Basic Linear Algebra Cemal Koc Pdf Pdf
Guy Clena
I have a record session for my application. When user started a record session I start collecting data from device's CMMotionManager object and store them on CoreData to process and present later. The data I'm collecting includes gps data, accelerometer data and gyro data. The frequency of data is 10Hz.
Currently I'm struggling to calculate the lean angle of device with motion data. It is possible to calculate which side of device is land by using gravity data but I want to calculate right or left angle between user and ground regardless of travel direction.
This problem requires some linear algebra knowledge to solve. For example for calculation on some point I must calculate the equation of a 3D line on a calculated plane. I am working on this one for a day and it's getting more complex. I'm not good at math at all. Some math examples related to the problem is appreciated too.
It depends on what you want to do with the collected data and what ways the user will go with that recording iPhone in her/his pocket. The reason is that Euler angles are no safe and especially no unique way to express a rotation. Consider a situation where the user puts the phone upright into his jeans' back pocket and then turns left around 90. Because CMAttitude is related to a device lying flat on the table, you have two subsequent rotations for (pitch=x, roll=y, yaw=z) according to this picture:
You see two different representations (90, 90, 0) and (-90, 0, 90) for getting to the same rotation and there are more of them. So you press Start button, do some fancy rotations to put the phone into the pocket and you are in trouble because you can't rely on Euler Angles when doing more complex motions (s. gimbal lock for more headaches on this ;-)
Now the good news: you are right linear algebra will do the job. What you can do is force your users to put the phone in always the same position e.g. fixed upright in the right back pocket and calculate the angle(s) relative to the ground by building the dot product of gravity vector from CMDeviceMotion g = (x, y, z) and the postion vector p which is the -Y axis (0, -1, 0) in upright position:
If you can't rely on having the phone at a predefined postion like (0, -1, 0) in the equations above, you can only calculate the total angle but not the specific ones alphaLR and alphaFB. The reason is that you only have one axis of the new coordinate system where you need two of them. The new Y axis y' will then be defined as average gravity vector but you don't know your new X axis because every vector perpedicular to y' will be valid.
So you have to provide further information like let the users walk a longer distance into one direction without deviating and use GPS and magnetometer data to get the 2nd axis z'. Sounds pretty error prone in practise.
If I understand your question, I think you are interested in getting the attitude of your device. You can do this using the attitude property of the CMDeviceMotion object that you get from the deviceMotion property of the CMMotionManager object.
There are two different angles that you might be interested in the CMAttitude class: roll and pitch. If you imagine your device as an airplane with the propeller at the top (where the headphone jack is), pitch is the angle the plane/device would make with the ground if the plane were in a climb or dive. Meanwhile, roll is the angle that the "wings" would make with the ground if the plane were to be banking or in mid barrel roll.
Assuming I understand what you want, the good news is that you may not need to understand any linear algebra to capture and use these angles. (If I'm missing something, please clarify and I'd be happy to help further.)
The attitude values in the CMAttitude object are relative to the ground (i.e., the default reference frame has the Z-axis as vertical, that is pointing in the opposite direction as gravity), so you don't have to worry about cancelling out gravity. So, for example, if you lie your device on a flat table top, and then roll it up onto its side, the roll property of the CMAttitude object will change from 0 to plus or minus 90 degrees (+- .5pi radians), depending on which side you roll it onto. Meanwhile, if you start it lying flat and then gradually stand it up on its end, the same will happen to the pitch property.
While you can use the pitch, roll, and yaw angles directly if you want, you can also set a different reference frame (e.g., a different direction for "up"). To do this, just capture the attitude in that orientation during a "calibration" step and then use CMAttitude's multiplyByInverseOfAttitude: method to transform your attitude data to the new reference frame.
