Wednesday, 31 January 2018

homework and exercises - Angular Velocity expressed via Euler Angles


On the top of the fourth page from here, the author trivially derives the components of angular velocity, expressed via Euler angles of the system. I fail to understand how the components of angular velocity were derived. May you please enlighten me!



Answer



I assume you know about rotation matrices, and so for a sequence rotations about Z-X-Z with angles ϕ, θ and ψ repsectively you have


ω=˙ϕˆz+T1(˙θˆx+T2(˙ψˆz))



The logic here is apply a local spin of ˙ϕ, ˙θ and ˙ψ on the local axes in the sequence.



  1. Apply spin ˙ϕ about local Z and then rotate by T1

  2. Apply spin ˙θ about local X (rotated by T1) and then rotate by T2

  3. Apply spin ˙ψ about local Z (rotated by T1T2).


Update


There is a way to formally derive the above using the identity ˙T=ω×T but it is rather involved for 3 degrees of freedom.


For two degrees of freedom it goes like this. With a rotation matrix T=T1T2 (defined as above) the time derivative is


dTdt=˙T1T2+T1˙T2=((˙ψˆz)×T1)T2+T1((˙θˆx)×T2)=(˙ψˆz)×(T1T2)+(T1(˙θˆx))×(T1T2)=(˙ψˆz+T1(˙θˆx))×(T1T2)=(˙ψˆz+T1(˙θˆx))×T=ω×Tω=˙ψˆz+T1(˙θˆx)



using the distributed property T(a×b)=(Ta)×(Tb).


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