# OSU ECE 5463 - Lecture-Spatial-Representation_FA2014 (9 pages)

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## Lecture-Spatial-Representation_FA2014

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## Lecture-Spatial-Representation_FA2014

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Pages:
9
School:
Ohio State University
Course:
Ece 5463 - Introduction to Real Time Robotics Systems
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Spatial Descriptions and Transformations Read Chapter 2 Spatial representation use a legged robot Use coordinate system frame to represent spatial positions and orientation of objects X E Y E Z E set of three orthogonal unit vectors used to define an earth fixed coordinate system X B Y B Z B set of three orthogonal unit vectors used to define a body fixed coordinate system original at the Center of Gravity COG X F Y F Z F set of three orthogonal unit vectors used to define a foot fixed coordinate system 30o ZF XB ZB XF YF 5 XE ZE YB 4 10 YE Fundamentals 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 another Homogeneous 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 values Position vectors A position vector may be represented by its coordinates in any given frame P 5X B 4Y B 3Z B 5 BP 4 3 A leading superscript indicates the coordinate system of reference B Homogeneous coordinates 10 5 8 BP 4 6 3 2 1 x y z P x X B y Y B z Z B 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 vector u uxX B uyY B uzZ B u v vxX B vyY B vzZ B w u v uxvx uyvy uzvz u v cos w uyvz uzvy X B uzvx uxvz Y B uxvy uyvx Z B w u v sin v 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 1 0 0 0 1 0 0 0 1 0 0 0 1 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 EP BORG Homogeneous transformation position continued Then we have EP 1 0 0 0 0 1 0 0 0 0 0 10 1 3 0 1 5 4 3 1 5 6 0 1 AP ATB BP Equivalent B A AP AP BORG BP AP BP APBORG Fourth vector of ATB is a vector from A to B expressed in frame A

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