - these are (1,8), (2, 3), (3,1), etc. – those that add up to numbers other than 12. Moreover, let, for example, the pair (7,5) appear in 27% of cases, while all other pairs that add up to 12 appear in 0.024%+0.8% +1%+1.5%+…=3% of cases. So, from the histogram we found out that numbers that add up to 12 appear in 30% of cases. Moreover, we know that if a 12 is rolled, then most often (27% of 30%) the reason for this is the pair (7,5). That is, if already If a 12 is rolled, we can say that in 90% of cases (27% of 30% - or, what is the same, 27 times out of every 30) the reason for the 12 being rolled is the pair (7,5). Knowing that most often the reason for receiving a sum equal to 12 is the pair (7.5), it is logical to assume that, most likely, it has fallen now. Of course, it’s still not a fact that in fact now the number 12 is formed by this particular pair, however, next time, if we come across 12, and we again assume the pair (7,5), then in about 90% of cases out of 100% we will be right. But if we guess the pair (2, 10), we will be right only in 1% of 30% of cases, which is equal to 3.33% of correct guesses compared to 90% when guessing the pair (7,5). That's it - that's the point of the Kalman filter algorithm. That is, the Kalman filter does not guarantee that it will not make a mistake in determining the summand by the sum, but it guarantees that it will make a mistake a minimum number of times (the probability of an error will be minimal), since it uses statistics - a histogram of the occurrence of pairs of numbers. It is also necessary to emphasize that the Kalman filtering algorithm often uses the so-called probability distribution density (PDD). However, it is necessary to understand that the meaning there is the same as that of a histogram. Moreover, a histogram is a function built on the basis of PDF and is its approximation (see, for example,).
In principle, we can depict this histogram as a function of two variables - that is, in the form of a certain surface above the xy plane. Where the surface is higher, the probability of getting the corresponding pair is higher. Figure 2 shows such a surface.
Fig.2
As you can see above the straight line x+y=12 (which has variants of pairs of giving in total 12), surface points are located at different heights and the highest height is for the variant with coordinates (7,5). And when we encounter a sum equal to 12, in 90% of cases the reason for the appearance of this sum is precisely the pair (7,5). Those. It is this pair, which adds up to 12, that has the highest probability of occurrence, provided that the sum is 12.
Thus, the idea behind the Kalman filter is described here. It is on this basis that all sorts of its modifications are built - one-step, multi-step recurrent, etc. For a more in-depth study of the Kalman filter, I recommend the book: Van Trees G. Theory of detection, estimation and modulation.
p.s. For those who are interested in explanations of the concepts of mathematics, as they say “on the fingers,” we can recommend this book and in particular the chapters from its “Mathematics” section (you can purchase the book itself or individual chapters from it).
Random Forest is one of my favorite data mining algorithms. Firstly, it is incredibly versatile; it can be used to solve both regression and classification problems. Search for anomalies and select predictors. Secondly, this is an algorithm that is really difficult to apply incorrectly. Simply because, unlike other algorithms, it has few customizable parameters. And it is also surprisingly simple in nature. And at the same time, it is amazingly accurate.
What is the idea behind such a wonderful algorithm? The idea is simple: let's say we have some very weak algorithm, say . If we make a lot of different models using this weak algorithm and average the results of their predictions, the final result will be significantly better. This is what is called ensemble learning in action. The Random Forest algorithm is therefore called “Random Forest”; for the received data, it creates many decision trees and then averages the result of their predictions. The important point here is the element of chance in the creation of each tree. After all, it is clear that if we create many identical trees, then the result of their averaging will have the accuracy of one tree.
How does he work? Let's assume we have some input data. Each column corresponds to some parameter, each row corresponds to some data element.
We can randomly select a certain number of columns and rows from the entire data set and build a decision tree based on them.
Thursday, May 10, 2012
Thursday, January 12, 2012
That's all. The 17-hour flight is over, Russia remains overseas. And through the window of a cozy 2-bedroom apartment, San Francisco, the famous Silicon Valley, California, USA, looks at us. Yes, this is the very reason why I haven’t written much lately. We moved.
