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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Least Square Error Estimation: Conditions for $ (A^TA)^{ -1} = A^{-1}(A^T)^{-1} $?
I am new to linear algebra and have this simple question...
in least sqaure estimation...the best estimation of the equation $Ax = b$ is $x_{Estimated} = A (A^t A )^{-1} A^t b$...the projection of $b$ ...
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Kalman Filtering with Unknown State Transition Matrix
I'm currently studying the use of Kalman filters for estimating linear systems. My current State Transition Matrix (STM) is the identity since so far I've been dealing with non time-varying systems. ...
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Kalman Filter with Accelerometer with DC Offset
Goal: For a particle moving uniaxially, to estimate position ($d$) and velocity ($v$) from noisy acceleration ($a$) and very noisy position (GPS) measurements using a Kalman filter.
Catch: The ...
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How to Combine / Fuse 2 Least Squares Estimates?
Say you want to compute the least squares estimate of $w$ from a data-set:
$$
\begin{bmatrix}d_1 \\d_2 \\\vdots\\d_N \end{bmatrix} =\begin{bmatrix} x_1 \\x_2 \\ \vdots \\x_N\end{bmatrix}w + \begin{...
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In what sense is the Kalman filter optimal?
The Kalman filter is a minimum mean-square error estimator. The MSE is defined as $E\left(||\hat{x}_k-x_k||^2\right)$ where $x$ is the state and $\hat{x}$ is the estimate. When $x$ is a vector, for ...
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Alternative to Extended Kalman Filter When Prediction Function Is Not Differentiable / Given by a Black Box
I am looking at a tracking problem. It can be modeled similarly to the Extended Kalman Filter:
$$
\begin{array}{rcl}
\mathbf{x}_k &=& \mathbf{f}(\mathbf{x}_{k-1}, \mathbf{u}_k) + \mathbf{w}_k\\...
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In Kalman filters why is it necessary to transform the systems dynamics matrix to the state transition matrix?
Previously when I have implemented Kalman filters I have used the transformation
$$
\mathbf{A(t)} = \mathcal{L}^{-1} \left( s \mathbf{I} - \mathbf{F} \right) ^{-1}
$$
to calculate the state ...
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Why is the Kalman gain constant when position and velocity change?
I used the measured position to estimate the velocity using a Kalman filter.
But in simulations, the Kalman gain changes quickly and then remains constant when position and velocity continue to ...
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Kalman Filter - How to combine data from sensors with different measurement rates?
I'm trying to implement a Kalman filter for tracking the position of a vehicle with the help of position data from GPS and Odometry measurements. The GPS data (WGS84 format collected from an app on an ...
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Capture legitimate huge increase/drop in fuel level from noisy measurements
I have a fuel sensor installed in a lorry fuel tank. The sensor measures the current fuel level (in percentage) of the tank. However, due to the fact that the fuel is sloshing in the tank, the fuel ...
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The role of GPS in INS/GPS navigation systems
Ideally, a gyroscope and an accelerometer would be enough for a complete navigation solution (attitude + position), using dead reckoning. This comprise the Inertial Navigation System, INS. In non-...
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Interpreting sensor noise data
Introduction :I am trying to create an EKF for an autonomous vehicle, and i need to model the error of the sensors. (I am a mechanical engineer so my knowledge on electric signals is limited).
The ...
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How to determine covariance matrices $\mathbf P$, $\mathbf Q$, and $\mathbf R$ in Extended Kalman Filter
I am implementing an Extended Kalman-Filter and an Unscented Kalman-Filter for state and parameter estimation of a conveyer belt system. The problem is that I don't really know how to determine the ...
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Alternatives to offline Kalman filtering
Recently I got into vehicle models and filtering in general and immediately faced with the following question.
I have the recorded GPS data from car driving on a highway. However, there is a ...
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Image Restoration using kalman filter
I have been trying use a Kalman filter to restore an image that was blurred with a known Point Spreading Function and corrupted with noise. I have looked at theory and have a basic understanding of ...
