Local and Global Accessibility Evaluation with Tool Geometry

Transcription

1 19 Local and Global Accessblty Evaluaton wth Tool Geometry Jnnan Wang 1, Chell A. Roberts 2 and Scott Danelson 3 1 Arzona State Unversty, wangn@asu.edu 2 Arzona State Unversty, chell.roberts@asu.edu 2 Arzona State Unversty, scott.danelson@asu.edu ABSTRACT Accessblty analyss s an mportant step for the automatc generaton of machnng plannng. One am of accessblty analyss s to fnd a doman n whch the tool can maneuver wthout colldng wth the workpece. Most approaches to accessblty analyss use rays or half-lne proectons that do not nclude tool geometry consderatons. In ths paper we present an approach for the global accessblty analyss that ncludes tool geometry. The paper demonstrates how local accessblty analyss forms an upper bound for the tool radus selecton and how a global accessblty approach determnes pont and part accessblty. The results n ths paper are lmted to ball-endmll tool geometry. The paper also only consders tool workpece nterference and collsons and does not consder tool holder geometry. Experments on sample parts are shown to valdate the approach. Keywords: accessblty evaluaton, manufacturng plannng, tool geometry 1. INTRODUCTION Automated machnng plannng ncludes machne selecton, tool selecton, part orentaton, fxture plannng, toolpath plannng, and parameter selecton operatons towards optmzng obectves, such as cost, tme and/or qualty [4]. Much of the automated manufacturng plannng research for machnng processes s based on manufacturng features (MFs). Over the last decade, the concept of MFs has emerged as a powerful and wdely adopted representaton to assst n the converson of desgn nformaton nto machnng nstructons. MFs have proved to be convenent because they characterze the capabltes of machnng processes such as 3-axs mllng and turnng farly well. However, the feature-based approach s not wthout ts dsadvantages [1]. Frst, any feature-based system s lmted by the extent of ts vocabulary. Secondly, MFs are not drectly avalable from CAD representatons. They must be extracted by a process referred to as feature extracton or they must be desgned nto the model. Whle there has been sgnfcant progress wth feature-based approaches, there stll reman problems to fully automated manufacturng plannng due to recognton of nteractng features and complex features wthn sculptured surfaces. Decades of machnng plannng research has resulted n some lmted successes of algorthmc plannng, but has faled to produce a complete and consstent approach. Ths s due, n large part, to the complexty of analyzng a computer product model relatve to a specfc manufacturng process and a specfc tool. Common problems nclude matng the geometry of a tool wth the geometry of the product model, determnng the set of possble tool orentatons for the model geometry and tolerances, and fttng nspecton data to the model geometry and tolerances. Key to the soluton of these problems s the avalablty of mathematcal models for tool/part accessblty. Both the feature-based approach and the prmtve-based approach need to solve the problem of tool/part accessblty. One am of accessblty analyss s to fnd a doman n whch the tool can maneuver wthout colldng wth the workpece [6]. There are two approaches to analyzng the accessblty of a workpece: (a) entre feature accessblty, where the accessblty of the whole workpece s evaluated, and (b) pont accessblty, where the drectons along whch the tool can reach a specfc pont are examned. The entre feature accessblty doman can be derved from the pont accessblty domans. The accessblty domans are ndependent of the part orentaton, and optmal part orentatons are determned after analyzng the accessblty domans of cuttng locaton (CL) ponts. Pont p s accessble to a tool n NC machnng f the tool does not penetrate the part when ts cuttng pont s placed at p [5]. Fxturng devces and other obstacles are not consdered n ths paper.