This all started back in April 2011 when I had a phone interview with Zynga. Then it all seemed like some kind of game unrelated to reality and I could not even imagine what it would lead to. In June 2011, Zynga came to Moscow and conducted a series of interviews, about 60 candidates who passed a telephone interview were considered and about 15 people were selected from them (I don’t know the exact number, some later changed their minds, others immediately refused). The interview turned out to be surprisingly simple. No programming problems, no tricky questions about the shape of hatches, mostly testing your ability to chat. And knowledge, in my opinion, was assessed only superficially.
And then the rigmarole began. First we waited for the results, then the offer, then the LCA approval, then the approval of the visa petition, then documents from the USA, then the queue at the embassy, then additional verification, then the visa. At times it seemed to me that I was ready to give up everything and score. At times I doubted whether we needed this America, after all, Russia is not bad either. The whole process took about six months, in the end, in mid-December we received visas and began to prepare for departure.
Monday was my first working day in a new place. The office has all the conditions for not only working, but also living. Breakfasts, lunches and dinners from our own chefs, a lot of varied food stuffed in every corner, a gym, massage and even a hairdresser. All this is completely free for employees. Many people commute to work by bicycle, and several rooms are equipped for storing vehicles. In general, I have never seen anything like this in Russia. However, everything has its price; we were immediately warned that we would have to work a lot. What “a lot” is, by their standards, is not very clear to me.
I hope, however, that despite the amount of work, in the foreseeable future I will be able to resume blogging and, perhaps, tell something about American life and working as a programmer in America. Wait and see. In the meantime, I wish everyone a Happy New Year and Christmas and see you again!
For an example of use, we will print out the dividend yield of Russian companies. As the base price, we take the closing price of the share on the day the register is closed. For some reason, this information is not available on the Troika website, but it is much more interesting than the absolute values of dividends.
Attention! The code takes quite a long time to execute, because... For each promotion you need to make a request to finam servers and get its value.
Result<- NULL for(i in (1:length(divs[,1]))){ d <- divs if (d$Divs>0)( try(( quotes<- getSymbols(d$Symbol, src="Finam", from="2010-01-01", auto.assign=FALSE) if (!is.nan(quotes)){ price <- Cl(quotes) if (length(price)>0)(dd<- d$Divs result <- rbind(result, data.frame(d$Symbol, d$Name, d$RegistryDate, as.numeric(dd)/as.numeric(price), stringsAsFactors=FALSE)) } } }, silent=TRUE) } } colnames(result) <- c("Symbol", "Name", "RegistryDate", "Divs") result
Similarly, you can build statistics for previous years.
The Kalman filter is probably the most popular filtering algorithm used in many fields of science and technology. Due to its simplicity and efficiency, it can be found in GPS receivers, sensor data processors, in the implementation of control systems, etc.
There are a lot of articles and books on the Internet about the Kalman filter (mostly in English), but these articles have a fairly high barrier to entry; there are many vague places, although in fact it is a very clear and transparent algorithm. I will try to talk about it in simple language, with a gradual increase in complexity.
What is it for?
Any measuring device has some error; it can be influenced by a large number of external and internal influences, which leads to the fact that the information from it is noisy. The more noisy the data, the more difficult it is to process such information.A filter is a data processing algorithm that removes noise and unnecessary information. In the Kalman filter, it is possible to specify a priori information about the nature of the system, the relationship of variables, and based on this build a more accurate estimate, but even in the simplest case (without entering a priori information) it gives excellent results.
Let's consider a simple example - suppose we need to control the fuel level in the tank. To do this, a capacitive sensor is installed in the tank; it is very easy to maintain, but has some disadvantages - for example, dependence on the fuel being filled (the dielectric constant of the fuel depends on many factors, for example, temperature), and the large influence of “bottleness” in the tank. As a result, the information from it represents a typical “saw” with a decent amplitude. These types of sensors are often installed on heavy mining equipment (don’t be confused by the tank volume):
Kalman filter
Let's digress a little and get acquainted with the algorithm itself. The Kalman filter uses a dynamic model of the system (for example, a physical law of motion), known control inputs, and multiple sequential measurements to form an optimal state estimate. The algorithm consists of two repeated phases: prediction and adjustment. At the first stage, the prediction of the state at the next moment in time is calculated (taking into account the inaccuracy of their measurement). On the second, new information from the sensor corrects the predicted value (also taking into account the inaccuracy and noise of this information):The equations are presented in matrix form; if you don’t know linear algebra, that’s okay, what follows is a simplified version without matrices for the case with one variable. In the case of one variable, matrices degenerate into scalar values.
