Derivation of Rodrigues’ Rotation Formula

In this video we explain and derive Rodrigues’ Rotation Formula. This functions describes how to rotate an arbitrary vector about another arbitrary axis of rotation. This has applications to orbital mechanics, computer graphics, rigid body kinematics, and dynamic system simulation. This is also known as Goldstein’s Rotation of a Vector Formula.
Topics and time stamps:
0:07 - History of the Formula
1:25 - Physical demonstration/example of setup
7:59 - Derivation of Rodrigues’ Rotation Formula
38:35 - Matlab simulation/demonstration
42:23 - Conclusions
Lecture notes and code can be downloaded from github.com/clu...
For more information see:
-Stevens, B. L. and Lewis, F. L., “Aircraft Control and Simulation, 2nd Edition”, John Wiley and Sons, 2003, pg. 7-8.
-Goldstein, Herbert, “Classical Mechanics 2nd Edition”, Addison-Wesley Longman, Inc., 1980.
- Olinde Rodrigues, "Des lois géometriques qui regissent les déplacements d' un systéme solide dans l' espace, et de la variation des coordonnées provenant de ces déplacement considérées indépendant des causes qui peuvent les produire", J. Math. Pures Appl. 5 (1840), 380-440.
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You can support this channel via Patreon at / christopherwlum or by clicking on the ‘Thanks’ button underneath the video. Thank you for your help!

[@ChristopherLum]

In case it is helpful, here are all the Flight Mechanics videos in a single playlist kzbin.info/aero/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!

[@edwardmau5877]

AE 512 The 3D example with the pvc pipes and labels was a useful visual to setting the stage for the formulation.

[@Colin_Baxter_UW]

AA516: The real-life model was helpful. Looking forward to seeing how this formula is used.

[@stein372]

Lum老师会说中文吗？ Dr.lum is the teacher I dream to have in the math and physical class of the highschool, detailing the steps and rediscovering the inherent beauty of mathematics. Thank you! It is really helpful.

[@ChristopherLum]

I'm glad it was helpful. Thanks for watching!

[@rockstarali99]

May you be blessed professor! The passion, the effort, the diligence to explain everything to the dot. I have so much respect and appreciation for you!

[@jackpascho2561]

AA516: I liked how you took what seemed to be a complicated derivation and spent time in the beginning breaking down what we we're really looking at and what we needed to solve for. This made it wayyy easier to follow and understand.

[@ChristopherLum]

Jack, great, I'm glad it was helpful. Let me know how the other videos go.

[@shavykashyap]

AA 516 Very easy to understand

[@JB-hr6yp]

Incredible explanation, finally made me actually understand it instead of just trying to memorize

[@bsgove]

AE512: It's crazy how many different building blocks underly flight mechanics, this being one of them.

[@jlol4951]

Just messing about. First day using 'Octave' (don't have Matlab), so apologies for the code. Super lectures, thank you very much. If one tries.... %Start of Script clear clc close all %Define unit vector n = [1;0;1]; n = (1/norm(n))*n; %n needs to be unit vector %define vector U u = [2;2;-1]; %Does not need to be unit vector %define range of angles of rotation mu_range =linspace(0,2*pi,50); %rotate u around vector a figure % Moved code JL clf legend ('u','n','v') xlabel('x') ylabel('y') zlabel('z') grid on title('Rotation of u around n') axis equal view([32 15]) axis([-norm(n)-2 norm(n)+5 -norm(n) norm(n)+5 -norm(n)-5 norm(n)+5]) %Slightly changed JL % End Moved code JL %Var to signal first time plotted JL VOldSet=0; for i=1:length(mu_range) %clf; %Added JL if (VOldSet==1) %If has been plotted then cover with white (with outline) plot3([0 VOld(1)],[0 VOld(2)],[0 VOld(3)], 'w-','Linewidth', 2); endif hold on %Get the current rotation angle mu=mu_range(i); %Apply Rodrigues rotation of a vector, formula (Equation 1.2-6 in %Stevens and Lewis) v = (1 - cos(mu))*dot(n,u)*n + cos(mu)*u - sin(mu)*cross(n,u); %Draw u plot3([0 u(1)],[0 u(2)],[0 u(3)], 'r-','Linewidth', 2); %Draw n plot3([0 n(1)],[0 n(2)],[0 n(3)], 'g-','Linewidth', 2); %Draw v plot3([0 v(1)],[0 v(2)],[0 v(3)], 'b-','Linewidth', 2); %Keep old V plot - Set flag for above.... VOld=v; VOldSet=1; ## legend ('u','n','v') ## ## xlabel('x') ## ylabel('y') ## zlabel('z') ## ## grid on ## title('Rotation of u around n') ## axis equal ## ## view([32 15]) ## axis([-norm(n) norm(n) -norm(n) norm(n) -norm(n) norm(n)]) drawnow pause(0.1); %Pause for animation end %End of Script

[@ojasvikamboj6083]

Ojasvi Kamboj - AA 516

[@charlesharmon9065]

Great derivation - It's fun to see things fall into place!

