Homogeneous Transformation Examples and PropertiesHomogeneous transformation - examplesConsider the left foot of the walking robot as #2 footXFYFXBYBXEYEZEZFZB30o5410#1 Foot#2 Foot-453 in ZB3/2 1/2 0 51/23/2 0 400001 301 = 0001=What is ? 5431 FrontLeg Front ViewZFXFYF#2#3#0We need • To obtain , inverse : ( = C1 = -(54 6.33C2 = - (54 0.96C3 = - (31= -3 = 3/2 1/2 0 11/23/2 0 200001 30 13/2 1/2 0 6.331/23/2 0 0.9600001 30 15431= 46.9201Can you find the answer by inspection?Relative (compound) transformation•Since and We have = 1 0 0 00 1 0 1000001 30 13/2 1/2 0 51/23/2 0 400001 30 146.9201= 51401Use of Homogeneous Transformation• To transform point vectors • To transform a free vector (free in the air, no origin)– Use rotation matrix only– Does not change magnitude E.g.: = 3/2 1/2 01/23/2 0001• To transform unit vectors= ^^^^^^Properties of Homogeneous Transformation• Column of R are unit vectors:|n| = |o| = |a| = 1• • Column of R are orthogonal: • 0; • 0; • 0• Dot product of any two rows (columns) i and j:= 0 when = 1 when • Cross product of any two rows (columns) give the components of the third row (column) ; ;
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