Robotics, Fall 2006 Dr. Jennifer Kay Lecture 1: Coordinate Frames Copyright © 2005, 2006 Jennifer Kay

Drawing 3 Dimensional Frames in 2 Dimensions Y points to the right Z points up X points ???

Two views of the same cube Viewed from below Viewed from above

Now Overlay the 3-D Coordinate Frame x y z x y z Viewed from above Viewed from below X points away from you! X points towards you!

We will always use a right-handed coordinate frame (x points out) z

We will not use a left-handed coordinate frame (x points in) z x y

Right Handed Coordinate Systems: Using the right hand rule Given two axes, determine the direction of the third axis Point right fingers along x Curl fingers towards y Thumb points to z

Right Handed Coordinate Systems: Using the right hand rule Given two axes, determine the direction of the third axis z x y x y x y

Right Handed Coordinate Systems: Practice z y x y y x z x y z

Rotating Axes: Direction of Positive Rotation If we want to rotate a frame, which way do we rotate it? Example: Rot z (90 degrees) x y z y z x ? y x z ?

Rotating Axes: Another Right Hand Rule Point right thumb in direction of axis you want to rotate Fingers curl in direction of positive rotation x y z y x z Rot z (90)

Plotting Points (Neatly) in 3-D Example: Plot (5,3,4) Step 1: Draw tick marks

Plotting Points (Neatly) in 3-D Example: Plot (5,3,4) Step 2: Draw two lines on the groundplane 5 3 4 3 5 3

Plotting Points (Neatly) in 3-D Example: Plot (5,3,4) Step 3: Draw a perpendicular with tick marks, and your point 5 5 3 4 4 4 3 3 5

Plotting Points: Example Plot (1,3,5) on a coordinate frame where: X points up and Y points into the paper Y points down and Z points out of the paper

Working with Multiple Coordinate Frames Tower Coordinates Plane Coordinates

Working with Multiple Coordinate Frames Location of the nose of the plane in tower coordinates? Location of the origin of the tower in plane coordinates? Location of the nose of the plane in plane coordinates? 50 meters 5m

Working with Multiple Coordinate Frames Location of the nose of the plane in tower coordinates? Location of the origin of the tower in plane coordinates? Location of the nose of the plane in plane coordinates?