via the WWW

Giuseppe Castanotto, USC-visiting

Ken Goldberg, UC Berkeley

## Contents
- Abstract
- Motivation
- Related Work
- Related Web Sites
- The Modular Fixturing Algorithm
- Quality Metric
- FixtureNet Interface
- Implementation
- Future Work
- Acknowledgments
- References
- About the Authors
## AbstractThe Internet offers tremendous potential for rapid development of mechanical products to meet global competition. In the past several years, a variety geometric algorithms have been developed to evaluate CAD models with respect to manufacturing properties such as feedability, fixturability, assemblability, etc. Unfortunately, most of these algorithms are tailored to a particular CAD system and format and so have not been widely tested by industry. The World Wide Web (WWW) may offer a solution: its simple interface language offers ade facto standard for the exchange of geometric data with
industry and research groups (e.g., to encourage verification of algorithms).
In this paper we describe a feasibility study for such an interactive system,
which can be tested online at:http://memento.ieor.berkeley.edu/fixture ## MotivationThe Internet has launched a revolution in the means of worldwide data transfer. WWW browsers such as Netscape have opened up new avenues in information and resource sharing. This has fueled the emergence of many commercial services and products that are marketed and accessed over the Net. This technology also offers potential for the design and manufacture of new products. For examples, see Enterprise Integration Technologies, Inc., Stanford's ACORN Project, and Autodesk's Mechanical Library:http://acorn.eit.com/acorn-info.html http://www.autodesk.com/products/datapub/mechlibr/mechlib.htmDigital communication over the Internet offers advantages in terms of speed, efficiency and automation. Fortunately, new geometric algorithms for design, simulation, and manufacture have been developed and reported in the research literature. Unfortunately, the impact of these advances on the manufacturing community has been limited despite a well documented need for improved communication during product development. At the same time researchers rarely have access to each others algorithms since implementations are difficult to port from one platform to another. In this paper we describe one model for interactive computer aided design (CAD) via the WWW that we believe holds promise for use both in industry (e.g., during the design cycle) and in research (e.g., to encourage verification of results). Our - Minimize data transfer
- Can be understood solely from on-line documentation
- Synchronize service requests from multiple simultaneous users.
FixtureNet demonstrates such an interface in the context of modular
fixturing. In automated manufacturing, parts undergoing fabrication
or assembly operations are often held or supported in fixtures.
Fixtures can either be custom designed or assembled from modular fixturing
kits. Modular fixtures are amenable to automated design, particularly if
the number of modular elemental types is small. Such is the case with the
three-locator-and-clamp planar modular fixturing set utilized in (Randy
Brost and Ken Goldberg, 1994 [2]). As a feasibility study, we implemented
an interactive WWW interface to that modular fixture design algorithm.
## Related WorkCommercially available "modular fixturing" systems typically include a square lattice of tapped and doweled holes with precise spacing and an assortment of precision locating and clamping elements that can be rigidly attached to the lattice using hardened bushings or expanding mandrels. Ordinarily, human expertise is required to synthesize a suitable arrangement of these elements to hold a given part (E. G. Hoffman. Modular Fixturing 1987 [5]). Besides being time consuming, if the set of alternatives is not systematically explored, the designer may fail to find an acceptable fixture or may settle upon a sub-optimal fixture. Our fixture design algorithm often generates counter-intuitive solutions that may be overlooked by even an experienced machinist, much as chess machines can play moves that look naive at first glance but lead the experienced chess player to explore new variations.Work related to ours includes user interface design, software testing, and CAD. The WWW provides an unprecedented opportunity for a large number of researchers to test experimental computer programs. Often the designers of a research algorithm cannot anticipate the kinds of inputs a variety of users in related disciplines might subject the program to. The automated design of fixtures is a challenging research area. The earliest work in this area is related to the necessary conditions for holding parts (work pieces) securely.
## Form ClosureReuleaux (F. Reuleaux 1963 [9]) in "The Kinematics of Machinery" first described form closure which captures the intuitive requirement of a fixture: a part is held in form closure if it can resist arbitrary forces and torques. Lakshminarayana (K. Lakshminarayana 1978 [5]) showed that seven frictionless contacts are necessary to hold a 3D part in form closure; Mishra (Bud Mishra 1987 [7]) showed that seven frictionless contacts are also sufficient.Goldman and Tucker (A. J. Goldman and A. W. Tucker 1956 [3]), in a purely
mathematical paper on linear algebra, described the necessary and sufficient
conditions for positively spanning an
