I have read somewhere that movement of a vehicle in cities is non-linear. It accelerates or de-accelerates frequently.
Can I use Extended Kalman filter for vehicle tracking moving on a road?
I am not able to understand the difference between KF and EKF.
The difference between a KF and EKF is in the model that is used, i.e. the equations used for propagation of the state (transition) and measurement update. If the model is linear, you can use a KF, EKF's are used for non-linear models.
For your case, the movement may not be linear, you can still create a linear state transition model:
Assume a vehicle with 1 dimensional motion. You could model this using a state [x,v,a] (position, speed, acceleration).
The state transition can be modeled as:
which is a linear model.
Related
I would like to factor in the CO2 emission in the Vehicle Routing Problem. The Arc Cost is not the distance anymore but the CO2 emission over each Arc
As a first order approximation, the CO2 emission of each arc can be modeled as a linear function of the current load and constant topographic characteristics of the arc (the slope, distance, altitude, ...)
arc_cost(arc,load) = K(arc) + alpha(arc)*load
K and alpha are functions of the arc only (stateless)
The load depends on the previous deliveries.
I haven't found how to model this with the ORTools Python API. Here is my current attempt : https://github.com/remisoulignac/scm_optim_problems/blob/main/SCM290-GreenVehicleRoutingProblem.ipynb
For now, I fall back on a two-level optimization approach:
A classical VRP optimization model. I parameterize this model by several control points at which I impose a certain weight of goods to be delivered to the chosen route.
A hyperparametric optimization level that will play with the different control points and optimize the overall real fuel consumption of the tour.
Thank you for your help,
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 implemented an easy mathematical model of a pendulum on simulink. I implemented a Kalman filter to estimate the velocity state having as input a zero torque, the initial position set to a certain angle (pi/18) and the initial velocity set to zero.
I suppose to have just a position measurement as the output of the system and the goal is to have a good estimation of the states. I expect to obtain a good estimation of the angle and the velocity but, as far as the velocity estimation is concerned, it is totally wrong.
I think there is something wrong in the block diagram. The question is: it is possible to have a "good" estimation of both the state using a Kalman filter for a mechanical linear Pendulum?
I am fairly new to Hidden Markov Models and I am trying to wrap my head around a pretty basic part of the theory.
I would like to use a HMM as a classifier, so, given a time series of data I have two classes: background and signal.
How are the emission probabilities estimated for each class? Does the Viterbi algorithm need a template of the background and signal to estimate prob(data|state)? Or have I completely missed the point?
To do classification with Viterbi you need to already know the model parameters.
Background and Signal are your two hidden states. With the model parameters and observed data you want to use Viterbi to calculate the most likely sequence of hidden states.
To quote the hmmlearn documentation:
The HMM is a generative probabilistic model, in which a sequence of
observable X variables is generated by a sequence of internal hidden
states Z. The hidden states are not be observed directly. The
transitions between hidden states are assumed to have the form of a
(first-order) Markov chain. They can be specified by the start
probability vector π and a transition probability matrix A. The
emission probability of an observable can be any distribution with
parameters θ conditioned on the current hidden state. The HMM is
completely determined by π, A and θ
.
There are three fundamental problems for HMMs:
Given the model parameters and observed data, estimate the optimal sequence of hidden states.
Given the model parameters and observed data, calculate the likelihood of the data.
Given just the observed data, estimate the model parameters.
The first and the second problem can be solved by the dynamic
programming algorithms known as the Viterbi algorithm and the
Forward-Backward algorithm, respectively. The last one can be solved
by an iterative Expectation-Maximization (EM) algorithm, known as the
Baum-Welch algorithm.
So we've got two states for our Hidden Markov model, noise and signal. We also must have something we observe, which could be ones and zeroes. Basically, ones are signal and zeroes are noise, but you get a few zeros mixed in with your signal and a few ones with the noise. So you need to know
Probablity of 0,1 when in the "noise" state
Probability of 0,1 when in the "signal" state
Probability of transitioning to "signal" when in the "noise" state.
Probability of transitioning to "noise" when in the "signal" state.
So we keep track of the probability of each state for each time slot and, crucially, the most likely route we got there (based on the transition probabilities). Then we assume that the most probable state at the end of the time series is our actual final state, and trace backwards.
The Viterbi Algorithm requires to know the HMM.
The HMM can be estimated with a Maximum-Likely-Estimation (MLE) and it's called the Baum–Welch algorithm.
