Spatial Descriptions and TransformationsRead Chapter 2Spatial representation (use a legged robot)• 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?Homogeneous 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