In case it is helpful, here are all the Flight Mechanics videos in a single playlist ruclips.net/p/PLxdnSsBqCrrEx3A6W94sQGClk6Q4YCg-h. 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!
Hello, thanks for the lecture, Im trying to run matlab code but seems like DrawFrame functşon is missing, i checked the github but couldnt find might u help me
AA516: I watched this last week but realized while watching this week's videos that I did not comment here. Good video, I like the Matlab demonstrations!
I've been troubled for a long time by the Euler angles and how to correctly perform the self-rotation in sequence. This video is very clear and solves almost all my problems. Thank you so much
Beyond the excellent content, it's evident that your enthusiasm, excitement, and eagerness to share your knowledge on the subject you're teaching shine through.
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
I have thoroughly enjoyed watching this video. As a well over eighty-year-old engineer, I lived through the early days of computer programming and about 50 years ago using my primitive computer I tried to program a " line shape of a cross with a rudder "as a substitute for a basic aircraft outline which needed to follow my flying instructions of pitching rolling and yawing and translation. I obtained the flying simulation by a sequence of " slow" multiple transformations and I could not deduce that one could do it with one transformation through selecting a particular axis. Thank you for taking me back so many enjoyable years when I could afford the time to see the beauty of matrix transformations.
Carmel, thanks for the great comment and historical perspective. It is great to hear your experience as it helps people appreciate the tools they have today as opposed to what was being done in the good old days of engineering. If you are interested, there is a follow in sequence of videos that show how some of this is implemented on some hardware, please feel free to check out the video and I'd love to hear your perspective. I hope to catch you at another video, thanks for watching!
@@ChristopherLum Thank you for your reply, I shall definitely look at your sequence of videos and try to catch up with my lacking knowledge. At the moment, as an old man being locked in with " Coronavirus threat" I have been looking into spinning and precessing gyros and their elegant behavior. I have analyzed them and came to some useful conclusions about the gyro losing its centrifugal force, and simulated them with two linear linked differential equations. Though it is only an approximation one gets the idea of how torque in the vertical direction links torques in a horizontal direction. I shall now try to simulate the "floating and rotating action" of a "two-line gyro Rg spinning and Ra precessing radii" as shown in this video using your angular transformations. Thank you for the information , I shall follow your work and let you know of my thoughts. Thanks again. ruclips.net/video/Pbb_RnxeYys/видео.html
AE512: Great explanation of the DCM. Rotations are much less confusing when broken down into the three axes. (I also was impressed with the hand-drawn planes you had for each axis!)
This is the clearest video describing a complex topic that I have ever watched. The way you motivate every discussion and clearly describe what's going on is INCREDIBLE. Thank you so much!!
I'm glad it was helpful. There are other similar videos on the channel please feel free to check them out and let me know what you think in the comments. Thanks for watching
AA516: Forgot to comment on this when I watched it the first time but fantastic content, super engaging, and really enjoyed learning the concepts this way
Hi Wil, great to hear from you. I've got a follow up to this video coming shortly, hopefully I'll see you there as well. I hope all is well on your end during these uncertain times!
@@ChristopherLum Looking forward to the next video in the series, particularly if it will cover quaternions...all good here, I hope you and your family are still healthy. Have fun, be safe.
I'm glad it was helpful. This is the start of a series of videos discussing vehicle attitude. There is a video on the channel discussing how to do this using quaternions. Please let me know what you think of that one as well. Thanks for watching!
Hi Raul, 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
AA516: Awesome lecture. Rotating around the eigenvector with Rodrigue's rotation was very interesting! I have only heard of quaternions and apparently they make the process of rotating vectors easier. I am looking forward to learning about quaternions. Thanks Professor!
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: Thanks for writing everything out and being so thorough. The additional time spent doing this is very helpful so I could verify my own understanding and calculations.
This video is very helpful, I'm trying to come up with a math project for highschool and your video gave me the idea to derive the equation of the vector after rotations.
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 OK so I have one question I understood most of the video but what does Proving that the DCM is unitary do. That's where I am a bit confused like does it prove that the DCM is reliable in telling the final vector of the aircraft or some other reason?Thanks
Hi David, 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
AA516 love the matlab animation and i think it makes us understand easier with those visual simulations. I would love to learn in this manner. Thank you!
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!
I'm glad it was helpful. There are other similar videos on the channel. Please feel free to check them out and let me know what you think. Thanks for watching!
AA516: While watching the first part I was thinking we could just define an axis in the very first frame and then an angle to rotate by to simplify it, but then I realized that's going back to the idea of the axis-angle representation. And it looks like we got to that in the second part!
