A two-link arm is controlled by two angles, but it moves in an x–y world. The Jacobian is the matrix that converts a small joint-velocity (θ̇1, θ̇2) into an end-effector velocity (ẋ, ẏ). Its two columns are the workspace velocities each joint produces alone — spinning them tells you every direction the hand can move, and when it can't.
…or drag the hand (red) directly and the arm follows via inverse kinematics.
arm links — lengths L₁, L₂col 1 of J — hand velocity from θ̇₁col 2 of J — hand velocity from θ̇₂Forward kinematics: x = L₁cosθ₁ + L₂cos(θ₁+θ₂), y = L₁sinθ₁ + L₂sin(θ₁+θ₂). Differentiate and the coefficients form J; each column is one joint's contribution to hand velocity, drawn as the orange and blue arrows. Their span is the yellow ellipse — a round ellipse means the hand moves equally well in all directions; a squashed one means it's fast one way, sluggish the other. When the arm stretches straight out (or folds fully back) the two columns line up, the ellipse collapses to a line, det J → 0, and the arm loses a degree of freedom — a kinematic singularity. There the hand cannot move radially at any joint speed, and inverse-kinematics controllers blow up. Detecting and steering around these is a core robotics problem; the same matrix underlies resolved-rate control and the manipulability measures used to plan good arm poses.