A Variational Approach for the Design of the Spatial Four-Bar Mechanism

An analytical formulation for computing kinematic sensitivity of the spatial four-bar mechanism is presented. The experimental code developed is used to compute an assembled configuration for the mechanism due to a design variation. The mechanism is modeled using graph theory where a body is defined as a node and a kinematic joint is defined as an edge. The spherical joint is cut to convert the model into a tree structure by cutting an edge and introducing constraints. The effect of variation in mechanism design using concepts of virtual displacement and rotation is introduced. The variation of the spherical constraint is computed while maintaining joint-attachment vectors and orientation matrices as variables. A system of equations that has more design variables than equations is then solved using the modified Moore-Penrose pseudo inverse. A recursive formulation is introduced to obtain the state variation of a body in terms of the state variation of a junction body and of the relative coordinates along the chain. The Jacobian matrix is then transformed from Cartesian coordinate space to joint coordinate space using velocity transformation matrices. Kinematic sensitivity analyses due to changing a joint-attachment vector and an orientation are presented. I. INTRODUCTION The planar and spatial four-bar mechanisms are some of the most used machine subsystems. Because the concept of these mechanisms is old and widely used, they have been extensively studied. In recent years, the emphasis has been on the synthesis using numerical techniques. For example, the finite position synthesis of a spherical four-bar mechanism using a mapping approach of the desired positions to points in an image space was presented by Bodduluri and McCarthy [1]. A solution of the nine-point path synthesis of four-bar linkages was presented by Wampler, et al. [2] where the problem is formulated and solved using a combination of classical elimination, multihomogeneous variables, and numerical continuation methods. Another method that uses the continuation method in the analysis of four-bar linkages was reported by Subbian and Flugrad [3]. Graphical synthesis of four-bar linkages were reported by Norton, et al. [4] and Khanuja, et al. [5]. Other synthesis methods are reported by Hwang and Chang [6].