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Y. Xiang, D. Marsh, C. G. Gibson A New Philosophy for Kinematic Design of Mechanisms
Abstract
A new philosophy for the design of IDOF pianar mechanisms using bifurcation curves js presented. The method is exemplified by a computer package coupler which facilitates the design of planar four-bar coupler curves. The philosophy can be extended to the design of multi-DOF mechanisms with a desired workspace boundary.
Introduction
This paper is concerned with mechanism design. For a one degree of freedom (IDOF) planar mechanism, there are a number of bifurcation curves in its coupler plane which represent boundaries between qualitatively distinct types of motion [16]. Bifurcation curves have a potential use as a design tool since they can be used to obtain trajectories with a desired shape and specific singularities [10]. Until recently, extremely little was known about these bifurcation curves, except for the case of the planar four-bar for which there are a number of isolated results [13].
For IDOF planar mechanisms there are three bifurcation curves: the moving centrode, the transition curve, and the triple-point curve. In addition, there is Ball-point curve which takes into account points of inflexions [10, 13]. The union of these curves stratifies the coupler plane into a number of connected regions each corresponding to a distinct type of coupler curve. To exemplify the new philosophy we have developed a package coupler with a graphical interface which can quickly locate any shape of planar four-bar coupler curve with cusps, inflexion points, Ball's points, crunodes, tacnodes, ramphoid cusps, self-osculating points, triple-points etc. The facilities demonstrated by coupler can be extended to any planar IDOF mechanism, and ultimately to the design of multi-DOF planar mechanisms with a prescribed workspace boundary [16].
Coupler curves: shapes and singularities
The shape of planar four-bar coupler curves have been described as eggs, pears, raindrops, bananas, figure-of-eight and double figure-of-eights, etc. [1, 2, 11]. However, it is difficult to predict the shape of curve generated by a specific coupler point. Though various attempts have been made [1, 2, II, 12, 14, 15] the practical problem of obtaining coupler curves of a particular shape with specific singularities has not been systematically solved. Indeed, the bifurcation curves are necessary and sufficient to determine the occurence of any generic coupler curve singularity.
A new philosophy: stratification of the coupler plane
Our work is based on recent theoretical work [3,10]. The theory utilizes the fact that the position of a configuration of a IDOF (or 2DOF) mechanism may be represented by a point on a mechanism curve (or surface) in a higher-dimensional configuration space. The family of coupler curves (or workspace boundaries) may be obtained by a linear projection of the mechamsm curve (surface) and a number of geometric results can be directly obtained from the geometry of the mechanism curve (or surface). Gibson and Marsh [4,5,6,7,8] apply this approach to study several widely used mechanisms. In general, coupler curves of a IDOF mechanism only exhibit a relatively small number of distinct singularity types. The possible singularities have been classified and their
geometry described [3]. A consequence of the theory is that a family of coupler curves arising from the motion of a particular mechanism can be partitioned or stratified into sets of coupler curves with the same set of singularities. The stratification can be visualized via computer graphics thus leading to a new general tool and design philosophy.
A four-bar design tool: COUPLER
To exemplify the philosophy, we have developed a graphics package coupler (coded in the C language, and implemented on SunSparc workstations using the SunPhigs library) to facilitate coupler curve design of both non-degenerate and degenerate planar four-bars. Coupler curves with cusps, inflexion points, Ball's points, crunodes, tacnodes, ramphoid cusps, or self-osculating points, etc. are easily obtained from a visualization of the stratification of the coupler plane. Table I summarizes the conditions for different types of singularities to occur) where centrode, transition, triple refer to the moving centrode, the transition curve, the triple-point curve respectively, and fe means intersections among these bifurcation curves. When a coupler point is placed at a specific position in a coupler plane given by the right column of Table I, a coupler curve traced by this coupler point will hold a singularity given by the corresponding left column.
Table 1: Conditions for singularities to occur
singularity type position of a coupler point
node occur generally
cusp on centrode
tacnode on transition
ramphoid cusp where centrode & transition tangent
fleenode at a cusp of transition
triple-point on triple
cusp-plus-line at centrode & transition & triple
tacnode-plus-line at transition fe triple
Package coupler is a useful tool for exploring the features of various planar four-bar mechanisms and understanding their motions. By using it one can easily achieve the following tasks which are not available before.
I. To trace the Ball-point curve, the moving centrode, the transition curve and the triple-point curve. Among these four curves, the former two are relatively easy to obtain, and algorithms for obtaining them can be found in [16] and elsewhere. The algorithms to trace the transition and the triple-point curves are new and are derived from the underlying geometry of the mechanism curve. The resulting algorithms have proved to be efficient and robust [9,16].
• To generate the stratification of the coupler plane. This is achieved by superimposing the bifurcation curves and the Ball-point curve in the coupler plane. The stratified plane is visualized via computer graphics where each curve Jf P*ot*ed by different colours on a computer screen. These curves are the boundaries of regions of coupler points which give rise to distinct types of motions. Any two coupler points in the same region give rise to coupler curves 3 W|th an identical set of singularities.
o trace a coupler curve with a desired singularity. COUPLER permits the user to
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choose a coupler point within any region of the stratification and to generate the corresponding coupler curve. The computer screen is divided into different viewports for rendering the stratifications and coupler curves etc. Whenever users choose a coupler point in the stratified coupler plane by clicking a mouse button in its viewport, the corresponding coupler curve is immediately generated and displayed in the coupler curve viewport. For example, according to Table I, when a cusp on a transition curve is chosen as the coupler point its coupler curve immediately appears on the screen holding a flecnode.
Further features of coupler include animation of the mechanism, zooming in or zooming out on particular features, Postscript output of the stratifications and coupler curves etc. Detailed descriptions for coupler can be found in Xiang [16].
