Numerical algorithm for quaternion integration based on three independent parameters with no need for re-normalization

The relative attitude change determined from the beginning of each time step to its end defines a relative quaternion. This relative attitude is characterized by only three independent quaterion parameters, with a closed form expression for a fourth dependent parameter as a function of the independent parameters. This procedure ensures that each estimate of attitude is automatically based on a unit quaternion without the need for re-normalization. An example of a complicated three-dimensional motion is used to show that there is no loss in accuracy relative to standard unit and non-unit formulations. Also, an example of Euler integration is used to demonstrate distortion of the rotation matrix due to re-normalization.