2 20 In collson-free machnng, both local and global gougng effects should be avoded. Local gougng means that n all arbtrary small neghborhoods of the current touchng pont, the mllng tool cuts nto the fnal surface of the workpece, thereby destroyng t. More complcated are global collsons occurrng at some dstance from the touchng pont [7]. Local accessblty means the tool s cuttng wthout local gougng; so does global accessblty. Fg. 1 demonstrates the dstncton between local and global accessblty. To perform automated machnng plannng, the accessblty nformaton wth tool geometry of the part should be evaluated. Ths can then be used for machne selecton, tool selecton, part orentaton, fxture plannng, tool path plannng and parameter selecton operatons. tool a. Global Accessblty Cone b. Fg.1: a. Global accessblty: tool relatve to entre part, and, b. Local accessblty: tool relatve to part surface. In ths paper, we present an approach for the global accessblty analyss wth tool geometry. In secton 2, we show how local accessblty analyss forms an upper bound for the tool radus selecton. Secton 3 presents an approach to analyze global accessblty wth tool geometry. Secton 4 presents the experments performed on the sample parts to verfy and valdate the approaches. Fnally, n secton 5, we draw our conclusons. 2. LOCAL ACCESSIBILITY ANALYSIS To fnd a feasble tool radus for a gven surface, both local and global accessblty are analyzed. In ths secton we consder the tool radus constrcted by the local accessblty sets an upper bound of the mnmum sze tool (whch means that ths tool can cut entre part). In the next secton, local accessblty s used as an ntal mnmum sze tool for global accessblty analyss. In mult-axs (4-axs and 5-axs) NC machnng, usng dfferent types of endmlls (e.g., flat-endmlls or torus-endmlls) for sculptured surface machnng s generally far more effcent than usng tradtonal sphercal endmlls on 3-axs machnes. Ths s due manly to the number of paths needed to obtan small cusps and to the cuttng neffcency of a sphercal endmll, whch has a low speed cuttng zone at ts bottom [3]. But n ths work, we only consder sphercal endmlls for sculptured machnng to smplfy the problem. The local accessblty of a pont on a free form surface for a sphercal endmll s equvalent to the maxmum curvature of that pont. We wll use both NURBS models and trangular mesh models to demonstrate our approach. We tessellate NURBS models by parttonng them nto small facets. Each facet s a trangular patch that s small enough to approxmate the curved surface by a plane trangle wthn a prescrbed tolerance. We set the centrods of the trangular facets as access ponts (the pont we wll consder for accessblty). The local accessblty of the entre feature can be derved from the local accessblty of these access ponts. On the NURBS model, the two prncple curvatures of the pont can be found by ts surface parameters. The maxmum prncple curvature s the maxmum curvature at the pont. On the trangular mesh model, an approxmated quadratc surface of a trangular facet s created from a group of neghborhood ponts. The maxmum curvature s then found by calculatng the prncple curvatures at the centrod accordng to the approxmated quadratc surface. The maxmum curvature of the feature s derved from the maxmum curvature of the maxmum curvatures of the access ponts. The upper bound of the tool radus s the recprocal of the maxmum curvature. A free form surface usually has some hgh curvature regons. In Fg. 2 the male face NURBS model and the female face trangular mesh model s shown. Each patch can also be coded wth dfferent rad to analyze the local accessblty. Ths provdes a way to fnd all the regons of the surface that can be cut out by a gven tool. The upper bound of the mnmum tool radus for the hghest curvature regons s also calculated. Fgure 2 also shows four sphercal endmll tools of dfferent rad and the hgh curvature regons of each face model. The tool wth the smallest radus s the mnmum sze tool needed to cover the part, the other three tools are 0.25 nch, 0.50 nch and 1.00 nch. The fgure s the result of usng all four tools where we frst cover all the facets of the model possble wth the largest tool, we then cover the porton of the model not covered by the largest tool wth the next largest tool and so on.