Let's first understand the notation: a subscript indicates a moment in time: k - current, (k-1) - previous, the minus sign in the superscript indicates that this predicted intermediate value.
Descriptions of the variables are presented in the following images:
You can describe for a long time and tediously what all these mysterious transition matrices mean, but it is better, in my opinion, to try to apply the algorithm on a real example - so that the abstract meanings acquire real meaning.
Let's try it in action
Let's return to the example with the fuel level sensor, since the state of the system is represented by one variable (the volume of fuel in the tank), the matrices degenerate into ordinary equations:Defining a Process Model
In order to apply the filter, it is necessary to determine the matrices/values of the variables that determine the dynamics of the system and the dimensions F, B and H:F- a variable describing the dynamics of the system, in the case of fuel - this can be a coefficient that determines fuel consumption at idle during the sampling time (the time between steps of the algorithm). However, in addition to fuel consumption, there are also gas stations... so for simplicity, we will set this variable to 1 (that is, we indicate that the predicted value will be equal to the previous state).
B- variable determining the application of control action. If we had additional information about engine speed or the degree of pressure on the accelerator pedal, then this parameter would determine how fuel consumption would change during the sampling period. Since there are no control actions in our model (there is no information about them), we accept B = 0.
H- a matrix defining the relationship between measurements and the state of the system; for now, without explanation, we will accept this variable also equal to 1.
Defining smoothing properties
R- measurement error can be determined by testing measuring instruments and determining the error of their measurement.Q- determining process noise is a more difficult task, since it is necessary to determine the variance of the process, which is not always possible. In any case, you can select this parameter to provide the required level of filtration.
Let's implement it in code
To dispel the remaining confusion, let’s implement a simplified algorithm in C# (without matrices and control actions):class KalmanFilterSimple1D
{
public double X0 (get; private set;) // predicted state
public double P0 ( get; private set; ) // predicted covariance
Public double F ( get; private set; ) // factor of real value to previous real value
public double Q ( get; private set; ) // measurement noise
public double H ( get; private set; ) // factor of measured value to real value
public double R ( get; private set; ) // environment noise
Public double State ( get; private set; )
public double Covariance ( get; private set; )
Public KalmanFilterSimple1D(double q, double r, double f = 1, double h = 1)
{
Q = q;
R = r;
F = f;
H = h;
}
Public void SetState(double state, double covariance)
{
State = state;
Covariance = covariance;
}
Public void Correct(double data)
{
//time update - prediction
X0 = F*State;
P0 = F*Covariance*F + Q;
//measurement update - correction
var K = H*P0/(H*P0*H + R);
State = X0 + K*(data - H*X0);
Covariance = (1 - K*H)*F;
}
}
// Application...
Var fuelData = GetData();
var filtered = new List();
Var kalman = new KalmanFilterSimple1D(f: 1, h: 1, q: 2, r: 15); // set F, H, Q and R
kalman.SetState(fuelData, 0.1); // Set the initial values of State and Covariance
foreach(var d in fuelData)
{
kalman.Correct(d); // Apply the algorithm
Filtered.Add(kalman.State); // Save the current state
}
The result of filtering with these parameters is shown in the figure (to adjust the degree of smoothing, you can change the parameters Q and R):
The most interesting part remains outside the scope of the article - applying the Kalman filter for several variables, specifying the relationship between them and automatically outputting values for unobserved variables. I will try to continue the topic as soon as I have time.