Figure I and Figure 2 present two examples generated by coupler for planar four-bar mechanisms. Figure I is for a non-degenerate four-bar with lengths 1.5, 1.2,2.1, l.S, where the left picture shows the stratification of the coupler plane generated by the moving centrode (dotted) and the transition curve (solid) and the right picture shows the coupler curve with a ramphoid cusp when the tangent point po between the two bifurcation curves is chosen as the coupler point. Figure 2 is for a degenerate four-bar with lengths 1.0, 1.0,0.8,0.8, where the left picture shows the stratification generated by the triple-point curve and the transition curve, and the right picture shows the coupler curve with a tacnode-plus-line point when the tangent point po between the two bifurcation curves is chosen as the coupler point.
Summary
In this paper we describe a new design philosophy for IDOF planar mechanisms based on the bifurcation curves which give rise to a stratification of the coupler plane. A computer package coupler has been developed which renders the stratification of the planar four-bar on a computer screen and provides various interactive tools for a user to obtain coupler curves with prescribed shapes and singularities. The philosophy has a natural generalization to the design of multi-degree of freedom mechanisms with a desired workspace boundary. Rigorous efforts have been directed to study them. After the success for IDOF planar mechanisms, we have obtained many important results for some 2DOF planar mechanisms, in particular, the planar five-bar mechanism and the planar double four-bar mechanism. The stratifications and other features of the mechanisms will be published in our forthcoming papers.
References
1. Artobolevskii et al.: Synthesis of Planar Mechanisms, Fizmatgiz, Moscow, 19S9.
2. Davies T. H.: Proposals for a Finite 5-Dimensional Atlas of Crank-Rocker coupler curves, Mechanism and Machine Theory, 16, (5), 1981.
3. Gibson, C. G. and Hobbs, C. A.: Local Models for General One-Parameter Motions of the Plane and Space, Proc. Royal Soc. Edinburgh, 125A, 639-656,1995.
4. Gibson, C. G. and Marsh: Concerning Cranks and Rockers, Mechanism and Machine Theory, 23, (5), pp. 355-360,1988.
5. Gibson, C. G. and Marsh, D.: On the Linkage Varieties of the Watt 6-Bar Mechanisms - 1: Basic Geometry, Mechanism and Machine Theory, 24, (2), pp. 106-113, 1989.
6. Gibson, C. G. and Marsh, D.: On the Linkage Varieties of the Watt 6-Bar Mechanisms - II: The Possible Reductions, Mechanism and Machine Theory, 24, (2), pp. 115-121,1989.
7. Gibson, C. G. and Marsh, D.: On the Linkage Varieties of the Watt 6-Bar Mechanisms - III: Topology of the Real Varieties, Mechanism and Machine Theory, 24, (2), pp. 123-126,1989.
8. Gibson, C. G. and Marsh, D.: On the Geometry of Geared 5-bar Motion, Journal
ofMechanical Design, 112, (4), pp. 620-627,1990. rJenerate the
9. Gibson, C. G., Marsh, D. and Xiang, Y, An Algorithm Xo Generate^the
Transition Curve of the Planar Four-bar Mechanism, Mechanism and Machi ry,
,nPri0.Gibson, C. G. and Newstead, P. E„ On the Geometry of the Planar 4-Bar Mechanism, Acta Applicandae Mathematicae, 7, pp. j 13-135, • inkace The
11.Hrones, J. A and Nelson, G. L.: Analysis of the Four-bar Linkage,
Technology Press of MIT and Wiley, New York, 1951. iTniwercitv Press 1978.
12. H5t, K. H.: Kinematic Synthesis of Mechan.sms OxtodUn,^
[13] Muller, R.: Translations, Kansas State Univeraty BuUetin 46 (6) Spec p No. 21,1962. [14] Torfason, L. E. and Armed, A.: Mechanism and Machine Theory, 13,
197813.Wang, D. and Xiao, D.: Distribution of c°Mer for Crank-Rocker
Linkages, Mechanism and Machine Theory, 28, (5), pp. 671- > • . Q ^ jw0
14.Xiang, Y.: Bifurcations of Singularities of Planar Lmkages with One and iwo Degrees of Freedom, Ph.D thesis, Napier University, Edinburgh w.
YAK 658.512.2
Chauncey Powis SOFTWARE PROCESS: Presentation and Use
The Software Process can be represented byfour strategic forces. Each force can be described by multiple factors. This model of the Software Process can be used to compare Software Engineering Methodologies. The description offerees byfactors can provide a means for analysis of existing or proposed Software Processes helping to ocumen strengths and weaknesses.
The process of developing engineering and business software has become a field in the world today. RAD (Rapid Application Development) and uu (Uiyeci Oriented) development methodologies have become common technical terms, ut ow well do these or other concepts fill our current and future software development neeasr This paper poses a frame work of forces based upon subordinate factors on which development methodologies may be compared. An effective Software Process is dependent on the presentation and use of any development methodology used today.
A desirable Software Engineering Methodology must consider and attempt to
reconcile four strategic forces to emerge and remain in the field software engineering, the
four forces are: Tangible Quality, Intangible Quality, Time/Amount, and Business
Economics. This is a complex relationship of forces where maximizing the Process
Completeness factor in the Tangible Quality force may limit the Early Usable Results
factor in the Time/Amount force. Each of the four Software Process forces will be briefly
discussed with their supporting factors. Suggested initial questions are also included with
each factor. These questions may be applicable prior to, during, or at the completion ot all projects.
Tangible Quality Force
thesJf6 TangibIe Q^lity Force contains factors that are semi-subjective in nature, but nro' /TSk arC cr't'ca* t0 the Software Process effort that is necessary to complete a fo-^ factors are more easily measured than the factors in the Intangible Quality
