The Kalman filter is a mathematical method using noisy measurements observed over time to produce values that tend to be closer to the true values of the measurements and their associated calculated values.

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Non-zero mean of Kalman innovation

I am using a Kalman filter to fuse gyro and inclinometer data. The prediction step is given by: $\hat{\alpha_{i}}^- = \hat{\alpha}^+_{i-1} + \omega_{i}$ Where $\hat{\alpha_{i}}^-$ is the prediction ...
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Will an unscented Kalman filter be “as good” as other optimisation algorithms for this problem?

I want to calibrate a tri-axis magnetometer when a tri-axis gyroscope is also available. I am fairly certain I can solve this problem using various optimisation algorithms, but I would prefer to use ...
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Can a (vanilla) Kalman Filter's Observation Matrix $H_k$ depend on the state vector $x_k$?

A vanilla Kalman Filter allows for a time varying observation matrix $H_k$. Is it allowable for $H_k$ to be a function of the system state $x_k$ in a vanilla Kalman filter? First, am I correct that ...
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Why Runge-Kutta for Quaternion integration in Kalman filter?

I'm reading up on Kalman filtering at the moment. In particular, I'm interested in using the "extended" and "unscented" variants for IMU sensor fusion and calibration. In A comparison of unscented ...
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Kalman filter : simple code example

I read lots of things about Kalman filtering, but in order to fully understand it, I would probably need to see it working on some data. Would you have a minimal example (Python code or any other ...
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Implementing a 1-D Kalman Filter Regression, Missing the smoothing action (getting the opposite)

I am implementing the 1D Kalman Filter in Python on a fundamentally noisy set of measurement data, and I should be observing a large amount of smoothing...but, instead, my Kalman Filter is doing the ...
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29 views

Tracking position and velocity using a kalman filter

I am using a kalman filter (constant velocity model) to track postion and velocity of an object. I measure x,y of the object and track x,y,vx,vy . Which works but if a add gausian noise of +- 20 mm to ...
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Kalman filter to filter noise from acceleration data

I would like to apply a Kalman filter to remove noise from my measured acceleration data. The idea is then to compare the results obtained in this way whit the results obtained by applying a low-pass ...
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Combining Kalman Filters

Say I want to use Kalman filters for predicting the price of items at a supermarket. I have a Kalman filter for each item (apple/beef/brooms/etc). I notice that some items are sort of related, like ...
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76 views

Kalman Filter for estimating position with nonconstant velocity & acceleration

I am trying to estimate the position & head direction of a rodent going through a 2D environment (a circular surface of 1m radius). Above his head is an overhead camera which records 4 LEDs ...
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Open source GPS+IMU sensor fusion?

Are there any Open source implementations of GPS+IMU sensor fusion (loosely coupled; i.e. using GPS module output and 9 degree of freedom IMU sensors)? -- kalman filtering based or otherwise I see a ...
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Observation Matrix in Kalman Filter

I've been trying to understand the concept of the Kalman Filter. I came across this great article which makes the concept sufficiently clear. However I could not understand the concept of the matrix ...
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What f() for unscented kalman filter for stock trading?

I am trying to estimate to "next" price of a stock, based on a group of 5 other correlated stocks. I believe this is a 6 state unscented Kalman problem. However, I do not know how to describe ...
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39 views

Why does my GAIN remain constant after a few cycles?

I am assuming that GAIN is the matrix P? From this example: ...
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Kalman Filter initial Q values

I have a 6 state Kalman Filter (Unscented). When I use a diagonal matrix only for Q (i.e only the diagonal has covariances), I get a "smooth" plot of estimate against actual. If I use the entire Q ...
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Is the Kalman Filter a Best Linear Unbiased Estimator (BLUE) for Heteroscedastic Noise?

According to the Gauss-Markov Theorem, a ordinary least squares estimator is BLUE if the noise entering a system is uncorrelated with zero mean and is homoscedastic (has a constant finite variance). I ...
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Can I model process noise as a known “error” in my dynamics while designing a Kalman Filter?