AE512: Super interesting that the eigenvector ends up being the axis of rotation, but this video finally connected why with my previous understanding of eigenvectors
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!
for those of you watching in 2025 and are trying Mathematica for the first time, not sure if it's just me but I encounter nightmares in defining matrices when following the notation in this video (C1v, C21, and Cb2). I think one should not use capital letters. Instead define them as c1v, c21, cb2. Took me a fat hour to figure this one out but now at least this worked for me.
AA516 - Great lecture as always. Also introduced me to some new Mathematica syntax - I now know how to simplify matrices instead of just leaving cos^2 + sin^2 in there, like I did for hw1 (oops)
If you are interested in a more in depth look at Mathematica, I have a whole series of videos on this topic, feel free to check them out. For example, you can learn about Simplify vs. FullSimplify.
Thanks for the great lecture. I am very new to this topic with little mathematical background, but I try to reason about "gimbal lock." So, I came across your video-very helpful! However, I have a few (maybe very naive) questions about the composition of the 3 "intrinsic" rotations around the axis in different frames to generate the final C(b/v) matrix. * First, are all those rotation matrices passive transforms that "leave points fixed but change the frame of reference or coordinate system relative to which they are described"? It seems yes from the value of the rotation matrices, but no from what you are demonstrating with the aircraft, as the object (body of the aircraft) rotates along its body frames? * Second, is the final C(b/v) matrix represented in the V frame? Is any conversion required to convert from intrinsic rotation to extrinsic rotation to be able to actually calculate the final results? It seems you could only multiply metrics defined/represented in the same frame? By reading the explanations here en.wikipedia.org/wiki/Davenport_chained_rotations#Conversion_between_intrinsic_and_extrinsic_rotations Thanks again for the great video.👍
you have advantage of writing with left hand...carry on..This concepts apply to robotics as well and are now quite clear to me ...thanks... and what should we do after we derive DCM. How it will help controlling flight !!
Hi Viren, Thanks for reaching out, I'm glad you enjoyed the video. Unfortunately I'm unable to respond to questions on RUclips due to the sheer volume of inquiries that I receive. That being said, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum or via the 'Thanks' button underneath the video as I'll be able to answer questions there. Given your interest in the topic, I'd love to have you as a Patron as I'm able to talk/interact personally with Patrons. Thanks for watching! -Chris
Hello Sir. Thanks for the wonderful lecture. I may need one clarification on how did you arrive at the atan2 equation for phi and chi (it starts at 49.11 in the video) from the DCM. May you point to any other lecture in case i need to understand this bit more.
AA516: Allie S, I am also interested in the quaternions discussion. I have attended lectures on it before and the concept behind them is so wild and somehow intuitive! I'm sure your explanations will be really helpful.
AA 516: How did you determine where to put the cosines and (negative) sines in C_2/1? Is it just the same proof from previous videos but about the y-axis instead of z? or is there a quicker way to determine?
Zach, yes, this is a similar proof as the previous but about a different axis. Unfortunately there isn't a real shortcuts as during the derivation you'll see where the negative signs go (it can be different depending on which axis).
first of all i want to thank you for all your work Christopher! amazing work! In second place i wanna share a dubt, to be exactly in 1:05:00 in the matlab code you use a function called "DrawFrame", and my question is how did you define the function? i tried to found it in internet but i couldn't.
Thanks for making this video. At 9:13, presenter says the vehicle starts aligned with the NED frame, but in the beginning of the video, he stated that the goal was to align the NED frame to the vehicle's Body frame. I was confused because the vehicle is already in the Body frame, and we want to align the NED frame to it, right? Maybe the presenter forgot to say that he was just using the vehicle/prop at 9:13 to show the sequence of euler rotations from NED to Body frame, is this right?
Hi, Thanks for reaching out, I'm glad you enjoyed the video. Unfortunately I'm unable to respond to questions on RUclips due to the sheer volume of inquiries that I receive. That being said, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum or via the 'Thanks' button underneath the video as I'll be able to answer questions there. Given your interest in the topic, I'd love to have you as a Patron as I'm able to talk/interact personally with Patrons. Thanks for watching! -Chris
First of all, thank you very much for such a nice lecture. I just have one question. In the lecture, you mentioned that the cosine matrix is not singular. I was wondering if this concept is always correct no matter how we choose angles of rotations!? The thing is I have seen in another lecture in which Tain-Bryant angles were used to present the cosine matrix. In that lecture, it was mentioned that for theta=90 the matrix is singular.
Sir, you covered all the necessary information just in 1 hours. Thank you for sharing. Which functions of matlab have you used in order to rotate frame with mouse ?
