The Jacobian and Velocity Kinematics

The Jacobian tells you how fast the hand moves for given joint speeds. Each column is the hand velocity that one joint would make by itself at unit speed. The total hand velocity is the sum of the columns, each scaled by its joint speed.

θ₁ θ₂ Joint 1 speed Joint 2 speed

The Jacobian

For hand position $\mathbf{x}(\boldsymbol\theta)$, the Jacobian is the matrix of partial derivatives, and it maps joint speeds to hand speed:

$$\dot{\mathbf{x}} = J(\boldsymbol\theta)\,\dot{\boldsymbol\theta}, \qquad J = \begin{bmatrix} -L_1 s_1 - L_2 s_{12} & -L_2 s_{12} \\ L_1 c_1 + L_2 c_{12} & L_2 c_{12} \end{bmatrix}$$

with $s_1 = \sin\theta_1$, $s_{12} = \sin(\theta_1 + \theta_2)$, and so on.

What to notice

Each joint's arrow is perpendicular to the line from that joint to the hand, because a joint can only swing the hand in a circle around itself. Joint 1's arrow is longer, since it is farther from the hand. The Jacobian also works backward: the transpose maps a force at the hand to the joint torques that hold it, $\boldsymbol\tau = J^\top \mathbf{F}$.