A systematic approach to the instantaneous duality of mechanisms and its application

The instantaneous duality (also known as symmetry) between serial chains and fully parallel manipulators is well known in the literature. This paper takes the idea one step further, by introducing a systematic method that transforms one mechanical system into another. This duality concept rests on the concept of dual graphs to define the kinematics of the dual system. The mechanism structure can be represented in two essentially different ways: its kinematic topology or its constraint system. The first is embodied by the topological graph and the second by the constraint graph. The dual to a topological graph is a constraint graph and vice versa. Hence, there are various ways to introduce a dual topology. The dual kinematics is defined by instantaneously identifying the twist screws in the original system with the wrench screws of the dual. This identification allows carrying over statements from the original to the dual system. In particular, it is shown that the singularities can be easily established in the dual if they are known in the original system. This concept of transference is a powerful tool since a variety of dual systems can be assigned to a given system. This idea is demonstrated for a Bricard mechanism that is instantaneously dual to a 6/6 Stewart platform at a singular position, and in another configuration (resembling the cyclohexane molecule) it is dual to the 6/3 Stewart platform at the singular position. This provides another perspective of the known mobility of this molecule.