Thank you for this video - it makes sense when the angular rotations about each axis are applied sequentially. However, a MEMS gyro doesn't tell you "here are the 3, sequential euler angles since the last control loop iteration that you can put into a DCM." So, how could you derive a rotation matrix that takes you from the orientation at the previous timestep to the one in the current timestep (which can then be chained onto a running rotation matrix that starts at the initial body frame aka global frame)? I ask because a MEMS gyro returns instantaneous angular velocity rates about each of the 3 orthogonal body axes - right? Do we assume the angular displacements (angular-velocity-rates-reported-about-each-axis-of-the-gyro * dt) about the 3 body axes are infinitesimal and thus we can kinda-sorta call then the 3 euler angles as if they were applied sequentially to the orientation at the last timestep and form a rotation matrix that way? I just saw ur video titled "Computing Euler Angles: The Euler Kinematical Equations and Poisson’s Kinematical Equations" which can give me the time rate of change of the 3 euler angles given the current euler angles and the angular rates in the body frame as reported by the gyro. Can the rate of change of these euler angles by integrated independently of the other euler angles to get the total euler angles?
AA 516: At the end of the video you described intrinsic versus extrinsic angles. Could you describe again why the psi, theta, phi angles we were working with are intrinsic? Also does this mean that the eigenvalue rotation angle is extrinsic? -Sydney B
Great question. This is a little interesting as if you rotate around the primary axis this is only a single rotation, let's chat at office hours to make sure I understand the question.
Hi Emin, Thanks for reaching out, I'm glad you enjoyed the video. Unfortunately I'm unable to respond to questions on RUclips due to the sheer volume of inquiries that I receive. That being said, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum as I'll be able to answer questions there. Given your interest in the topic, I'd love to have you as a Patron as I'm able to talk/interact personally with Patrons. Thanks for watching! -Chris
AA516 - Certainly a highly engaging lecture. Connecting the Euler angles to the aerospace industry, is their definition based on the definition of the stability axes for an aircraft? Perhaps more specifically, is there a non-casual connection between the fact that the term C_13 term used to calculate the pitch angle from the direction cosine matrix is only dependent on the pitch angle, just as the longitudinal aircraft dynamic modes are uncoupled from motion in the other stability axes?
just a thing missing is, given a rotated coordinate frame, how to find those tait bryan(euler) angles wrt to world coordinate frame. It basically includes defining a Nodal line(xy plane intersecting Y'Z' plane) and then finding those rotating angles based on the sequence (XYZ, YZX, etc). I am quite clear about Classic euler angles(ZYZ, XYX, ZXZ, YZY, YXY, XZX) but finding for tait bryan angles tend to haunt me.
Thanks Dr Lum. At the end of video, you briefly mentioned intrinsic euler angle vs. extrinsic Euler angle, have you discuss extrinsic Euler angle in some of your video?
Superb explanation. That's awesome Chris. Thank you for sharing the playlist. If I create book on Spacecraft dynamics, I will make sure I will include your contribution. Could you please let us know if any Matlab simulations will accompany the flight dynamics playlist someday?
AA 516: To determine the positive right handed rotation do you point your thumb in the direction of the axis of rotation and the direction your fingers curl is a positive rotation?
AE512: I didn't realize until later in the class how exactly this would fold into our modeling of aircraft. It's clearly essential to mediate between different reference frames. It's unfortunate that DCM formulations across software packages are not always the same.
I'm a little lost here. What is the advantage of euler over dcm? Dcm being unitary means it can't go into a gimbal lock? If dcm doesn't have gimbal lock issue then what's the advantage of using quaternion?
Hello Sir as an enthusiast I would like to thanks you for those amazing videos ! I have a question concerning the calculus of the Rotation matrix from v to b at 31:45: Since its a combinaison of linear transformations and because the product of several matrices are not commutative should not we were supposed to do : C1v.C21.Cb2 instead of C2b.C21.C1v ? Greetings from France !
Hi, Thanks for reaching out, I'm glad you enjoyed the video. Unfortunately I'm unable to respond to questions on RUclips due to the sheer volume of inquiries that I receive. That being said, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum or via the 'Thanks' button underneath the video as I'll be able to answer questions there. Given your interest in the topic, I'd love to have you as a Patron as I'm able to talk/interact personally with Patrons. Thanks for watching! -Chris
Thanks for reaching out, I'm glad you enjoyed the video. Unfortunately I'm unable to respond to questions on RUclips due to the sheer volume of inquiries that I receive. That being said, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum as I'll be able to answer questions there. I'd love to have you as a Patron as I'm able to talk/interact personally with Patrons. Thanks for watching!
Thanks Dr Lum. Very clear explainations. But question is: at 1:07:41, you did not mention how we calculate the rotation angle mu. Could you explain it? Many thanks!
