Spades: Software for Synthesizing Spatial 4c Mechanisms

In this paper we present SPADES, an interactive graphics based software package for SPA tial mechanism DES ign. The program provides a platform for the synthesis of a mechanism that guides a body through either three or four prescribed positions in space. The purpose of this work is to assemble the current spatial 4C synthesis theory into a software package that is useful for spatial mechanism design and research. INTRODUCTION The synthesis of planar mechanisms is inherently a two dimensional problem. Therefore, the design techniques are well suited to a drafting table, blackboard, etc. This is not true of spatial mechanisms. The inherent three dimensionality of these mechanisms makes such two dimensional graphical constructions difficult. For these mechanisms it is useful for the designer to be able to visualize the entire problem in its three dimensions. Modern computer workstations provide the high speed graphics capabilities which make possible real-time visualization of spatial mechanisms. SPADES uses the three dimensional graphics capabilities of a Silicon Graphics Indigo2 desktop workstation to provide the interactive environment needed to design spatial 4C mechanisms. Previous efforts which utilized three dimensional graphics in computer-aided mechanism design are summarized in Erdman (1995) and Erdman (1993) and are presented in Rubel and Kaufman (1977), Barris, Kota, Riley, and Erdman (1988), 1 Thatch, and Myklebust (1988), and Larochelle et al (1993). SPADES is a computer graphics based interactive program for designing spatial mechanisms formed by a closed chain consisting of four cylindric(C) joints. Cylindric joints allow both translation along and rotation about an axis, hence, they are two degree of freedom joints. The result of rigidly connecting four C joints is a two degree of freedom spatial closed chain mechanism, referred to as a spatial 4C mechanism. SPADES facilitates the synthesis of spatial 4C mechanisms to guide a body through three or four finitely separated positions in space. The theory for the design of spatial mechanisms for four position rigid body guidance is analogous to that for planar mechanisms. In the planar case the designer specifies four positions in the plane and computes the set of points in the moving body which have four positions on a circle, see Hartenberg and Denavit (1964) and Burmester (1888). These points are the moving pivots of planar RR dyads compatible with the four positions and they form a circular cubic curve called the circle point curve. The points that are the centers of these circles are the corresponding fixed pivots of the planar RR dyads and they form a cubic curve called the center point curve. In the case of spatial 4C synthesis the designer specifies four positions in space. The set of lines in the moving body which maintain a constant distance(i.e. normal distance and twist) from a fixed line form a congruence of lines called the moving line congruence. The set of corresponding fixed lines of the CC dyads form the fixed line congruence, see Larochelle (1995) and Bottema and Roth (1979) for further disCopyright  1998 by ASME cussions about these line congruences and their properties. The major difference between planar and spatial finite position synthesis is the essential three dimensionality of spatial mechanisms. While synthesis curves for planar mechanisms may be sketched or plotted in two dimensions the synthesis congruences for spatial mechanisms and the mechanisms themselves must be viewed in their full three dimensional form. The designer requires the ability to manipulate the synthesis congruences and view them in an arbitrary orientation in order to gain an understanding of the available choices of fixed and moving axes. Furthermore, once the linkage has been specified the evaluation of its motion also requires the ability to view the linkage in a three dimensional environment. The remainder of the paper proceeds as follows. First, we review the underlying synthesis theory utilized by SPADES to perform synthesis for three and four position rigid body guidance. This is followed by a summary of the kinematic analysis tools employed by SPADES and a discussion regarding the organizational structure of the software. Finally, we conclude with a design case study which illustrates the utility of SPADES as a research and design tool and discuss the planned future development of SPADES. THREE POSITION SYNTHESIS The methodology utilized for performing the dimensional synthesis of spatial 4C mechanisms for three position rigid body guidance involves synthesizing two CC dyads separately and then joining them with a coupler to form a complete closed chain mechanism. This algorithm is based upon the works of Larochelle (1994), Suh and Radcliffe (1978), and Tsai and Roth (1973). We consider one CC dyad of the spatial 4C mechanism as shown in Fig. 1. Let the axis of the fixed joint be specified by the dual vector û measured in the fixed reference frame and let the moving axis be specified by λ̂ measured in the moving frame M, see Bottema and Roth (1979) for further discussion of dual vectors. In a CC dyad there is a link which connects the moving line and the fixed line. This link is assumed to be rigid and therefore the twist and normal distance between the fixed and moving lines of the dyad remain constant. The constant twist condition may be expressed as,