Unified Robotics I: Actuation

Linkage Analysis

Coordinate System Transformations

Let’s now look at some simple coordinate system transformations. We’ll start by defining coordinate frame F0 which is defined by the unit vectors x0 and y0. We'll also define coordinate system or frame F1 which is rotated with respect to coordinate system F0. The representation of this system can be seen in the image below:

Figure 1 - Coordinate Frames

Notice that the two coordinate systems share a common origin. Geven how these coordinate systems are defined, its possible to express the unit vector for frame F1 in terms of the unit vectors of frame F0 and the angle θ1 which is the angle frame F1 is rotated with respect to frame F0.

Now suppose we have a vector u given by its coordinates with respect to the frame F1 as shown in the following figure.

Figure 2 - Coordinate Frames

The vector u can be given by the equation: u = x1x1+y1y1. An interesting equestion to ask is: What would the coordinate of the end of the vector u be relative to the F0 coordinate system (which has the same origin as frame F1 but is rotated by the andgle θ1 as shown)?

  • Show Answer
    • Figure 3 - Vector u with respect to x0

To answer that question we could simply substitute the values we previously calculated for the unit vectors x1 and y1.

This would give us the following results:

  • u=x1x1+y1y1
  • u=x1[cos(θ1)x0+sin(θ1)y0]+y1[-sin(θ1)x0+cos(θ1)y0]

Where,

  • x1=cos(θ1)x0+sin(θ1)y0
  • y1=-sin(θ1)x0+cos(θ1)y0

We can rearrange this a bit to get the following:

  • u=[x1cos(θ1)-y1sin(θ1)]x0+[x1sin(θ1)+y1cos(θ1)]y0

which simplifies down to: u=x0x0+y0y0

Where,

  • x0=x1cos(θ1)-y1sin(θ1)
  • y0=x1sin(θ1)+y1cos(θ1)

You can see now that it is quite straightforward to express the coordinates of a position vector with respect to multiple different coordinate systems.

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