I have accelerometer readings of three axis X, Y and z, will be getting data in a frequency of (62 records per second). Could you please suggest me how can I calculate the displacement.
Data in hand:
Accelerometer readings with respect to time.
Do I need to calculate the displacement using time domain data or need to convert into frequency domain. Which one will give a accurate results?
You can double integrate the acceleration vector over time to obtain the displacement. In theory this is a perfectly sensible solution.
But in practice, there will always be a component of g (acceleration due to gravity) acting on at least one of the axes all the time. Let's say you subtract the g component from your xyz vectors. The problem is that any slight error in readings (even off by a small order of magnitude) when double integration will lead to the error adding up over time rendering the displacement wildly inaccurate.
According to the integrated values, you will most likely see even an idle object fly off into space. You'll need an additional sensor to tell you the orientation - like a gyroscope, and have some point of reference (the Wiimote does this with an IR sensor).
This is primarily a time domain problem, but you could have a frequency domain stage where some amount of filtering is done to remove measurement error or process error.
tl;dr Positional tracking with acceleration sensors alone is a hard problem.
Related
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 have ten measurements, containing 4 signals, with a length of 6-8 hours each at a sample rate of 10Hz. (200k-300k samples per measurement)
3 signals are the x-, y- and z-axis of an accelerometer (measuring the acceleration m/s^2). This sensor is positioned on a sphere. The sphere is inflated and deflated. The air-fluctuation in the sphere is the 4th signal.
The accelerometer values, are thus a representation of fluctuation of air in the sphere. If the sensor is positioned on top of the sphere, a good correlation is seen between air fluctuation and Z-axis of the accelerometer. The X- and Y- axis should in a perfect situation be 0, but they are not and contain noise.
I have tried many methods of combining the three signals into one, and comparing that to the air-fluctuation signal. At some points this does give a good outcomes, but when the accelerometer is not moving over one axis, the signal-noise ratio seems to be too high (e.g. when the sensor is at a 45 degree angle).
I am wondering if computer machine learning in Matlab can be used here, to automatically generate a model that makes the best fit: combine the 3 axis signals into one, that best represents the 4th signal. I assume this is what computer machine learning is about?
I can provide the signals in filtered format, the integrated signals, the angle of the sensor at a given time, full signal or just a minute of data, etc.
I have however no idea how to start and which toolbox to use to tackle this problem. Also what signals to feed into the algorithm? at what length (full vs couple of seconds/minutes)
Can you help me getting started on the computer machine learning process, to use 3 signals (and/or formatted versions of them) into one signal that closely matches the fourth signal*.
I'm not asking for full code, but what steps to take to tackle this problem. Like have a look at that toolbox, and function x.
Over the past year I have used many different methods of combining Accelerometers, gryos and Magnetometers to get accurate readings of Head angles.
I have also started looking into using a Kalman filter to further improve these readings.
Yet I am still to find a method of removing external magnetic field influences using the other sensors, for example;
If my heading angle was accurate, and suddenly an external magnetic field approaches, my heading angle will be influenced, but to my gyro and accelerometer I haven’t moved.
Is there any algorithms or calculations anyone can think of to override the magnetometer in a way that can determine whether you have moved or not?
Any help would be much appreciated!
One simple solution is to use the gyro/accelerometer as you mentioned, and combine that with delayed filtering, where you wait for a couple of seconds before providing an estimate of the attitude.
Compute the short term attitude from gyro/accel only (start with any arbitrary heading) using gyro integration with accel measurements, and then compute the short term attitude from the magnetometer/accel only using, say TRIAD. Compute the error between these two quantities and decide on a threshold. If the you exceed the threshold, it means there is a magnetic disturbance, and you can thus stop using it in your attitude solution. If they are within threshold, you can continue using the magnetometer.
If you think of more metrics to decide whether you are in a magnetic disturbance or not (such as the magnetometer norm rising to a ridiculous number), then you can add those metric to an HMM, which will combine these metrics and give you an estimate of whether you are in a disturbance or not.
Can any one guide me to do an app like this link
i.e How to calculate G-Force,Speed and Distance.
g-force can be calculated as a length of the vector given by accelerometer (Core Motion framework, see CMAccelerometerData for details).
As for distance & speed you can get them from GPS/GLONASS (Core Location framework, see CLLocation class)
Don't try to calculate distance and speed from acceleration - it's impossible to do this reliably.
For example to calculate speed you have to know acceleration of the car. Given Aa is a measurements from accelerometer, car's own acceleration would be Ac = Aa - Ga where Ga is a gravity acceleration vector in car's coordinates. To know this value one can use CoreMotion services (if gyro is present) or apply low-pass filter to extract gravity vector from accelerometer measurements.
Accelerometers used in iDevices are not very precise. Following integration will accumulate measurement errors steadily. Filters and such are second source of errors.
Now we can calculate our speed:
V(T[n]) = Ac[n] * dt[n] + V(T[n-1])
where V(T[n-1]) is the previous speed of the car. To determine first value (V(T[0])) we have to either force user to stop her car (to ensure zero speed, bad) or force her to input this value (unbelievably bad).
Distance is calculated by the same process
S(T[n]) = V(T[n]) * dt[n] + S(T[n-1])
which means already extremely imprecise speed value is now integrated over time. Happy, happy, joy, joy. BTW you can enjoy the results without coding all this - just read user comments for the app mentioned by OP.
Using just the rotationRate property of Core Motion or Gyroscope is it possible to extract how much radians (or degrees if you like) the device rotated?
I have tried to do a timed sampled of Core Motion data, for example, sampling it 5 times per second so I know that there's 0.2seconds between each reading. Then if I have a rotationRate of 0.5 radians per second from one read to another, in theory I could divide this by 5 and know now much radians the device rotated since the last time.
This seems logical, but the results have nothing to do with reality. Rotating the device 90 degrees will produce results telling me that the device rotated 100 times less than that.
Is it possible to extract how much the device rotated just by looking at rotationRate?
What am I missing?
Need more space than a comment ;-) iOS retrieves all data in radians and if your other calculations are correct, I thought it might be the angle measured in radians.
In general your approach seems to be alright: Take every signal's angle velocity, multiply it with time delta and you will have the angle delta for this timeframe. Then sum all your angles and the result should be the covered distance as angle in radians. Angle phi is the integral of angle velocity omega over the elapsed time and doing numerical integration with the trapezoidal rule (i.e. like described) is OK for gyroscope's data (not for accelerometer).
In general I would recommend to use the timestamp delivered by core motion instead of the defined period (1/5) as recommended by Apple, because device motion data is often delivered in a lower frequency than expected (see What is the official iPhone 4 maximum gyroscope data update frequency and or Push method for core motion and frequency of Accelerometer/Gyroscope Data.
Furthermore you should take a higher frequency to avoid errors in your numerical integration.
[Update from comments section:]
If your are interested in integrating via extended Simpson's Rule I recommend this paper (German only, p. 173 ff.) and An Extension to Newton-Cotes Formulae. Some sample code as extracted snippet taken from an existing project can be found here: DevicePosition.m Note that it might not compile, no warranty, as is, ... you know this from other sites ;-)
A free app displaying sensor input as graphs for iPhone: Sensor Monitor