Model approach:
I am modelling on Matlab-Simulink a very thin flexible structure. All points of the model are link with each other with springs and dampers this way (without the tethers in the center):
Mesh description
The general equation of my model applied at each point of the mesh is the following:
Dynamic formula of mass/spring/damper system
With k the springs stiffness, and c the dampers damping.
To adapt the physical properties of the material I want to model, the spring stiffness has been set to a very high value, around k = 5000. This mean that my spring links are highly reactive to any deformation.
Problem:
This leads to my problem: High stiffness links induce high frequency displacement that I can consider as noise in the simulation.
The simulation is much slower as the variable time step, I am using must be very low.
This high-frequency displacements (around 160 Hz, which the resonance frequency of the springs) stays all along the simulations.
Here is a simulation of my structure rotating at a constant angular speed:
In-time evolution of a random point of my structure in spherical coordinates
We can see that R is vibrating at a very high frequency. However, the displacement amplitude is clearly negligible.
To speed up the simulation, I want to suppress those vibrations!
Investigation:
To suppress them, I investigate on signal filtering techniques, mainly low-pass filtering. On every loop of our simulation, and what should enter our filter are data of all my points in all my axis.
Simulink low-pass filter block
The continuous version of low-pass filter in Simulink library has been tested on the acceleration, the speed and the position, with several cut-off frequencies from 100 Hz to 500 Hz.
For example, for a cut-off frequency of 200Hz and filtering the position at t=0.6 sec I have:
In-time filtered evolution of a random point of my structure in spherical coordinates
It is an in-plane movement so I don’t have any elevation angle, but azimuth angle and point distance from the center are completely diverging.
The problem might come from:
The fact that I am in a closed-loop system
The fact that for the mesh we have, the filter receives 81 vectors of 3*1 at each time step and maybe the filter block is not made to function with that.
The fact that for the mesh we have, the filter receives 81 vectors of 3*1 at each time step and maybe the filter block is not made to function with that.
Main question:
Are there filtering techniques for closed-loop and multiple inputs system that could solve my problem?
Digital filter designer works with SISO signals. Just demux your signals and apply some lowpass filters. You gave lots of info that made it harder to understand the core problem, if there is anything else you can re-iterate. I'd start with a 3rd order Butterworth LPF wc at around 100Hz for your needs.
Related
I've successfully plotted the signal strength coverage map for a generic narrow band (read single frequency) horn antenna using MATLAB's in-built functions design(), txsite() and coverage().
MATLAB uses the 'Longley-Rice' propagation model when terrain data is present which I downloaded and introduced using addCustomTerrain().
However, I don't want my antenna to be narrow band operating at a single frequency.
I want to model the coverage map I would get on location with a known ultra-wide band (UWB) transient pulsed signal. I have the time domain E-field of this waveform as well as the FFT and energy spectral density.
My plan was to loop over many tx antennas, each having an operating frequency equal to one of the ~1000 frequency bins in the UWB spectral content and an output power equal to the scaled energy spectral density (ESD) multiplied by the frequency step size (df) and divided by the total time period of the measured pulsed signal (to get power). P = ESD * (df/T).
However, when I ran this looped code, I got:
"Error using em.EmStructures/savesolution
The calculated result is invalid; possible cause is a coarse mesh. Please consider refining the mesh
manually."
I assume this means MATLAB can't model 1000 different antennas on the same exact location.. but any idea on this error?
Is what I'm trying to do possible in MATLAB?
Are there alternative methods?
Thank you for any help in advance!
I am currently working on a mission to fuse GNSS and IMU for a more accurate navigation system for autonomous vehicles. I am very familiar with using GNSS to get the accurate position, however I'm a newbie in using IMU sensor. I've read several kinds of literature but am still confused about which better way should I do to remove bias from the accelerometer and gyroscope measurement.
I have 2 kinds of raw measurement data using MPU-9250, they are acceleration data (m/s2) in the x,y, z-axis and angular velocity data (deg/s) also in the x,y, z-axis. I have tried to input these data into my sensor fusion program. Unfortunately, I got unsatisfied with accuracy.. Hence I think firstly I should correcting (removing bias) of raw data IMU, and then the corrected IMU data can be input to my fusion program.
I couldn't find an answer that my brain could understand or fit my situation. Can someone please share some information about this? Can I use a high-pass filter or a low-pass filter in this situation?
I would really appreciate if there is someone could explain in detail to me without using complex math formulas/symbols, I'm not a mathematician and this is one of my problems when looking for information.
Thank you in advance
Accelerometer and Gyroscope have substantial bias usually. You could break the bias down to factors like,
Constant bias
Bias induced by temperature variation.
Bias instability
The static part of bias is easy to subtract out. If the unit starts from level orientation and without any movement, you could take samples for ~1s, average it and subtract it from your readings. Although, this step removes a big chuck of bias, it cannot still fully remove it (due to level not being perfect).
