Analytical vs. Geometric Jacobians

If we parametrize the position and orientation of the end-effector as

\chi_E(\mathbf q) = \begin{bmatrix} \chi_{pos,E} \\\ \chi_{rot,E} \end{bmatrix},

then the rotational velocity \dot\chi_{rot} is not in general the same as the angular velocity \omega:

\dot \chi_{rot} = \frac{\partial\chi_{rot}}{\partial \mathbf q} \dot \mathbf q = J_{A,R}(\mathbf q)\dot \mathbf q \neq \omega.

Instead, we define the analytical Jacobian, which maps generalized velocities \dot\mathbf q to changes in the parameters \dot \chi:

\boxed { J_A(\mathbf q) = \begin{bmatrix} J_{A,P} \\\ J_{A,R} \end{bmatrix} = \begin{bmatrix} \frac{\partial\chi_{pos}}{\partial \mathbf q} \\\ \frac{\partial\chi_{rot}}{\partial \mathbf q} \end{bmatrix} }

If, however, we want to directly map generalized velocities \dot\mathbf q to end-effector angular and linear velocities, we must rely on the geometric Jacobian (also called basic Jacobian), defined as:

\boxed{ J(\mathbf q) = \begin{bmatrix} J_P \\ J_R \end{bmatrix} = \begin{bmatrix} \mathbf n_1 \times \mathbf r_{1,E} & … & \mathbf n_n \times \mathbf r_{n,E} \\ \mathbf n_1 & … & \mathbf n_n \end{bmatrix} }

where \mathbf n_i is the unitary vector representing the axis of rotation of joint j, and \mathbf r_{i,E} is the vector from joint i to the end-effector E. Bear in mind these depend on the current joint configuration \mathbf q_t and should be expressed in the same coordinate system (usually the inertial frame \mathcal I).

In conclusion, we have the following relations between the Jacobians and generalized velocities:

\chi_E(\mathbf q) = \begin{bmatrix} \chi_{pos,E} \\\ \chi_{rot,E} \end{bmatrix} = J_A(\mathbf q) \dot\mathbf q,
\textbf{w}_E = \begin{bmatrix} \textbf v_E \\\ \omega_E \end{bmatrix} = J(\mathbf q)\dot\mathbf q.