Inverse Kinematics
How it works
Analytical methods derive closed-form joint-angle equations from the manipulator geometry (possible for robots with suitable structure, e.g. a spherical wrist). Numerical methods iteratively minimize the end-effector pose error using the manipulator Jacobian — e.g. Jacobian transpose, Jacobian pseudoinverse, Damped Least Squares (Levenberg–Marquardt) or CCD (Cyclic Coordinate Descent). For redundant robots, extra degrees of freedom serve secondary goals (collision, joint-limit and singularity avoidance). Near singularities, regularization (DLS) is used to prevent solution blow-up.
Problem solved
Robot tasks are usually defined in Cartesian space (where the gripper should go), while the robot is actuated in joint space. IK bridges this gap by converting a task-space goal into joint commands.
Components
A description of robot geometry (e.g. Denavit–Hartenberg parameters) defining the joint-to-end-effector-pose relation.
A matrix relating joint velocities to end-effector velocity; the basis of numerical IK methods.
The algorithm producing the solution: closed-form equations or iteration (pseudoinverse, DLS, CCD).
Implementation
Near singularities the Jacobian loses rank and naive pseudoinverse yields very large joint velocities.
Redundant robots have infinitely many solutions — a selection criterion must be defined.
Evolution
The standard formalism for describing kinematic chains, a foundation for IK analysis.
Popularization of damped least squares (Wampler; Nakamura/Hanafusa) for stable IK near singularities.
Hyperparameters (configurable axes)
Analytical vs numerical (pseudoinverse, DLS, CCD, optimization-based).
Regularization near singularities in Damped Least Squares.