In case you observe that the temperature of IMU die varies during operation (even 5-10 deg matters), note down the bias and temperature (MPU9250 has an inbuilt temperature sensor). Fit a linear or quadratic curve that captures bias against temperature. Later on, use the temperature reading to estimate bias and subtract it out.
Even after implementing 1 and 2, there will still be some stubborn bias left. If the same is used in a fusion algorithm like Kalman filter (that is not formulated to estimate bias, the resulting position and orientation estimates will be biased too).
Bias can be estimated along with important states (like position) using some external reference/sensor like GNSS, Camera.
Complementary filter (low pass + high pass) or a Kalman filter can be formulated for this purpose.
Kalman filter approach:
Good amount of intuition along with some mathematics is needed to use this approach. Basically the work involves formulating prediction & measurement model and then provide rough noise variances for your measurements and prediction. An important thing to understand is that, Kalman filter assumes that the errors follow normal distribution without any bias. So the formulation should deliberately put bias terms as unknown states that should be estimated too (Do not assume that the sensor is bias free in the formulation)..
You could checkout my other answer to gain a detailed understanding of this approach.
Complementary filter approach
Complementary filter is simpler for simpler problems :P
The idea is that we use low pass filter on noisy measurement and high pass filter on biased measurement. Then add them up and call it a day.
Make sure that both the LPF and HPF are complements of each other (Transfer function of HPF should be 1-LPF). Typically first order filters with same time constants are used. Additionally the filter equations have to be converted from continuous laplace domain to discrete form (Read about ZOH, Tustins approximation...).
The final form is scattered around the internet too.
Personally I would use a Kalman filter for this purpose, but complementary filter can be used with same amount of effort. You could do this,
Assume that the body is not accelerating on average in long term (1-10 s or so). Then you could say that the accelerometer measures the direction of gravity in long term relative to the IMU. Then arctan(accy, accz) can be used to obtain an estimate of pitch and roll. But this pitch and roll readings will suffer from substantial noise. Implement a low pass filter on it with time constant ~5 seconds or so. Additionally add the latest pitch/roll with dt*transformationMatrix*gyroscope to get another pitch and roll. But these suffer from bias. Implement a HPF over gyro based Pitch and Roll. Add them together to get Pitch and Roll. Lets call these IMU_PR.
Now forget our original acceleration assumption. accelerometer gives specific force (which is net acceleration - gravity). Since we have Pitch and Roll angles (IMU_PR), we know gravities direction. Add gravity to accel readings to get an estimate of acceleration. Apply proper frame conversion to bring this acceleration to same coordinate frame as GPS (you will need an estimate of Yaw to do so. Fuse a magnetometer with gyroscope for this purpose). Then do vel = vel + acc*dt. Integrate it again to get an estimate of position from IMU. But this will drift due to the bias in accelerometer (and pitch, roll). Implement a high pass filter over this position and low pass filter over GPS position to get a final estimate.
I'm dealing with CWT, and I have a big problem converting scales to frequencies. In the MAtlab Wavelet Tutorial they use this expression to convert scales to frequencies
But if i use the default function scal2freq I obtain different result.
I don't understand the role of the Morlet Fourier Factor
Thanks in advance
It is a pretty complicated concept, which I somewhat understand it. I'll write some points here so that you might figure it out yourself, rather easier.
A simple fact is that:
Scale is inversely proportional to frequency.
For example, imagine we have a 1-100 Hz range of frequencies in some time series data such as stock markets data or earthquake data. Scale is "supposed to be" the inverse of that. For instance, if scale would be in range of 1 to 100, we'd have had:
Scale(1/Hz) Frequency (Hz)
1 100
50 50
100 1
Therefore,
The frequency is not the real frequency of those time series data (e.g., stock market, earthquake) that we know of. They are only related, inversely.
And we can safely say that here we are calculating some "pseudo-frequencies", which MATLAB does that (by approximating that). You can read about the approximation process in the documentation in the section pseudo-frequencies:
MATlAB does calculate those pseudo-frequencies based on:
In wavelet analysis, the way to relate scales to frequencies is to determine the center frequency of the wavelet function:
which you can visually see in this image and of-course it would differ, when we would change the types of our function in the calculation. Thus, that center frequency will change everytime in our approximation process:
That "MorletFourierFactor" is a variable to approximate a constant so that when you would do the 1/scale, it would closely approximate those "pseudo-frequencies".
I thought this image about shifting (time axis) and scaling (frequency axis) might be a little helpful to look into as well:
The bottom line is that don't worry about pseudo-frequencies, you wouldn't probably need those. If you would want any frequency spectrum, you can likely go towards applying some of those frequency methods (such as Fast Fourier Transform) on whatever time series data that you have.
