A Unified Approach to Dyad and Triad Synthesis for Planar Mechanisms for Motion Generation

In this paper, we present a unified approach to the synthesis of planar dyads and triads found in planar closed-loop mechanisms for the motion generation problem. By decomposing a planar mechanism in terms of dyads and triads as building blocks, the problem of mechanism synthesis is reduced to the identification of such entities. By using planar quaternions and kinematic mapping, a common algebraic form of the geometric constraints of the dyads and triads is derived and a two-step algorithm then finds the type and dimensions of dyads and triads. The input to the problem is a set of constraints on poses, pivot locations, or other practical constraints, which can be represented in an algebraic form. The two-step algorithm first assembles these constraints as a set of linear equations, uses Singular Value Decomposition to obtain a candidate solution space, which is further subjected to constrained Lagrangian optimization to find the exact or approximate dyads or triads. This algorithm outputs a set of dyads and triads as solutions, which can be combined to obtain four-bar and six-bar mechanisms as well as planar parallel manipulators.