Using a Homogeneous Transformation Matrix to Combine Rotation and Translation

[@ChristopherLum]

In case it is helpful, here are all the Flight Mechanics videos in a single playlist kzbin.info/aero/PLxdnSsBqCrrEx3A6W94sQGClk6Q4YCg-h. Please let me know what you think in the comments. Thanks for watching!

[@Gholdoian]

AE512: Interesting to see the breakdown of translation and rotation in a single matrix, I've usually performed the operations separately but nice to see you can easily perform successive transformations.

[@anthonysu1712]

- no questions; just wanted to say I'm so thankful you derived the 2D case and extended it to 3D instead of directly deriving the 3D case :)

[@manitaregmi6932]

AA 516 - You made it real easy to understand by Developing the homogeneous transformation in 2D and then doing the Extension to 3D. Great explanation on material as always!

[@aimeepak717]

AE512: Great lecture. I was able to do the homework following this step by step.

[@milesrobertroane955]

AA516: Good video! Augmenting the matrix makes total sense after seeing the composite transformations

[@zaneyosif]

AE512: Great lecture, especially the 3D example. Really helped put things into perspective with different objects with different reference frames (translationally/rotationally) and how to relate them all together via transformation matrices.

[@sianhoan833]

AA 516: Another great lecture! Glad to see that our aircraft can start flying around instead of only rotating in one place and excited for what comes next.

[@ChristopherLum]

Yes, and it only gets better, we'll start flying in earnest soon!

[@charlesharmon9065]

AE 512 - Great video. Sketches of frames/coordinate systems and inclusion of numerical examples were extremely helpful in making the material 'stick.'

[@boeing797screamliner]

AA516 - I really like how you incorporate real-world examples into these lectures. It really helps me grasp the material

[@ChristopherLum]

Gary, thanks for the feedback. And thanks for actively participating in the discussion, you are rocking the participation points 😀!

[@jackpascho2561]

AA 516: One of my favorite lectures so far. Doing a multi-stage chain transformation like this has always been something I thought about particularly in regard to air combat with all of the sensor and radar data that gets distributed between aircraft, and has been something I wish I could do on a variety of projects. Very excited that I understand the fundamentals behind this process!

[@ChristopherLum]

Jack, great, I'm glad it was simulating. This comes into play a lot when working with robotics.

[@SayedTorak]

AE 512: great video prof. Lum

[@AlexandraSurprise]

AA516: Allie S, this is really neat! I haven't thought about this type of transformation being possible and I am excited to learn even more uses for it.

[@edwardmau5877]

AE 512 Very impressive the measurements on the board matched the mathematical calculations, nice steady drawing skills.

[@reesetaylor3506]

AE 512: Nice lecture, the physical examples with measurments definitely help with my understanding, thanks!

[@koshiroyamaguchi9613]

AA516 - So far I really appreciate Prof. Lum's lecture styles with various real world/computaitonal examples. Looking forward to learning more lectures!

[@steahrcase8379]

AE512: I absolutely love your real world examples and enthusiasm

[@ChristopherLum]

Great, I'm glad it is entertaining, hopefully it applies to HW as well :)

[@Kumky605]

AA516: Forgot to comment on this when I watched it the first time but great video! These types of transformations were super interesting and I was excited to go more indepth

[@rowellcastro2683]

AA516: I was going to ask this in Office Hours but I believe I figured it out. For anyone that might be wondering in 33:10 why the rotation matrix is different from what we learned in the past rotation lecture (Flight Mech 02), it is because the rotation from frame b to frame 1 is left handed! That is why it is transposed/inversed.

[@rowellcastro2683]

That also might be why Dr. Lum wrote it incorrectly in the beginning.

[@PatrickGalvin519]

AA 516: Great video, very informative lecture.

[@shavykashyap]

AA 516 - Great Lecture

[@petermay6090]

AA516: Very useful!

[@timproby7624]

[AE 512] 42:40 You are drawing the vector from car to origin because you have to keep the vector addition "tip to tail". Basically, however many vectors you add, you have to make sure they are tip to tail to keep the equations valid?

[@ChristopherLum]

Yep, that is a good way to think of it. Maybe add this to your notation cheat-sheet :)

[@yaffetbedru6612]

AA516: Even though the reference frames started adding up, it still was easy to follow and see how powerful the homogeneous transformation matrix is.

[@NikolajKuntner]

Good topic, I'm also very much into spatial transformations atm.

[@ChristopherLum]

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!

[@steahrcase8379]

AE512: at 46:30, why do you measure theta from x2 to xr all the way around? In the previous case, x1 to xr, it was the smaller angle. Why is it different here (why do we not measure from xr to x2)?

[@ryancusack5909]

Are you sure you wrote that rotation matrix wrong at 16:38? It looks like the same type of rotation in your video Expressing Vectors in Different Frames Using Rotation Matrices at 19:16 in that video. It is a right hand angle, rotating from the reference frame to a 1 frame. If the angle is positive, it looks like what you wrote is correct, or matches the other video.