Consider I am modelling the dynamics of a robot and using a Kalman filter to obtain estimates of some state. I have certain terms in my equation which correspond to data not accessible to this robot ( ...
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Optimality of Kalman Filter for Process Noise dependent on magnitude of state

Consider I have a dynamical system $\dot{x} = Ax + w(t)$, $x \in \mathbb{R^2}$ where $w(t)$ is a Gaussian random variable with mean $E(w(t)) = C\|x\|^2$ where $C \in R^2$ is a constant and covariance ...
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85 views

Kalman Filter - Velocity [Matlab]

I have a quite typical Kalman filter to design. I really read a lot of articles about the design of this filter but the performances of my filter are still quite bad. Here is my situation. I have a ...
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Convergence analysis and Kalman Gain

I have a general question in regard to stability and convergence analysis of filtering algorithm like Kalman filter and its non-linear version - Extended and Unscented for parameter estimation. In ...
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108 views

Kalman Filter to estimate 3D position of a node

Code given on this link works for 1D: More on: Kalman filter for position and velocity In my problem I need to estimate 3D position.What is the criteria ? How F, G ,H,Q and R change in 3D case. ...
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53 views

Q matrix and updating times in a Kalman filter

The context of the problem is that I have several robots located remotely which give their position (x,y coordinates) every x seconds and send it to a centralized remote server. The value of the ...
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Kalman filter restricted by areal constraints

I'm developing a device for tracking pets in a room with several motion detectors. I use a Kalman filter to estimate the position, which is based on the active outputs of the motion detectors. If an ...
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Purple noise modeling / Differentiated white noise (Kalman Filtering)

I am designing a Kalman Filter for a signal which features a certain kind of noise and I do not know how to model it properly in the filter. The noise is constructed from a white noise source, called ...
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Correct form for State Space Equation for Kalman Filter

In this paper: http://www.ssc.upenn.edu/~fdiebold/papers/paper55/DRAfinal.pdf in eqns 3,5 the state eqn has the mean removed. $(z_t-\mu)=A(z_{t-1}-\mu) + \epsilon_t$ $y_t=C z_t + \delta_t$ ...
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Fusing data from multi rate accelerometers

I am working on a project that requires combining multiple accelerometers with different frequencies. What is the best way of combining signals from these accelerometers to obtain dynamic behavior? ...
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reverse engineering - RT frequency estimation

I need to maintain code I inherited. The code is for RT frequency estimation. There is not a single line of comment in the code. I was hoping someone here will recognize which technique is being used ...
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Smoothing process of Kalman filter

I have a question about the smoothing (backward) process of Kalman filter. Is it correct to say $E[x_{t|T}] = x_{t|t}$ where $x_{t|t}$ is the estimated result from forward process? I am struggling ...
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68 views

How to represent the nonlinear model as a state space in Unscented Kalman Filter

There is an Autoregressive model of order 1 (AR(1)) that is excited by a non-linear signal as the input: $$x_t = \rho x_{t-1} + u_t \tag{1}$$ The time series $u_t$ is generated from a nonlinear map, ...
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113 views

Wrong estimation of derivatives with an extended Kalman filter

I am trying to implement an extended Kalman filter (EKF) in MATLAB for the estimation of joint trajectories (angular position, angular velocity and angular acceleration) from noisy motion capture ...
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31 views

Sensor Data Fusion with Orientation Sensors in 3D Euclidian Space

Preconditions For measuring the position of a mobile device in 3D space, I utilize two sensors with different characteristics that measure device orientation. Sensor A (a combined sensor of ...
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Identifiability of a state space model (Dynamic Linear Model)

Take a general linear Gaussian state space model (SSM)(aka Dynamic Linear Model DLM): $X_{t+1}=FX_t + V_t$ $Y=HX_t+W_t$ $V_t \sim N(0,Q)$ $W_t \sim N(0,R)$ I am interested in the ...
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EM algorithm and Kalman filter

I am new to the subject of Kalman filtering and therefore my question might seem trivial. I see that there is a tight connection between Kalman filter and EM algorithm when one wants to predict the ...
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Kalman Filter for particle tracking

I am new to Kalman and trying to use it for a particle tracking technique for velocity measurments in Multiphase systems. I have particle locations but there are error associated with these locations. ...
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71 views

What is pulse train filter?

