Spherical Wrist

A wrist structure where a manipulator's last three joint axes all intersect at a single point.

A spherical wrist means the axes of a 6-DoF manipulator's last three revolute joints all intersect at one point (the wrist center), behaving much like a single ball joint: it only changes the end effector's orientation without moving the wrist center's position. Donald Pieper proved in his 1968 Stanford PhD thesis that a serial manipulator with 6 revolute joints, three of which are adjacent and share a common intersection point, has a closed-form inverse kinematics solution — a result later named the Pieper criterion. The reason is that position and orientation decouple: the wrist center's required position can be worked out first from the target pose, and the first three joints alone can place the wrist center there; the last three joints then supply the required orientation. This means the inverse solution can be computed directly with formulas instead of iterating, and every solution can be enumerated (up to 8 solution sets for this kind of configuration). Many industrial six-axis arms use an orthogonal-parallel-base-plus-spherical-wrist layout, with the PUMA 560 as a classic example.

ExampleGiven a target pose for this kind of arm: first step back along the tool's approach direction by a fixed distance to find where the wrist center should be, use the first three joints to send the wrist center there, then solve for the last three joint angles to orient the gripper as required.

Also called

Pieper Criterion, Wrist-Partitioned Manipulator

Related

Inverse Kinematics (IK) · Analytical Inverse Kinematics · 6-Axis Robot Arm · IKFast · Denavit-Hartenberg (DH) Parameters · Singular Configuration (Kinematic Singularity)

Sources

Wikipedia: Donald L. Pieper
Wikipedia: Inverse kinematics