4 - Direction Cosine Matrices

[@brunobucciarati2711]

This is very helpful! Thank you for making this video.

[@flengineering4020]

Hey Bruno! You’re very welcome. Glad you found the video useful.

[@TheLostPig]

Absolutely amazing explanation. Thank you. Also, it'd be nice if we can have a copy of the slides you presented

[@flengineering4020]

Thanks. Glad it helped. Part of the reason I haven't posted the slides is because I feel there's value in you yourself physically writing it down. I find it gave me another level of understanding to slow down and actually write out the notes. When I was in college, I would watch all of my professor's lectures twice. Once to just watch and absorb (no note taking), and then once to take notes and write down important points the professor would say. But I get that everyone's different so maybe one day I'll post pdfs but for now I'm going to force some good 'ol fashioned note taking.

[@astrophage381]

Great videos. Subscribed to your channel. You should continue making videos on similar topics.

[@flengineering4020]

Thank you! And I will. I've just been so busy this year and haven't made one in a while.

[@salmantechnologies282]

I think every one who one to know flight dynamics from the scratch should watch this series absolutely amazing love it Buddy thanks do you do some coding stuff or embedded work

[@flengineering4020]

Thanks Salman. That's really nice of you to say. And I do a lot of coding in Python which I use in some of my videos.

[@salmantechnologies282]

@@flengineering4020 did you done some working on how to apply direction cosine matrix with accelerometer i did not find any video who apply direction Cosine Matrices on accelerometer such as MPU6050 , Adafruit ADXL345 i experimented on these sensor but i did not find a way to solve i seen many videos they apply quaternion but not direction cosine matrices thank you

[@flengineering4020]

@@salmantechnologies282 I have not done any videos working with any sensors. Just pure theory so far.

[@nemonobady485]

Very cool explanation! But I'm still wondering why in other sources on the internet the rotation matrices are transposed to the matrices used in the video.

[@flengineering4020]

Thanks Nemo. To answer your question, it just has to deal with the direction you're going (which basis you want to represent as the other basis). For example, skip to 11:09 in the video. You can see that we are representing the green basis in terms of the red basis. e1 = E1 + 0 E2 + 0 E3 e2 = 0 E1 + cos E2 + sin E3 e3 = 0 E1 - sin E2 + cos E3 Now, we could have just as easily gone the other way where we represent the red basis in terms of the green basis. You can work this out yourself using the exact same approach I covered in this video except instead of locating the green vectors with respect to the red vectors, you would be locating the red vectors with respect to the green vectors. This is equivalent to taking a transpose of the DCM and switching your bases so that they would look like: [Red] = [DCM]T * [Green] E1 = e1 + 0 e2 + 0 e3 E2 = cos e2 - sin e3 E3 = sin e2 + cos e3 Does that make sense?

[@nemonobady485]

​​@@flengineering4020 Yes! Thanks a lot, now I understand. The DCMs in the video rotate only coordinate system, but not the whole plain. Whereas the DCMs on the wikipedia (Rotation matrix) rotate the whole plain, and not only the coordinate system. As I explained to myself. Suppose, for instance, there is a point A(sqrt(2) / 2, sqrt(2) / 2, 0). As explained in the video: If we rotate around the z-axis with the angle of θ, we should use following formulas: e1 = E1 cos(θ) + E2 sin(θ) + 0 E2 e2 = - E1 sin(θ) + E2 cos(θ) + 0 E2 e3 = 0 E1 + 0 E2 + E3 ...and when θ = 45, we will get: e1 = 1 e2 = 0 e3 = 0 These are the coordinates of the same point A, but the terms of the new coordinate system, which was formed after the primarily coordinate system was rotated to an angle of 45 degrees. If we use the same conditions: A(sqrt(2) / 2, sqrt(2) / 2, 0) , θ = 45 and use the formulas from wikipedia: e1 = E1 cos(θ) - E2 sin(θ) + 0 E2 e2 = E1 sin(θ) + E2 cos(θ) + 0 E2 e3 = 0 E1 + 0 E2 + E3 ...we will get: e1 = 0 e2 = 1 e3 = 0 These are the coordinates of point A' in terms of fixed coordinate system. The point A' obtained after the whole plain with the point A itself was rotated by the 45 degrees. (Or just a vector A(sqrt(2) / 2, sqrt(2) / 2, 0) was rotated by the 45 degrees and we got a new vector A'(0, 1, 0)). So, this is all from which perspective we are looking from: whether the coordinate system rotates, and we determine coordinates of one point in term of rotating coordinate system, or we rotate the entire coordinate plain with the point itself and measure the coordinates of the new point obtained after the rotation in terms of the fixed coordinate system.

[@flengineering4020]

@@nemonobady485 Yup! That's a great explanation. I think you got it!

[@sohamtripathy3458]

Thank you so much for making this. This helped me alot

[@flengineering4020]

Hey! You're welcome. Glad it was helpful.

[@cheseapeakebaykayakfisher1385]

Well done. Thank you.

[@flengineering4020]

Hey! You're welcome!

[@sluffnut7507]

WOOOOOOOO!

[@flengineering4020]

Woooooo indeed

[@ThePodPod]

nice

[@flengineering4020]

nice

[@Spaghetti-man256]

Woooooooooh

[@flengineering4020]

WOOOOOOO!!!!

[@michaelstevenson2517]

Awe4some Vid, thanks. Would be cool if you posted a link to the slides.

[@flengineering4020]

Thanks Michael. I'm glad you liked the video. With regards to posting the slides, I'm going to copy my reply I gave to another person asking that I post the slides: "Part of the reason I haven't posted the slides is because I feel there's value in you yourself physically writing it down. I find it gave me another level of understanding to slow down and actually write out the notes. When I was in college, I would watch all of my professor's lectures twice. Once to just watch and absorb (no note taking), and then once to take notes and write down important points the professor would say. But I get that everyone's different so maybe one day I'll post pdfs but for now I'm going to force some good 'ol fashioned note taking."

[@mohammedmomaya3963]

Could you please elaborate on how the rotation about the different fundamentals are done. For example when going from fundamental 3 rotation to a fundamental 2 rotation the E2 and e2 unit vectors were now pointing out of the page as opposed to before when it was the E3 and e3 unit vectors - Is this something you have to follow when taking the rotation about different fundamentals such that the reference frames has to be rotated in such a way where both would be point out of the page?

[@flengineering4020]

You have to use the appropriate rotation matrix which depends on which axis you're rotating about. If you're rotating about the 3-axis, you use the fundamental 3 rotation DCM. If you rotate about the 2-axis, you use the fundamental 2 rotation DCM. The derivations for each of the fundamental rotation DCMs are different because you're rotating about different axes. I'm not sure if that answers you're question. Maybe try asking it a different way or point to a specific part of the video and I can help you out a little better?

[@mohammedmomaya3963]

@@flengineering4020 in simpler terms i wss asking how the diagram for different fundamental rotation were draw. So for example at 1:30 the diagram of the axis is for the rotation about E3 however at 7:31 the diagram has now been changed for the rotation about the E2 axis. I was jus asking how to alter the diagram when you rotste it about a different axis as its quite hard to visualise it.

[@flengineering4020]

@@mohammedmomaya3963 The axis I have pointing out of the page is the axis the basis is rotating about. So at 1:30, in order to align the red basis to the green basis, we have to rotate the red basis about its 3-axis by theta. Same idea at 7:31, in order to align the red basis to the green basis, we have to rotate the red basis about its 2-axis by theta. Does that make sense?