1. Rotations

2. Space and Body  Space coordinates are an inertial system. Fixed in spaceFixed in space  Body coordinates are a non- inertial system. Move with rigid body x1x1 x2x2 x3x3 x1x1 x2x2 x3x3

3. Matrix Form  A linear transformation connects the two coordinate systems.  The rotation can be expressed as a matrix. Use matrix operationsUse matrix operations  Distance must be preserved. Matrix is orthogonalMatrix is orthogonal Product is symmetricProduct is symmetric Must have three free parametersMust have three free parameters x1x1 x2x2 x3x3

4. Axis of Rotation  An orthogonal 3 x 3 matrix will have one real eigenvalue. Real parameters Cubic equation in s  The eigenvalue is unity. Matrix leaves length unchanged  The eigenvector is the axis of rotation. x1x1 x2x2 x3x3 +1 for right handedness

5. Single Rotation  The eigenvector equation gives the axis of rotation. Eigenvalue = 1Eigenvalue = 1  The trace of the rotation matrix is related to the angle. Angle of rotation Angle of rotation  Trace independent of coordinate systemTrace independent of coordinate system

6. Rotating Vector  A fixed point on a rotating body is associated with a fixed vector. Vector z is a displacement Fixed in the body system  Differentiate to find the rotated vector. x1x1 x2x2 x3x3

7. Angular Velocity Matrix  The velocity vector can be found from the rotation.  The matrix  is related to the time derivative of the rotation. Antisymmetric matrixAntisymmetric matrix Equivalent to angular velocity vectorEquivalent to angular velocity vector

8. Matching Terms  The terms in the  matrix correspond to the components of the angular velocity vector.  The angular velocity is related to the S matrix.

9. Body Rotation  The angular velocity can also be expressed in the body frame. Body version of matrixBody version of matrix x1x1 x2x2 x3x3