Jacobian Matrix

A matrix of partial derivatives that maps joint velocities to end-effector velocities: ẋ = J(q)·q̇. The Jacobian is central to robot control — it enables Cartesian velocity control, force mapping (τ = Jᵀ·F), singularity analysis, and redundancy resolution. For an n-DOF arm with a 6D task, J is 6×n. The Jacobian is configuration-dependent and must be recomputed at each timestep.