Differential Kinematics

The kinematics that relates joint velocities to end-effector velocity through the Jacobian matrix.

Forward kinematics answers “where is the end effector given these joint angles”; differential kinematics answers “how fast does the end effector move given these joint speeds.” The bridge between them is the Jacobian matrix J(q), the partial derivative of forward kinematics with respect to joint angles q: V = J(q)·q̇, where V is the end effector's 6-dimensional spatial velocity (3D linear velocity plus 3D angular velocity) and q̇ is the vector of joint velocities. Running this backward is differential inverse kinematics: given a desired end-effector velocity V_d, the pseudoinverse (a generalization of “inverse” to non-square matrices) gives q̇ = J⁺V_d, which is then integrated into joint-angle commands, repeated every control cycle. It avoids having to solve a full analytical inverse-kinematics problem each time, which suits teleoperation and policies that output end-effector deltas well — but near a singular configuration (where J loses rank), the pseudoinverse can blow up, so practical implementations often use damped least squares instead, or frame the problem as a quadratic program with joint limits built in. Pink and mink are libraries built around this approach.

ExampleDuring VR teleoperation, each control cycle converts the controller's displacement into a desired end-effector velocity, and q̇ = J⁺V_d gives the speed each of the 7 joints should turn at, so the arm's end effector smoothly tracks the operator's hand.

Also called

Velocity Kinematics, Differential Inverse Kinematics (Diff IK)

Related

Jacobian Matrix · Inverse Kinematics (IK) · Jacobian Pseudoinverse · Damped Least Squares · Singular Configuration (Kinematic Singularity) · mink (MuJoCo inverse kinematics)

Sources

Robotic Manipulation (MIT, Russ Tedrake) - Basic Pick and Place: Differential kinematics
kevinzakka/mink - GitHub
stephane-caron/pink - GitHub