3 21 Fg. 2: Models: NURBS, tessellated, tool mappngs, and tessellaton mappngs. 3. GLOBAL ACCESSIBILITY ANALYSIS The global accessblty analyss s easy to start as usng a pencl of rays of zero radus emanatng from a selected pont. The accessblty can be evaluated by checkng to see f any rays ntersect other portons of the part and buldng up a pont vsblty cone (PVC) or bnary sphercal map (BSM) to represent the global accessblty (Fg. 3). Kang & Suh [2] created an approach to tessellate the product model nto n patches, wth a unt sphere placed n the centrod of each patch. Each unt sphere s also tessellated nto m patches. Each patch of the unt sphere represents a normal or a drecton from that patch. All patches of the product model are then mapped to each unt sphere. If there are n patches on the surface, n proectons need to be mapped onto the bnary sphere, whch would ental the samplng of m normals or drectons wthn sphere. If n such ponts exst on the surface under evaluaton, then the order of the algorthm that can determne the surface vsblty would be m.n 2. Generatng the BSMs by patch proecton s computatonally expensve. Kang & Suh s approach s a typcal one to generate the global accessblty nformaton wthout a known tool radus. Fg. 3: Pont vsblty cone (PVC), bnary sphercal map (BSM), and a BSM embedded n a facet. Sptz & Requcha [5] presented an approach to nclude tool radus nto the global accessblty analyss. The approach can verfy that a tool radus r penetrates an obstacle X ff the nongrown half-lne penetrates X r, where X r denotes X grown by r. In ths approach, an obect s grown by the current tool radus r, ncludng all the ponts that are at a dstance no greater than r from another pont n the obect, and then the global accessblty analyzed for the grown obect wth rays. The result of the analyss ncludes the current tool radus r. Unfortunately, computng the sold model of a grown obect s an expensve and nontrval task prone to precson errors and t produces curved obects even when the nput s polyhedral. Woo et al [8] presented the concept of sphercal and vsblty maps where surface normals where enhanced wth cuttng tool geometres. Woo et al. recognze the concept of partal vsblty whch they presents as an equvalent to the global accessblty noton of Spyrd. Woo categorzes a wde range of manufacturng processes by ther pont, lne and surface vsbltes besdes consderng the varous degrees of freedom n the mechansm of the machnes that gve a computatonal structure n terms of a pont great crcle and a sphercal rectangle. Our approach s to develop an augmented surface model where each access pont on the surface s assocated wth a BSM, a reflecton of the pont accessblty of the entre model. Frst we create a set of BSMs wth rays. We then develop an approach to modfy all of the BSMs wth the tool radus. 3.1 Global Accessblty Analyss The frst step s to evaluate the global accessblty wth rays by creatng BSMs for all access ponts and generate the augmented surface model. A BSM s a unt sphere created wth ts surface decomposed nto m trangular facets accordng to the global coordnate system. The m s then calculated by the resoluton of the accessblty

4 representaton that s the grd number of the unt sphere, m= 12 grd (as an example, f grd = 8, m= 12 8 = 768 ). Each BSM s translated to each access pont by placng the center of the BSM on the access pont, whch allows the accessblty of the facets can be represented n a unfed algebrac fashon. However, the sphercal trangular facets of a BSM under, or ntersectng wth the tangent plane of the access pont, would be marked as naccessble before the patch proecton. The patch proecton only performs on the facets of the BSM above the tangent plane. Fg. 3 depcted a tessellated model of a BSM and a BSMs on a sngle facet wth ts accessble orentaton vectors. The darker shaded areas on the BSM represent naccessble facets where a proecton vector would ntersect the part model. The followng algorthm detects and marks each of these facets as beng naccessble f a ray emanatng from the center of the facet sphere ntersects any facet of the part. The procedure s as follows. Step 1. Create tessellatons on the surface, F, [1, n]. The access ponts are the centrods of the facets, P, [1, n]. Step 2. For pont P, [1, n], calculate the coordnates of P, and calculate the surface normal n, at P. Step 3. Create a unt sphere wth ts surface decomposed nto m trangular facets accordng to the global coordnate =, t [ 1, m]. a represents a sphercal trangular facet. t a represents a drecton from center of system, { } A a t the unt sphere to the centrod of the sphercal trangular facet a t. Step 4. Translate the unt sphere to each P as The center of the unt sphere s concdent wth A. A { a } t t =, [1, n], t [ 1, m] and ntalze a = 0 (accessble). P. Check the sphercal trangular facet under the tangent plane of access pont P (whch means f one of the three vertces of t a, t t [ 1, m] a s under the t. If a s t tangent plane, calculated by dot product wth the patch normal), then mark t as naccessble (set a t = 1). Step 5. Proect all the surface trangular facets F,, [1, n] that le above the tangent plane of the access pont P to the unt sphere (whch means f one of the three vertces of by dot product wth the patch normal). p F s the proecton of Step 6. Check the sphercal trangular facet a, t [ 1, m], f a = 0 and as naccessble (set a t = 1). If only one and t p F causes t P s calculated and recorded. If more than one P and dstances between t F les above the tangent plane, calculated F on the unt sphere A. a fall nto t p F,, [1, n], then mark t a to become naccessble, then the dstance between P p F causes a t to become naccessble, then all the P are calculated but only the shortest one s recorded. In ths procedure, we tessellate the surface nto n facets. A unt sphere also tessellated nto m facets s then placed on the facet such that the centrod of the sphere and the centrod of the facet are calculated. Fnally all the facets of the part surface are mapped to each of the unt spheres to create the bnary sphere map (BSM) on each facet of the surface. Durng the facet mappng, we add a dstance calculaton to Kang s approach. We record the shortest dstance of the naccessble orentaton n Step 6. The dstance data wll be used for modfyng the BSM wth the tool radus. The mappng of one facet onto a BSM d depcted f Fg. 4. Fg. 4: The mappng of one facet onto the BSM.

