Task-Space Control

Computing error and issuing commands based on the end-effector's position and orientation, rather than individual joint angles.

Task-space control writes the goal in terms of the end-effector's (gripper or tool) pose rather than each joint's angle: given a desired end-effector trajectory, it computes the deviation between the actual and desired end-effector state, then converts that into joint commands. There are two common routes for the conversion. At the velocity level, the pseudoinverse of the Jacobian matrix J (which maps joint velocity to end-effector velocity) is used: θ̇ = J⁺(θ)·V, where θ̇ is joint velocity and V the desired end-effector velocity. At the torque level, τ = Jᵀ(θ)·F converts a desired end-effector force F into joint torque τ; Khatib's 1987 operational space control belongs to this category and additionally accounts for the end-effector's effective inertia. It matches task descriptions like ‘move the cup here’ more naturally than joint-space control, at the cost of having to handle singularities (poses where some directions suddenly can't move) and redundant degrees of freedom.

ExampleIn Diffusion Policy's Franka setup, the policy outputs a desired end-effector pose at 10 Hz, and a mid-level controller running at about 1 kHz solves a quadratic program each step to find the joint velocities that best track that target end-effector velocity, integrating them into joint positions handed to the arm's built-in joint controller; obstacle avoidance and joint limits are written as constraints, and any leftover degrees of freedom are used in the null space.

Also called

Cartesian-Space Control, End-Effector Control, Cartesian Control

Related

Task Space · Joint-Space Control · Operational Space Control · Jacobian Pseudoinverse · Inverse Kinematics (IK) · Cartesian Impedance Control

Sources

Lynch & Park, Modern Robotics (2017 preprint), 11.3.3 / 11.4.3 Task-Space Motion Control
Khatib, A unified approach for motion and force control of robot manipulators: The operational space formulation (IEEE J. Robotics & Automation, 1987)
Diffusion Policy: Visuomotor Policy Learning via Action Diffusion（附录 Franka Robot Station） (Chinese)