Principle:ARISE Initiative Robosuite Inverse Kinematics Control
| Knowledge Sources | |
|---|---|
| Domains | Robotics, Motion Planning, Kinematics |
| Last Updated | 2026-02-15 07:00 GMT |
Overview
Inverse kinematics control converts desired end-effector poses (position and orientation) into joint-space configurations that achieve those poses, enabling task-space reasoning for articulated manipulators.
Description
Inverse kinematics (IK) is a fundamental technique in robotics that solves the mapping from Cartesian task-space targets to joint-space configurations. Given a desired end-effector position and orientation, the IK solver computes the set of joint angles that place the end-effector at the target pose. Because this mapping is generally nonlinear and may admit multiple solutions (or none), iterative numerical methods are typically employed.
Differential inverse kinematics linearizes the forward kinematics around the current configuration using the manipulator Jacobian. The velocity-level relationship dX = J(q) dq is inverted using a damped pseudo-inverse to obtain joint velocity commands dq = J^T (J J^T + lambda^2 I)^{-1} twist, where lambda is a damping factor that prevents numerical instability near kinematic singularities. The resulting joint displacements are integrated to produce new joint position targets.
When the manipulator has more degrees of freedom than the task requires (a redundant manipulator), the null space of the Jacobian can be exploited. A secondary objective, such as maintaining a preferred posture, is projected into the null space via (I - J^+ J) K_n (q_0 - q), where q_0 is the rest posture and K_n are null-space gains. This ensures the secondary objective does not interfere with the primary task-space tracking. The resulting joint position targets are then fed into a lower-level joint position controller that applies PD torques to track them.
Usage
Apply inverse kinematics control when the task is most naturally expressed in end-effector coordinates (e.g., reach a Cartesian target, follow a trajectory in world space) and the robot has a reliable joint position tracking layer underneath. IK is particularly suited for teleoperation and policy-driven control where actions are specified as delta position/orientation commands. It is best used on robots whose kinematic models are well-characterized and for which Jacobian computation is efficient.
Theoretical Basis
The core algorithm proceeds as follows:
Given:
- Current joint configuration q
- Desired end-effector displacement dX = [dp; d_theta]
- Jacobian J(q) = d(FK(q)) / dq
- Damping factor lambda
- Null-space gain K_n and rest posture q_0
1. Compute the twist from the desired displacement:
twist[:3] = K_pos * dp / dt
twist[3:] = K_ori * d_theta / dt
2. Compute the Jacobian at the current configuration:
J = J(q) restricted to controllable joints
3. Solve for joint velocity via damped least-squares:
dq = J^T * (J * J^T + lambda^2 * I)^{-1} * twist
4. Add null-space posture control (for redundant robots):
dq += (I - pinv(J) * J) * K_n * (q_0 - q)
5. Clamp dq to maximum joint velocity:
if |dq|_max > v_max: dq *= v_max / |dq|_max
6. Integrate to get desired joint positions:
q_des = q + dq * dt
7. Feed q_des to a joint position controller (PD control with gravity compensation)
Key design decisions:
- Damping factor (lambda): A nonzero damping factor trades off tracking accuracy for numerical stability near singularities. Larger values increase robustness but reduce precision.
- Null-space gains: Weight the secondary posture objective per-joint to preferentially constrain certain joints (e.g., lower gains on wrist joints).
- Integration step (dt): Controls sensitivity of the solver. Smaller values yield finer-grained motion but require higher control frequency.
- Multi-site IK: When controlling multiple end-effector sites simultaneously (e.g., on a humanoid), a unified solver stacks the Jacobians and site errors into a single optimization, using null-space projections to manage redundancy across all sites.