I hope the description was not too tedious and complicated, if you have any questions or clarifications, welcome to the comments)
Transcript
1 # 09, September 2015 UDC Application of the Kalman filter for processing a sequence of GPS coordinates Listerenko R.R., bachelor Russia, Moscow, MSTU. N.E. Bauman, Department of Computer Software and Information Technologies Scientific supervisor: Bekasov D.E., assistant Russia, Moscow, MSTU. N.E. Bauman, Department of Computer Software and Information Technologies The task of filtering GPS coordinates Currently, GPS tracking services are widely used, the task of which is to track the routes of observed objects in order to save them and further reproduce and analyze them. However, due to the error of the GPS sensor due to a number of reasons, such as loss of signal from the satellite, changes in satellite geometry, signal reflection, computational errors and rounding errors, the final result does not exactly correspond to the route of the object. There are both minor deviations (within 100 m), which do not impede the perception of visual information about the route and its analysis, and very significant ones (up to 1 km, in the case of loss of satellite signal and the use of base stations up to several tens of km). To demonstrate the result of the algorithm presented in the article, a route containing deviations from the actual location exceeding several kilometers is used. In order to correct such errors, an algorithm is being developed that transforms a sequence of coordinates. The input data for the algorithm is a sequence of GPS coordinates. Each coordinate contains the following information received from the sensor: Latitude Longitude Azimuth in degrees Instantaneous speed of the object at a given point in m/s
2 Possible deviation of object coordinates from the true value in meters Time for receiving the coordinate by the sensor The result of the algorithm is a sequence of coordinates with corrected latitude and longitude. It was decided to use the Kalman filter as the basis for constructing the algorithm, since it allows us to separately take into account measurement errors and random process errors, as well as use the object movement speed obtained from the sensor. Construction of a mathematical model using the Kalman filter To use the Kalman filter, it is necessary that the process under study be described as follows: = + + (1) = + (2) In formula (1) - the state vector of the process, A - matrix of dimension n n, describing the transition of the observed process from state to state. The vector describes the control influences on the process. Matrix B of dimension n l maps the vector of control actions u into a change in state s. is a random variable describing the errors of the process under study, and ~0, where Q is the covariance matrix of the process errors. Formula (2) describes the measurements of a random process. - vector of the measured state of the process, matrix H with dimension m n maps the state of the process into the process dimension. - a random variable characterizing measurement errors, and ~0, where P is the covariance matrix of measurement errors. Since the process of motion of an object is being studied, the equation of state is compiled based on the equation of motion of the body = + +!" #$ % & ". In addition, there is no additional information about the movement process, therefore it is considered that the control action is equal to 0. The vector = + () *, - is taken as the state of the process. +, where x, y are the coordinates of the object, and are the projections of the object’s velocity. Thus, for the process under consideration, equation (1) takes the following form: = + /!, (3) Youth Scientific and Technical Bulletin of the FS, ISSN
3 where = ! = 3! + 7 " 0 ; 6 2: 6 " / = : 6 0: 6 2: 6 0: , (4)!,4, (5) (6) In this model, the acceleration of an object is considered as a random error of the process. The following assumptions are made: a) Accelerations along different axes are independent random variables.),* b)
4 = AB = C. C E. = C/!!. /. =/C!!. /. Since the components of the vector ak (5) are independent random variables, then C!!. = " 0 " G. Consequently, formula (7) takes the following form: = / " (8) The measurement vector zk for this problem is represented as follows: H I = 0 + J, J (7) 2, (9) where H, I - the coordinates of the object received from the sensor, J +, J - the speed of the object received from the sensor. The matrix H in formula (2) is taken to be equal to the identity matrix of dimension 4 4, since within the framework of this problem it is considered that the measurement is a linear combination state vector and some random errors. The covariance matrix of the measurement error R is considered given. One of the possible options for its calculation is the use of data on the expected measurement accuracy obtained from the sensor. Application of the Kalman filter to the constructed model To apply the filter, it is necessary to introduce the following concepts: - a posteriori estimate state of the object at moment k, obtained from the results of observations up to and including moment k. L - uncorrected a posteriori estimate of the state of the object at moment k. - a posteriori covariance matrix of errors, specifying an estimate of the accuracy of the obtained estimate of the state vector and including an estimate of the error variances of the calculated state and covariances, showing the identified relationships between the parameters of the system state. L is the unadjusted posterior covariance error matrix. Matrix P0 is set as zero, since it is assumed that the initial position of the object is known. Youth scientific and technical bulletin of the FS, ISSN
5 One iteration of the Kalman filter consists of two stages: extrapolation and correction. a) At the extrapolation stage, the estimate L is calculated from the estimate of the state vector L and the covariance matrix of errors L according to the following formulas: L =, (10) L =. +, (11) where matrix Ak is known from formula (4), matrix Qk is calculated using formula (8). b) At the correction stage, the matrix of gain factors Kk is calculated using the following formula: M = L. L. + (12) where R, H are considered known. Kk is used to correct the object state estimate L and the error covariance matrix L as follows: = L + M L, (13) = N M L, (14) where I is the identity matrix. It should be noted that in order to use the above relations, it is necessary that the units of measurement be consistent for the object parameters involved in the calculations. However, in the source data, latitude and longitude are given in angular coordinates, and speed in metric coordinates. In addition, it is also more convenient to specify the acceleration for calculating process error in metric units. Vincenti's formulas are used to convert velocity and acceleration into angular units. Result of the filter In Fig. 1 shows an example of a route before processing. It can be noted that in this example there are several coordinates with a high degree of error, which is expressed in the presence of “peaks” of coordinates significantly removed from the main route. In Fig. Figure 2 shows the result of the filter working with this route.
