Two Remarks On Rotation Matrix

296. THREE INTERPRETATIONS OF HOMOGENEOUSTRANSFORMATION:A. Two REMARKS ON ROTATION MATRIXThere are only three independent parameters in a rotation matrix.Rotation matrices are proper orthonormal matricesProper-------unit magnitude.Orthonormal---All columns of R are manually orthogonal and havedet(R)= = +1 (Non-Proper det (R) = -1)Cayley's formula for orthonormal matrices states that forS, such thatproper orthonormal matrix R, there exists a skew-symmetric any matrix-ST) of dimension 3, is specified by three parameters (Sx, Sy, Sz) asWhere I3 is a 3x3 identity matrix. A skew-symmetric matrix (i.e. S =$ 10 -S2 SyAny 3x# rotation matrix can be specified by just three parametersAnother Explanation:Since a rotation matrix is always orthogonal namely, if we denoteR=[X Y ], thenX=8=2=1Only 3 independent parametersOperations of rotation do not commute in general, i.e.,R1R2 # R2R1 30B. DESCRIPTION OF AN ORIENTATION:Roll, Pitch and Yaw AnglesZYawRollPitchXYConsider two frames {A} and {B} which are initially identical(1). Rotate {B} about X by r (roll) to get {B'}ZAZR'YYYARollXA XA 31(2). Rotate {B'} about Y by B (pitch) to get {B"}ZR*ZAZ"YRBYR"YABPitch(X) X4XR" (3). Rotate {B"} about ZA by a (yaw) to get {B"}ZAaZR"YRaYR"ZRYAYawXAXRXR" 33given byThe equivalent rotation matrix AR to the above ordered rotations isCa0Sa0Ca-Sy 34Pick an arbitrary vector say AP, and rotate it 3 times as describedabove to get àP2{A}P2P1{B}Claim:I. Form {B} such that ^P,=B P2II. ^P2 IfIII.AP=ARBPI & IIIAP2=ARAP, Compare with II=CaCB CaCBSy-SYCYAR= SaCB -SB SaSBSY+Cacy CaSBCY+SoLSy text)= SaSBCY-CaSy (2.64 of Page 46 of theCBSYCBCY