Coordinate Frames

This section describes Camera, Grid/Matrix/Image, and Robot coordinate frames.

Grid Pixel Center

Values on a grid \(a\) are stored such that the grid cell stores the value at the coordinate \(i \times d\ \)where \(d\) is the spacing between sample points. This means \(a\left\lbrack i \right\rbrack = f\left( i \times d \right)\). Thus if \(y = f\left( x \right)\) shall be retrieved in “nearest” interpolation mode, \(x\) is rounded to the nearest index \(i\) and thus \(y = a\left\lbrack i \right\rbrack\). In “linear” interpolation mode \(x\) is mapped to the lower index \(i0\) and thus \(y = \left( 1 - p \right) \times a\left\lbrack i0 \right\rbrack + p \times a\left\lbrack i0 + 1 \right\rbrack\).

../../../_images/coord-frame-gmi-frame.png

Grid/Matrix/Image Coordinate Frame

In a 2-dimensional frame, the vector order is simple:

  • 0: row

  • 1: column

    ../../../_images/coord-frame-simple.png

Camera Coordinate Frame

Note: specifically a 3D point (x,y,z) in camera frame is represented as depicted in this diagram. If the 3D point is projected onto the image the coordinates are switched to image coordinates which follow the (row/col) order as shown in the following diagram.

The vector order is:

  • 0: right

  • 1: down

  • 2: forward

    ../../../_images/coord-frame-camera-frame.png

Robot Coordinate Frame

The robot coordinate frame is actually placed inside the robot. In the Carter robot the center of the frame is between the wheels.

The vector order is:

  • 0: forward

  • 1: left

  • 2: up

    ../../../_images/coord-frame-robot-frame.png