6 Fig. 1. Object route Fig. 2. The route of the object after applying the filter As a result, there are practically no “peaks”, with the exception of the largest one, which was noticeably reduced, and the rest of the route was smoothed out. Thus, using the above algorithm, it was possible to reduce the degree of distortion of the route and increase its visual quality. Conclusion This paper examined an approach to correcting GPS coordinates using a Kalman filter. Using the above algorithm, it was possible to eliminate the most noticeable distortions of the route, which demonstrates the applicability of this method to the problem of smoothing the route and eliminating peaks. However, to further improve the quality of the algorithm, additional processing of the sequence of coordinates is necessary for the purpose of Youth Scientific and Technical Bulletin of the FS, ISSN
7 elimination of redundant points that arise when there is no movement of the observed object. References 1. Yadav J., Giri R., Meena L. Error handling in GPS data processing // Mausam Vol. 62.No. 1. P Kalman R. E. A New Approach to Linear Filtering and Prediction Problems // Transactions of the ASME Journal of Basic Engineering Vol. 82.No. Series D. P. P. Welch G., Bishop G. An Introduction to the Kalman Filter: Tech. Rep. TR Available at: accessed Vincenty T. Direct and Inverse Solutions of Geodesics on the Ellipsoid with application of nested equations // Survey Review apr. Vol. 23. No PP
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National Technical University of Ukraine "Kiev Polytechnic Institute" Department of Instruments and Orientation and Navigation Systems Guidelines for laboratory work in the discipline "Navigation
UDC 629.78.018:621.397.13 METHOD OF PAIRED DISTANCES IN THE PROBLEM OF FLIGHT ADJUSTMENT OF ASTRO SENSORS OF A SPACE CAR SYSTEM B.M. Sukhovilov As the accuracy and reliability of astronomical data improves,
UDC 629.05 Solving the navigation problem using a strapdown inertial navigation system and an air signal system Mkrtchyan V.I., student, department of “Instruments and systems for orientation, stabilization and navigation”
MODEL OF THE VISUAL SYSTEM OF THE HUMAN OPERATOR IN OBJECT IMAGE RECOGNITION Yu.S. Gulina, V.Ya. Kolyuchkin Moscow State Technical University named after. N.E. Bauman, Explains the mathematical
ROCKET AND SPACE INSTRUMENT ENGINEERING AND INFORMATION SYSTEMS 2015, volume 2, issue 3, p. 79 83 UDC 681.3.06 SYSTEM ANALYSIS, SPACE VEHICLE CONTROL, INFORMATION PROCESSING AND TELEMETRY SYSTEMS
Linear dynamic systems. Kalman filter. Educational program: some properties of the normal distribution Distribution density.4.3.. -4 x b.5 x b =.7 5 p(x a x b =.7) - x p(x a,x b) p(x a) 4 3 - - -3 x.5
UDC 621.396.671 O. S. Litvinov, A. A. Gilyazova ASSESSMENT USING THE METHOD OF OWN DIAGRAMS OF THE INFLUENCE OF INTERFERENCE GROUPS ON THE RECEIPT OF USEFUL SIGNAL BY A LINEAR EQUIDISTANT ADAPTIVE ANTENNA
UDC 681.5.15.44 PRONOSIS OF PIECELE-STATIONARY PROCESSES E.Yu. Alekseeva Discrete random processes containing parameters changing abruptly at random times are considered. For
UDC 63966 LINEAR OPTIMAL FILTERING FOR NON-WHITE NOISE G F Savinov In this work, an optimal filter algorithm is obtained for the case when the input influences and noises are random Gaussian
Determination of oscillatory motions of non-rigid satellite elements using video image processing D.O. Lazarev Moscow Institute of Physics and Technology Scientific supervisor, Ph.D.: D.S. Ivanov, Institute