[@chayweaver.2995]

[AE 512]: @16:41 you say that C_1/b is a 2x2 rotation matrix with the angle being psi, where psi is defined as FROM the 1 axes TO the b axes. However, we want to rotate FROM the b axes TO the 1 axes, so isn't the rotation matrix as you've defined it here incorrect? Shouldn't C_1/b actually be the transpose of what's on the board?

[@habesharockets7460]

AA516: Hi professor! The video is great as always. At 19:00 min, we used r(t/b) frame 1 instead of frame r (reference frame). Is that because r(t/b) origin aligns with frame 1? (Daniel Teshome) Thank you!

[@ChristopherLum]

Hi Daniel, yes, sometimes we need to translate (but not rotate) the frames to make the downstream rotations work.

[@divinadileep9464]

tnku so much sir

[@thomasstiglich3484]

Really enjoying your videos, hope to get through each one. I did find one thing confusing though, as may others. In a previous video, "Expressing Vectors in Different Frames Using Rotation Matrices", when a vector is known in Fr and desired in F1, the angle Theta was calculated in the RHS from Fr to F1, resulting in [cos(theta) sin(theta); -sin(theta) cos(theta)]. In this video, the rotation from Fb to F1 was [cos(alpha) -sin(alpha); sin(alpha) cos(alpha)] as the result of the angle being defined as the RHS angle from F1 to Fb. If the angle was from Fb to F1 as in the prior video, the prior rotation matrix could be used. Wolfram Mathworld uses [c(psi) s(psi); -s(psi) c(psi)] where psi is the RHS angle from the reference frame to the rotated frame, as in your first video. Do you have a preference as to how the angle is defined?

[@AliG.G]

As far as I understand is both r same. Here is a short video demonstration LHS and RHS referencing kzbin.info/www/bejne/gZaWmWqEh7GcaK8. If I remember right in the other video Z was facing down i.e. NED frame of reference and in this video, Z comes out of the screen towards you in ENU frame.

[@nathanlipshutz3140]

AE512: During a rigid body dynamics calculation, I have previously done something where I used a combination of a vector addition and rotation matrices to accomplish something similar. I think this could have expedited my workflow.

[@BryanLiberman]

AE512: Thanks Dr. Lum! Off the top of your head, do you have any resources for how to concatenate matrices in mathematica? I think it might be easier to build up the zeta matrix one piece at a time, and put it all together!

[@ChristopherLum]

Yes, try the Join function. I have a video on the channel showing matrices in Mathematica if you would like more details

[@tonymcdaniel6878]

AA 516: On the last statement made in the video, when you went to the 3D rotation and translation matrix, what if the target reference frame was at two different rotation ie a different psi and theta

[@davidtelgen8114]

AE512: now I won't be confused when I hear a GNC engineer talk about 'pose'!

[@prathammaheshwari4489]

sir can you upload dynamics also like forces and moments

[@kaikim8402]

Question : @16:14 Please correct me if i am wrong. The rotational matrix C1/b(psi) from Fb to F1, shouldn't the psi in the -ve direction since the Z^r is coming out of the plane?

[@ChristopherLum]

Hi Kai, Great to hear from you again. I've started answering questions directly through my Patreon page at www.patreon.com/christopherwlum. Please free to submit your question there and we can discuss. Thanks for watching and for supporting the channel!

[@sethwhittington28]

AE512: these transformations keep building on eachother and getting more complicated! I think you’re doing a great job breaking it down piecewise though.

[@ChristopherLum]

These come in handy, especially in robotics applications.

[@sethwhittington28]

@@ChristopherLum @28:10 you measure a 40 degree angle to define the rotation matrix that takes you from the b frame to the 1 frame C1/b(psi). In the diagram on the right side of the board, it looks like this 40 degree angle is actually the rotation that would take you from the 1 frame to the b frame, but in your homogenous matrix at the bottom of the screen you define C1/b which i think is the rotation that takes you from the b frame to the 1 frame. Should you have used NEGATIVE 40 degrees here to go from b to 1...?

[@bsgove]

AE512: It's amazing how someone can keep all of these building blocks in mind while trying to create a dynamic simulation where this is only one small part.

[@DonutDude206]

AA516 are there other applications of this transformation where we could expand to dimensions greater than 3D?

[@ChristopherLum]

For most of our applications 3D is probably the largest dimension we can deal with.

[@ToanLuu158]

AE512: I wondered what is the maximum frames that someone had been developing in the real world using this transformation matrix method.

[@princekeoki4603]

AA516 : Cool how successive transformations stack on top of each other.

[@WalkingDeaDJ]

Jason-AE512: This is helpful to understand the homework question 1

[@RNeilen216]

???? why are there so many letters, and what does this have to do with physical posistions

[@ojasvikamboj6083]

A A 516: Ojasvi Kamboj

[@h3ngy1ng68]

"I dont wanna get the s i g n of s i n wrong" hahaha (thr goes my 1st ever comment in yt) 33:59

[@AndersonYuan-j7g]

AA 516: Celeste Yuan

[@Colin_Baxter_UW]

AA 516, no questions

[@edengebretsadik7793]

AE512

[@PoHuang-n4e]

AA516