1. Body System

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 fixed handedness

5. Translated Body  The origin of the body system may not coincide with the space system. Offset vector aOffset vector a  Six parameters are needed to describe the body. Three for origin of bodyThree for origin of body Three for orientationThree for orientation x1x1 x2x2 x3x3

6. Rotation and Translation  The rotation may also occur at an arbitrary point. Rotate through point wRotate through point w Rotation SRotation S Translation bTranslation b  Equate to the origin-based translation. Vector b uniquely determined by a, S and wVector b uniquely determined by a, S and w One is free to select wOne is free to select w  A general displacement is a rotation around an axis and a translation parallel to the axis.

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

8. Body System Tensor  The moment of inertia can be expressed as a tensor. Components are not constant in space systemComponents are not constant in space system Convert to body systemConvert to body system  This can be diagonalized. Choice of rotationChoice of rotation Positive valued elementsPositive valued elements

9. Principal Axes  The inertia tensor is a symmetric matrix. Three real positive eigenvaluesThree real positive eigenvalues  The eigenvectors can be used as body system coordinates. Matrix is diagonalMatrix is diagonal  These are the principal axes. next