UDC 004 ON METHODS OF TRACKING AND TRACKING AN OBJECT ON A VIDEO STREAM IN APPLICATION TO A VIDEO ANALYTICS SYSTEM FOR COLLECTING AND ANALYZING MARKETING DATA Chezganov D.A., Serikov O.N. South Russian State
Electronic journal "Proceedings of MAI". Issue 66 www.ma.u/scence/tud/ UDC 69.78 Modified navigation algorithm for determining the position of an artificial satellite using GS/GLONASS signals Kurshin A. V. Moscow Aviation
UDC 621.396.96 Study of the algorithm for establishing and confirming trajectories according to the criterion M from N Chernova T.S., student of the department of “Radio-electronic systems and devices”, Russia, 105005, Moscow, MSTU. N.E.
THEORY AND PRACTICE OF NAVIGATION DEVICES AND SYSTEMS UDC 531.383 INFLUENCE OF STAND ROTATION ERROR ON THE ACCURACY OF CALIBRATION OF THE BLOCK OF GYROSCOPES AND ACCELEROMETERS Avrutov V. V., Mazepa T. Yu. National Technical
Lecture 6 Characteristics of portfolios In previous lectures, the term “portfolio” was repeatedly used. For the mathematical formulation of the problem of choosing an optimal portfolio, a strict definition of this is necessary
IDENTIFICATION OF TIME SERIES WITH GAPS BASED ON STATE SPACE MODELS R. I. Merkulov V. I. Lobach Belarusian State University Minsk Belarus e-mail: [email protected] [email protected]
DEVICES AND SYSTEMS OF AUTOMATIC CONTROL UDC 51971 V N ARSENYEV, A G KOKHANOVSKY, A S FADEYEV MATHEMATICAL MODEL OF CONNECTION OF ISOCHRONIC VARIATIONS OF CONTROL SYSTEM STATE VARIABLES WITH PARAMETERS PERTURBATIONS
Proceedings of MAI. Issue 89 UDC 629.051 www.mai.ru/science/trudy/ Calibration of a strapdown inertial navigation system when turning around a vertical axis Matasov A.I.*, Tikhomirov V.V.** Moskovsky
Analytical geometry Module 1 Matrix algebra Vector algebra Text 4 (independent study) Abstract Linear dependence of vectors Criteria for linear dependence of two, three and four vectors
UDC 62.396.26 L.A. Podkolzina, K.. Drugov INFORMATION PROCESSING ALGORITHS IN NAVIGATION SYSTEMS OF LAND MOVING OBJECTS FOR THE CHANNEL OF DETERMINING POSITION COORDINATES To determine coordinates and parameters
STATISTICAL ANALYSIS OF PARAMETRIC TIME SERIES WITH GAPS BASED ON STATE SPACE MODELS S. V. Lobach Belarusian State University Minsk, Belarus e-mail: [email protected]
Mathematical methods for processing data UDC 6.39 S. Ya. Zhuk.. Kozheshkurt.. Yuzefovich National Technical University of Ukraine “KP” ave. Pobeda 37 356 Kyiv Ukraine Institute of Information Registration Problems NAS
Construction of MM statics of technological objects When studying the statics of technological objects, objects with the following types of structural diagrams are most often encountered (Fig: O with one input x and one
Estimation of spacecraft orientation parameters using the Kalman filter Student, Department of Automatic Control Systems: D.I. Galkin Scientific supervisor: A.A. Karpunin, Ph.D., Associate Professor
5. Meleshko V.V. Strapdown inertial navigation systems: Textbook. allowance / V.V. Meleshko, O.I. Nesterenko. Kirovograd: POLYMED-Service, 211. 172 p. Received before the editor 17th quarter 212 roku ÓKostyuk
UDC 004.896 Application of geometric transformations for anamorphizing images Kanev A.I., specialist Russia, 105005, Moscow, MSTU. N.E. Bauman, Department of Information Processing and Management Systems
4. Monte Carlo methods 1 4. Monte Carlo methods To simulate various physical, economic and other effects, methods called Monte Carlo methods are widely used. They owe their name
