Questions tagged [kalman-filter]

For questions about Kalman filter.

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55
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An Explanation of the Kalman Filter

In the past 3 months, I've been trying to understand the Kalman Filter. I have tried to implement it, I have watched YouTube tutorials, and I have read some papers about it and its operation (update, ...
11
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2answers
1k views

What is the difference between optimal control and robust control?

What is the difference between optimal control and robust control? I know that Optimal Control have the controllers: LQR - State feedback controller LQG - State feedback observer controller LQGI - ...
9
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2answers
3k views

Scaling factor and weights in Unscented Transform (UKF)

I'm trying to implement the UKF for parameter estimation as described by Eric A. Wan and Rudolph van der Merwe in Chapter 7 of the Kalman Filtering and Neural Networks book: Free PDF I am confused by ...
7
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2answers
9k views

How to estimate variances for Kalman filter from real sensor measurements without underestimating process noise.

As the title says, I want to estimate the variances needed for a Kalman filter from real sensor measurements only. For example we can take a temperature sensor, but the solution shall be as ...
7
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1answer
588 views

Kalman filtering correlated measurements

I would like to run a Kalman filter over a set of measurements which may (will) be correlated. Essentially, each new measurement contains (say) 90% of the same information from the previous ...
7
votes
2answers
2k views

Differences between Quaternion integration methods

I've implemented a Quaternion Kalman filter and i have the choice between multiple way to integrate angular velocities. The goal is to predict futur orientation $q^{n+1}$ from current orientation $q^{...
6
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2answers
5k views

What are “Filtering” and “Smoothing” with regards to Hidden Markov Models?

The Wikipedia article about Hidden Markov Models mentions "filtering" and "smoothing" tasks, see here: http://en.wikipedia.org/wiki/Hidden_Markov_model#Filtering. It gives a brief explanation but no ...
5
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0answers
159 views

Forecast equivalence between two steady-state Kalman filters

I have two related steady-state Kalman filter problems that I want to prove satisfy a condition associated with their respective Kalman gains. I am not really looking for a complete proof since this ...
5
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0answers
101 views

How to derive the distribution of measurement noise in discrete Kalman filter which is transformed from continuous one?

With sampling time $T$, and a continuous measuring model: $$ \begin{align} y(t) &= Cx(t)+v(t) \\ v(t) & \sim \text{N}(0,R_c) \end{align} $$ we can change it into a practical discrete one, ...
5
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0answers
136 views

Kalman Filter Derivation - Why does taking the derivative of the trace give minimum error and not maximum?

When deriving the Discrete Kalman Filter, there is an intermediate step where you take the derivative of the trace of $P_k$ and set it equal to 0: $P_k = E[e_k e_k^T]$ $ = P_k^- + K_kH_kP_k^-H_k^-...
4
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1answer
14k views

Kalman Filter to determine position and attitude from 6DOF IMU (accelerometer + gyroscope)

I'm going to describe the problem I'm trying to solve and walk through what I understand so far about the Kalman Filter. I have an IMU which gives me the following measurements every time interval t: ...
4
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4answers
256 views

In what ways is a Kalman-filter a filter?

In what ways is a Kalman filter a filter? I think about a filter like a system that takes an input signal and outputs a signal with certain missing components from the input. With this understanding ...
4
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2answers
845 views

What is the significance of multiplying 2 Gaussian PDFs?

I've been reading on Kalman Filter and came across the following statement: The best estimate we can make of the location is achieved by multiplying the 2 corresponding PDFs together. But I don't ...
4
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1answer
184 views

Kalman Filter constant steady state value

Without explaining you the details of my problem I would like to ask you a theoretical question. I think I know the answer but I would like to be sure. I have a set of $n$ agents moving in 3D space. ...
4
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4answers
3k views

Kalman filter with missing measurement inputs

I am a newby to Kalmar filters, but after some study, I think I understand how it works now. For my application, I need a Kalmar filter that combines the measurement input from two sources. In the ...
4
votes
1answer
329 views

Connection between the Kalman filter and the multivariate normal distribution

Consider at dynamic linear model where $$ \theta_{1} \sim N(\mu_{1}, W_{1}), $$ $$ \theta_{i}=G\theta_{i-1} + w_{i}, w_{i}\sim N(0,W), $$ $$ Y_{i} = F\theta_{i} + v_{i}, v_{i}\sim N(0,V) $$ and $ \...
4
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1answer
147 views

Estimating the input to a system from a system state

[ Cross-posted to: https://dsp.stackexchange.com/questions/3098/estimating-the-input-to-a-system-from-a-system-state-using-ekf ] I have a system for which I have obtained a non-linear time-varying ...
4
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1answer
434 views

Why is the square root of Cholesky decomposition equal to the lower triangular matrix?

