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Jacobian inversion method

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 $X$ and the joint space of the joint angles $\theta$. The Jacobian, in fact, transforms the differential angle changes to the differential motions of the end effector [Mck91].

\begin{displaymath}\dot X = J(\theta)\dot \theta \eqno (6) \end{displaymath}

The vector $\dot X$ represents the linear velocity $(dx, dy, dz)$ and rotational velocity $(\delta_x, \delta_y, \delta_z)$ of the end effector and $\dot \theta$ represents time derivative of the state vector (rotational velocity for each joint).

Since the unknown is $\dot \theta$, the Jacobian inversion is needed. Hence, the equation (6) is transformed to the form:

\begin{displaymath}\dot \theta = J^{-1}(\theta) \dot X \eqno (7) \end{displaymath}

Lukas Barinka 2002-03-21