The Use of Matrices in Space Shuttle Control Systems

Rotations in Three Space:
The rotation matrix for the 3*3 case is defined as follows:

- Rotation Around the x-axis:
A roll is a counterclockwise rotation of gamma about the x-axis; the rotation matrix is given by:

- Rotation Around the y-axis:

A pitch is a counterclockwise rotation of beta about the y-axis; the rotation

matrix is given by:

- Rotation Around the z-axis:

A Yaw is a counterclockwise rotation of alpha about the z-axis; the rotation

matrix is given by:

Rotation matrices can be multiplied with each other, which leads to a series of rotations of an object around x-, y-, and z-axes. There is one crucial fact to note when dealing with rotation matrices: The order of rotations is important, which means it is not commutative.

In the picture below, the figure at left is first rotated 90 degrees about the z-axis, then rotated 30 degrees about the x-axis. The result is shown in the top right figure. Next, the same operations are performed, but in opposite order, as shown in the bottom right view. It is now obvious that these operations are not commutative.

Putting it All Together:
Q: What does this Mean for the space Shuttle?
One can now expand these principles to derive a sequence of control operations to use for directing 3-dimensional objects such as the space shuttle. This is one of the greatest benefits to be had from the application of Linear Algebra.

P.S.: If you visit the link below and scroll down to the bottom of the page, you will find an application that helps you visualize how rotation matrices can help in determining yaw, pitch and roll. Just put in your angles of choice and watch the magic happen. Hope you liked my contribution to this blog.
http://prt.fernuni-hagen.de/lehre/KURSE/PRT001/course_main_sh3d/node11.html<!--[if gte mso 9]> Normal 0 false false false EN-US X-NONE AR-SA

Data retrieved from: http://www.google.com/url?sa=t&source=web&cd=1&ved=0CBYQFjAA&url=http%3A%2F%2Fwww.math.washington.edu%2F~king%2Fcoursedir%2Fm308a01%2FProjects%2Fm308a01-pdf%2Fpilawski.pdf&rct=j&q=the%20use%20of%20matrices%20in%20space%20shuttle&ei=Ad2tTYe4F4WVOurs_ekL&usg=AFQjCNE-LArQYRSxKUqTuhFp6UQqc2b9RQ&cad=rja

By George Hamoush