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A GEOMETRIC ALGORITHM FOR FINDING THE LARGEST MILLING CUTTER

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ISR develops, applies and teaches advanced methodologies of design and analysis to solve complex, hierarchical,heterogeneous and dynamic problems of engineering technology and systems for industry and government.ISR is a permanent institute of the University of Maryland, within the Glenn L. Martin Institute of Technol-ogy/A. James Clark School of Engineering. It is a National Science Foundation Engineering Research Center.Web site http://www.isr.umd.eduIRINSTITUTE FOR SYSTEMS RESEARCHTECHNICAL RESEARCH REPORTA Geometric Algorithm for Finding the Largest Milling Cutterby Zhiyang Yao, Satyandra K. Gupta, Dana S. NauTR 2001-321A GEOMETRIC ALGORITHM FOR FINDING THE LARGEST MILLING CUTTERZhiyang YaoMechanical EngineeringDepartment and Institute forSystems ResearchUniversity of MarylandCollege Park,MD-20742Email:[email protected] K. Gupta∗Mechanical EngineeringDepartment and Institute forSystems ResearchUniversity of MarylandCollege Park,MD-20742Email:[email protected] S. NauComputer ScienceDepartment and Institute forSystems ResearchUniversity of MarylandCollege Park,MD-20742Email:[email protected]: In this paper, we describe a new geometric algorithm to determine the largest feasible cutter size for2-D milling operations to be performed using a single cutter. In particular:1. We give a general definition of the problem as the task of covering a target region without interfering with anobstruction region. This definition encompasses the task of milling a general 2-D profile that includes bothopen and closed edges.2. We discuss three alternative definitions of what it means for a cutter to be feasible, and explain which of thesedefinitions is most appropriate for the above problem.3. We present a geometric algorithm for finding the maximal cutter for 2-D milling operations, and we show thatour algorithm is correct.KEYWORDS: Computer-Aided Manufacturing, Process Planning, Cutter Selection for Milling1 INTRODUCTIONNC machining is being used to create increasingly complex shapes. These complex shapes are used in a variety ofdefense, aerospace, and automotive applications to (1) provide performance improvements, and (2) create highperformance tooling (e.g., molds for injection molding). The importance of the machining process is increasing dueto latest advances in high speed machining that allows machining to create even more complex shapes. Complexmachined parts require several rouging and finishing passes. Selection of the right sets of tools and the right type ofcutter trajectories is extremely important in ensuring high production rate and meeting the required quality level. Itis difficult for human planners to select the optimal or near optimal machining strategies due to complex interactionsamong tools size, part shapes, and tool trajectories.Although many researchers have studied cutter selection problems for milling processes, there still exist significantproblems to be solved. Below are two examples:• Most existing algorithms only work on 2-D closed pockets (i.e., pockets that have no open edges), despite thefact that open edges are very important in 2-D milling operations.• Since there are several different definitions of what it means for a cutter to be feasible for a region, differentalgorithms that purport to find the largest cutter may in fact find cutters of different sizes.In this paper, we present an algorithm for finding maximal cutter for general 2-D milling operations to be performedusing a single cutter.• We formulate the general 2-D milling problem in terms of a target region and an obstruction region. Thisproblem formulation encompasses the general problem of how to mill a 2-D region that has both open edges(edges that don’t touch the obstruction region) and closed edges (edges that touch the obstruction region) (see ∗ Corresponding author.2Section 3 for definitions). Our formulation allows arbitrarily complex starting stocks. Therefore, it can be usedto model machining of geometrically complex castings such as engine blocks.• We analyze three different definitions of what it means for a cutter to be feasible, and explain why one of thesedefinitions is more appropriate than the others.• We describe a new “region covering” algorithm that finds the largest cutter that can cover the target regionwithout interfering with the obstruction region, and we give the correctness proof for our algorithm.In practice, quite often multiple cutters are used to machine a complex milling feature. Our region-coveringalgorithm is also useful as one of the stage in finding an optimal sequence of cutters for machining a complexfeature (for our subsequent work on this subject, please see [12]).2 RELATED WORKBecause of the wide range of the complexity of products, requirement for machine accuracy, different machiningstages, selecting optimal cutter size is an active research area. Below we provide a summary of previous research inthe area of milling cutter selection.Bala and Chang presented an algorithm to select cutters for roughing and finishing milling operations [2]. Theirwork encompasses almost all the features found on prismatic parts, such as slots, steps. Their algorithm is based ongeometric constraints. The basic idea is trying to fit the possible large circle into contours to select possible largecutter to save processing time. They take both cutter change time and geometric constraints into consideration. Theystated the problem as follows. If there exists a set of cutters, and after machining with these cutters, only finishingmachining is needed for fillet radii corners, so the problem is to determine the cutter with the largest radius in thisset. The main concern is to make sure that the material left behind by the cutter at each of the convex vertices can beremoved by one pass along the boundary of the finishing cutter. A convex vertex is defined as a vertex at which theinterior angle is less than 180°. For each convex vertex, the radius of the circle touching the edges forming thevertex as well as the fillet circle is found. Then they check to see if this circle intersects any of the edges of thebounding polygon or any of the islands. If an edge that violates the circle is found, the circle has to be modified orthe radius has to be reduced to remove this violation. The aim is to make the circle tangential to this edge. Afterthat, the reflex vertices have to be handled. Reflex vertices are those vertices at which


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