Description of a frame

6III. Description of a frame:A frame is characterized by the position of its origin and theorientation of the frame relative to a reference coordinate system(frame). Say {A}, i.e. a reference coordinate systemZA & {A}{B}ABBORG8XTwo special cases:1. If AR=1,{B} represents a position2. If AB B BORG =0{B} represents an orientation. 73. CHANGING DESCRIPTIONS FROM FRAME TO FRAME:(Homogeneous Transformations)a. MAPPING INVOLVING TRANSLATED FRAMESFix a position P, in {A}. the position is described by vector AP,while in {B}, it is described by vector BP. What is the relation of ^Pto BP?(1)Note: Since {B} is obtained by a translation of {A}, the projectionof BP onto the principal axes of {B} = the projection of Bp onto theprincipal axes of {A}[%: Equation (1) holds when vector involved is expressed withreference to {A}]{B}{A}^pBPA P'AA P BORG 8b. MAPPINGS INVOLVING ROTATED FRAMESGiven frames {A} and {B} where {B} is obtained by rotating {A}by certain angle(s) about certain axis (axes).PZBZA{A}BpApYBYXBXFix a point P, there exists one position vector in this case, but it hasTWO representations. i.e. ^P in {A} and BP in BB 9Problem: Relation of AptoBP?Solution:Aside: Use inner product to represent a projectionLet Ap = 8924 AP2 AP,APx(2)Qawhere AP, = the projection of Ap onto==If expressed in {B}If Q1 is a unit vectorNote Projection (i.e. inner product) is FRAMESimilarlyINDEPENDENTAP,(3)SinceBX(4)BZAImplying (2), (3) and (4)AP,BZA(5)Similarly(Since (AR)(5') 10Example:{B} is obtained by rotating {A} about Z axis by 30°, given0/32201230°YB30°30°YXX802 11c. MAPPINGS INVOLVING GENERAL FRAMES(HOMOGENEOUS TRANSFORMATION)Translation only:'P="P+" BOBGRotation only:A P=AR.B PGeneral Frame:P=AR.B P+1 P BORG{B}{A}XBPBPApYBA P BORGYAX 12^PHomogeneous coordinates)(6)Where 4x4 matrix AT is called the homogeneous transform just asrotation matrix is used to specify an orientation, AT can be used todescribe frames. The description of frame {B} relative to {A} is AT. 13Example:10 {B} units is obtained by rotating {A} about ZA axis by 30° and translatingin X direction and 5 units in Y direction.Given:7 3 find BPthe = 100.866-0.50:100.50.866000 140.866-0.501039.089[A PBP0.50.866=AT0 5712.56211001000019.089AP=12.56208YBPXBBPAp{B}PBORGXA{A}