Spatial Descriptions and Transformations (1) Read Chapter 2Spatial representation• Use coordinate system (frame) to represent spatial positions and orientation of objects–(XE, YE, ZE) set of three orthogonal unit vectors used to define an earth-fixed coordinate system–(XB, YB, ZB) set of three orthogonal unit vectors used to define a body-fixed coordinate system, original at the Center of Gravity (COG)–(XF, YF, ZF) set of three orthogonal unit vectors used to define a foot-fixed coordinate system  XFYFXBYBXEYEZEZFZB30o5410Fundamentals• When we manipulate objects as we do in robotics, we need a way of describing positions and orientations of objects and the spatial relationship between them: body ground; leg body• Positions and orientations are equally important• Our approach to describe the position/orientation of objects:– Attach a coordinate system (frame) to each object– Vectors which position its original in space to give directions of its unit vectors– Frame is a description for each object which carries all the position/orientation information– Define the position/orientation of the frame with respect to anotherHomogeneous Transformation• Use a 4 × 4 matrix• Gives position/orientation information of one frame with respect to another• First used in graphics, also computer vision• Applied in robotics to describe spatial relationship• A free body in space is said to have 6 degrees of freedom (DOF)•A homogeneous transformation in general has 6 independent pieces of information for specifying these 6 valuesPosition vectors• A position vector may be represented by its coordinates in any given frame:•P = 5XB+4YB+3ZBBP = 543– A leading superscript indicates the coordinate system of reference {B}• Homogeneous coordinates:BP = 5431= 10862= xyzP = xXB+ y YB+z ZB   Position vectors (continued)• H.C. are handy because multiplication by a constant does not change the associated vector (we will use w=1 always)• Dot product u and v: u • v = ? a scalar• Cross product of u and v: u × v = ? a vectoru= uxXB+ uyYB+uzZBv= vxXB+ vyYB+vzZB• |u•v| = uxvx+ uyvy+uzvz= |u||v|cos•w = (uyvz-uzvy)XB+ (uzvx-uxvz)YB+ (uxvy-uyvx)ZB• |w| = |u||v|sin  uvwHomogeneous transformation - position• A point represented in one frame carries information in 4 vectors– 3 for directions of unit vector and 1 for P origin of the frame• Assume the coordinates of the point P in the body frame to be determined in the earth-fixed frame. A 4 × 4 matrix Bwill do the job: B • What is B? – see the board note in the classHomogeneous transformation – position (continued)B = 100 010 001    000 1 EPBORG= position vector from the origin of the earth-fixed frame to the origin of body-fixed frame expressed in the earth frame.For the diagram shown earlier, EPBORG = [0 -10 -3 1]T.Homogeneous transformation – position (continued)• Then we have:EP 1000100010 103000 1 5431= 5601{A}{B}APBORGBPAPAP = BP+APBORGAP = ATBBPEquivalentFourth vector of ATB is a vector from A to B expressed in frame