I have a question regarding the transformation matrix

- How can I derivative the rotational matrix from Nacelle / Yaw Coordinate System (d1,d2,d3) to Rotor-Furl Coordinate System (rf1,rf2,rf3) ? Where is the origin location of Rotor-Furl Coordinate System (rf1,rf2,rf3)?

Please provide me with the required papers to understand this part

Thanks

Dear Mohamed,

I can’t seem to find the derivation of the transformation matrix between the nacelle and rotor-furl coordinate system published in any report. However, the transformation can be derived by multiplying several simpler 3x3 matrices together i.e.

{rf} = [TransMat] * {d}

with

[TransMat] = [RFrlSkew]^T * [RFrlTilt]^T * [q_RFrl] * [RFrlTilt] * [RFrlSkew]

where,

{} represents a 3x1 vector

[] represents a 3x3 matrix

^T represents a matrix transpose

[RFrlSkew] represents the transformation matrix associated with the single rotation RFrlSkew

[RFrlTilt] represents the transformation matrix associated with the single rotation RFirlTilt

[q_RFrl] represents the transformation matrix associated with the single rotation of RFrlDOF

The origin of the rotor-furl coordinate system is not defined nor used in FAST.

I hope that helps.

Best regards,

Thank you for your explanation. However, I have a question

Why it is necessary to use Similarity Transformations and express the [q_RFrl] rotational matrix in Yaw frame?

It seems that the rotations [RFrlSkew] and [RFrlTilt] are made with respect to current frame concept (not fixed frame concept).

Dear Mohamed,

I’m not sure I understand your question, but [TransMat] in this case is defined such that {d} and {rf} coordinate systems are parallel {rf} = {d} when q_RFrl = 0, i.e. when [q_RFrl] = the 3x3 identity matrix.

Best regards,

Thank you for kind consideration and clarification.

However, I mean that why the transformation between rotor furl frame {rf} and Yaw frame{d} was not like this

{rf} = [TransMat] * {d}

[TransMat] =[q_RFrl] * [RFrlTilt] * [RFrlSkew]

Also, I agree with you partially, {d} and {rf} coordinate systems are parallel {rf} = {d} only when q_RFrl = 0, RfrlSkew =0 and RfrlTilt=0

So, Why it is necessary to add( [RFrlSkew]^T * [RFrlTilt]^T ) to the [TransMat] ?

Dear Mohammed,

[RFrlSkew]^T * [RFrlTilt]^T are included in [TransMat] so that {rf} = {d} for any value of RFrlSkew and RFrlTIlt, as long as q_RFrl = 0.

Best regards,