Hi Gustavo, I have code and other benefits available on my Patreon page at www.patreon.com/christopherwlum. Given your interest in the topic, I'd love to have you as a Patron as I'm able to talk/interact personally with Patrons. Thanks for watching! -Chris
Thank you so much for this lecture and the other lectures you have on flight mechanics. You're a great person for sharing these for free. I do have a question, though, which I hope you will find the time to answer. How come your sines in the rotation matrices for the DCM have opposite signs of what I find online when searching for rotation matrices? Is it due to positive being defined as down in the NED coordinate system?
Thanks for reaching out, I'm glad you enjoyed the video. Unfortunately I'm unable to respond to questions on RUclips due to the sheer volume of inquiries that I receive. That being said, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum as I'll be able to answer questions there. Given your interest in the topic, I'd love to have you as a Patron as I'm able to talk/interact personally with Patrons. Thanks for watching!
Helpful lecture, Professor, thank you! Though, in the section where we talked about expressing a DCM as a single rotation about some axis, I was surprised that we didn't compute the angle that you need to rotate about that axis. Is that coming in a later video? Or is deriving that angle left as an exercise for the reader?
Justin, you are absolutely correct, your spidey sense should have been tingling. We don't have the mechanisms in place to directly compute that right now. However, once we get to quaternions, we will directly compute this. I just posted the video a few hours ago if you're chomping at the bit and wanted to get to this early.
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!
I think there is a mistake here. Here you mention ruclips.net/video/GJBc6z6p0KQ/видео.html the rotation sequence is in ZYX sequence but you have multiplied in wolfram in XYZ sequence. The answer I got multiplying ZYX sequence is different to yours.
Yes he did the multiplication as it's a "fixed" euler sequence rotation. I don't know if he's using a different convention but i guess he also made a mistake in Mathematica when defining the rotation matrices, the (-) sign in front of the sin(angle) in all matrices is not placed in the correct position.
Hi Logan, Thanks for the kind words, I'm glad you enjoyed the video. If the find the these videos to be 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
THIS IS FOR MY REF NOTE ONLY AND NOT A COMMENT FOR THE ABOVE VIDEO :parallelly go through the concepts of Inverse, singularity, eigen vector and eigen values concepts of a matrix, what are they how they are usefull in real world scenarios
In case it is helpful, here are all the Flight Mechanics videos in a single playlist ruclips.net/p/PLxdnSsBqCrrEx3A6W94sQGClk6Q4YCg-h. 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!
Hello, thanks for the lecture, Im trying to run matlab code but seems like DrawFrame functşon is missing, i checked the github but couldnt find might u help me
AA516: I watched this last week but realized while watching this week's videos that I did not comment here. Good video, I like the Matlab demonstrations!
I've been troubled for a long time by the Euler angles and how to correctly perform the self-rotation in sequence. This video is very clear and solves almost all my problems. Thank you so much
One of the best instructors I got online... Thank you so much for your simplification of such important concepts.
Beyond the excellent content, it's evident that your enthusiasm, excitement, and eagerness to share your knowledge on the subject you're teaching shine through.
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
Came to understand more about euler rotations. Did not expect to also be so entertained! :D
I'm glad it was helpful. Thanks for watching!
I have thoroughly enjoyed watching this video. As a well over eighty-year-old engineer, I lived through the early days of computer programming and about 50 years ago using my primitive computer I tried to program a " line shape of a cross with a rudder "as a substitute for a basic aircraft outline which needed to follow my flying instructions of pitching rolling and yawing and translation. I obtained the flying simulation by a sequence of " slow" multiple transformations and I could not deduce that one could do it with one transformation through selecting a particular axis. Thank you for taking me back so many enjoyable years when I could afford the time to see the beauty of matrix transformations.
Carmel, thanks for the great comment and historical perspective. It is great to hear your experience as it helps people appreciate the tools they have today as opposed to what was being done in the good old days of engineering. If you are interested, there is a follow in sequence of videos that show how some of this is implemented on some hardware, please feel free to check out the video and I'd love to hear your perspective. I hope to catch you at another video, thanks for watching!
@@ChristopherLum Thank you for your reply, I shall definitely look at your sequence of videos and try to catch up with my lacking knowledge.
At the moment, as an old man being locked in with " Coronavirus threat" I have been looking into spinning and precessing gyros and their elegant behavior. I have analyzed them and came to some useful conclusions about the gyro losing its centrifugal force, and simulated them with two linear linked differential equations. Though it is only an approximation one gets the idea of how torque in the vertical direction links torques in a horizontal direction.
I shall now try to simulate the "floating and rotating action" of a "two-line gyro Rg spinning and Ra precessing radii" as shown in this video using your angular transformations. Thank you for the information , I shall follow your work and let you know of my thoughts. Thanks again.
ruclips.net/video/Pbb_RnxeYys/видео.html
That sounds like a great project, keep me posted!
