D1___Rotations.pdf - EE106A Discussion 1 Rotations 1 Frame-specific representations Points and vectors are described by coordinates that are only

# D1___Rotations.pdf - EE106A Discussion 1 Rotations 1...

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EE106A Discussion 1: Rotations 1 Frame-specific representations Points and vectors are described by coordinates that are only meaningful with respect to a correspond- ing coordinate frame. Figure 1: Two coordinate frames A and B Problem 1.Write the representation of pointqwith respect to the coordinate framesAandB, whichwe denoteqaandqbrespectively.2Rotation matricesLet’s first think solely about the mathematical definition of a rotation matrix before discussing howthey are used in practice. A rotation matrix is a matrix that is defined according to two coordinateframes.Definition 1.Say we have coordinate frameA, defined by its principal axes{xa,ya,za}, and frameB, with principal axes{xb,yb,zb}. Then, we define a rotation matrixRabto beRab:= [xabyabzab]where{xab,yab,zab}are orthonormal principal axes of frame B expressed in the coordinates of frameA.Problem 2.Find the rotation matrixRab= [xabyab]for an arbitrary 2D rotation (as depicted inFig. 1) 1 Problem 3.Work out whatRz(θ)is. Problem 4.Find the rotation matrixRbagiven the same 2D coordinate frames in Fig. 1. What doyou notice about the relationship betweenRabandRba?Commutative rule: Say we now have three framesA,B, andC. I tell you whatRbcis, (ie. theorientation ofCfrom the reference ofB), and you’ve already calculatedRAB. How can we expressRac, that is, the orientation ofCfrom the reference ofA? We simply combine rotation matrices to  • • • 