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

Abstract

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.

A Task-Driven Approach to Unified Synthesis of Planar Four-Bar Linkages Using Algebraic Fitting of a Pencil of G-Manifolds

“…(6). In particular, when a 0 6 ¼ 0, we obtain the constraint manifold of a RR dyad whose projection onto the hyperplane Z 4 ¼ 1 is a hyperboloid of one sheet [7,9,14,21]. Figure 3 shows an example of such a hyperboloid.…”

“…This includes the definition of the approximation error (called structural error) in the image space, formulation of a least squares problem and application of appropriate numerical methods to find values of the design variables for minimization of the error. Pursuant to Ravani and Roth's kinematic mapping approach for mechanism synthesis, further research has been done by Bodduluri and McCarthy [15], Bodduluri [16], Larochelle [17,18], Ge and Larochelle [19], Husty et al [20], and more recently by Wu et al [21], Purwar and Gupta [22], and Hayes et al [23,24]. Schrcker et al [25] applied the kinematic mapping approach to detect branch defect in the planar four-bar linkage synthesis-a result that can be used in this work as well.…”

“…Their method is limited to RR type dyads only and involves solving bivariate cubic equations. More recently, Wu et al [21] and Purwar and Gupta [22] have demonstrated a visual, computer graphics approach for multidegrees-of-freedom mechanism design that exploits the constraint manifold geometry and its apparent effect on the parameters of a mechanism to interactively perform kinematic synthesis. Hayes et al [23,24] have presented preliminary results for combining type and dimensional synthesis of planar mechanisms for multipose rigid body guidance.…”

A Unified Approach to Exact and Approximate Motion Synthesis of Spherical Four-Bar Linkages Via Kinematic Mapping

“…This includes the definition of the approximation error (called fitting error) in the image space, formulation of a least-squares problem, and application of appropriate numerical methods to find values of the design variables for minimization of the error. Pursuant to Ravani and Roth's kinematic mapping approach for mechanism synthesis, further research has been done by Bodduluri and McCarthy [15], Bodduluri [16], Ge and Larochelle [17], Husty et al [6], Wu et al [18], and Purwar et al [19], both of which have demonstrated a visual, computer graphics approach for multi-degrees-of-freedom mechanism design that exploits the connection between constraint manifold geometry and its apparent effect on the parameters of a mechanism to interactively perform kinematic synthesis. Hayes and Rusu [20] have presented preliminary results for combining type and dimensional synthesis of planar mechanisms for multipose rigid body guidance.…”

Synthesis of Spatial RPRP Closed Linkages for a Given Screw System

“…See Ravani and Roth [14] and more recent applications by Hayes [15], Schröcker [16] and Wu and Ge [17]. For spatial motion, Study's kinematic mapping is used to obtain simplified equations for analysis and synthesis, see Husty et al [18] and [19].…”