Interactive Dimensional Synthesis and Motion Design of Planar $6R$ Single-Loop Closed Chains via Constraint Manifold Modification

In this paper, we present an interactive, visual design approach for the dimensional synthesis of planar $6R$ single-loop closed chains for a given rational motion using constraint manifold modification. This approach is implemented in an interactive software tool that provides mechanism designers with an intuitive way to determine the parameters of planar mechanisms, and in the process equips them with an understanding of the design process. The theoretical foundation of this work is based on representing planar displacements with planar quaternions, which can be seen as points in a special higher dimensional projective space (called the image space), and on formulating the kinematic constraints of closed chains as algebraic surfaces in the image space. Kinematic constraints under consideration limit the positions and orientation of the coupler in its workspace. In this way, a given motion of a mechanism in the Cartesian space maps to a curve in the image space that is limited to stay within the bounds of the algebraic surfaces. Thus, the problem of dimensional synthesis is reduced to determine the parameters of equations that describe algebraic surfaces. We show that the interactive approach presented here is general in nature, and can be easily used for the dimensional synthesis of any mechanism for which kinematic constraints can be expressed algebraically. The process of designing is fast, intuitive, and especially useful when a numerical optimization based approach would be computationally demanding and mathematically difficult to formulate. This simple approach also provides a basis for students and early designers to learn and understand the process of mechanism design by simple geometric manipulations.