If you really really want to map that, you can also try to design some methods to approximate it yourself.
Source
Harvard Seismology
I was using PCA for dimensionality reduction of MD (molecular dynamics) trajectory data of some protein simulations. Basically my data is xyz coordinates of protein atoms which change with time (that means I have lot of frames of this xyz coordinates). The dimension of this data is something like 20000 frames of 200x3 (atoms by coordinates). I implemented PCA using princomp command in Matlab.
I was wondering if I can do FFT on my data. I have experience of doing FFT on audio signals (1D signal). Here my data has both time and space in picture. It must be theoretically possible to implement FFT on my data and then filter it using a LPF (low pass filter). But I am unsure.
Can someone give me some direction/code snippets/references towards implementing FFT on my data?
Why people are preferring PCA more often compared to FFT and filtering. Is it because of computational efficiency of algorithm or is it because of the statistical nature of underlying data?
For the first question "Can someone give me some direction/code snippets/references towards implementing FFT on my data?":
I should say fft is implemented in matlab and you do not need to implement it by your own. Also, for your case you should use fftn (fft documentation)to transform and after applying lowpass filtering by dessignfilt (design filter in matalab), the apply ifftn (inverse fft in matlab)to inverse the transform.
For the second question "Why people are preferring PCA more often compared to FFT and filtering ...":
I should say as the filtering in fft is done in signal space, after filtering you can't generalize it in time space. You can more details about this drawback in this article.
But, Fourier analysis has also some other
serious drawbacks. One of them may be that time
information is lost in transforming to the frequency
domain. When looking at a Fourier transform of a
signal, it is impossible to tell when a particular event
has taken place. If it is a stationary signal - this
drawback isn't very important. However, most
interesting signals contain numerous non-stationary or
transitory characteristics: drift, trends, abrupt changes,
and beginnings and ends of events. These
characteristics are often the most important part of the
signal, and Fourier analysis is not suitable in detecting
them.
I found a Matlab implementation of the LKT algorithm here and it is based on the brightness constancy equation.
The algorithm calculates the Image gradients in x and y direction by convolving the image with appropriate 2x2 horizontal and vertical edge gradient operators.
The brightness constancy equation in the classic literature has on its right hand side the difference between two successive frames.
However, in the implementation referred to by the aforementioned link, the right hand side is the difference of convolution.
It_m = conv2(im1,[1,1;1,1]) + conv2(im2,[-1,-1;-1,-1]);
Why couldn't It_m be simply calculated as:
it_m = im1 - im2;
As you mentioned, in theory only pixel by pixel difference is stated for optical flow computation.
However, in practice, all natural (not synthetic) images contain some degree of noise. On the other hand, differentiating is some kind of high pass filter and would stress (high pass) noise ratio to the signal.
Therefore, to avoid artifact caused by noise, usually an image smoothing (or low pass filtering) is carried out prior to any image differentiating (we have such process in edge detection too). The code does exactly this, i.e. apply and moving average filter on the image to reduce noise effect.
It_m = conv2(im1,[1,1;1,1]) + conv2(im2,[-1,-1;-1,-1]);
(Comments converted to an answer.)
In theory, there is nothing wrong with taking a pixel-wise difference:
Im_t = im1-im2;
to compute the time derivative. Using a spatial smoother when computing the time derivative mitigates the effect of noise.
Moreover, looking at the way that code computes spatial (x and y) derivatives:
Ix_m = conv2(im1,[-1 1; -1 1], 'valid');
computing the time derivate with a similar kernel and the valid option ensures the matrices It_x, It_y and Im_t have compatible sizes.
The temporal partial derivative(along t), is connected to the spatial partial derivatives (along x and y).
Think of the video sequence you are analyzing as a volume, spatio-temporal volume. At any given point (x,y,t), if you want to estimate partial derivatives, i.e. estimate the 3D gradient at that point, then you will benefit from having 3 filters that have the same kernel support.
For more theory on why this should be so, look up the topic steerable filters, or better yet look up the fundamental concept of what partial derivative is supposed to be, and how it connects to directional derivatives.
Often, the 2D gradient is estimated first, and then people tend to think of the temporal derivative secondly as independent of the x and y component. This can, and very often do, lead to numerical errors in the final optical flow calculations. The common way to deal with those errors is to do a forward and backward flow estimation, and combine the results in the end.
One way to think of the gradient that you are estimating is that it has a support region that is 3D. The smallest size of such a region should be 2x2x2.
if you do 2D gradients in the first and second image both using only 2x2 filters, then the corresponding FIR filter for the 3D volume is collected by averaging the results of the two filters.
The fact that you should have the same filter support region in 2D is clear to most: thats why the Sobel and Scharr operators look the way they do.
You can see the sort of results you get from having sanely designed differential operators for optical flow in this Matlab toolbox that I made, in part to show this particular point.