If you have trouble with the Viterbi Algorithm there's a working implementation here
I have been trying to implement a navigation system for a robot that uses an Inertial Measurement Unit (IMU) and camera observations of known landmarks in order to localise itself in its environment. I have chosen the indirect-feedback Kalman Filter (a.k.a. Error-State Kalman Filter, ESKF) to do this. I have also had some success with an Extended KF.
I have read many texts and the two I am using to implement the ESKF are "Quaternion kinematics for the error-state KF" and "A Kalman Filter-based Algorithm for IMU-Camera Calibration" (pay-walled paper, google-able).
I am using the first text because it better describes the structure of the ESKF, and the second because it includes details about the vision measurement model. In my question I will be using the terminology from the first text: 'nominal state', 'error state' and 'true state'; which refer to the IMU integrator, Kalman Filter, and the composition of the two (nominal minus errors).
The diagram below shows the structure of my ESKF implemented in Matlab/Simulink; in case you are not familiar with Simulink I will briefly explain the diagram. The green section is the Nominal State integrator, the blue section is the ESKF, and the red section is the sum of the nominal and error states. The 'RT' blocks are 'Rate Transitions' which can be ignored.
My first question: Is this structure correct?
My second question: How are the error-state equations for the measurement models derived?
In my case I have tried using the measurement model of the second text, but it did not work.
Kind Regards,
Your block diagram combines two indirect methods for bringing IMU data into a KF:
You have an external IMU integrator (in green, labelled "INS", sometimes called the mechanization, and described by you as the "nominal state", but I've also seen it called the "reference state"). This method freely integrates the IMU externally to the KF and is usually chosen so you can do this integration at a different (much higher) rate than the KF predict/update step (the indirect form). Historically I think this was popular because the KF is generally the computationally expensive part.
You have also fed your IMU into the KF block as u, which I am assuming is the "command" input to the KF. This is an alternative to the external integrator. In a direct KF you would treat your IMU data as measurements. In order to do that, the IMU would have to model (position, velocity, and) acceleration and (orientation and) angular velocity: Otherwise there is no possible H such that Hx can produce estimated IMU output terms). If you instead feed your IMU measurements in as a command, your predict step can simply act as an integrator, so you only have to model as far as velocity and orientation.
You should pick only one of those options. I think the second one is easier to understand, but it is closer to a direct Kalman filter, and requires you to predict/update for every IMU sample, rather than at the (I assume) slower camera framerate.
Regarding measurement equations for version (1), in any KF you can only predict things you can know from your state. The KF state in this case is a vector of error terms, and thus you can only predict things like "position error". As a result you need to pre-condition your measurements in z to be position errors. So make your measurement the difference between your "estimated true state" and your position from "noisy camera observations". This exact idea may be represented by the xHat input to the indirect KF. I don't know anything about the MATLAB/Simulink stuff going on there.
Regarding real-world considerations for the summing block (in red) I refer you to another answer about indirect Kalman filters.
Q1) Your SIMULINK model looks to be appropriate. Let me shed some light on quaternion mechanization based KF's which I've worked on for navigation applications.
Since Kalman Filter is an elegant mathematical technique which borrows from the science of stochastics and measurement, it can help you reduce the noise from the system without the need for elaborately modeling the noise.
All KF systems start with some preliminary understanding of the model that you want to make free of noise. The measurements are fed back to evolve the states better (the measurement equation Y = CX). In your case, the states that you are talking about are errors in quartenions which would be the 4 values, dq1, dq2, dq3, dq4.
KF working well in your application would accurately determine the attitude/orientation of the device by controlling the error around the quaternion. The quaternions are spatial orientation of any body, understood using a scalar and a vector, more specifically an angle and an axis.
The error equations that you are talking about are covariances which contribute to Kalman Gain. The covariances denote spread around the mean and they are useful in understanding how the central/ average behavior of the system is changing with time. Low covariances denote less deviation from the mean behavior for any system. As KF cycles run the covariances keep getting smaller.
The Kalman Gain is finally used to compensate for the error between the estimates of the measurements and the actual measurements that are coming in from the camera.
Again, this elegant technique first ensures that the error in the quaternion values converge around zero.
Q2) EKF is a great technique to use as long as you have a non-linear measurement construction technique. Be very careful in using EKF if their are too many transformations in your system, i.e don't try to reconstruct measurements using transformation on your states, this seriously affects the model sanctity and since noise covariances would not undergo similar transformations, there would be a chance of hitting singularity as soon as matrices are non-invertible.
You could look at constant gain KF schemes, which would save you from covariance propagation and save substantial computation effort and time. These techniques are quite new and look very promising. They actively absorb P(error covariance), Q(model noise covariance) and R(measurement noise covariance) and work well with EKF schemes.