## Modular FixturingHoffman's text (E. G. Hoffman 1987 [5]) provides an overview of conventional practice with modular fixtures. Research on Modular Fixturing includes basic questions about the existence of solutions (Y. Zhuang, K. Goldberg, and Y. C. Wong 1994 [15]), practical extensions to 3D (Wagner, Zhuang, and Goldberg 1995 [10]) and the problem of fixture loading (Kyeonah Yu and Ken Goldberg 1995 [14]). We are also studying how the model can be extended to curved parts (Aaron S. Wallack and John F. Canny 1994 [12]).Asada and By (H. Asada and Andre B. By 1985 [1]) in "Kinematic Analysis of Workpart Fixturing for Flexible Assembly with Automatically Reconfigurable Fixtures" describe an automatic fixture reconfiguration system using a robot manipulator and a CAD system to provide a systematic method for designing fixtures. They also provide an analytic test for form closure and suggest how contact points might be applied, but they did not consider how a restricted set of modular elements could be used to reach those points. They call fixture synthesis "designing a fixture layout," which is in keeping with the mechanical drawing practice of calling a drawing that gives the locations of parts a "layout" drawing. They develop analytic tools for designing fixture layouts using a set of hardware primitives implemented at MIT. They also considered loading and unloading of their fixtures. Wolter and Trinkle (J. D. Wolter and J. C. Trinkle 1994 [13]) describe a non-modular fixture synthesis that uses analysis of frictionless stability in "Automatic Selection of Fixture Points for Frictionless Assemblies." This is an impressive paper because it applies to both 2D and 3D fixtures, but it is "non-modular" because the fixture points selected are from a continuum in space and not from a discrete set of locations. In their problem, frictionless elements of assemblies need to be held together by a fixture. They analyze fixtures for "stability" in terms of virtual work. Their fixture synthesis algorithm uses a "shotgun" approach: they scatter fixels about the assembly and solve a linear program to minimize contact forces at the fixels by having fixel location on the part boundary be a system variable. Fixels that have reaction forces of zero get discarded. This is an effective approach, but it is not guaranteed to find an optimal solution. Also, it is not applicable to modular fixturing hardware sets as currently available. Brost and Goldberg (Randy Brost and Ken Goldberg 1994 [2]) have demonstrated
a complete algorithm for synthesizing 2D fixtures that forms the basis
of Until recently there have been few papers describing modular fixture
synthesis algorithms in 3D. However, Wagner, Yzhuang, and Goldberg have
described a new modular strut hardware set and a complete algorithm for
automated fixture synthesis with these primitives in (Rick Wagner, Yan
Zhuang, and Ken Goldberg 1995 [10]), also the subject of an extension to
Recently, Ponce has described "immobilizing" grasps (Jean Ponce, Joel Burdick, and Elon Rimon 1995 [8]), and has proposed their possible application in fixturing in both two and three dimensions. Immobilizing fixtures require only three contact points in the plane and four contacts in three dimensions. The practical application of immobilizing fixtures is somewhat limited, however, in that when they are evaluated in terms of the quality metrics generally applied to form closure fixtures, they will be ranked below fixtures with form closure. This is because an immobilizing fixture generates very large reactions (assuming no friction and rigid parts and fixture elements) with the application of a moment load. Immobilizing fixtures may be a very attractive alternative for light duty applications with friction.
## Related Web SitesRelated Web sites include the following. All of these can be reached from ourFixtureNet "Related Links" page:
- Raju Mattikalli and Pradeep Khosla's online fixturing system at Carnegie Mellon University. Currently it supports analysis of a given set of 3D wrenches (to determine if they provide form closure), and it will soon be expanded to automatically synthesize fixture designs. Mattikalli and Khosla are also working with Mark Cutkosky at Stanford on a Java front-end for defining 3d parts.
http://www.cs.cmu.edu:80/afs/cs.cmu.edu/user/rajum/www/fix4.html -
University of Minnesota's
*Geometry Center*has a great collection of interactive geometric algorithms.
http://www.geom.umn.edu:80/apps/ -
David Eppstein's
*Discrete and Computational Geometry*page.
http://www.ics.uci.edu/~eppstein/geom.html - Jeff Erickson has been maintening a small collection of computational geometry World Wide Web pages.
http://www.cs.berkeley.edu/~jeffe/compgeom.html -
*Prof. Antonio Bicchi's Non-Holonomic Motion Planning Site*at University of Pisa allows users to define obstacles for path planning and even sets up a standard for others to submit algorithms for comparison.
http://www.piaggio.ccii.unipi.it/prova/motion.html -
*The AutomationNET!*In December 1996*PC Computing*as one of the best engineering Web sites.
http://www.AutomationNET.com/
## The Modular Fixturing AlgorithmBrost and Goldberg (Randy Brost and Ken Goldberg 1994 [2]) considered a class of modular fixtures that prevent a part from translating and rotating in the plane based on three round locators, each centered on a lattice point, and one translating clamp that must be attached to the lattice via a pair of unit-spaced holes, thus allowing contact at a variable distance along the principle axes of the lattice (see Figure 1). Brost and Goldberg gave an algorithm that accepts part geometry as input and synthesizes the possibly empty set of all fixture designs in this class that achieve form closure for the given part. If the part hasn edges and its maximal
diameter (in units of lattice spacing) is d, the algorithm runs
in time O(n5d5). The paper describes extensive experiments
run on a Lisp machine. This is the first fixture synthesis algorithm that
is complete in the sense that it guarantees to find an admissible fixture
if one exists.
## Quality MetricTheFixtureNet algorithm is complete in the sense that it
will find all solutions to a fixturing problem if any exist and report
failure if no solutions exist. FixtureNet often finds a large number
of solutions. The user generally cares only about a subset of the best
solutions. The issue of measuring the quality of a fixture is a research
topic in its own right. FixtureNet provides two very different quality
metrics, a default metric that minimizes part interface reactions under
a "general" load, and an optional metric that minimizes locator interface
reactions for a fixed clamp load. The "best" fixtures selected under these
two metrics are usually quite different sets.
Recently, we developed a WWW server to provide modular fixture design alternatives
to engineering users around the world. We refer to this service as |

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For questions or comments, please send e-mail to: Prof. Ken Goldberg