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

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