# OSU ECE 5463 - Lecture-Spatial-Representation_FA2014-2 (7 pages)

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## Lecture-Spatial-Representation_FA2014-2

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- Pages:
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- School:
- Ohio State University
- Course:
- Ece 5463 - Introduction to Real Time Robotics Systems

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Spatial Descriptions and Transformations 2 Read Chapter 2 Homogeneous transformation rotation The H T matrix has the structure as T 0 0 0 1 where P is a 3 1 position vector and R is a 3 3 rotation matrix For example the orientation of the foot is at 30o angle with respect to X B Y B Z B T X B X F 30o o o XB 30 sin30 0 XF Y B sin30o 30o 0 Y F Z B 0 0 1 Z F Z Y F Y B Homogeneous transformation rotation continued That is X B cos 30o X F sin 30o Y F Y B sin 30o X F cos 30o Y F Z B Z F Use H T we may have where 3 2 0 0 0 1 1 2 0 0 1 2 0 5 3 2 0 0 0 4 1 3 0 1 Homogeneous 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 1 0 RX 0 c 0 s 0 s c c 0 s 1 0 RY 0 s 0 c c RZ s 0 s 0 c 0 0 1 cos and s sin X B X F Y B Z Y F Note the direction for the angle A general rotation A general rotation is to provide successive rotations as RZ RY RX 1 2 3 4 Start with frame C Rotate about Z yaw Rotate about Y pitch Rotate about X roll There are other ways to represent general rotations read Section 2 8 Homogeneous transformation inverse Homogeneous transformation can be back and forth between two frames if homogeneous If we know we can find transformation is known What about the other way We need to know 1 so we have Naturally 1 What is the relationship between and Homogeneous transformation inverse 0 0 Dot product 0 1 Transpose 0 0 0 1

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