Questions tagged [kalman-filters]

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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56
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5answers
17k views

What is the relationship between a Kalman filter and polynomial regression?

What is the relationship, if any, between Kalman filtering and (repeated, if necessary) least squares polynomial regression?
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4answers
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How to understand Kalman gain intuitively?

The Kalman filter algorithm works as follows Initialize $ \hat{\textbf{x}}_{0|0}$ and $\textbf{P}_{0|0}$. At each iteration $k=1,\dots,n$ Predict Predicted (a priori) state estimate ...
23
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1answer
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Kalman filter for position and velocity: introducing speed estimates

Thanks to everyone who posted comments/answers to my query yesterday (Implementing a Kalman filter for position, velocity, acceleration ). I've been looking at what was recommended, and in particular ...
19
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3answers
897 views

Should the input of a Kalman filter always be a signal and its derivative?

I always see the Kalman filter used with such input data. For example, the inputs are commonly a position and the correspondent velocity: $$ (x, \dfrac{dx}{dt}) $$ In my case, I only have 2D ...
17
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2answers
9k views

Is a Kalman filter suitable to filter projected points positions, given Euler angles of the capturing device?

My system is the following. I use the camera of a mobile device to track an object. From this tracking, I get four 3D points that I project on the screen, to get four 2D points. These 8 values are ...
16
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4answers
8k views

Intuitive explanation of tracking with Kalman filters

I would much appreciate an intuitive explanation for (visual) tracking with Kalman filters. what I know: Prediction step: Dynamic system state $\mathbf x_t$: target location at time $t$ Measurement $...
16
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1answer
6k views

Kalman filter in practice

I have read the description of the Kalman filter, but I am not clear on how it comes together in practice. It appears to be primarily targeted at mechanical or electrical systems since it wants linear ...
14
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1answer
2k views

How to derive the stationary Kalman filter predictor?

In its chapter on Kalman filters, my DSP book states, seemingly out of the blue, that the stationary Kalman filter for a system $$\begin{cases} x(t+1) &= Ax(t) + w(t) \\ y(t) &= Cx(t) + v(t) ...
13
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3answers
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When to use EKF and when Kalman Filter?

I'm learning Kalman Filter for a week now. I just discovered that EKF (extended Kalman Filter) might be more appropriate for my case. Le't suppose I'm applying KF/EKF for variometer (the device that ...
13
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1answer
903 views

Kalman Filter - Optimal Way to Handle “Derived” Measurements?

Ie, if you have as state variables position (p) and velocity (v), and I make low-frequency measurements of p, this also indirectly gives me information about v (since it's the derivative of p). What ...
11
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7answers
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Good book or reference to learn Kalman Filter

I am totally new to the Kalman filter. I've had some basic courses on conditional probability and linear algebra. Can someone suggest a good book or any resource on the web which can help me can ...
11
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2answers
14k views

have position, want to calculate velocity and acceleration

This is simple i thought, but my naive approach led to a very noisy result. I have this sample times and positions in a file named t_angle.txt: ...
10
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0answers
620 views

Will an Unscented Kalman Filter Be “As Good” as Other Optimization 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 ...
10
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0answers
178 views

Estimating the input to a system from a system state using EKF [closed]

[ Cross-posted from: https://math.stackexchange.com/questions/164169/estimating-the-input-to-a-system-from-a-system-state ] I have a system for which I have obtained a non-linear time-varying state-...
9
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1answer
18k 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 ...
9
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5answers
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Kalman filter - implementation and deciding parameters

First of all, this is the first time I try to make a Kalman filter. I earlier posted this thread on stackoverflow which describes the background for this post. This is a typical sample of values I'm ...
9
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2answers
207 views

Statistical properties of the Kalman estimates under Gaussian noise

For a linear state-space model with independent Gaussian state and output noises and perfect guess for initial state, do Kalman estimates have the following properties: $$ E(\hat{x}_{k|k} - x_k) = 0 $$...
9
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1answer
25k views

Implementing a Kalman filter for position, velocity, acceleration

I've used Kalman filters for various things in the past, but I'm now interested in using one to track position, speed and acceleration in the context of tracking position for smartphone apps. It ...
8
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2answers
11k views

Estimating velocity from known position and acceleration

I am stuck at modeling a system model, i.e. getting my state vector and input vector. My guess is that position and velocity are state vector and acceleration is input vector. My 2nd guess is that all ...
8
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2answers
5k views

Different state-space representations for Auto-Regression and Kalman filter

I see that there are different ways to write an AR model into a state-space representation, so that we can apply Kalman filter to estimate the signal. See Example 1, 2 and 3 here. I wonder what ...
8
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2answers
1k views

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 ...
7
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2answers
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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 ...
7
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1answer
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How do I choose the parameters of a Kalman filter?

Suppose I want to track the position of a car in 2D. What I get as sensor data is my current position. So my state is $$\mathbf{x} = \begin{pmatrix}x\\y\\\dot{x}\\\dot{y}\end{pmatrix}$$ where $x \in \...
7
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1answer
196 views

Can a state space model have changing state size over time?

I have worked with state space models in relation to Kalman estimation. Here I have always seen state space models with fixed state size over time, i.e. the state transition matrix is square. Let us ...
6
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1answer
509 views

How do I model a system to use with a Kalman Filter?

