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In this work we present several parallel robots with reduced mobility whose platforms can change their subgroups of displacement when the robot is displaced continuously from one set of positions to another one. In some cases, also the number of degrees of freedom of the platform may change, in other cases, only the group of displacement or its invariant properties are modified. By using some results on mobility of single-loop kinematic chains based on the theory of the displacement groups, the way to synthesize these robots is discussed.

This paper presents a systematic theory for metric relations between the invariant properties of displacement groups, and shows this theory application to mechanism kinematics. Displacement groups, their invariant properties and operations are briefly described. Kinematic constraints are then introduced as tools for relating abstract group properties to actual mechanism constraints. Criteria and operating rules to employ metric relations for the generation of a meaningful set of closure equations for kinematic chains are detailed.

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