Analytical Inverse Kinematics

An inverse-kinematics method that plugs a target end-effector pose into pre-derived formulas to get all joint angles directly.

Inverse kinematics — finding joint angles from a target end-effector pose — has two solving approaches. Numerical methods iterate toward an answer; analytical methods derive formulas ahead of time and plug in the target pose to get every joint angle in one shot. Closed-form solutions only exist for specific arm geometries; the classic enabling condition for a 6-axis arm is that the last three joint axes all intersect at a single point (a spherical wrist), which lets position and orientation be solved separately — this is exactly the PUMA-style arm layout. The benefit is speed, numerical stability, and the ability to enumerate every solution: Modern Robotics notes that a general 6R serial arm has at most 16 solutions, while a PUMA-style arm with offsets has 4 solutions just for the position sub-problem. The downside is that a different arm geometry requires re-deriving everything from scratch. The tool IKFast can automatically analyze a kinematic chain and generate C++ code for its analytical solution, solving in just a few microseconds per call.

ExampleA PUMA-style 6-axis arm first solves the first three joints from the wrist-center position (shoulder left/right times elbow up/down gives 4 combinations), then solves the last three wrist joints from the end-effector orientation, with the wrist able to flip as well; an arm with this spherical-wrist layout has up to 8 solutions for a given end-effector pose, and the controller typically picks whichever is closest to the current joint angles.

Also called

Closed-Form IK, Analytic IK

Related

Inverse Kinematics (IK) · Numerical Inverse Kinematics · Forward Kinematics (FK) · Spherical Wrist · IKFast · Kinematic Redundancy

Sources

Modern Robotics: Mechanics, Planning, and Control（Lynch & Park, 2017 预印本）第 6 章 Inverse Kinematics (Chinese)
Inverse kinematics - Wikipedia
IKFast Kinematics Solver - MoveIt Documentation