Expressing Vectors in Different Frames Using Rotation Matrices
Вставка
- Опубліковано 9 лют 2025
- In this video we develop notation to express a vector in different reference/coordinate frames. We then investigate how to use rotation matrices to translate from a vector expressed in one frame to this same vector expressed in another frame.
Topics and time stamps:
0:43 - Introduction using drones
2:49 - Notation for expressing vectors in different frames
5:03 - Rotation matrices
27:32 - Inverse of a rotation matrix
30:57 - Eigenvalues of a rotation matrix
34:00 - Example
Lecture notes and code can be downloaded from github.com/clu...
For more information see:
-en.wikipedia.o...
-Stevens, B. L. and Lewis, F. L., “Aircraft Control and Simulation, 2nd Edition”, John Wiley and Sons, 2003, pg. 25.
All Flight Mechanics videos in a single playlist ( • Flight Mechanics )
#FlightMechanics
You can support this channel via Patreon at / christopherwlum or by clicking on the ‘Thanks’ button underneath the video. Thank you for your help!
In case it is helpful, here are all the Flight Mechanics videos in a single playlist ua-cam.com/play/PLxdnSsBqCrrEx3A6W94sQGClk6Q4YCg-h.html. You can support this channel via Patreon at www.patreon.com/christopherwlum. Please let me know what you think in the comments. Thanks for watching!
Christopher is a great teacher for engineers in an adjacent discipline. He doesn't drag you back to sixth grade, at the same time, he doesn't fly over important details you need. He is the teacher's teacher. I subscribed half way through my first video of his.
AE 512: it's always useful to review how the handy rotation matrices were actually derived!
Dave, good to see you are getting a head start on the lectures. Keep me posted on how things go in the future!
Best 45 minutes I've ever spent on UA-cam.
I'm glad it was helpful. There are other similar videos on the channel please feel free to check them out. Thanks for watching!
AA 516: Very interesting properties of the rotation matrix, definitely a good addition to the video content.
It is great to hear that you found the rotation matrix properties interesting! By the way, please send me an email so I know who this is and can record for the future, thanks!
AA 516: During my bachelor's degree, I took a Mechatronics course that introduced these fundamental concepts. Initially, I found them quite perplexing, but your clear and engaging explanation has now illuminated my understanding. Thanks for making those concepts accessible and easy to grasp!
Thank you for all this series Dr Lum! It has really help me to be able to understand this topics, even when i alredy viewed them in my flight dynamics course
Amazing explanation of a very complex concept especially for the freshers in this field. Thank you Dr. Lum.
Hi Prabhjeet,
Thanks for the kind words, I'm glad you enjoyed the video. If you find these videos helpful, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum. Given your interest in this topic, I'd love to have you a as a Patron as I'm able to talk/interact personally with all Patrons. Thanks for watching!
-Chris
The best in-depth explanation that I've ever found. Thank you so much!
You're very welcome, this is a part of a series of videos on the channel. Please feel free to check them out and let me know what you think. Thanks for watching!
AE 512: The notation used for expressing rotations of vectors has always confused me and this video really helped clarify/simplify it, thanks!
AA516 - Loved your demo at the beginning, and the derivation was concise and clear
Great, I'm glad it was helpful, keep me posted on how the rest of the videos go!
AA 516: Very clear examples of how rotation matrices work, thanks!
I believe that it is wonderful that I was able to hop on youtube and search for this without thinking of cost. Thank you!
I hope Fred appreciates us going from his base to his tip 1:22 lol. But seriously these videos are absolutely amazing their very well explained thank you
AA516: Very nice derivation of the rotation matrix
Thank you Mr. Lum.
After your explanation I was finally able to see how to deduce rotation matrix!
Hi Nikita,
Thanks for the kind words, I'm glad you enjoyed the video. If you find these videos helpful, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum or via the 'Thanks' button underneath the video. Given your interest in this topic, I'd love to have you a as a Patron as I'm able to talk/interact personally with all Patrons. I can also answer any questions, provide code, notes, downloads, etc. on Patreon. Thanks for watching!