[@Gholdoian]

AE512: Very interesting concept, seems similar to the foundations of a quaternion, wonder how much influence these concepts had on each other

[@PatrickGalvin519]

AA 516: I enjoyed following the derivation and thought the props at the beginning did a great job illustrating what was happening in 3D.

[@milesrobertroane955]

AA516 Good one, I always love imagining rotating vectors. Ever since calc 3 doing volume integrals of rotated lines - the mental image is so cool.

[@yaffetbedru6612]

AA516: Shows many applications in orbits with multiple reference frames

[@tonymcdaniel6878]

AA516: Came back for a rewatch, I appreciate the coding in matlab in the end, the graphic really helped

[@WalkingDeaDJ]

Jason-AE512: The matlab part help me understand the material better.

[@SayedTorak]

AE512: Learned something new today. Thanks!

[@AndersonYuan-j7g]

AA 516: Celeste Yuan (Late due to name discrepancies discussed)

[@vishnunair6416]

Thank you so much for taking time out of your life so that we get a better understanding of the field we are pursuing. Utmost respect.

[@ChristopherLum]

I'm glad it was helpful. If you are interested, this is part of my series of videos on Flight Mechanics (kzbin.info/aero/PLxdnSsBqCrrEx3A6W94sQGClk6Q4YCg-h), feel free to check these out. Thanks for watching!

[@umedina98]

Amazing! Wow you truly show your passion for teaching. Thanks so much!

[@ChristopherLum]

My pleasure!

[@jasonhe6947]

omg, this is such a complete explanation for angle axis rotation. thank you so much

[@ChristopherLum]

I'm glad it was helpful, thanks for watching!

[@ashhadtoseef1834]

pausing the video just to appreciate how well you have explained everything. you sir saved a life.

[@ChristopherLum]

I'm glad it was helpful thanks for watching!

[@sethwhittington28]

AE512: this one was a bit harder to follow than the prior two videos as I got a little lost in all the linear algebra notation. The Matlab show and tell sections are very helpful in helping me understand what’s going on.

[@DharambirPoddar]

you are the great prof of this century, huge thanks sir

[@sunnysarkar7643]

AE 512: This was a great and easy to follow derivation. I am a little confused on how/when this will be applied in this course. An example of that would be nice throughout the lecture.

[@ChristopherLum]

Yes, this is coming up in a future discussion

[@aimeepak717]

AE512: I found the Rodrigues derivation easy to follow. I noticed that the lecture notes in the github was only the cover page when I downloaded it. Maybe there was an error in the upload?

[@ChristopherLum]

Hi Aimee, good observation. This set of notes is actually only supposed to be 1 page and our just refers you to the hw solution. We will release this once hw1 is due

[@davidtelgen8114]

AE 512 Cool formula, very useful!

[@zachsmith1973]

AA 516: This was an awesome video, I really appreciate the visuals. I'm curious, what is so special about making it a left handed rotation (since almost everything we do is right handed)? I assumed that would be arbitrary because switching it just adds a negative, but wondering if there was something more to it.

[@ChristopherLum]

Zach, good observation. This is purely historical. Rodrigues derived it using a left handed rotation and we keep this to be consistent with his original derivation. As you mention, most angles a right handed nowadays to be consistent.

[@yaffetbedru6612]

AA 516: Great to see detailed steps on the derivation

[@burningbush2009]

AE512: Thanks for the detailed derivation! One question - why do Stevens and Lewis (and Goldstein) use the left hand rule? Comparing to other sources on the internet, Rodrigues appears to use the right hand rule, which causes a sign switch on the sine term (en.wikipedia.org/wiki/Rodrigues%27_rotation_formula)

[@ChristopherLum]

Ryan, great point. I'm not sure why they derived it using a left handed rotation, it is probably historical but as you mention, it just requires a sign change.

[@marshawnlynch6310]

I'm just here so I won't get fined.

[@rowellcastro2683]

AA516: Great lecture video! You explained the Goldstein rotation formula more in depth than what is mentioned in the book and I really apreciate your visualizations with the pipes and MatLab simulation. Thanks Professor!

[@ChristopherLum]

Glad you enjoyed it!

[@philipcasey8361]

AE512: Thanks for the great visuals/diagram! Is this formula related to quaternions?

[@ChristopherLum]

Ah, you're a few weeks ahead of the game, we'll hit quaternions in a few weeks.

[@princekeoki4603]

AA516 : Cool Matlab Script!

