Three Interpretations of Homogeneous Transformation

235. THREE INTERPRETATIONS OF HOMOGENEOUSTRANSFORMATION:TRANSFORMATION ARITHMETICA. 3 INTERPRETATIONS OF HOMOGENEOUS TRANSFORMATIONSA homogeneous transformation is described byR : Q0 1 14x4where R is 3 X 3 orthogonal (i.e.RTR=Is)and Q is a 3-D vectorIt describes a frame {B} with reference to a given frame {A}Example:P BORG = Q[18,18" ^]=R(A)8XIt is a transformmapping that mapsPRORGBp to Ap, i.e., 24Example:BORG =Q^ZB]= R{A}ApBp{B}PRORGIt is a transform operator that moves AP to AP2 where AP2 isQ. obtained by rotating AP, by R (i.e. RAP) and then translating it by 25B. TRANSFORMATION ARITHMETICCompound Transformations:Given{B} in reference to {A} and{C} in reference to {B}then ATwhereAR:AB B BORGBRBT11ARERRB CORG +1 P BORGAT=01(B){A}P CORG{C}A B BORGA P CORG 26Inverse of a Transformations:()'=BR 1-ARTAP BORG01sinceand8(APBORG)=0P BORRBRAPBORO=-ART.APBORG 27C. A WORKING EXAMPLEPart I: Assume we knowBT- B - Location and orientation of the gripperBT- - Location and orientation of the table (station)ST - Location and orientation of the boltTask:To find the location and orientation of the bolt inreference to {G} i.e. T(T)(B){G{S}B:Base frameT:Tool frameS:Station frameG:Goal Frame 28Solution:Part II: Letand(i.e. {S} is obtained by rotating {B} about 2 axis by 60°and then translating it 6 units in X, 10 in YB.)the(i.e. {G} is obtained by rotating {S} about Z axis byST=-45° and then translating it 2 units in Xs, 5 in Ys.)3/41/410BT-1/23/43/410(i.e. {T} is obtained by rotating (B) about Z axis by-1/23/2-30° and about X by -30° and then translating it 10 units10in X, 10 in Y and 10 in 2)0001Using MATLAB to set0.707 -0.707 0 : -8.4640.61230.6123-0.55.0IT0.35350.35350.866-8.660001