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Improving Velocity Estimation Using Multiple Sensors in a Dynamic System
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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Kalman Filter Algorithm for Unknown Process and Measurement Noise
I want to be use kalman filter for the state estimation but I don't know about process and measurement noise. How can I estimate process and measurement noise and use this information for kalman ...
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When is it necessary to use a Kalman Filter, and not a simple estimation method?
I know its a stupid question, but I got understand this very fundemental point. Say we have a sinousodial signal, which we want to extract from a noisy (known variance and mean) measurement. It is ...
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Kalman Filter - Deriving state transition function
I am relatively new to using Kalman Filtering. Currently I am trying to understand it and how to implement it in Matlab. I found a website with some nice examples that I would like to rewrite in ...
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Kalman Filter State Covariance Matrix
If I have a discrete time process model of the form:
$$x_{k+1} = x_{k} + v_{k}\cos(\theta_{k})dt$$
$$y_{k+1} = y_{k} + v_{k}\sin(\theta_{k})dt$$
$$v_{k+1} = v_{k} $$
$$\theta_{k+1} = \theta_{k}$$
...
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Question About Kailath's Paper - An Innovations Approach to Least Squares Estimation Part I: Linear Filtering in Additive White Noise
I'm reading the paper at the link below and I was following it for about 2 pages until I hit a road block on the bottom of page 648 where the author says:
putting together 9-11, we obtain
and ...
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Naive Kalman filter for 3D position
I looked at posts that discusses 3D kalman filter.
Kalman Filter to estimate 3D position of a node
Help with Kalman Filter implementation for estimating 3D position
Both from my understanding, both ...
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Simulating Range Bearing Sensor with MATLAB with Gaussian Noise (Generating Gaussian Colored Random Vector)
I would like to simulate a sensor that provides range and direction of a beacon. This is for EKF localization, so the noise must be Gaussian (i.e. $\mathcal{N}(0, \sigma^{2})$. Also, I would like to ...
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Kalman Filter and Generalized Least Squares
I have read in many places how Kalman Filter is related to generalized least squares algorithms. But there is still a bit I found a bit counterintuitive.
Kalman gain solution is ${K}_k = {{P}}_{k\mid ...
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Computationally Efficient Implementation of Kalman Filter
I know there are many formulations of the Kalman Filter. A few I can name are:
Classical Covariance Form
Informational Filter Form
Square-Root Form or Factor Form
But somehow it's hard for me to ...
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How to define Q matrix (covariance matrix for process noise) in EKF
I am implementing an Extended Kalman Filter (EKF). I have developed the model for my system and I know must find the Q matrix. I have searched all over the web and noone explains how to define this ...
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Unscented Kalman Filter for Parameter Estimation (Tracking) of Amplitude Frequency and Phase of a Multi Component Harmonic Signal
I'm trying to implement an Unscented Kalman Filter that tracks the amplitude, frequency, and phase of a multi-component oscillatory signal. Below is an attempt using the ...
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What do I measure in a Sound Sample Buffer to remove noise from an audio file using the Kalman Filter?
I am developing a computer program that removes or reduces the background noise from an audio file using the Simple Kalman Filter. I have implemented the Kalman Filter and a way of obtaining the "...
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State estimation in HyperSonic Missiles
I'm curious, what would be the correct approach for state estimation for hypersonic Missiles?
Would it be exclusively GPS and IMU? Historically these are what was used, but I have also seen ground ...
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How to compute (in non adaptative way based on MMSE FIR) the feedback and feedword filter for DFE equalization?
System Model
Consider the single-input single-output (SISO) communication system in the figure below:
where the information bit stream is encoded and mapped such that the output of the encoder is a ...
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Kalman Filter for position : introducing acceleration estimates
I want to estimate the position on a 3D environment by introducing only acceleration estimates. Is that possible? If I use the extended Kalman Filter and introduce these estimates will I have the ...
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Converting an FIR Filter Model to a State Space Model for Kalman Filtering
I want to try and determine the true value of a quantity $\alpha[k]$ from observations of a related quantity $\vartheta[k]$ using a Kalman filter. The observations are of the following FIR filter form:...
