Principle:ARISE Initiative Robosuite Operational Space Control
| Knowledge Sources | |
|---|---|
| Domains | Robotics, Control Theory, Dynamics |
| Last Updated | 2026-02-15 07:00 GMT |
Overview
Operational space control (OSC) computes joint torques that produce desired end-effector forces and torques by leveraging the robot's full dynamic model, enabling compliant task-space impedance behavior.
Description
Operational space control, originally formulated by Oussama Khatib (1987), provides a principled framework for controlling a robot manipulator directly in task space (the Cartesian space of the end effector) rather than in joint space. Unlike kinematic approaches that compute desired joint positions, OSC computes the joint torques required to achieve a desired task-space wrench (force and torque) while respecting the manipulator's inertial dynamics.
The central idea is to define a task-space impedance relationship F = K_p * e_pos + K_d * e_vel where e_pos is the position/orientation error between the desired and actual end-effector pose, e_vel is the velocity error, and K_p and K_d are the stiffness and damping gain matrices respectively. The damping is typically set as K_d = 2 * sqrt(K_p) * zeta where zeta is the damping ratio (zeta = 1 for critical damping). The desired task-space wrench is then projected through the dynamically-consistent pseudo-inverse of the Jacobian to obtain the corresponding joint torques.
A key aspect of OSC is the use of the operational space inertia matrix Lambda = (J M^{-1} J^T)^{-1} where M is the joint-space mass matrix and J is the manipulator Jacobian. This matrix decouples the task-space dynamics from the joint-space dynamics, ensuring that torques applied in task space do not produce unintended accelerations. Position and orientation control can optionally be decoupled by computing separate Lambda matrices for the translational and rotational components. Null-space torques maintain a preferred joint posture without affecting the end-effector motion.
Usage
Apply operational space control when compliant, force-aware behavior at the end effector is required, such as contact-rich tasks, impedance-controlled manipulation, or when task-space dynamics decoupling is important. OSC is the preferred choice when the robot dynamics model is available and the task benefits from variable impedance (adjustable stiffness and damping). It supports both delta and absolute position/orientation commands and can operate in either a world reference frame or the robot base frame.
Theoretical Basis
The operational space control law is derived as follows:
Given:
- Position error: e_pos = x_des - x_act
- Orientation error: e_ori = orientation_error(R_des, R_act)
- Velocity errors: e_vel_pos, e_vel_ori
- Gains: K_p (stiffness), K_d = 2 * sqrt(K_p) * damping_ratio
1. Compute desired task-space wrench:
F_pos = K_p[:3] * e_pos + K_d[:3] * e_vel_pos
F_ori = K_p[3:] * e_ori + K_d[3:] * e_vel_ori
2. Compute operational space matrices:
M = joint-space mass matrix
J_full = full 6xN Jacobian (position + orientation)
Lambda = (J * M^{-1} * J^T)^{-1} (task-space inertia)
J_bar = M^{-1} * J^T * Lambda (dynamically-consistent pseudo-inverse)
N = I - J_bar * J (null-space projector)
3. Decouple position and orientation (optional):
Lambda_pos = (J_pos * M^{-1} * J_pos^T)^{-1}
Lambda_ori = (J_ori * M^{-1} * J_ori^T)^{-1}
F_decoupled = [Lambda_pos * F_pos; Lambda_ori * F_ori]
4. Compute joint torques:
tau = J^T * F_decoupled + gravity_compensation
5. Add null-space posture torques:
tau += N^T * (K_null * (q_0 - q) - D_null * dq)
Key design decisions:
- Impedance mode: Can be fixed (constant gains), variable (gains are part of the action space), or variable_kp (only stiffness varies, damping ratio fixed at 1.0 for critical damping).
- Position/orientation decoupling: Separating Lambda into positional and orientational components prevents cross-coupling between translational and rotational control.
- Reference frame: Actions can be specified relative to the robot base frame or the world frame, with automatic coordinate transformations applied.
- Goal update mode: Goals can be updated relative to the achieved state or relative to the previously desired goal, trading off responsiveness for tracking fidelity.