# What is Kalman filter: Definition and 49 Discussions

In statistics and control theory, Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe. The filter is named after Rudolf E. Kálmán, who was one of the primary developers of its theory.
The Kalman filter has numerous applications in technology. A common application is for guidance, navigation, and control of vehicles, particularly aircraft, spacecraft and dynamically positioned ships. Furthermore, the Kalman filter is a widely applied concept in time series analysis used in fields such as signal processing and econometrics. Kalman filters also are one of the main topics in the field of robotic motion planning and control and can be used in trajectory optimization. The Kalman filter also works for modeling the central nervous system's control of movement. Due to the time delay between issuing motor commands and receiving sensory feedback, the use of the Kalman filter supports a realistic model for making estimates of the current state of the motor system and issuing updated commands.The algorithm works in a two-step process. In the prediction step, the Kalman filter produces estimates of the current state variables, along with their uncertainties. Once the outcome of the next measurement (necessarily corrupted with some amount of error, including random noise) is observed, these estimates are updated using a weighted average, with more weight being given to estimates with higher certainty. The algorithm is recursive. It can run in real time, using only the present input measurements and the previously calculated state and its uncertainty matrix; no additional past information is required.
Optimality of the Kalman filter assumes that the errors are Gaussian. In the words of Rudolf E. Kálmán: "In summary, the following assumptions are made about random processes: Physical random phenomena may be thought of as due to primary random sources exciting dynamic systems. The primary sources are assumed to be independent gaussian random processes with zero mean; the dynamic systems will be linear." Though regardless of Gaussianity, if the process and measurement covariances are known, the Kalman filter is the best possible linear estimator in the minimum mean-square-error sense.Extensions and generalizations to the method have also been developed, such as the extended Kalman filter and the unscented Kalman filter which work on nonlinear systems. The underlying model is a hidden Markov model where the state space of the latent variables is continuous and all latent and observed variables have Gaussian distributions. Also, Kalman filter has been successfully used in multi-sensor fusion, and distributed sensor networks to develop distributed or consensus Kalman filter.

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1. ### I Noise Whiteness hypothesis in Kalman filtering

In Kalman filter mathematical treatment I have always read that a foundamental hypothesis is represented by the whiteness of the process noise. I have tried to do again the mathematical steps in the Kalman filter derivation but I can't see where such hypothesis is crucial. Could you help me...
2. ### Engineering Extended Kalman Filter Jacobians

Hi, I have a question about calculating the Jacobian matrices for the Extended Kalman filter. Question: If we have a system of the form: \begin{align*} x_{k+1} =f_k (x_k , u_k) + w_k \\ y_k = h_k (x_k , u_k ) +v_k \end{align*} where the state x_k comprises of the three variables p_1 ...
3. ### Comp Sci Kalman filter in one dimension to track a moving object

Below is the whole code. I can't change the whole code, I can only change the "Kalman class". The Kalman class in the code below is my attempt to solve the problem. But the code doesn't work well. I have written these 5 equations in the Kalman-filter algorithm: State Extrapolation Equation...
4. ### Unit of Velocity from a Random Walk measured by an accelerometer

Hi, I am working on a kalman filter where my measurement equation involves "-g + v" , where g is in m/s^2 and v is velocity random walk given in m/s/sqrt(hr). Feels like a stupid question, but how can I transform the unit of velocity random walk so I can do the calculation correctly?
5. ### Can the Kalman filter handle variations in sampling interval?

Dear Community, I want to use a Kalman- filter approach to calculate inclination angles (pitch and roll) obtained from accelerometer- and gyroscope- raw data. However the problem is that my sampling interval dt is not constant – it has a significant jitter instead. To investigate whether the...
6. R

### Kalman filter unstable in closed loop application

I have implemented the closed-loop motor control system as above in a Matlab simulation (pic Kalman.png). Here, the Kalman filter estimates the torque disturbance and angular speed of the motor and those are feed to the RLS algorithm for parameter identification, here it estimates the combined...
7. ### Problems with Unscented Kalman Filter

Hi guys, I have problems understanding UKF and its environment. I will ask a couple of questions and see if you can help me. I have a nonlinear system (SYS) with one input called P (scalar) and one output called IP (also scalar). I would like to estimate the future output (the IP), so in my...
8. ### Kalman Filter States: What Do They Represent?