AE512: Great explanation of the DCM. Rotations are much less confusing when broken down into the three axes. (I also was impressed with the hand-drawn planes you had for each axis!)
This is the clearest video describing a complex topic that I have ever watched. The way you motivate every discussion and clearly describe what's going on is INCREDIBLE. Thank you so much!!
I'm glad it was helpful. There are other similar videos on the channel please feel free to check them out and let me know what you think in the comments. Thanks for watching
I couldn't be more agree with you. It was a great explanation about Euler Angles.
cant thank you enough!!!im unable to express my gratitude in words
AA516: Awesome discussion on Euler angles and rotation!
AA516: Forgot to comment on this when I watched it the first time but fantastic content, super engaging, and really enjoyed learning the concepts this way
Thanks Dr Lum, I was hoping you'd cover this topic....great info as always! I hope you and yours are healthy and safe.
Hi Wil, great to hear from you. I've got a follow up to this video coming shortly, hopefully I'll see you there as well. I hope all is well on your end during these uncertain times!
@@ChristopherLum Looking forward to the next video in the series, particularly if it will cover quaternions...all good here, I hope you and your family are still healthy. Have fun, be safe.
Another great video! Thank you for taking the time to present these videos.
I'm glad it was helpful. This is the start of a series of videos discussing vehicle attitude. There is a video on the channel discussing how to do this using quaternions. Please let me know what you think of that one as well. Thanks for watching!
You explain so good these topics ,that if not well explained could be a struggle. But,you make it so interactive. good job.
Hi Raul,
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
AA516: Awesome lecture. Rotating around the eigenvector with Rodrigue's rotation was very interesting! I have only heard of quaternions and apparently they make the process of rotating vectors easier. I am looking forward to learning about quaternions. Thanks Professor!
AA 516: Really interesting video, It's very satisfying watching the animation about a single axis at the end!
AE512: great explanations, and it even helped me understand eigenvalues better too.
why you are my hero is that you make those topic that sounds like so complicated, eaisier to understand
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
AA 516: Great MATLAB visualizations, very interesting lecture!
AA516: Thanks for writing everything out and being so thorough. The additional time spent doing this is very helpful so I could verify my own understanding and calculations.
Love your videos, having knowledge is different from having skill/talent to share the knowledge, you have both.
Thanks for the kind words. I'm glad it was helpful. Please let me know what you think of any of the other videos as well. Thanks for watching!
AA 516: A lot of useful information explained relatively quickly and explained clearly with examples, very nice lecture.
Many thanks .Very helpful and finally I understood the concept of Euler parameters
Glad it was helpful! Thanks for watching!
AA516: the matlab graphs were really helpful!
This video is very helpful, I'm trying to come up with a math project for highschool and your video gave me the idea to derive the equation of the vector after rotations.
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 OK so I have one question I understood most of the video but what does Proving that the DCM is unitary do. That's where I am a bit confused like does it prove that the DCM is reliable in telling the final vector of the aircraft or some other reason?Thanks
Thank You, Christopher. Very well done!
Hi David,
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
AE512: Appreciate the eigen vector explanation, it's an interesting connection back to the linear systems theory.
This channel is priceless. Thank you
AA516 love the matlab animation and i think it makes us understand easier with those visual simulations. I would love to learn in this manner. Thank you!
AE 512
The MATLAB animation really helped with breaking down the concept into the 3 angles within their independent frames.
thank you very much!
Greeting from Dominican Republic
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!
This so amazing!
You've put in so much effort!
This video is amazing! I loved every minute! Thank you Sir!
Thanks you so much for this informative class Christopher. It made my life easier .
Best regards
Aditya (a student from India)
I'm glad it was helpful. There are other similar videos on the channel. Please feel free to check them out and let me know what you think. Thanks for watching!
great video, thank you. clear explanation with the right level of detail
Clear explanation ... thanks for sharing ... Christopher...
I really appreciate you for making such an exelent video, It was really helpful for me to do my homework.
AA516: While watching the first part I was thinking we could just define an axis in the very first frame and then an angle to rotate by to simplify it, but then I realized that's going back to the idea of the axis-angle representation. And it looks like we got to that in the second part!
I'm not quiet following, are you suggesting that the axis is the same for all 3 rotations?
AE512: Super interesting that the eigenvector ends up being the axis of rotation, but this video finally connected why with my previous understanding of eigenvectors
This is great :) Thanks for detailed explanations :)
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!
AA516 thanks for the lecture!
Understood very, very well sir.
Glad it was helpful, there are several other similar videos on the channel, please feel free to check them out. Thanks for watching!
Nicely explained..I appreciate the efforts .