I came across this as I was learning unscented Kalman filters. Suppose I have a symmetric and positive definite matrix P. I want to take its square root. After I perform the Cholesky decomposition ...
4
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1answer
224 views

The role of the extraction matrix in a Kalman filter

The extraction matrix shown as $H_k$ below, transforms the state vector into a form that can be subtracted from the measurements vector: $\hat{X}_k = \hat{X}_k^- + K_k ({z}_k - H_k \hat{X}_k^-)$ ...
3
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2answers
883 views

Prerequisites for learning kalman filtering

What would be the prerequisites needed to implement a kalman filter (and of course, understand it)? I'm currently on my fifth year on engineering in Brazil, and I know: Multivariable Calculus Linear ...
3
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2answers
141 views

Derivation of Kalman Gain for the Unscented Kalman Filter (UKF)

I recently went through the mathematical derivations of the Kalman filter (KF), the extended Kalman filter (EKF) and the Unscented Kalman filter (UKF). My question is concerned with some detail ...
3
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1answer
1k views

Kalman Filter Process Noise Covariance

I want to model the movement of a car on a straight 300m road in order to apply Kalman filter on some noisy discrete data and get an estimate of the position of the car. In a Kalman filter the matrix ...
3
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0answers
121 views

Confusion regarding usage of Mahalanobis distance for update rejection in Kalman filtering

I recently came across some material that discussed a method for performing update rejection in Kalman filters when bad measurements are received. [Paper 1] [Paper 2: see Section III(E)] This method ...
3
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1answer
40 views

Difference of Stochastic Filtering and Stochastic Smoothin

I know the stochastic filtering problem estimates the dynamics of the density $\pi_t(\phi)$ of the random variable $$ \mathbb{E}\left[ \phi(X_t) \mid \mathfrak{F}_t^Y \right]^o, $$ where $X_t$ is the ...
3
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0answers
127 views

Is this $H_\infty$ robust control?

This is going to be a long question. To answer this question, you need to undestand the following: Sensitivity transfer function matrix $S(s)$ Complementary sensitivity function matrix $T(s)$ ...
3
votes
2answers
480 views

Covariance matrix $P$ for an Extended Kalman Filter not symmetric

I am trying to implement an Extended Kalman Filter (EKF) and it is becoming harder than I thought. I have one question. I noticed that the covariance matrix which should get updated over each ...
3
votes
1answer
90 views

Extended Kalman filter for the model x_dot=f(x,u,w)

There is a lot of info about EKF out there but everything I find explains it for the simplified model of the form x_dot = f(x,u) + w; i.e. the process noise is a ...
3
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1answer
480 views

Quaternion Kalman Filter Process Noise

I'm implementing a extended Kalman filter using quaternions. I've extended this paper to deal with my custom observations. My state space is analogous to the one in the previous paper : $ \mathbf{...
3
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0answers
136 views

Example of Kalman filter with irregular time steps/weights to observations

I am trying to find a Kalman filter implementation which does not make the assumption that each observation is equally spaced and of equal weight. In particular, I have measurements of a process ...
3
votes
0answers
59 views

Conditional distribution [closed]

I am trying to figure out the derivation of Kalman filter based on Bayesian estimator. As we know, the assumption of Gauss-Markov model is used, then, the conditional distribution p(x(t)|Y(t-1))can be ...
3
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0answers
237 views

Why is a Complementary Filter a Good Approximation?

Can someone help me understand what the full expression for the original complementary filter should be and why the one proposed by Colton is a good approximation? Context: Having found some ...
3
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0answers
340 views

How to solve the recursive relation in Kalman filter?

I was wondering how to solve the Kalman filter's recursive equation (also see the appendix at the end of this post) for the estimated state $\hat{\textbf{x}}_{n|n}$ at time $n$, over discrete times $k=...
2
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3answers
752 views

13 DOF Kalman filter

I'm trying to develop a system with the following characteristics: Inputs: 3-axis accelerometer [3 DOF] 3-axis gyroscope [3 DOF] GPS with three parameters (lat, lon, altitude) [3 DOF] Barometric ...
2
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3answers
1k views

Sensor fusioning in Kalman filter

I'm interested, how is the dual input in a sensor fusioning setup in a Kalman filter modeled? Say for instance that you have an accelerometer and a gyro and want to present the "horizon level", like ...
2
votes
1answer
2k views

extended kalman filter equation for orientation quaternion

I have a body pose data sampled with a given frequency. Using model with constant velocity motion between frames i filter position with EKF. State equation is given by: $$ \begin{pmatrix} x_{k+1} \\ ...
2
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1answer
3k views

How to derive the process noise co-variance matrix Q in this Kalman Filter example?

