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Rotation matrices are used to define a rotation in a given axis. The most important properties a rotation matrix has are:

They have unitary vectors in all of their rows or columns They have a determinant of 1 the normalization of a rotation matrix is always 1 Its orthogonal If its orthogonal its transpose is equal to its inverse.

We can denote a rotation matrix as:

matriz de rotacion

The next image shows what a rotation by an angle of theta does to a coordinate frame. rotacion

Now that you know what a rotation matrix is and does you have to ask yourself “What purpose does it have in robotics?”, the answer to that question is that the rotation matrix defines a rigid objects rotation on a given axis. If we dig further in we can define what local and global frames are.

The global frame is the reference coordinate frame in which the rest of the robot rotates or translates. in the case of a robot arm the local frames would be the joints that rotate and translates and finally give the end effector its position due to its total spatial displacement.

The transofrmation of a global coordinate frame to a local frame depends of the axis the rotation. The transformations for x,y,z axis rotations are shown below:

Rotacion a lo largo del eje X

Rotacion a lo largo del eje Y

Rotacion a lo largo del eje Z

As I previously stated rotation matrices are unitary so its transformation is also unitary therefor its transpose is also equal to its inverse. The process of the transformation from global to local coordinate frames is given below in which R is the rotation matrix along an axis and Puvw is the local coordinate frame point in space.

transformacion de locales a globales

When we clear R we would normally get 1/R but we actually know that 1/(any matrix) is equal to the inverse of the matrix. Recalling that the inverse is equal to the transpose in unitary matrices we get the next transformation.

Locales a globales final

David Rodriguez

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