AA516 - Great discussion
AA516 good use of Mathematica here, might be convinced to try it out
for those of you watching in 2025 and are trying Mathematica for the first time, not sure if it's just me but I encounter nightmares in defining matrices when following the notation in this video (C1v, C21, and Cb2). I think one should not use capital letters. Instead define them as c1v, c21, cb2. Took me a fat hour to figure this one out but now at least this worked for me.
AA516 - Great lecture as always. Also introduced me to some new Mathematica syntax - I now know how to simplify matrices instead of just leaving cos^2 + sin^2 in there, like I did for hw1 (oops)
If you are interested in a more in depth look at Mathematica, I have a whole series of videos on this topic, feel free to check them out. For example, you can learn about Simplify vs. FullSimplify.
Thanks for the great lecture. I am very new to this topic with little mathematical background, but I try to reason about "gimbal lock." So, I came across your video-very helpful!
However, I have a few (maybe very naive) questions about the composition of the 3 "intrinsic" rotations around the axis in different frames to generate the final C(b/v) matrix.
* First, are all those rotation matrices passive transforms that "leave points fixed but change the frame of reference or coordinate system relative to which they are described"? It seems yes from the value of the rotation matrices, but no from what you are demonstrating with the aircraft, as the object (body of the aircraft) rotates along its body frames?
* Second, is the final C(b/v) matrix represented in the V frame? Is any conversion required to convert from intrinsic rotation to extrinsic rotation to be able to actually calculate the final results? It seems you could only multiply metrics defined/represented in the same frame? By reading the explanations here en.wikipedia.org/wiki/Davenport_chained_rotations#Conversion_between_intrinsic_and_extrinsic_rotations
Thanks again for the great video.👍
AA516 Great video
priceless explanation.thanks a lot
you have advantage of writing with left hand...carry on..This concepts apply to robotics as well and are now quite clear to me ...thanks... and what should we do after we derive DCM. How it will help controlling flight !!
Hi Viren,
Thanks for reaching out, I'm glad you enjoyed the video. Unfortunately I'm unable to respond to questions on RUclips due to the sheer volume of inquiries that I receive. That being said, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum or via the 'Thanks' button underneath the video as I'll be able to answer questions there. Given your interest in the topic, I'd love to have you as a Patron as I'm able to talk/interact personally with Patrons. Thanks for watching!
-Chris
another wonderful lecture, thank you!
AA516 Very important topic with very interesting and clear explanation. Po
AE512: Great to have the hand drawn diagram of the aircraft to show what the euler angles and DCM actually represent.
A A 516: Really liked the MATLAB visuals
Hello Sir. Thanks for the wonderful lecture. I may need one clarification on how did you arrive at the atan2 equation for phi and chi (it starts at 49.11 in the video) from the DCM. May you point to any other lecture in case i need to understand this bit more.
AA516: Allie S, I am also interested in the quaternions discussion. I have attended lectures on it before and the concept behind them is so wild and somehow intuitive! I'm sure your explanations will be really helpful.
AA 516: How did you determine where to put the cosines and (negative) sines in C_2/1? Is it just the same proof from previous videos but about the y-axis instead of z? or is there a quicker way to determine?
Zach, yes, this is a similar proof as the previous but about a different axis. Unfortunately there isn't a real shortcuts as during the derivation you'll see where the negative signs go (it can be different depending on which axis).
first of all i want to thank you for all your work Christopher! amazing work!
In second place i wanna share a dubt, to be exactly in 1:05:00 in the matlab code you use a function called "DrawFrame", and my question is how did you define the function? i tried to found it in internet but i couldn't.
Thanks for making this video. At 9:13, presenter says the vehicle starts aligned with the NED frame, but in the beginning of the video, he stated that the goal was to align the NED frame to the vehicle's Body frame. I was confused because the vehicle is already in the Body frame, and we want to align the NED frame to it, right? Maybe the presenter forgot to say that he was just using the vehicle/prop at 9:13 to show the sequence of euler rotations from NED to Body frame, is this right?
Hi,
Thanks for reaching out, I'm glad you enjoyed the video. Unfortunately I'm unable to respond to questions on RUclips due to the sheer volume of inquiries that I receive. That being said, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum or via the 'Thanks' button underneath the video as I'll be able to answer questions there. Given your interest in the topic, I'd love to have you as a Patron as I'm able to talk/interact personally with Patrons. Thanks for watching!
-Chris
First of all, thank you very much for such a nice lecture. I just have one question. In the lecture, you mentioned that the cosine matrix is not singular. I was wondering if this concept is always correct no matter how we choose angles of rotations!? The thing is I have seen in another lecture in which Tain-Bryant angles were used to present the cosine matrix. In that lecture, it was mentioned that for theta=90 the matrix is singular.