5 Mathematcal Model of Global Accessblty Analyss wth Tool Geometry In ths secton, we demonstrate how a BSM can be used to capture global accessblty wth a tool radus. In Kang s approach, BSMs are created wth rays (no tool sze). A BSM (tessellated unt sphere) s composed of trangles wth each trangle representng a potental tool orentaton where the tool orentaton s a proecton of the vector from the centrod of the BSM through the centrod of the facet. The process of mappng surface facets to the BSMs creates a map of all naccessble tool orentatons for each of the BSMs. The proecton of a surface facet onto the BSM s equvalent to sayng that a tool cannot be orented n that same proecton orentaton to access the part because t would have to pass through the proected surface n order to reach the BSM. When the tool geometry s ncluded n the mappng, the global accessblty wll ether reman unchanged or the accessblty map (BSM) wll decrease. That means some or all of the accessble orentatons evaluated wth rays mght become naccessble orentatons when evaluated usng the tool radus. Fgure 5 shows BSMs wth and wthout tool geometry ncluded. Fg. 5: Global accessblty changed by ncludng tool geometry. The frst step of the BSM modfcaton approach s to pck one BSM of the model that has been created usng ray ntersectons. The second step s to pck an naccessble orentaton of the upper hemsphere of the BSM (whch s above the tangent plane of the access pont). Fgure 5 shows an orentaton evaluated wth a ray that we wll consder to be naccessble (AP). Pont A s an access pont. Pont P represents a trangular surface facet of the other porton of the part (the facet s small enough to be approxmated as a pont). The ncluson of tool radus wll create a small naccessble regon for a tool radus ( r > 0 ). The unt hemsphere on the left represents the upper hemsphere of the orgnal BSM. Its center s on pont A. Pont P s on a lne passng through the centrod of a trangular facet of the surface of the BSM (sphere facets are not shown). The unt hemsphere on the rght represents the BSM after the modfcaton wth a sphercal endmll. Its center s on pont O (the center of the sphercal end of the tool). To avod a collson of the tool shaft wth the other porton of the part, pont P, an naccessble regon, was marked on the unt sphere. The boundary of the naccessble regon s formed by the orentatons of the tool when ts shaft ust touches pont P. Snce an naccessble orentaton from the ray approach becomes an naccessble regon wth the tool radus, ths change wll affect the other orentatons that are ray accessble. All naccessble orentatons of the upper hemsphere of the orgnal BSM would go through the same process to complete one BSM modfcaton wth the tool radus. And all the BSMs would be modfed by sequence. The small gray regon shown n Fg. 5 s the general case. It could approxmately be defned by four crtcal orentatons, a, b, c and d. Orentaton a s the orentaton of the tool that s closest to the normal vector of Pont A. Orentaton b s the orentaton of the tool that s furthest from the normal vector of Pont A. Orentatons c and d are two orentatons n the same plane wth the naccessble orentaton, and ths plane s perpendcular wth the plane that conssts of normal vector of pont A and orentaton a. To demonstrate the change of BSM wth tool radus, the problem can be stated as: that gven a tool radus and an naccessble orentaton wth a dstance recorded to fnd the four crtcal orentatons, a, b, c and d. After a small gray regon s defned by these four orentatons, we change the status of all the orentatons that are ray accessble wthn the small gray regon to be tool naccessble. We adopt the followng model notatons: A - An access pont, centrod of a trangle facet of the surface. O - Center of the sphere of the sphercal endmll. P - A pont representng a facet of the other porton of the part, the reason causes the orentaton to be naccessble. L - Dstance between pont A and pont P. 2 2 L - Dstance between pont O and pont P. L = L + r 2Lr cosθ

