Orientation Kinematics

Abstract

Any finial orientation of a rigid body with a fixed point O, after a finite number of rotations is equivalent to a unique ϕ about a fixed axis $\hat {u}$. Also any rotation ϕ of a rigid body with a fixed point O about a fixed axis $\hat {u}$ can be decomposed into three rotations about three given non-coplanar axes including the global or body principal exes. Determination of the angle and axis is called the orientation kinematics of rigid bodies.

There are several methods to express rotation of a body frame B with respect to the global frame G. The most applied method is the Rodriguez rotation formula that provides us with a 3 × 3 transformation matrix made of three independent elements.

The objectives of this chapter are

• 1. To determine the transformation matrix ^G R _B between two Cartesian coordinate frames B and G with a common origin when B is turning ϕrad about a given axis indicated by the unit vector $^{G}\hat {u}= \left [ \begin {array}{ccc} u_{1} & u_{2} & u_{3} \end {array} \right ] ^{T}$.

• 2. To determine the axis $^{G}\hat {u}$ and angle ϕ of rotation for a given transformation matrix ^G R _B.