-Chris
Thank you Dr Lum, this was very well done, and I'm very much looking forward to the rest of the videos in this series, as well as the control theory stuff. Thank you for taking the time to share this with us.
Great lecture professor..it helped me a lot in understanding this topic. thanks a lot
Every morning, while i am travelling on the bus i bus your videos and now i beleive my quadcopter project will be better with your helps. Again you are literally hero ☺️
Hi Mehmet,
Thanks for the kind words, I'm glad you enjoyed the video. If you find these videos helpful, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum or via the 'Thanks' button underneath the video. Given your interest in this topic, I'd love to have you a as a Patron as I'm able to talk/interact personally with all Patrons. Thanks for watching!
-Chris
AA516: A great refresher on reference frame and awesome visual understanding to frame rotations.
AA516 2024 Wi : Really good explanation. Intuitive diagrams! WOrk on drones, so this was insightful and refreshing.
AA 516: Clear and easy to follow! I appreciate the explanations on the subscript notations of the rotation matrices since I always mix those up.
Yes, the notation is going to be critical especially when we start chaining together multiple rotations.
Easy to follow, rotation matrices make a lot more sense now
AA 516. I appreciate the effort you put into making these videos
you got my thumsup after the first example with the drone in the air :D :D like so much!
Thanks, I've got a few other similar videos on the channel with drones, please feel free to check them out. Thanks for watching!
AA 516 Thanks for the deep dive into rotation matrices! Super helpful.
you are the best! that was the best tutorial in this topic I've ever seen. thank you
Awesome lecture. Really helped me to understand what was going on. Great examples. Thank you!
I'm glad it was helpful thanks for watching!
Thank you Christopher for making it easy.
Always nice to have notation clarified.
AA516: I was also going to mention multiple rotations about different axes until you mentioned it last minute. I appreciated the reminder on definition of Eigenvalues, I don't use it often enough to remember it but it also pops up often enough that I'd want to keep it in mind.
clear and easy to understand explanation! 45 minutes well spent ! thanks for the effort >
AE 512: The references to dihedral and anhedral were fun and nice for an aero lecture
Very easy to follow; makes spatial reasoning easy!
AA516 : thank you for the clear explanation
Your videos are very useful. Great job man!
Glad you like them, thanks for watching!
Amazing explanation! I will do great during exam thank you
I'm glad it was helpful, thanks for watching!
AA516: Watching the application of rotation matrices to aircraft motion was an interesting part as an AA student.
hey brother, I love you stuff... and I find your over-use of the word "here" hilariously awesome!!!! I am thinking of teaching my self some machine learning and I am planning to build a program that can detect and count the number of times you say "here" LOL. Since you have lots of videos I have loads of data to train the network.
(i hope you aren't offended by what I said, it just goes to show how much I love and watch your videos!)
Hi DJ,
No worries, I know I say that I lot, it seems to be my tic when lecturing. I'm glad you're enjoying the videos, thanks for watching!
-Chris
@@ChristopherLum Whatever it is, its working and I'm loving the content. If I can be so bold to suggest some topics? I am current building an EDF-powered hovering rocket that gimbals the motor (instead of using fins for control). It would be awesome to see you discuss actually implementing an LQR Controller on a microcontroller. I understand the concepts decently well but when I go to implement my software from the simulation, I get a bit overwhelmed. Also, on a more theoretical note, I would love to see you explain sensor fusion in great detail, specially combining a gyro/accel (eg. MPU6050) with a magnetometer to calculate a complete 9-DOF state space. Thanks in advance.
@@djredrover
I am able to take requests, talk about concepts, and answer questions via my Patreon page at www.patreon.com/christopherwlum. Given your interest in this topic, I'd love to have you a as a Patron as I'm able to talk/interact personally with all Patrons. Thanks for watching!
-Chris
@@ChristopherLum just became your patreon! You are my first!
AA516: Forgot to comment on this when I watched it the first time but very clear examples and applicatioon of rotation matrices
amazing, i see how Rotation Matrices are created, really thanks alot
AA516: great explanation!