Even though your question only mentioned capturing the "lean angle" (with the ground), you will probably want to capture at least 2 of the 3 attitude angles (e.g., pitch and either roll or yaw, depending on what they are doing), potentially all three, if the device is going to be in a person's pocket. (The device could rotate in the pocket in various ways if the pocket is baggy, for example.) For the most part, though, I think you will probably be able to rely on just two of the three (unless you see radical shifts in yaw throughout the course of a recording session). So for example, in my jeans pocket, the phone is usually nearly vertical. Thus, for me, pitch would vary a whole bunch as I, say, walk, sit or run. Roll would vary whenever I change the direction I'm facing. Meanwhile, yaw would not vary much at all (unless I do kart-wheels, which I can't!). So yaw can probably be ignored for me.
Depending on what you want to do, you may want to use this math. You would want to use the linear algebra and gravity if you need a standardized way of "talking about" and/or comparing attitudes over the course of your recording session. If you just want to visualize them, you can probably still get away with not using the increased complexity. (For example, visualizing (pitch=90, roll=0, yaw=0) should be the same as visualizing (pitch=0, roll=90, yaw=90).) In my approach above, while you could have multiple ways of referring to the "same" attitude, none of them is actually wrong, per se. They will still give you the angles relative to the ground.
But the fact that the gyroscope can switch from one valid description of an attitude to another means that what I wrote above about getting away with only 2 of the 3 components needs to be corrected: because of this, you will need to capture all three components, no matter what. Sorry.
I have seen the following theorem on the book of Linear Algebra by Cemal Ko, and gave a counterexample, but this book is used as the main textbook for linear algebra courses in my university, so I'm assuming I'm missing something.
The makeup exam will be on Wednesday, May 29 at 13:30 in M203. You are eligible to take the makeup if you have missed only one exam. The excuse for not attending an examination must be proved with documents. The content of the makeup exam is cumulative, i.e. covers all the topics from the beginning till the end.
Important Announcement: Possession of any kind of active electronic device during the exam is prohibited. If you need to bring an electronic device to the exam room, it is your responsibility to turn it off AND place it somewhere you cannot reach. Otherwise, it will be treated as cheating.The content of the final exam is cumulative, i.e. covers all the topics from the beginning till the end. However there will be an emphasize on those topics below that are not covered in the midterms. These sections are from the first textbook, namely Friedberg, Insel, Spence - Linear Algebra, 4th edition:
Important Announcement: Possession of any kind of active electronic device during the exam is prohibited. If you need to bring an electronic device to the exam room, it is your responsibility to turn it off AND place it somewhere you cannot reach. Otherwise, it will be treated as cheating.Click here to see your exam location.
Linear algebra is a central subject of mathematics which hasapplications not only in mathematics itself but also in statistics, economics, physics and engineering (just to name a few).By the end of the course, the students should have a detailed understanding of how to use the eigenvalue and eigenvector concept, the diagonalization process, the canonical forms of linear operators, the structure of inner product spaces.
The second textbook is Cemal Ko - Topics in Linear Algebra which is the classical linear algebra book of our department for the second semester. Tentative Course OutlineYou can find the tentative course outline below. For each week, we will attempt to cover the content of the indicated pages of Friedberg, Insel, Spence - Linear Algebra (4th edition).
The makeup exam will be on Wednesday, January 16 at 16:00 in M102. You are eligible to take the makeup if you have missed only one exam. The excuse for not attending an examination must be proved with documents unless you have already informed me about an exam conflict. The content of the makeup exam is cumulative, i.e. covers all the topics from the beginning till the end.
Important Announcement: Possession of any kind of active electronic device during the exam is prohibited. If you need to bring an electronic device to the exam room, it is your responsibility to turn it off AND place it somewhere you cannot reach. Otherwise, it will be treated as cheating.The final exam will be on Friday, January 11 at 17:00 in P1, P2 and P3. The exam duration is 120 minutes. You cannot exit from the exam room in the first 20 minutes and in the last 20 minutes. Note the unusual location, namely the Department of Physics building, of the exam.
The content of the final exam is cumulative, i.e. covers all the topics from the beginning till the end. However there will be an emphasize on those topics below that are not covered in the midterms. These sections are from the first textbook, namely Friedberg, Insel, Spence - Linear Algebra, 4th edition:
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