It will sound a little bit stupid, but Ill ask anyways: I have two series systems, the second is an UKF (Uscented Kalman Filter). I was told to reduce the first system in order to that the UKF estimate less states. My question is: What exactly do the states represent? I mean, if I have a...
9. ### Kalman filter and measurements

Hello, I am using a Kalman filter with a PI controller. The goal is to control a mirror to stabilize the position of a laser spot: I am sending a disturbance on the mirror and the PI sends command to the same mirror so the spot stays fix. With a camera I am recording the image of the laser and...
10. ### State prediction equation for Kalman filter

I have to filter data with Kalman filter. I know process error covariance Q and measurment error covariance R. Problem is with state transition matrix A, control matrix B and observation matrix H. First of all, data goes through this transfer function: ##W(s) = \frac{4s}{4s+1}## I can't get it...
11. ### How Are GM and Auto-Regression Models Used in GPS/INS Systems?

Hi I'm using INS with GPS. and I have some question about noise. After denoising measures from high frequency noise with wavelet. the short frequency will stay on measures. we generally GM or Auto-regression to model the remain error. Is this models used only for kalman filter or it can be...
12. ### Kalman filter a posteriori estimate covariance matrix

I am having some trouble deriving the a posteriori estimate covariance matrix for the linear Kalman filter. Below I have shown my workings for two methods. Method one is fine and gives the expected result. Method two is the way I tried to derive it initially before further expanding out terms to...
13. ### GNS/INS (Inertial Navigation System) integration

Hi I'm working on a GPS/INS model. Now I developped the mecanization equations for the INS. I have a kalman filter which estimate the error state (position error with respect to north and east). it takes as an input the diffrence between GPS and INS and gives its estimation . I have...
14. ### How does the Kalman filter calculate derivatives?

Suppose we have a Kalman filter. We have a position sensor, for example GPS. We use the filter to estimate position. However in all examples I see higher derivatives in the state vector: speed, acceleration and sometimes jerk. There is no sensor that calculates these values directly, so they...
15. ### Noise modeling with Markov modeling

Hi I'm using accelerometer & horizontal gyroscope in order to replace GPS. Now, I'want to model the noise with first order markov process, to use it in kalman filter. I recorded measurement on all axes and computed auto-correlation. This picture represents auto-correlation on one of axes...
16. ### Kalman filter, model to matrix

I have my state vector containing $$[X, Y, v_x, v_y, \theta, r, a_x, a_y, b_{\theta}]^T$$ and I have them related by $$dX = v_x cos \theta - v_y sin \theta\\ dY = v_x sin \theta + v_y cos \theta\\ dv_x = a_x\\ dv_y = a_y\\ d\theta = r\\ dr = 0\\ da_x = 0\\ da_y = 0\\ db_\theta = 0\\$$ Now...
17. ### How to use KALMAN with accelerometer?

hi I found an article that shows how to use kalman filter to models error of accelerometer. they used markov process as stochastic error, then output of this model will be used as input of KALMAN filter. Now, I don't what to do. I'm not sure, but I think that I need to double integrate the...
18. ### Kalman filter - help me to set up a state equation

I would like to find the distance that a vehicle travels using a Kalman filter. The vehicle is a car that travels the road between two positions. The vehicle has a GPS/barometer/accelerometer device that collects position data, which I converted from a longitude and latitude to a North, East...
19. ### How Can a Kalman Filter Estimate Distance Using GPS and Velocity Data?

I want to get the distance between points. I have a measurement of latitude, longitude and height from GPS/barometer systen (which I converted into north east down coordinates) and I have another measurement of the vehicle's velocity in the north east and down direction. To find the distance, I...
20. ### Kalman filter with mixed frequency

Hi first post hoping for some advise or guidance. Let's say I have GPS data on a robot every 10 Hz (every 0.1 second), I have accelerometer data every 40 Hz and steering wheel data every 80 Hz. I would like to use Kalman filter to produce estimates of where the robot is every 40 Hz given the...
21. ### MATLAB How Do System and Measurement Noise Covariance Values Affect Kalman Filtering?