Sir, you covered all the necessary information just in 1 hours. Thank you for sharing. Which functions of matlab have you used in order to rotate frame with mouse ?
Thank you for this video - it makes sense when the angular rotations about each axis are applied sequentially. However, a MEMS gyro doesn't tell you "here are the 3, sequential euler angles since the last control loop iteration that you can put into a DCM." So, how could you derive a rotation matrix that takes you from the orientation at the previous timestep to the one in the current timestep (which can then be chained onto a running rotation matrix that starts at the initial body frame aka global frame)? I ask because a MEMS gyro returns instantaneous angular velocity rates about each of the 3 orthogonal body axes - right? Do we assume the angular displacements (angular-velocity-rates-reported-about-each-axis-of-the-gyro * dt) about the 3 body axes are infinitesimal and thus we can kinda-sorta call then the 3 euler angles as if they were applied sequentially to the orientation at the last timestep and form a rotation matrix that way?
I just saw ur video titled "Computing Euler Angles: The Euler Kinematical Equations and Poisson’s Kinematical Equations" which can give me the time rate of change of the 3 euler angles given the current euler angles and the angular rates in the body frame as reported by the gyro. Can the rate of change of these euler angles by integrated independently of the other euler angles to get the total euler angles?
AA 516: At the end of the video you described intrinsic versus extrinsic angles. Could you describe again why the psi, theta, phi angles we were working with are intrinsic? Also does this mean that the eigenvalue rotation angle is extrinsic?
-Sydney B
Great question. This is a little interesting as if you rotate around the primary axis this is only a single rotation, let's chat at office hours to make sure I understand the question.
Is there a sequence to rotate a frame from an NED to an ENU coordinate system? Or is it easier to just set them X' = Y, Y' = X, and Z' = -Z?
Hi Emin,
Thanks for reaching out, I'm glad you enjoyed the video. Unfortunately I'm unable to respond to questions on RUclips due to the sheer volume of inquiries that I receive. That being said, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum as I'll be able to answer questions there. Given your interest in the topic, I'd love to have you as a Patron as I'm able to talk/interact personally with Patrons. Thanks for watching!
-Chris
AA516 - Certainly a highly engaging lecture. Connecting the Euler angles to the aerospace industry, is their definition based on the definition of the stability axes for an aircraft? Perhaps more specifically, is there a non-casual connection between the fact that the term C_13 term used to calculate the pitch angle from the direction cosine matrix is only dependent on the pitch angle, just as the longitudinal aircraft dynamic modes are uncoupled from motion in the other stability axes?
Daniel, sort of. We'll talk about the stability axis later and we'll see they are different but related to Euler angles.
thank you, gooooood lecture
[AE 512] I've had some experience with DCMs before but did not know about the eigenvector/value correlation to the "single" axis of rotation.
How to calculate, using plane equation to find roll pitch yaw angle
AA516: Does the vector that we can rotate about to get to the final frame in one rotation have a name?
Sometimes this is called the principle axis of rotation
just a thing missing is, given a rotated coordinate frame, how to find those tait bryan(euler) angles wrt to world coordinate frame. It basically includes defining a Nodal line(xy plane intersecting Y'Z' plane) and then finding those rotating angles based on the sequence (XYZ, YZX, etc). I am quite clear about Classic euler angles(ZYZ, XYX, ZXZ, YZY, YXY, XZX) but finding for tait bryan angles tend to haunt me.
great video！
Thanks Dr Lum. At the end of video, you briefly mentioned intrinsic euler angle vs. extrinsic Euler angle, have you discuss extrinsic Euler angle in some of your video?
Superb explanation. That's awesome Chris. Thank you for sharing the playlist. If I create book on Spacecraft dynamics, I will make sure I will include your contribution. Could you please let us know if any Matlab simulations will accompany the flight dynamics playlist someday?
AA 516: To determine the positive right handed rotation do you point your thumb in the direction of the axis of rotation and the direction your fingers curl is a positive rotation?
Yes that is correct
AE512: I didn't realize until later in the class how exactly this would fold into our modeling of aircraft. It's clearly essential to mediate between different reference frames. It's unfortunate that DCM formulations across software packages are not always the same.
AE512: Why do we see rotations that only cover 2 axis in industry, like a xyx or a zyz ratoation
I'm a little lost here. What is the advantage of euler over dcm? Dcm being unitary means it can't go into a gimbal lock? If dcm doesn't have gimbal lock issue then what's the advantage of using quaternion?
Hello Sir as an enthusiast I would like to thanks you for those amazing videos !
I have a question concerning the calculus of the Rotation matrix from v to b at 31:45:
Since its a combinaison of linear transformations and because the product of several matrices are not commutative should not we were supposed to do : C1v.C21.Cb2 instead of C2b.C21.C1v ?
