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MEAM 620 Project 2, Phase 2-Inertial Odometry for a Quadrotor Solved
This assignment investigates the problem of estimating the pose of a flying robotic platform using information obtained from its onboard sensors, in this case an onboard inertial measurement unit (IMU), and an onboard stereo pair. In this assignment we will be fusing both sources of information together in an Error State Kalman Filter based on the notes and the paper we discussed in class. Please download the latest version of the lecture slides since changes have been made to improve clarity. Note that we are also providing a link to the underlying paper which describes the techniques you will be implementing.
In order to further ground the assignment we are having you apply the code you write to real data acquired by a quadrotor platform. For this we are using portions of the EuRoc dataset collected at ETH Zurich. We are providing a paper describing this dataset and the platform on which it was collected as one of the documents associated with this assignment. I would definitely reccomend taking a quick look at this paper.
2 Code Organization
The code packet that you are being provided with as part of this assignment is organized in the usual manner. In the top level directory there is a file entitled setup.py which you should run in order to install all of the needed packages. The proj2 2 package is divided into 3 subdirectories.The util directory contains a set of tests you can run on your code via the unit test script test.py. The dataset directory contains a small portion of the EuRoc dataset which is comprised of stereo and IMU data.
The code directory contains a set of code files and sandbox files which constitute the coding assignment. The file stereo.py contains a series of utilities that we are providing for reading information from the dataset, you should not modify this file. The file vio.py contains function stubs to implement the key steps in the ESKF filter algorithm described in class. This is the only file that you should modify, it contains comments indicating the parts we expect you to fill in.
The sandbox vio.py script runs your completed ESKF implementation against the provided stereo and IMU dataset to estimate the pose of the aircraft. After it runs it produces a plot showing the estimated position and orientation over time. Note that this script will produce garbage results until you complete the code. Note that running this script can take 20 seconds or more since it runs against several thousand stereo and IMU measurements. We show you what the final plot should look like in the PDF files that we provide in the util directory.
3 Task 1: Nominal State Update
One core step in the ESKF algorithm involves updating the nominal state based on the prevailing measurements from the IMU. This function takes in a tuple which represents the current nominal state, two 3 × 1 vectors that represent the angular velocity and linear acceleration and a scalar value dt indicating the duration of the timestep. The tuple that you are provided contains all of the elements of the state: the current position, p ∈ R3, the current velocity, v ∈ R3, the current rotation, q, which is provided as a Rotation object, the accelerometer bias, ab ∈R3, the gyroscope bias, ωb ∈R3, and the current estimate for the gravity vector, g ∈R3. Note that all of these inputs, except q, are provided as 3 × 1 vectors.
The output of this function should be a new tuple in the same format indicating the nominal state of the system at the end of the timestep.
4 Task 2: Covariance Update
The next function is used to update the error state covariance matrix in response to an IMU update. The inputs to this function include all of the inputs to the previous function as well as an initial error state covariance matrix in R18×18. In order to apply the covariance updates specified in the slides you will need a few parameters that define the noise covariance matrix.
The following table gives the name of the parameter and the quantity in the slides it refers to:
accelerometer noise density
σa˜n
gyroscope noise density
σω˜n
accelerometer random walk
σaw
gyroscope random walk
σωw
The mysterious term RT{(ωm − ωb)∆t} refers to the transpose of the rotation matrix formed by taking the vector, (ωm − ωb)∆t, and viewing it as specifying a rotation about that axis by an angle corresponding to the vector length in radians. Note that the Rotation class provides a member function called from rotvec that you may find useful for this purpose.
The notation [am − ab]× denotes the 3 × 3 skew symmetric matrix associated with the vector (am − ab)
5 Task 3: Measurement Update Step
Your final task is to implement a function that updates the nominal state and the error state covariance matrix based on a single image measurement provided as a 2×1 vector in the parameter uv. The parameter Pw ∈R3 denotes the coordinates of the point with respect to the world coordinate system.
The first thing that you should compute is the innovation vector which is simply the difference between the measured image coordinates, uv, and the predicted value, (Xc/Zc,Yc/Zc). The result should be a 2 × 1 vector. In order to account for outliers we compare the magnitude of this vector to a specified error threshold, if the vector magnitude is greater than this quantity no update is performed and the original state and covariance are returned, otherwise we compute updated versions of the nominal state and error state covariance according to the prescription given in the lecture slides. Basically we end up performing a modified version of the Extended Kalman Filter updating step. Note that once we calculate the error state update, δx, we use this vector to update the nominal state estimate.
