A DH-parameter based condition for 3R orthogonal manipulators to have 4 distinct inverse kinematic solutions

TL;DR

This paper presents a precise condition based on DH-parameters for 3R orthogonal manipulators to have four distinct inverse kinematic solutions, aiding in manipulator design.

Contribution

It derives a necessary and sufficient condition for 3R orthogonal manipulators to have four inverse kinematic solutions, linking link lengths and DH-parameters.

Findings

The transition between 2 and 4 solutions is characterized by quadruple roots.

An explicit inequality involving link lengths and DH-parameters is established.

The condition is useful for designing manipulators with desired kinematic properties.

Abstract

Positioning 3R manipulators may have two or four inverse kinematic solutions (IKS). This paper derives a necessary and sufficient condition for 3R positioning manipulators with orthogonal joint axes to have four distinct IKS. We show that the transition between manipulators with 2 and 4 IKS is defined by the set of manipulators with a quadruple root of their inverse kinematics. The resulting condition is explicit and states that the last link length of the manipulator must be greater than a quantity that depends on three of its remaining DH-parameters. This result is of interest for the design of new manipulators.