[@Po-ChihHuang]

AA516 very clear explanation

[@binyulong5995]

AA 516: Are we going to use this left-hand rotation version for all of our analyses in the future instead of the right-hand rotation?

[@ChristopherLum]

We'll use this form of the Rodrigues' rotation formula with its left handed rotation but for all other rotations we will use right handed rotations.

[@vaughanrocketry863]

Chris, your videos are great, any suggestions with respects to books that follow your material?

[@ChristopherLum]

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. Thanks for watching! -Chris

[@burhanmuhyiddin4372]

Thank you very much for your effort. Both the derivation of the formula and the MATLAB demonstration were perfect :)

[@ChristopherLum]

I'm glad it was helpful. I have other similar videos at kzbin.info/aero/PLxdnSsBqCrrEx3A6W94sQGClk6Q4YCg-h and I'm just about to post a new video that uses this result to look at Euler angles. Please feel free to check them out. Thanks for watching!

[@chayweaver.2995]

AE512: You mention at 28:10 that we have to add d as a unit vector to "give this thing a direction". The cross product yields a vector, not a scalar, so it already has a direction, right? Does multiplying by d just scale the resultant cross product vector such that it results in a magnitude of 1? If that's the case, then could another mathematically identical way of doing this be (instead of multiplying by d) to divide |u| |n| sin(phi) by the magnitude of |u| |n| sin(phi)?

[@ChristopherLum]

Correct, the cross product needs to return a vector, not just a scalar which is why I'm adding the direction. The magnitude of a general cross product does not need to be 1 but perhaps we can chat about this more at office hours. Good question!

[@chayweaver.2995]

@@ChristopherLum But a cross product will never return a scalar, right? I guess what's really throwing me off is that the final equation for d is U cross N over |u||n|sin(phi) which to me seems like it should just simplify to equal 1 because the numerator and denominator are the same thing written differently...

[@petermay6090]

AA 516: Clear lecture, was able to follow each step easily!

[@ChristopherLum]

Great, hopefully this helps with the HW. Keep me posted!

[@zaneyosif]

AE512: Super helpful to go through the derivation of the Rodrigues Rotation Formula, especially having the two different views of the circle. And how we were able to determine the desired vector based on a piece meal formulation of more or less projections in the frame we wanted.

[@reesetaylor3506]

AE 512: The color coding and drawings in multiple perspectives makes this long derivation much easier to follow. Great video!

[@wqh9720]

this files associated with this lecture(pdf) what can i be downloaded ?

[@ChristopherLum]

Hi, 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 and provide code/downloads/notes on Patreon. Thanks for watching! -Chris

[@sarakhan3851]

Amazing! proficient explanation!

[@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 in the comments. Thanks for watching!

[@JeremyLucier]

Great video! Super clear and informative -- definitely helped me out. Thanks!

[@ChristopherLum]

Glad it helped, thanks for watching!

[@Kumky605]

AA516: Forgot to comment on this when I watched it the first time but really great to see all the steps broken down into easy to digest parts

[@edengebretsadik7793]

AE512

[@edengebretsadik7793]

AE512

[@beritsyltebo3940]

AA516: Great, thoroughly description of such an important tool

[@sahnouna.fettah3240]

great video sir why can't express vector NW as (cos Mu ) and vector WV as( sin Mu) ?

[@AlexandraSurprise]

AA 516

[@morganwalker940]

I liked the derivation on the board. I also liked the part where you showed my eyes what was happening with matlab. But my favorite part was the transition at 42:22

[@jamesdavey3101]

Might have been interesting to see what happens in MATLAB when n is not a unit vector.

[@Bommis3]

Very nicely explained, helped a lot. Thank you!

[@kenrickchan1165]

Handy new formula with many applications!

[@zacharywilliams2436]

In depth derivation, easy to follow along with

[@ThomasNiemisto]

AE512: Words can't describe how helpful the props are in explaining these concepts, thanks for going the extra mile 🙂

[@ChristopherLum]

Glad you like them!

[@joshcole667]

This is amazing!!! Thank you so much for putting this together. As someone who does engineering as a hobby, resources like this are invaluable. So clear and easy to understand.

[@ChristopherLum]

Hi Josh, 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

[@helenkuni9039]

Great walkthrough of the derivation, thanks.

[@robwhitlock5030]

Excellent. Thank you!

[@imrak34]

Thanks a lot :)

[@ChristopherLum]

Hi, 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

[@jamesferrese2350]

Carter loves winglets

[@marilynjasmer8365]

Nice and clear derivation

[@sidraman6950]

Interesting derivation

[@Pluto4Planet2024]

Marker revealed @7:05

[@Hannah08LV]

thanks for the lecture!

[@mikeandrewputerakurnia5572]

interesting

[@jacobcarroll3933]

Great derivation!

[@triskitman124]

very easy to follow the derivation!