I have recently seen some code written in modelling tool like Simulink, ASCET which uses PT1 filter? Why do we use a pulse train filter? What is the difference between FIR, IIR, Chebyschev, ...
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55 views

Improving velocity estimation

I have a sensor reduction model which gives me a velocity estimate of a suspension system(velocity 1) . This suspension system estimate velocity is used to calculate another velocity(velocity 2) via ...
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77 views

How to form Kalman filtering matrices for a problem with variable acceleration?

Assuming we have time vector $T$ with constant time step $dt$ position vector $X$ velocity vector $V$ acceleration vector $A$ All vectors $X, V, A$ have noise on their measurement ( $n_x$ , $n_v$ ...
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73 views

Discrete sinusoidal to state space

I'm looking to apply an optimal LQR filter to a discrete signal of the form $x[n]=A\sin[\omega_0n + \phi]+ v[n]$ The amplitude $A$ and the phase $\phi$ are unknown variables I want to estimate using ...
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198 views

Kalman filter - understanding the noise covariance matrix

What is the significance of the noise covariance matrices in the Kalman Filter framework? I am referring to: process noise covariance matrix Q, and measurement noise covariance matrix R at any ...
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Relationship between a Kalman filter and a PLL?

For tracking a single variable frequency signal in noise, what is the relationship, if any, between using a Kalman filter and implementing a generic Phase Locked Loop?
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Hybrid Kalman Filter and State Space Augmentation

I'm having trouble combining state space augmentation and Hybrid Kalman Filter. Let me first clarify what I mean by these two terms: Hybrid Kalman Filter uses a continuous time model. ODEs need to be ...
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How to filter noise from vibrating particles?

Each particle has its position (x,y,z) tracked in the system over the course of a couple seconds. After analyzing the data, there is quite a bit of noise. I've done some reading and concluded that a ...
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129 views

Kalman filter, defining the measurement model

I would like to implement a Kalman filter to estimate the velocity and position of an object. I have an accelerometer, therefore the acceleration is known. The approach is same as: More on: Kalman ...
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Kalman filter - mathematical model for signal strength based tracking

I'm currently trying to wrap my head around Kalman filters. I've read some tutorials/introductions on this subject and am trying to apply it to a problem to get a better understanding of the mechanics ...
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Estimation of Hit Time Using Kalman

I have the following model for Kalman Filter. The Dynamic Equation: $$ \begin{bmatrix} {r}_{k} \\ {v}_{k} \end{bmatrix} = \begin{bmatrix} 1 & -T \\ 0 & 1 \end{bmatrix} \begin{bmatrix} ...
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System Identification: smoothing in E-step of EM algorithm

My question is, do the smoothed values affect the estimates obtained from the Maximization Step? If not then we could eliminate the smoothing. Once the estimates are obtained, they do not change apart ...
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can measurements go out of the uncertainty bounds?

In the below picture, the measurements are inside the $\pm 3 \sigma$ bounds. In my experiment, the measurements sometimes go out of the uncertainty bounds. This is a snapshot of my plot where the ...
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Kalman filter for velocity estimation from acceleration and displacement: constant-acceleration assumption

I have implemented a Kalman filter to estimate the velocity knowing the acceleration and the positon measurements as explained in this Q/A: Estimating velocity from known position and acceleration ...
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Question about Q matrix (noise process covariance) in Kalman filter

I am implementing my own discrete Kalman filter to estimate velocity from acceleration and position measurements (using Matlab ). I think I managed to deal with the $R$ matrix (measurements noise ...
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when is a kalman filter different from a moving average?

this thread asks when a discrete-time Kalman filter is better/different from a simple moving average of the observations: ...