Incredible, get exactly what I was searching
Nice content ,will help me along with a dued project .
AE512: I never knew what a rotation matrix was! Thank you for clarifying that!
This gets much more interesting later on down the road.
Very nice explanation 👌
Thanks so much for the deep explanation :)
AE 512: Good refresher!
AA 516: Great lecture to illustrate rotation matrices and their uses.
Thank you for your wonderful work !
Hi,
Thanks for the kind words, I'm glad you enjoyed the video. If you find these videos helpful, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum or via the 'Thanks' button underneath the video. Given your interest in this topic, I'd love to have you a as a Patron as I'm able to talk/interact personally with all Patrons. I can also answer any questions, provide code, notes, downloads, etc. on Patreon. Thanks for watching!
-Chris
Very informative! I was very interested in quadcopter control and I am glad I signed up for this class. AA516
Multirotor control has become very popular lately and is a great application of control theory!
What a wonderful example.
Excellent! Very useful
I'm glad it was helpful. There are several related videos on the channel. Please feel free to check them out and I would love to hear what you think in the comments. Thanks for watching!
@@ChristopherLum I just saw the video about Euler Angles and it was amazing too!!. Christopher, I am facing some challenges dealing with segment-fixed coordinate systems, but my field of study is Biomechanics..... Any suggestion to overcome my challenges?
Thank you Dr. Lum!
Very well explained.
anhedral/dihedral pop ups made me laugh. that was a nice touch
Hi Brian, great to hear from you, I'm glad it was entertaining. I'll try to have more popups in future videos.
Informative and easy to follow
You've got a very nice backyard! Make sure to water those plants though
also nice video too
nice Fred and Joe hats
rubbed my eyes several time for clearing vision and each time I noticed that it was camera getting blurred of a moment and clearing up :D
How much effort goes into making these lecture videos?
Chris: Yes.
Haha, yes you are correct, they do take some effort but there are also fun to make. Thanks for watching!
AE512: Good refresher here
AE 512
The tip for keeping notation in check when using rotation matricies is useful, esepcially the note that we’re assuming a right hand angle rotation.
Yes, the notation is going to be helpful in some of the future discussions.
AA516: Just a clarifying question in the topic of switching between the rotation matrix indices (42:20), Do we use C_(1/r), the original rotation matrix, for rotating clockwise; and then C_(r/1), transposed rotation matrix, for rotating counter-clockwise? Thanks!
We use this notation to help keep track when we chain multiple rotations together. I can talk more about this at office hours if you like.
AA516 What helps me with the rotation notation is thinking of the subscripts/superscripts as units. So the r basis on the left side 1/r * r --> cancels to leave us in frame 1
That's a good way to think of it, great observation!
AA516 - thanks for the good video sir
In which video in the playlist where gimbal lock is mentioned or explained?
AA516 : Interesting how rotation matricies are orthogonal. Are there any physical scenarios where you would need to find the rotation matrix when given the input and output vectors?
Jason-AE512: Thanks for the explanation.
AA516: Thank you for the refresher. Will we be covering quaternions in this course and, if not, what benefits do they have over DCMs?
You're a few weeks ahead of the game but you are correct, we'll be looking at quaternions in week 3. Stay tuned!
The QUT drip goes hard in this video. That's where I study.
AA516 great video, thanks!
I'm glad you found it helpful!
AA516 great video!
thanks for the lecture!
Tahnk you soooooo much!
Super Dr Lum,,,,Awesome lecture,,just one suggestion if you could do this on MATLAB to show some graphical and coding skills ,I hope you will do it for lazy or lame students like me
very good video!
AA516 Amazing explanation! Po
i am honoured,thank you very much
Hi Mentamir,
Thanks for the kind words, I'm glad you enjoyed the video. If you find these videos helpful, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum or via the 'Thanks' button underneath the video. Given your interest in this topic, I'd love to have you a as a Patron as I'm able to talk/interact personally with all Patrons. Thanks for watching!