On the mathworkd website, they have a case example of Kalman filtering here. What is the system and measurement noise covariance in this example?
22. ### MHB Kalman filter and simulating Gaussian noise

Water level assumed constant. Static state model $$P_0 = 1000$$, system noise $$Q = 0.0001$$, measurement noise $$R = 0.1$$. I want to use Kalman filtering to solve this problem. I know how to do this but I need to generate $$\mathbf{y} = y$$ using Gaussian random noise. $$P_0$$ is also call...
23. ### MHB Kalman filter where does y_1 come from?

In this presentation, on page 7, they say due to noise $$y_1 = 0.9$$. How or where did they get this value? It isn't an article just a beamer presentation so going from page 1 - 7 is quick and easy.
24. ### How Should Gyro Measurements Be Integrated in Kalman Filter Models?

While trying to learn kalman filters, I see contradictory info for a simple gyro State Space representation. Please be patient with this post, I've put a fair bit of time trying to make it easy to follow and read! My goal is to understand how a moving IMU's orientation can be found. I'd...
25. ### Kalman Filter EM decrease log likelihood

Hi all I am using the Kalman Filter with an EM algorithm, (schumway and stoffer). From my understanding the log likelihood should monotonically increase. In some instances however I obtain a decrease in the log likelihood, What can I infer from this? Has anyone experience of these...
26. ### Kalman Filter - Acceleration Measurements

I am using a extended Kalman filter for the state estimation of a nonlinear system. The real system has accelerometers, so I need to include the acceleration of the system as part of my measurements with noise added (position and angular rates are other measurements). All of the Kalman...
27. ### Question concerning the extended kalman filter

Good day, i read a lot about the kalman filter and the extended kalman filter, but some things are still not clear to me. E.g. I have one question concerning the jacobian matrix of the measurement matrix h. I want to point out my problem with a concrete example: A vehicle is represented by...
28. ### Calculating Q Matrix for Extended Kalman Filter Identification

Dear people, I want to implement the Extended Kalman Filter for identification. I have a model of the system and I want to estimate the states. My question is, how can give the Q Matrix, I mean, how can I calculate the elements of the covariance matrix related to the dispersion of the system...
29. ### : Lyapunov Equation for backward continuous-time Kalman Filter

URGENT: Lyapunov Equation for backward continuous-time Kalman Filter Hi, Consider a continuous Kalman filter running backward in time as desired in a "two-filter" smoother. What would be the form of Lyapunov equation for this backward-time filter? Given a system: dx/dt = Fx + Gv, and...
30. ### Data fusion using extended kalman filter

I am presently working with data fusion of redundant sensor data, basically trying to put together data from an IMU, a gyro and odometry... My question is whether the accuracy of the final state be greater than the accuracy of the most accurate sensor used in the ekf? Thanks in advance for...
31. ### Kalman Filter Derivation with Non-Diagonal Q Matrix

Hi all, I have a standard local level model, but the disturbances are not independent: y_t=μ_t+ε_t, μ_t+1=μ_t+η_t, E(ε_t η_t) =/= 0 In order to derive the Kalman filter, I rewrite this model in state space form y_t=Z_t α_t+ε_t, ε_t~NID(0,H_t ), α_(t+1)=T_t α_t+R_t η_t...
32. ### Extended Kalman Filter : Measurement Matrix

Hi, I am currently learning and trying to build an extended kalman filter. However I am not sure how to created the measurement matrix. The measurement matrix H is described using formula 3.6 on page 18 of this document http://www.scribd.com/doc/54702247/UAV-Design My problem is that...
33. ### Is Including Pressure in a Kalman Filter for Wind Forecast Causing High Error?

Hi, I am now using multiple parameters kalman filter to remove the systematic error of the output from the numerical weather prediction model. when applying kalman filter to the wind forecast, there is high error if pressure is included to be the predictors. I would like to know if there any...
34. ### Prediction covariance matrix with Kalman filter

Hello all. I have set up a model using the Kalman filter to estimate automobile prices. I'm having difficulty in figuring out how to formulate a prediction covariance matrix based on the model, i.e. given a set y_{new} = y_1, \ldots, y_N of N cars, finding the covariance matrix based on the...
35. ### How to Determine Correct Values for Q and R in Designing a Kalman Filter?

