Rotation Matrices 1

An Introduction

robot frames.png

In this page, I will be introducing some of the basic concepts around Rotation Matrices, in how they are formulated and how they can be used to measure changes in orientation between frames. This will be crucial in later parts of the Robot Kinematics series.

This uses the theory of Projections, a topic that will be covered in more depth in the general mathematics series. A list of useful Youtube Videos covering projections and this topic can be found in the list below.

Useful Videos On Projections And Rotation Matrices:

1. Introduction to projections | Matrix transformations | Linear Algebra | Khan Academy​​

https://www.youtube.com/watch?v=27vT-NWuw0M

2. Dr Peyum |  Rotation Matrix

https://www.youtube.com/watch?v=Ta8cKqltPfU

3. Angela Sodemann | Robotics 1 U1 (Kinematics) S3 (Rotation Matrices) P1 (Rotation Matrices)

https://www.youtube.com/watch?v=lVjFhNv2N8o&list=PLT_0lwItn0sDBE98BsbaZezflB96ws12b&index=10

 

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In the last page, we introduced frames and capturing their rotations around an axis. The Animation below shows where we left off.

To start with this topic I want to introduce Projections.  A Projection is like a 'shadow' of one line projected onto another from a certain viewpoint. The maths behind projections can vary in complexity depending on the method, so on this page, I will more summarise to make it easier to move on.

rotation matirx animations 15 slide 16 s

In the animation below shows one method for calculating the lengths inside of a projection, this example uses a Unit Circle. In a unit circle, the radius is equal to 1, so any lines from the centre to the circumference is also equal to one. 

 

I believed this would be a useful example as the axis of a frame are also conventionally considered to be equal to the value of 1. 

rotation matirx animations 15 slide 17.g

The animation below shows a slightly different method based on how projections of other lengths may be calculated. Through having the lines in question have a unit length of 1, the process is quickly simplified.

rotation matirx animations 15 slide 18.g

The next animation shows how the lengths just discovered can now be used to create a rotation vector. This can be applied to a starting position to create a desired rotation.

rotation matirx animations 15 slide 19.g

This concept will be developed on further in the next page, where we will combine rotation vectors into a rotation matrix.