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Inconsistant Variation in BLE Beacon RSSI Values for Distance Measurement
I am developing an application to estimate the distance to a BLE Beacon using its RSSi values measured from a Mobile Phone. But when I started to collect data I could see that they varied so much that ...
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Role of Riccati Equations in Kalman Filter Design
I am working on a Kalman Filter (KF) design problem and I am struggling to understand the role of the Riccati equations in the design process of a KF.
Some sources discuss the importance of Riccati ...
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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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(Unscented) Kalman Filter with variable state dimensions
i have to estimate a process which changes over time, not with respect to the system-evolution or the measurement-function, but regarding the number of objects that have to be estimated. So every ...
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Optimal inference for nonlinear state space models
When considering a linear-Gaussian state space model, it is often referred that, optimal inference is tractable which is very rare in state space models. When considering a nonlinear state space model,...
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Optimal measurement model for Kalman in Augmented Reality
I am developing an augmented reality SDK that uses Kalman for tracking a planar marker. My state is composed of 3D position, a quaternion, velocity and angular velocity.
\begin{bmatrix}{\vec{p}}\\{\...
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Situations where complexity is too big to exceed linearity and gaussianity
I'm studing about Kalman Filter and Particle Filter in multiple target tracking in computer vision (tracking pedestrians). Reading sientific papers I'm colliding with a lot of sentences like:
The ...
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Kalman filter after lowpass filter: bad idea?
I am new to Kalman filter and am enjoying playing with it. However, I generated some random velocities and acceleration and Kalman filter (with the covariance matrices I have chosen) is comparable ...
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Recommendation for courses / studies on digital signal processing
I hold a master's degree in mechanical engineering. However at my job I am more and more diving into topics of signal processing and data science. I find it great to discover about new topics and to ...
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Extended Kalman Filter (EKF) Linearization of Non Linear Functions
I have more general question about the extended Kalman filter usage. What is not clear to me is why the EKF uses non-linear functions $f$ and $h$ for state prediction and estimate, while in other ...
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How to choose the "best" measurment (from a given set) as input for a kalman filter?
Problem:
For an object tracking scenario with multiple objects and multiple tracks, I want to choose the "best" assignment object<->track. Therefore, I need a parameter indicating how suitable an ...
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Why is this matrix invertible in the Kalman gain?
In the wikipedia article about Kalman filters, the well-known expression of the matrix of Kalman gains is given:
$$ \mathbf {K} _{k}=\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\text{T}}\mathbf {S} _{k}^...
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How to handle a logarithmic term in Kalman filter?
I am trying to implement a Kalman filter for an echo pulse detection application as similar to this paper. (an open source version is here (pg 16))
The measurement variable is $h(x,t)=A_0 (\dfrac{t-\...
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Questions on Weighted Least Square Estimation
This is a page from the book linear algebra,geodesy and gps by Gilbert Strang....
the page explains about the justification of the inverse of the of the co variance matrix of measurement vector $b$ in ...
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Gyroscope sensor drift problem at speed
I have an iPhone mounted on a bike to measure the slope of the road in real-time. This works pretty well at first but as soon as I speed up, the reported pitch starts to drift. I'm using CoreMotion ...
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Kalman filter with multiple sensors
I have been reading more about EKF and I am a bit confused on how you handle predictions with multiple sensors.
Ex, I have IMU, GPS, Odom and Stereo Camera.
Each can be used to predict location, how ...
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Error in using Kalman Filter for 2D Position Estimation in Python
This is my first question on DSP Stack exchange, so I apologise if it is poorly worded. I have some positioning data from a vehicle (GPX Format, collected through Strava) and want to use a Kalman ...
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If a Kalman filter can only receive information on $(x, y)$ position, is there any reason to have acceleration as part of the model?
I'm new to working with Kalman filters. I have a Kalman filter that's working well for modelling the 2D position of an object. The model at the moment holds information on position ($x$ and $y$) and ...
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