# 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:

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