We also require that when computing the updated covariance matrix that you use the form specified in the slides that is guaranteed to produce a symmetric positive definite output namely:
Σt = (I − KtHt)Σ0t(I − KtHt)T + KtQtKtT
The final output is a tuple containing the new nominal state, the new error covariance and the computed innovation vector. Note that this is returned whether or not the innovation magnitude is within bound, if no update is performed the resulting nominal state and covariance matrix are the same as the ones provided as inputs.
In order to further ground the assignment we are having you apply the code you write to real data acquired by a quadrotor platform. For this we are using portions of the EuRoc dataset collected at ETH Zurich. We are providing a paper describing this dataset and the platform on which it was collected as one of the documents associated with this assignment. I would definitely reccomend taking a quick look at this paper.
2 Code Organization
The code packet that you are being provided with as part of this assignment is organized in the usual manner. In the top level directory there is a file entitled setup.py which you should run in order to install all of the needed packages. The proj2 2 package is divided into 3 subdirectories.The util directory contains a set of tests you can run on your code via the unit test script test.py. The dataset directory contains a small portion of the EuRoc dataset which is comprised of stereo and IMU data.
The code directory contains a set of code files and sandbox files which constitute the coding assignment. The file stereo.py contains a series of utilities that we are providing for reading information from the dataset, you should not modify this file. The file vio.py contains function stubs to implement the key steps in the ESKF filter algorithm described in class. This is the only file that you should modify, it contains comments indicating the parts we expect you to fill in.
The sandbox vio.py script runs your completed ESKF implementation against the provided stereo and IMU dataset to estimate the pose of the aircraft. After it runs it produces a plot showing the estimated position and orientation over time. Note that this script will produce garbage results until you complete the code. Note that running this script can take 20 seconds or more since it runs against several thousand stereo and IMU measurements. We show you what the final plot should look like in the PDF files that we provide in the util directory.
3 Task 1: Nominal State Update
One core step in the ESKF algorithm involves updating the nominal state based on the prevailing measurements from the IMU. This function takes in a tuple which represents the current nominal state, two 3 × 1 vectors that represent the angular velocity and linear acceleration and a scalar value dt indicating the duration of the timestep. The tuple that you are provided contains all of the elements of the state: the current position, p ∈ R3, the current velocity, v ∈ R3, the current rotation, q, which is provided as a Rotation object, the accelerometer bias, ab ∈R3, the gyroscope bias, ωb ∈R3, and the current estimate for the gravity vector, g ∈R3. Note that all of these inputs, except q, are provided as 3 × 1 vectors.
The output of this function should be a new tuple in the same format indicating the nominal state of the system at the end of the timestep.
4 Task 2: Covariance Update
The next function is used to update the error state covariance matrix in response to an IMU update. The inputs to this function include all of the inputs to the previous function as well as an initial error state covariance matrix in R18×18. In order to apply the covariance updates specified in the slides you will need a few parameters that define the noise covariance matrix.
The following table gives the name of the parameter and the quantity in the slides it refers to:
accelerometer noise density
σa˜n
gyroscope noise density
σω˜n
accelerometer random walk
σaw
gyroscope random walk
σωw
The mysterious term RT{(ωm − ωb)∆t} refers to the transpose of the rotation matrix formed by taking the vector, (ωm − ωb)∆t, and viewing it as specifying a rotation about that axis by an angle corresponding to the vector length in radians. Note that the Rotation class provides a member function called from rotvec that you may find useful for this purpose.
The notation [am − ab]× denotes the 3 × 3 skew symmetric matrix associated with the vector (am − ab)
5 Task 3: Measurement Update Step
Your final task is to implement a function that updates the nominal state and the error state covariance matrix based on a single image measurement provided as a 2×1 vector in the parameter uv. The parameter Pw ∈R3 denotes the coordinates of the point with respect to the world coordinate system.
The first thing that you should compute is the innovation vector which is simply the difference between the measured image coordinates, uv, and the predicted value, (Xc/Zc,Yc/Zc). The result should be a 2 × 1 vector. In order to account for outliers we compare the magnitude of this vector to a specified error threshold, if the vector magnitude is greater than this quantity no update is performed and the original state and covariance are returned, otherwise we compute updated versions of the nominal state and error state covariance according to the prescription given in the lecture slides. Basically we end up performing a modified version of the Extended Kalman Filter updating step. Note that once we calculate the error state update, δx, we use this vector to update the nominal state estimate.
We also require that when computing the updated covariance matrix that you use the form specified in the slides that is guaranteed to produce a symmetric positive definite output namely:
Σt = (I − KtHt)Σ0t(I − KtHt)T + KtQtKtT
The final output is a tuple containing the new nominal state, the new error covariance and the computed innovation vector. Note that this is returned whether or not the innovation magnitude is within bound, if no update is performed the resulting nominal state and covariance matrix are the same as the ones provided as inputs.
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