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The Jacobian is the multidimensional extension to the differentiation
of a single variable [WW92]. As it was mentioned in the
introduction, there is a relation between the Cartesian space of the
end effector and the joint space of the joint angles .
The Jacobian, in fact, transforms the differential angle changes to the
differential motions of the end effector [Mck91].
The vector represents the linear velocity
and rotational velocity
end effector and represents time derivative
of the state vector (rotational velocity for each joint).
Since the unknown is , the Jacobian inversion
is needed. Hence, the equation (6) is transformed to the form: