Another Description Of Orientation (Euler's Angles)

37C. ANOTHER DESCRIPTION OF ORIENTATION (EULER'S ANGLES)I. Z-Y-X Euler AnglesInstead of using axes of {A} in rotating {B}, one may rotate {B}about its own principal axes.1. Rotate {B} about -axis by a to get {B'}ZA ()aYYXXB 382. Rotate {B'} about YB axis to get {B"}XBXB 393. Rotate {B"} about Xg-axis by Y to get {B"'} - {B} (new)/83YBYXrXBThe angles (a, B, y) such rotated are called Z-Y-X Euler angles. 40Denote the rotation matrix AR by AR=AR273-Ca,B,YRo(yBo(py)=THE ca0C-Sa-SaTCBST10=-SCBOSCyCa.CB Ca.SB.Sy - Sa.Cy Ca.SB.Cy+Sa.Sy(2.71) (Page. 49)Sa.CB Sa.SB.Sy + Ca.Cy Sa.SB.Cy - Ca.Sy-spCBSCB.CyNoteX1821Describing the rotation of frame{B} relative to a fixed=frame {A} in opposite orderThe inverse problem : Similar to roll-pitch-yaw case 41II. Another set of Euler Angles: [Z-Y-Z Euler Angles ]1. Rotate {B} about ZB by a to get {B'}2. Rotate {B'} about YB by e to get {B"}3. Rotate {B"} about ZB" by Y to get {B"'} ={B} = (new)Ca.CB.Cy-Sa.Sy-SB.Cy -Ca.CB.Sy-Sox.Cy -Sa.CB.Sy+Ca.Cy SBS CaSB Sa.S.B(2.72)(Page. 49)CBThere are totally 12 set of fixed angles and 12 sets of Euler angles.See Appendix B for list.Notation :ROT(x, 0)Rx(0)(Notes)(Text) 42D. EQUIVALENT ANGLE-AXISGiven an arbitrary rotation matrix R in frame {A}, 3 (exits) unitvector AK and angle 0 such thatR=ROT(*K,e)A general formula for ROT(*K,0)Set up another frame {C} where Zo -axis is aligned with AK, thenwe can showLook at an arbitrary vector Ap in {C}, the vector is expressed byCP=SR^PZ,{A}PAp2cXc{C}YTcX 43When the vector is rotated about by 0, the vector becomesC P1In {A}, the resulting vector is expressed as= ROT(AK,e)^PwhereTaddc f k2 44with cf=0d6 conditionsSo we can express a, b, c, d, e, f in terms of kx, ky,k,k,v0+c0(2.80) Page 52k,k,v0+k,s0k,k2v0+c0where 10=1-cos0Next assume an arbitrary rotation matrix is given byY33 45then the equivalent angle-axis can be found through equation:AR=ROT(AR,e)0If 0 = 0, then no rotation occurred and K is then arbitrary.If 0 # 0 and 0# 180°, thenY32 -Y23If0=180°2k,k2k,k22k,k,2k,2-1, 46Example:{B} is initially coincident with {A}. Rotate {B} about0.707and passes through point ^P 3 30°, Find ATSolution:Setup an intermediate frame {A'} as shown in the figure:ZAZAYYAXKXr 0=3000^P0.707dr0.707 47THERotate A' about K by 30° to obtain {B'}Notice {B'} and {B} differ only by a translation, i.e.,THEThereforeROT(^R,30)010.933 0.067 0.354 : -1.129=1/0.0670.933-0.354 : 1.1290.3540.3540.866:0.048(2.88)0001