Orientation of a 3-D body

3II. Orientation of a 3-D body: (Rotation Matrix)In order to describe the orientation of a body, we will attach acoordinate system to the body in a known way and then give adescription of this coordinate system relative to the referencesystem.Given referencecoordinate system {A}, the orientation of the body in the figure canbe describe by frame {B}, i.e., by three unit vectors along itsprincipal axes'XB AYBwhich are three row vector of the rotation matrixZA{B}ZB{A}YBZBXBYAYBXBX 4Example:{B} is obtained by rotating {A+ about ZA by 30° as shown in thefigure. Find AR.Solution:3'XB = ^P, = 2ZA {A}ZB {B}YB30°I30°32 24SPZBYBA 'P =PAX XA +PaylA +PaziXBfisp BFYAXACoordinates frame attached to a rigid bodyLet { denote the standard outshormalbasis in SAS Thus, RA, 2/A. in are waite vectors, alongthe XA, , YA, ZA axes, respectively.Asimilarly let. { XB , Yns zr an denote the standard!orthnormal basis in FBS. Thus, Sis. Yis, 1 1 Z.z are writeVentors along the X. A.AKA = PAX KA + PAY YA PAZZAAlso"I's : Pisx V3 + Psy Is PBR EgSince AP. andlo are response representation ofbee save vertor P. &XA ^dot productSA + Pay Vs. is + For S.RPwe have simila 2 formulas for PAY. PAZ, namelyPAY=Pisn is is + PBY TS in + Pist. Es FAPAZ =Pass To SA of Pay I's EA + Parts a EAAp=GR'?whereAn AT is : in similarly, we care writePAXKA + PAYPBM=Psz it"P=SRAp1WhereVA.INiR&Ca.% VA.YO 2RA.ES in.z.Erage09Awe seeAK ARTwe Care confirusART-AR=Is = -sin 30" = 100190A Rtoo)YA A A 10 10330 = sir 30 -30AZA ty =1 = cos(o)and all the other products are zave. ThisBR Ap it10330 -sinzi Uwaszei( 5Property:R=;R'XBARTRIL=AYBXBAYB^ZB]AR="R"="R"equal to its transpose.From algebra it is known that the inverse of an orthogonal matrix is