-Chris
AE512: Chris, for some reason I usually use a rotation matrix about z that is
Rz = [cos(theta) -sin(theta) 0 ; sin(theta) cos(theta) 0 ; 0 0 1]
as opposed to the one you derived
Rz = [cos(theta) sin(theta) 0 ; -sin(theta) cos(theta) 0 ; 0 0 1]
in matlab syntax.
Can you explain the discrepancy? I believe they are both attempting to perform the same transformation.
Often this is a matter of which frame you are rotating to/from. The order of the subscripts matters a lot in these situations
Ah I think I see. It seems that somehow they are doing the opposite transformation.
AE512: what is the difference between a frame, a basis, and a coordinate system? Thanks for the video!
Hi Dave, these are good questions. A frame and coordinate systems are sometimes used interchangeably. Some authors consider a coordinate system to be a frame with an origin attached to it. A basis is typically a higher order concept as you can have a basis of functions instead of just directions. This might be easier to chat about at office hours, please remind me if you'd like to talk at our next meeting, thanks.
Great master,😃
At 17:07, why is it we can assume that P_1 = A_1 + B_1 + C_1?
Thanks for reaching out. If you have questions or would like to request a video, please consider supporting the channel via Patreon at www.patreon.com/christopherwlum. I interact personally with Patrons at all levels. Thanks for watching!
AA516 Is there a more efficient way to derive the rotation matrix (I'm thinking of something that can be implemented repeatedly in software), or is deriving by hand using angles such as psi the main method? Thanks Dr. Lum
Hi Ken,
Good observation. In practice, we'll just want to derive these once and then memorize or implement these rotation matrices in a software package. You can code it yourself or leverage the Matlab Aerospace blockset/toolbox for some of this.
-Chris
thank you
Thanks! Has anyone ever told you that you look like Donnie Yen?
Hi,
Haha, no I've never heard that but thanks for the kind words. Also let me say thank you for your generous support of the channel, it is very much appreciated!
Do you have any particular interests in terms of videos or topics? I try to prioritize request made by interested parties as much as possible as I plan future videos.
If you are interested, I interact personally with all Patreon members at www.patreon.com/christopherwlum. Given your interest in the topic, I'd love to have you as a Patron.
In any event, I want to again say thank you for your contribution and for supporting the channel. I hope to hear from you at a future UA-cam video or on Patreon!
@@ChristopherLum Hi, I found your channel through your Mathematica videos. I found the explanations to be very thorough and practical, so I explored more. Turned out you already have a lot of videos that match my interest.
I'm interested in computer-sciency stuff as well as (practical) engineering & mathematics, because I'm making a 3D animation engine so videos that explain linear algebra in a practical way like these always help.
I'm interested in pretty much all of the stuff you do, so I'll check out your patreon and maybe in the future I can suggest some topics.
@@khoavo5758 Awesome, please let me know if you have any suggestions for future videos or if you'd like to discuss any of the videos. I hope to hear from you again soon!
Excuse me, could you give an example of a frame in linear algebra, please?
Hi Olga,
Thanks for reaching out. If you have questions or would like to request a video, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum. I'd love to have you as a Patron as I'm able to talk/interact personally with Patrons. Thanks for watching!
-Chris
cool stuff
May I know the author name of the book you are referring? It would be helpful.
I use the textbook by Stevens and Lewis. If you have specific questions I can interact more via Patreon at www.patreon.com/christopherwlum, thanks for watching!
@@ChristopherLum thank you so much.
awesome
AA 516: Celeste Yuan (Late due to name discrepancies discussed)
i think it's easier if you were to project onto the former frame instead of onto the newer frame.
i cant really get my head around with the sign convention @20:00 and @21:37 ... anyone?
Think of it as just XY plane. As frame Fr rotates to F1, the components on a_hat rotate (phi) clockwise which results in y_axis (in Frame1) on a_hat(in Frame1 ) to be -alpha*cos(phi).
omg, It took me 7 months to understand this.
AA516 Thank you
Ojasvi Kamboj - AA 516
Hello Everyone!
First comment!
AE512
well the z axis goes into the center of the page or Earth ? Well then next video journey to the center of the earth.
AA516