Euler Angles for Aerospace | Yaw, Pitch, Roll

[@jgkostas]

I can think of two important reasons for preferring to work with angular velocity in the body frame as opposed to the inertial frame. 1.) In aerospace work, the gyro(and accelerometer) instruments are fixed on the vehicle and aligned with the principal(usually) axes of the vehicle. 2.) For simulation, working in the body frame means the inertia properties of the body are constant(or nearly so; fuel burn, etc). The inertia properties, expressed in the inertial frame, are functions of attitude. Also, thank you Professor Ross for making these excellent videos and lecture notes freely available on KZbin.

[@ProfessorRoss]

You're welcome. And yes, exactly! #1 and #2 are indeed the reasons. At first, students wonder why so many variables are shown in the body frame, because they come from an earlier course where everything was transformed into the inertial frame first. But as we go through the material in the course, they see the reasons why.

[@jgkostas]

Professor Ross -Thank you for the excellent lecture. Playing around with the D-C-M kinematic diff-Eqns in Octave(MATLAB clone). Used a simple Euler-integration scheme and a few of Schaub's code files. Got the same results as with the Euler-Angle B-matrix method. I did not include the plotting code. Cheers. % From Problem 3.12 of Schaub/Junkins % Using Direction-Cosine-Matrix e0 = [30;40;80]*pi/180; C = Euler3212C(e0); % From Schaub download site dt = 0.02; % Run Updates at 50Hz t = 0:dt:60; e = zeros(3,length(t)); % pre-allocate Euler 3-2-1 Angles e(:,1) = e0; # Body rates in rad/s w1 = sin(0.1*t)*pi/9; w2 = 0.01*ones(length(t))*pi/9; w3 = cos(0.1*t)*pi/9; % Update [C] : d[C]/dt = -omega_tilde*[C] for i = 1:(length(t)-1) C = C - dt*[0, -w3(i), w2(i);w3(i), 0, -w1(i);-w2(i), w1(i), 0] * C; e(:,i) = C2Euler321(C); % From Schaub download site end

[@ProfessorRoss]

Thank you for watching and for the code. I may want to include this as an example in future courses to illustrate a simple and effective numerical method. In the limit of small step-size dt, this numerical integrator result should be the same as other approaches. In fact, since it gets around the use of Euler angles, and their singularities, the d[C]/dt = - omega_tilde*[C] approach is more robust. Technically, it involves calculating 9 linear ODEs instead of 3 nonlinear ODEs (the B-matrix method), so for historical reasons related to limited computing power, it hasn't been used as much, but it's more straightforward.

[@bradbirney5964]

This video was exactly what I wanted! Thank you!

[@ProfessorRoss]

Glad it was helpful!

[@AlifuddinAkbar]

Thank You Professor Shane Ross for this video. Currently I'm working on Research in Unmanned Surface Vehicle (USV) and this video help a lot for me to have more depth understanding Kinematics Equation of USV. I Have a question related to the Convention. will the Yaw-Pitch-Roll (3-2-1 Convention) sequence yield the same results or conditions if the order of "Pure Rotation" is changed to Roll-Pitch-Yaw or Pitch-Roll-Yaw?

[@brendawilliams8062]

Thankyou