Band-pass filtering 1 Band-pass filtering In the previous sections, filtering of fast signal variations (smoothing) and its slow variations (detrending) was discussed. Sometimes you need to highlight
[NOTES] Explaining Kalman Filter Basics Using Simple and Intuitive Derivation Ramsey Faraher's article provides simple and intuitive Kalman filter derivation for the purpose of teaching
UDC 004.932 Fingerprint classification algorithm Lomov D.S., student Russia, 105005, Moscow, MSTU. N.E. Bauman, Department of Computer Software and Information Technologies Scientific supervisor:
Lecture NUMERICAL CHARACTERISTICS OF A SYSTEM OF TWO RANDOM VARIABLES -DIMENSIONAL RANDOM VECTOR PURPOSE OF THE LECTURE: to determine the numerical characteristics of a system of two random variables: initial and central moments covariance
The dynamics of the birth rate in the Chuvash Republic Contents Introduction 1. The general trend in the birth rate of the population of the Chuvash Republic 2. The main trend in the birth rate 3. The dynamics of the birth rate in urban and rural areas
IN 1990-5548 Electronics and control systems. 2011. 4(30) 73 UDC656.7.052.002.5:681.32(045) V. M. Sineglazov, Doctor of Engineering. Sciences, Prof., Sh. I. Askerov OPTIMAL COMPREHENSIVE DATA PROCESSING IN NAVIGATION
UDC 004.896 Features of the implementation of the algorithm for displaying anamorphic results Kanev A.I., specialist Russia, 105005, Moscow, MSTU. N.E. Bauman, Department of Information Processing Systems and
177 UDC 658.310.8: 519.876.2 USE OF ESTIMATION ACCURACY IN RESERVATION OF SENSORS L.I. Luzina The article discusses a possible approach to obtain a new sensor redundancy scheme. Traditional
COLLECTION OF SCIENTIFIC WORKS OF NSTU. 28.4(54). 37 44 UDC 59.24 ABOUT A COMPLEX OF PROGRAMS FOR SOLVING THE PROBLEM OF IDENTIFICATION OF LINEAR DYNAMIC DISCRETE STATIONARY OBJECTS G.V. TROSHINA A set of programs was considered
UDC 625.1:519.222:528.4 S.I. Dolganyuk S.I. Dolganyuk, 2010 INCREASING THE ACCURACY OF A NAVIGATION SOLUTION WHEN POSITIONING SHUNTERING LOCOMOTIVES THROUGH THE USE OF DIGITAL MODELS OF TRACK DEVELOPMENT
UDC 531.1 ADAPTATION OF THE KALMAN FILTER FOR USE WITH LOCAL AND GLOBAL NAVIGATION SYSTEMS A.N. Zabegaev ( [email protected]) V.E. Pavlovsky ( [email protected]) Institute of Applied Mathematics named after.
AUTOMATION AND CONTROL UDC 68.58.3 A. G. Shpektorov, V. T. Fam St. Petersburg State Electrotechnical University "LETI" named after. V. I. Ulyanova (Lenina) Analysis of the use of micromechanical
BASICS OF REGRESSION ANALYSIS CONCEPT OF CORRELATION AND REGRESSION ANALYSIS To solve problems of economic analysis and forecasting, statistical, reporting or observed data are often used
Lecture 4. Solving systems of linear equations using the method of simple iterations. If the system has a large dimension (6 equations) or the system matrix is sparse, indirect iterative methods are more effective for solving
58th scientific conference MIPT Section of dynamics and motion control of spacecraft System for determining the motion of control system models on an aerodynamic table using a video camera
Lecture 3 5. METHODS OF APPROXIMATION OF FUNCTIONS PROBLEM STATEMENT We consider grid tabular functions [ a b] y 5. defined at grid nodes Ω. Each grid is characterized by steps h uneven or h
1. Numerical methods for solving equations 1. Systems of linear equations. 1.1. Direct methods. 1.2. Iterative methods. 2. Nonlinear equations. 2.1. Equations with one unknown. 2.2. Systems of equations. 1.