Greetings from France !
Hi,
Thanks for reaching out, I'm glad you enjoyed the video. Unfortunately I'm unable to respond to questions on RUclips due to the sheer volume of inquiries that I receive. That being said, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum or via the 'Thanks' button underneath the video as I'll be able to answer questions there. Given your interest in the topic, I'd love to have you as a Patron as I'm able to talk/interact personally with Patrons. Thanks for watching!
-Chris
Thanks a lot professor
Excellent videos I have ever seen, would you please post (upload) the MATLAB scripts you used in the lecture.
where can i find the code?
Can you direct me to where we calculate the euler principal rotation angle?
Thanks for reaching out, I'm glad you enjoyed the video. Unfortunately I'm unable to respond to questions on RUclips due to the sheer volume of inquiries that I receive. That being said, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum as I'll be able to answer questions there. I'd love to have you as a Patron as I'm able to talk/interact personally with Patrons. Thanks for watching!
Thank you Prof, Please, could you send me the Matlab code? Thank you
Thanks Dr Lum. Very clear explainations. But question is: at 1:07:41, you did not mention how we calculate the rotation angle mu. Could you explain it? Many thanks!
For the given initial angles, he probably captured the moment when three axis aligned. ( Longer pause time to capture obviously :) )
Jason-AE512: I'm still a little bit lost, but I will try my best. Thanks professor.
Let me know if you have any questions and I'm happy to help
@Christopher You should put all your videos into a book, like Steve Brunton. I will buy it.
Really nice video
I wondering if we can get the code of Matlab, thanks in advance Mr.
Hi Gustavo,
I have code and other benefits available on my Patreon page at www.patreon.com/christopherwlum. Given your interest in the topic, I'd love to have you as a Patron as I'm able to talk/interact personally with Patrons. Thanks for watching!
-Chris
AE512: interesting, these seem reminiscent of the rotation matrices that we derived as part of our homework assignment this week.
Good observation, some of that comes into play here
Thank you so much for this lecture and the other lectures you have on flight mechanics. You're a great person for sharing these for free.
I do have a question, though, which I hope you will find the time to answer. How come your sines in the rotation matrices for the DCM have opposite signs of what I find online when searching for rotation matrices? Is it due to positive being defined as down in the NED coordinate system?
Thanks for reaching out, I'm glad you enjoyed the video. Unfortunately I'm unable to respond to questions on RUclips due to the sheer volume of inquiries that I receive. That being said, I hope you'll consider supporting the channel via Patreon at www.patreon.com/christopherwlum as I'll be able to answer questions there. Given your interest in the topic, I'd love to have you as a Patron as I'm able to talk/interact personally with Patrons. Thanks for watching!
Helpful lecture, Professor, thank you! Though, in the section where we talked about expressing a DCM as a single rotation about some axis, I was surprised that we didn't compute the angle that you need to rotate about that axis. Is that coming in a later video? Or is deriving that angle left as an exercise for the reader?
Justin, you are absolutely correct, your spidey sense should have been tingling. We don't have the mechanisms in place to directly compute that right now. However, once we get to quaternions, we will directly compute this. I just posted the video a few hours ago if you're chomping at the bit and wanted to get to this early.
Please make some videos on How to calculate orientation error using euler angles in Robotic arm.
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!
thank you
The three people who disliked this video failed their courses.
48:50 you can't use Atan2 to find out what quadrant you're in the way you did here. Atan2(sin(x), cos(x)) is not the same as Atan2(-sin(x), -cos(x)).
Nevermind! Theta is always between -90° and +90° so cosine of theta will always be positive.
I think there is a mistake here. Here you mention ruclips.net/video/GJBc6z6p0KQ/видео.html the rotation sequence is in ZYX sequence but you have multiplied in wolfram in XYZ sequence. The answer I got multiplying ZYX sequence is different to yours.
Yes he did the multiplication as it's a "fixed" euler sequence rotation. I don't know if he's using a different convention but i guess he also made a mistake in Mathematica when defining the rotation matrices, the (-) sign in front of the sin(angle) in all matrices is not placed in the correct position.
Here for AA 516
AA 516: Celeste Yuan
tyyyyyyyyy
:)
Hi Logan,
Thanks for the kind words, I'm glad you enjoyed the video. If the find the these videos to be 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
THIS IS FOR MY REF NOTE ONLY AND NOT A COMMENT FOR THE ABOVE VIDEO :parallelly go through the concepts of Inverse, singularity, eigen vector and eigen values concepts of a matrix, what are they how they are usefull in real world scenarios
EIGEN value concepts are stored in PID playlist, inverse matrx concpt as pdf in PID in Ddrive folder