Hey, I designed a kalman filter for my project. But how can I determine the correct values for Q and R? Thx!
36. ### Using Kalman Filter to Estimate Motion of Object Along Line Segment

I want to use a Kalman filter to estimate the motion of an object. However, the catch is, the measurements I have only tell me that the object is somewhere along a particular line segment. Typically Kalman filters require normally distributed measurements. I'm trying to work out how best to...
37. ### Kalman Filter for State Estimation

Hi, I am not sure if this is the right forum to post this. I am working on a Kalman Filter for battery state of discharge estimation during discharging process. I am using a Linearized Discrete Time Kalman Filter to estimate the State of Discharge/SOD (how much % of charge has been taken...
38. ### Extended Kalman Filter: Helping hand to a newbie

I trying to learn extended kalman filter. I am trying to simulate positions of point by angle mearsurments taken by an observer moving on a straight line. The EKF implementation does not converge. Below the MATLAB script clear all; close all; clc; %Generate staright line positions T=300...
39. ### Inertial navigation system using kalman filter

Hello, Im working on estimation of inertial navigation system using kalman filtering. Can anyone help me finding the state space model of inertial navigation system used to estimate the states via kalman filter. An early response is needed.Thanks in advance.
40. ### Sensor fusion using extended kalman filter

hi all, m trying to implement sensor fusion of two three axis accelerometers data.can anyone help me in modelling this sensor and fusing for state estimation using EKF? thanks in advance.!
41. ### Help with understanding kalman filter

I am trying to implement a kalman for use in sensor data fusion, for 3 axis accelerometers, 3 axis gyros, and 3 axis magnetometers for a helicopter UAV project. The roll, pitch, and yaw need to be known. Gyros do a great job of doing this, but the problem is that they drift of the zero value...
42. ### Extended Kalman Filter based SLAM for orientation

Dear all, I'm trying to implement an Extended Kalman Filter based SLAM for orientation and angular rates. To ease up things I discarded the estimation of position and linear velocities. I'm using a quaternion based approach with translation vectors of 3D-points/landmarks as measurements...
43. ### A question about Extended Kalman Filter

Hi, I'm currently implementing a visual 3D model-based vehicle tracking system as my undergrad dissertation. I've implemented the vehicle localization algorithm and now have an estimate of the x, y location on the ground plane as well as the orientation angle of the vehicle which would like...
44. ### Would you introduce me a tool about 'Kalman Filter'?

I need a tool to simulate 'Kalman Filter'. I know Matlab maybe do it, but it's not so convenient. Would you introduce me another one?
45. ### Exploring Kalman Filter Applications for Position Sensor Data Analysis

Hey everyone, I have a system that I know the x,y,z position and the alpha,beta,gamma euler rotation of an object in space at known intervals. I need to use a filter on this data to reduce the error of the measurements, and it would be nice to be able to predict future movement. I understand...
46. ### Extended Kalman Filter - Process noise and covariance

Dear all, I'm trying to implement an Extended Kalman Filter for position, velocity and orientation tracking of a rigid body and I am using quaternions for representing the orientation in the state vector. As this is the first time I have to work with a Kalman Filter, and the project is on a...
47. ### The Extended Kalman Filter

I've been asked to design a Kalman Filter where we can observe several states of a process (some of which being related to each other) and to use the Kalman filter to combine related observations to get a better estimate of each. Some texts I've been reading seem to indicate instead of making a...
48. ### Kalman Filter Help: Get Started Estimating Future Location of Mobile Device

Hi all, I am trying to estimate the future location of a mobile device and I was thinking of using the Kalman Filter to do this. Problem is I don't really know where to start. All the literature I have been reading gets much to advanced to quickly. What i have is basically, I am measuring...
49. ### Understanding the Extended Kalman Filter

Hi there, I'm wasn't sure whether I should post this question in the maths section or here, but since this technique is used a lot in aerospace I thought I'd try here first. I've grasped the idea of an ordinary Kalman Filter and created a program that tracks a noisy signal. However when...