UDC 621.396 RESEARCH OF ALGORITHMS FOR SECONDARY INFORMATION PROCESSING OF MULTI-POSITION RADAR SYSTEM FOR ELIZATION ANGLE CHANNEL Borisov A.N., Glinchenko V.A., Nazarov A.A., Islamov R.V., Suchkov P.V. Scientific
Topic Numerical methods of linear algebra - - Topic Numerical methods of linear algebra Classification There are four main sections of linear algebra: Solving systems of linear algebraic equations (SLAEs)
UDC 004.352.242 Restoration of blurred images by solving an integral equation of convolution type Ivannikova I.A., student Russia, 105005, Moscow, MSTU. N.E. Bauman, Department of Automated
AEROGRAVIMETRIC SURVEYING WITH STANDARD GPS OPERATION MODE Mogilevsky V.E. JSC State Research and Production Enterprise Aerogeofizika The most important element determining the success of airborne geophysical research is high-quality navigation
ANALYSIS OF ACOUSTIC SIGNALS BASED ON THE KALMAN FILTRATION METHOD I.P. Gurov, P.G. Zhiganov, A.M. Ozersky The features of dynamic processing of stochastic signals using discrete
UDC AA Minko IDENTIFICATION OF A LINEAR OBJECT BY RESPONSE TO A HARMONIC SIGNAL A generalized identification algorithm is proposed based on integral two-parameter Gaussian transformations of a linear stationary
LECTURE. Estimation of complex signal amplitude. Estimation of signal delay time. Frequency estimation of a signal with random phase. Joint estimation of delay time and frequency of a signal with random phase.
Computational Technologies Volume 18, 1, 2013 Identification of parameters of the anomalous diffusion process based on difference equations A. S. Ovsienko Samara State Technical University, Russia e-mail:
1 FORECASTING THE MARKET CONDITIONS FOR PETROCHEMICAL ENTERPRISES Kordunov D.Yu., Bityutsky S.Ya. Introduction. In modern economic conditions, which are characterized by the rapid development of global integration
Problem of simultaneous localization and mapping (SLAM) Roboschool-2014 Andrey Antonov robotosha.ru October 10, 2014 Plan 1 Basics of SLAM 2 RGB-D SLAM 3 Robot Andrey Antonov (robotosha.ru) SLAM problem
UDC 004.021 T. N. Romanova, A. V. Sidorin, V. N. Solyakov, K. V. Kozlov SYNTHESIS MONOCHROME IMAGE FROM A MULTI-RANGE PALETTE CONSTRUCTION USING SOLVING THE POISSON EQUATION
National Technical University of Ukraine "Kiev Polytechnic Institute" Department of Instruments and Orientation and Navigation Systems Guidelines for laboratory work in the discipline "Navigation
Digital Signal Processing /9 UDC 69.78 ANALYTICAL METHOD FOR CALCULATING ERRORS IN DETERMINING ANGULAR ORIENTATION BY SIGNALS OF SATELLITE RADIONAVIGATION SYSTEMS Aleshechkin A.M. Introduction Detection Mode
FEATURES OF FORMATION OF A COMPUTER MODEL OF A DYNAMIC OPTICAL-ELECTRONIC SYSTEM Pozdnyakova N.S., Torshina I.P. Moscow State University of Geodesy and Cartography Faculty of Optical Information
Proceedings of ISA RAS 009. T. 46 III. APPLIED PROBLEMS OF DISTRIBUTED COMPUTING Stationary states in the nonlinear model of charge transfer in DNA * Stationary states in the nonlinear model of charge transfer in