The 3-D rotation is defined as the relative rotation between the
object reference frame and the current camera reference frame. This
is represented using a unit quaternion, from which the rotation matrix
(R) can be generated:
The four elements of the unit quaternion only have three degrees of freedom due to the normality constraint. Thus, all four cannot be estimated independently; only a nonlinear constrained minimization will work to recover the quaternion directly. Since the EKF utilizes a linearization at each step, the nonlinear normality constraint cannot be rigidly enforced within the EKF computational structure.
However, a 3-parameter incremental rotation representation, similar to that used in [11], can be used in the EKF to estimate interframe rotation at each frame. Incremental Euler angles centered about zero (or discrete-time ``rotational velocity'') do not overparameterize rotation and are approximately independent and therefore can be used reliably in a system linearization.
The incremental rotation quaternion is a function of these three
parameters:
Thus, we have,