So I have an accelerometer, gyroscope, magnetometer and GPS. I would like to use a Kalman filter to optimally measure speed, position, acceleration and orientation. I have done research and I need to ...
6
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1answer
9k views

How to determine covariance matrix $Q$ and $R$ in Kalman-filter

I am implementing getting orientation from smartphone. I want to use Kalman filter and should determine process noise covariance matrix $Q$ and measurement noise covariance matrix $R$. (newbie to ...
6
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1answer
168 views

How to combine a perfect signal with a limited dynamic range with a poor one with high dynamic range?

I have two sensors that measure speed $v(t)$ of a moving vehicle. The first sensor produces a signal $f(t)$ which is a very accurate estimation of speed. However, it only works for slow to moderate ...
5
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4answers
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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 $...
5
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3answers
6k views

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: https://math.stackexchange.com/questions/173901/why-use-a-kalman-filter-...
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2answers
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How to model the noise in Kalman Filter?

Background: I am a newbie in DSP. I am implementing a simple Kalman Filter that estimates the heading direction of a robot. The robot is equipped with a compass and a gyroscope. My Understanding: I ...
5
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1answer
52 views

Model-based Kalman filtering a noisy signal

In a healthcare application, I need to calculate urine flow by differentiating the mass of urine emitted by a person over time. The measuring instrument consists of a load-cell under a fluid container,...
5
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1answer
123 views

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 ...
5
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2answers
311 views

Particle filter Tracking

I am tracking a car using particle filter by making a rectangle around it, and I am using the state vector $[x, y, u, v, a, h]$, where: $x$, $y$ is the position of the body $u$ and $v$ are velocity ...
4
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4answers
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Question About $ Q $ Matrix (Model 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 ...
4
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2answers
935 views

Can Kalman Filter be used to track Randomly Moving Target?

i want to track random moving object with a camera using kalman filter...i have the following questions... Randomly moving target means $Corelation(t) = E[ x(T)x(T+t) ]$ is very low...where $x(T)$ is ...
4
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2answers
2k 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 ...
4
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2answers
2k views

Kalman Filter Covariance

I've recently started playing with the Kalman filter for a simple 2D (x,y,dx,dy) tracking toy problem. But I seem to have some misunderstanding on what I can expect from the filter. I'm interested in ...
4
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1answer
727 views

Kalman filter for tracking sinusoidal motion

I am wanting to create a Kalman filter that can be used to track an object undergoing sinusoidal (lets assume simple harmonic) motion. I have seen many examples and implemented my own python code for ...
4
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2answers
4k views

Removing drift from integration of accelerometer data

I am trying to get a positional data from the accelerometer data using the following steps: Re-zero the accelerometer value Removing mean from accelerometer value First integration of accelerometer ...
4
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3answers
2k views

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 ...
4
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3answers
1k views

How do you apply Kalman Filter to track a signal?

The example that I've seen on state estimation involves deriving the ABCD matrix of a physical system (i.e. falling object) and tracking that object. I would like to use Kalman Filter for signal ...
4
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1answer
207 views

Extended Kalman Filter for linear systems with non-linear measurements

I'm successfully using an Extended Kalman Filter for object tracking. My state vector ($x, y, v_x, v_y$) needs to be in cartesian coordinates. The measurement data is transmitted in polar coordinates. ...
4
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1answer
66 views

When Kalman's assumptions are relaxed

From Kalman's seminal paper "A New Approach to Linear Filtering and Prediction Problem", it is clear that Kalman's exposition is based on the following fundamental assumptions: Measurements that are ...
4
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1answer
204 views

Is acceleration noise modelled differently in EKF and UKF Kalman Filters?

In a lecture on basic Kalman filter, I came across the following assumption about acceleration noise. Every component of the noise vector $\nu$ is itself a product of time and acceleration values. ...
4
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2answers
1k views

State Estimation by steady state Kalman Filter

I am working with a discret Kalman filter on a System $x_{k+1}=A_k x_k+B_k u_k+\omega_k$ $y_k=C_k x_k+\upsilon_k$ $E[\omega_k\omega_k^T]=Q$ $E[\upsilon_k \upsilon_k^T]=R$ I have estimated the ...
4
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2answers
288 views

Intertial navigation on android phone with Kalman filter

Ok, so suppose I've got a phone with gyroscope, compass and 3-d accelerometer. I wanted to track position of the moving phone for about 1 minute with let's say 50 mm accuracy. Actually let's say that ...
4
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1answer
200 views

Kalman filter for random observation matrix $G_t$

I have a problem that is similar to the state space model in Kalman filter but the observation matrix $G_t$ of $$y_t=G_tx_t +w_t,$$ is random. The elements of $G_t$ are i.i.d. random variables with a ...
4
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1answer
295 views

Is Kalman smoother symmetric in time?

Suppose my model is reversible in time (e.g. GPS + accelerometers for a vehicle), so that I can run Kalman filter forwards or backwards. Kalman filter, of course, cannot be symmetric, because it is ...
4
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1answer
129 views

Filtering difference of correlated measurements

I have an application with two separated GPS receivers giving live positions and I'm deriving a heading/displacement from the vector between them. Each set of position measurements is noisy and has ...
3
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1answer
3k views

Tracking a Sine Wave with Kalman Filter - How to Account for Offset (DC Signal)?

I am attempting to create a Kalman filter to track a sine wave (I am using a linear Kalman filter example assuming I already know the frequency of the sine wave) - the example I am using is derived on ...