Spatial Descriptions and Transformations (2) Read Chapter 2Homogeneous transformation - rotation• The H.T. matrix has the structure asT = 0001where P is a 3 ×1 position vector and R is a 3 × 3 rotation matrix. • For example, the orientation of the foot is at 30oangle with respect to [XB, YB, ZB]T:XBYBZB= 30osin30o0sin30o30o0001XFYFZFXBXFYBYF30oZHomogeneous transformation – rotation (continued)• That is:XB= cos 30oXF–sin30oYFYB= sin 30oXF+ cos 30o YFZB= ZF• Use H.T., we may have where = 0001= 3/2 1/2 0 51/2 3/2 0 400001 30 1Homogeneous transformation –rotation about all the axes• In general we use the following rotation matrix to relate frames with rotation about one of the coordinate axes:RX(10 00c s0s cRY(c 0s010s 0cRZ(c s 0s c 0001XBXFYBYFZNote the direction for the angle= cosand s= sinA general rotation• A general rotation is to provide successive rotations as1) Start with frame C2) Rotate about Z (yaw)3) Rotate about Y (pitch)4) Rotate about X (roll)• There are other ways to represent general rotations (read Section 2.8) = RZ(RY()RX() = α β α β γ α γ α β γ α γα β α β γ α γ α β γ α γβ β γ β γHomogeneous transformation - inverse• Homogeneous transformation can be back and forth between two frames• If we know , we can find if homogeneous transformation is known: • What about the other way? We need to know • Naturally = ( -1so we have ( −1 • What is the relationship between and ?Homogeneous transformation - inverse = 00 0 1= • • 00 • 0 1TransposeDot
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