How to understand the process co-variance matrix Q in the example below ( I extracted it from Wikipedia http://en.wikipedia.org/wiki/Kalman_filter ) Consider a truck on perfectly frictionless, ...
2
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1answer
19 views

Noisy Brownian Noise

Suppose that $W_t$ and $B_t$ are independent Brownian motions, and define the process $X_t\triangleq W_t + B_t$. What is the conditional expectation of $W_t$ given the $\sigma$-algebra generated by $...
2
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2answers
66 views

Bounding the solution to a Riccati equation

I have the following continuous-time matrix Riccati equation $$A X + X A' - X b b' X + Q = 0$$ where $Q>0$, $A$ is a diagonal matrix with strictly negative eigenvalues and $b$ is a (column) ...
2
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1answer
915 views

Covariance of the Kalman Filter innovation

I am trying to fully understand the derivation of the covariance of the innovation vector, however I am stucked conceptually at a point. I will show you my reasoning and where I am stuck (if someone ...
2
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1answer
155 views

Does probabilistic graphical models include kalman filters?

Do filters (specifically Kalman filters) come under probabilistic graphical models? If not where do they fit in? Thanks
2
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1answer
87 views

Is Wikipedia page on Kalman Filter's wrong?

I was reading the wikipedia page on Kalman filter Snippet from wikipedia There, when estimating the co-variance matrix, the Q matrix is used. When calculating Kalman gain, R matrix is used. In most ...
2
votes
2answers
269 views

How does Kalman Filter provide information regarding the accuracy of the current estimate?

I am a beginner in Kalman Filter and have been reading quite a lot on the Internet and books. I am stuck on how can the P matrix provide the accuracy information regarding the current estimate. Below ...
2
votes
1answer
286 views

Kalman Filter with two observations

I have a Kalman filter,$x_{k+1}=Ax_{k}+w_{k}$ for the state equation and $y_{k}=Cx_{k}+v_{k}$ for the observation. I have tried to implement this in Matlab, and I believe I have understood the ...
2
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0answers
35 views

How do I set the weights of an Unscented Kalman Filter?

In most of the papers I have read, calculation of weights is done by the following formula: $$W_m=\lambda/(L+\lambda)$$ $$W_c=\lambda/(L+\lambda)+(1−\alpha^2+\beta)$$ Where $L$ is the dimensionality ...
2
votes
1answer
76 views

Show that minimizing $Tr(Q)$ equals minimizing $x_0^{T}\:Q\:x_0$

In two different textbooks about Kalman Filter, the so-called Estimator Gain Matrix G is obtained as result of two different minimization problems, i would like to show or at least giustify that the ...
2
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0answers
33 views

LQG with bias rejection for quadcopter attitude control

We're trying to design an attitude controller for a quadcopter. The system dynamics are given: $$ \boldsymbol{\dot{q}} = \frac{1}{2} \boldsymbol{q} \otimes \begin{pmatrix} 0 \\ \vec{\omega} \end{...
2
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0answers
71 views

Continuous Kalman Filter optimization

The discrete Kalman Filter (assuming a linear time invariant system with uncorrelated Gaussian process and sensor noise and finite observation interval) solves the problem for minimizing the mean ...
2
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0answers
149 views

What is the difference between Kalman Filter and Recursive Least Squares? [closed]

I wish to understand the difference between Kalman Filter and Recursive Least Squares since both of them use prediction and correction approach. In Kalman filter, the value of existing state vector ...
2
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0answers
465 views

A simulation: What's the difference between linear MPC and LQG?

I thought it would be good to do a cooperation between basic MPC and basic LQR. So i hook up a state space model for simulation. $$\dot x = Ax(t) + Bu(t) \\ y(t) = Cx(t) + Du(t)$$ Code: ...
2
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0answers
22 views

Kalman tuning rules on States process noises

experts on Estimation, on Kalman, I'm considering a temporal linear KF estimation problem, the whole system is as follows: $$Y_{2n*1}(t) = [Y1;Y2]$$ $$Y1 = Ax +\epsilon_{1}$$ $$Y2 = Ax +Bb+\epsilon_{...