Robot Jacobian (velocity kinematics)

Level AdvancedDifficulty ★★★★★Application⌖ Open in the map

What is it?

Forward kinematics $p = f(q)$ gives the hand position from the joint angles; its Jacobian $J(q) = \partial f/\partial q$ maps joint velocities to hand velocities, $\dot p = J(q)\dot q$ , and joint torques to hand forces, $\tau = J^{\mathsf T}F$ . Where $\det J = 0$ the arm is singular.

Why does it exist?

Controllers command joint motors, but tasks are specified for the tool: "move the gripper 1 cm left". The Jacobian is the local dictionary between the two languages, recomputed at every control cycle because it depends on the pose.

Formulas

$\dot p = J(q)\,\dot q, \qquad J(q) = \frac{\partial f}{\partial q}$; $J(q) = \begin{pmatrix} -\ell_1\sin q_1 - \ell_2\sin(q_1 + q_2) & -\ell_2\sin(q_1 + q_2) \\ \ell_1\cos q_1 + \ell_2\cos(q_1 + q_2) & \ell_2\cos(q_1 + q_2)\end{pmatrix}$
two-link planar arm: $\tau = J(q)^{\mathsf T}F$
statics: forces at the hand ↔ joint torques

Example

Two-link arm with $\ell_1 = \ell_2 = 1$ , $q = (0, \pi/2)$ : the hand is at $(1, 1)$ and $J = \begin{pmatrix}-1 & -1\\ 1 & 0\end{pmatrix}$ , $\det J = 1$ . Rotating only the elbow ( $\dot q = (0, 1)$ ) moves the hand with velocity $(-1, 0)$ : straight left.

Why does it matter?

Every industrial arm, surgical robot and animation rig uses Jacobians for velocity control, force control and inverse kinematics.

The mathematics behind it

Jacobian matrix★★★★★fundamental
Maps joint velocities to end-effector velocities; its singularities are configurations where the arm loses mobility.
Trigonometric functions★★★★★frequent
Forward kinematics of a revolute arm is sums of $\cos$ and $\sin$ of joint angles.
Chain rule★★★★★frequent
End-effector velocity is the chain rule through the arm: $\dot p = \frac{\partial p}{\partial q}\,\dot q$ .

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

⚙ Robotics and control

Inverse kinematics★★★★★

What depends on it

Inverse kinematics