6 24 θ - Angle between the access pont s normal and the orgnal naccessble orentaton. θ - Angle between the access pont s normal and the tool s orentaton. R - Dstance between pont P and pont A s normal vector. R= L snθ H - Dstance between pont P and pont A s tangent plane. H = Lcosθ R - Radus of the sphercal endmll tool. And the ntal ranges of some notatons are gven as follows. L > 0 Dstance should be greater than zero. 0 θ < π Inaccessble orentaton s upon the tangent plane of the access pont. 2 0 θ < π Tool orentaton s upon the tangent plane of the access pont. 2 In order to create the mathematcal model of BSM modfcaton and solve t completely, the model has been consdered n many cases accordng to the condtons set byl, R, H and r. In table 1, the frst column shows the math relatonshp, the second column presents the fgure (the plane shown n each fgure s the tangent plane of pont A), and the thrd column s the unt sphere wth gray regon sgnfed as the naccessble regon caused by the tool. Math relatonshp Case 1: L <r Fgure Inaccessble regon Math relatonshp Case 4: L > r and 0< H r Fgure Inaccessble regon Case 2: L = r and H < r Case 5: L > r and r< H < 2r Case 3: L =r and H > r Case 6: > and L r H 2r Tab. 1: Sx cases for modfcaton of BSM s when consderng tool geometry. For llustraton, we show the modfcaton for case 6. Detals for the other cases can be found n [6]. Fg. 6 (Case 6) shows that when L > r & Lcosθ 2r, the naccessble regon of the unt sphere s defned by four crtcal orentatons, a, b, c and d. In Fg. 6, the thrd drawng shows how to fnd orentatons a and b. The last two drawngs show how to locate orentatons c & d. The drawng on the rght s the A-A vew of the drawng n the mddle.

7 BSM Modfcaton Procedure The procedure to modfy the BSM s follows. Step 1. Fg. 6: Case 6 of the BSM modfcaton wth tool geometry. Retreve an access pont P, [1, n] wth ts BSM A = { a }, t [ 1, m] of the global accessblty and surface normal n. a that are accessble and all Step 2. Pck all t two lsts A = { a }, A = { a }, t [ 1, m]. t t t a t that are naccessble and fallng above the tangent plane to form Step 3. By sequence take an naccessble orentaton a t n naccessble lsta, and name t the naccessble orentaton n. t n and n t. Step 4. Calculate the naccessble angle θ between Step 5. Use the formulatons to calculate the naccessble regon wth the tool radus. Step 6. Check f any a t whch s n the accessble orentaton lst A = { a } falls nto the naccessble regon. If yes, change the status of a to naccessble and take t out of the lst. If the accessble orentaton lst { } t A = a s empty (whch means the whole BSM s naccessble), stop and go to the next access pont. t Step 7. Repeat 3 through 6 untl all naccessble orentatons a t n t A have been checked. Step 8. By sequence, retreve the next access pont P and go to Step 2 untl all the BSMs on the access ponts have been modfed. To fnd a feasble tool radus for the hgh curvature regons of the surface, we use the tool radus found n the local accessblty analyss as the ntal tool radus to modfy the set of BSMs created by the global accessblty analyss wth rays. If any of BSMs become totally naccessble, we would shrnk the tool radus to redo the modfcaton untl we fnd a tool radus wth all the BSMs whch have some accessble orentatons.

8 26 4. EXPERIMENTAL RESULTS We demonstrate the accessblty approach on three models. The models nclude a male face model and a a bunny model at one facet resoluton (Tab 2). We use only one resoluton of the bunny model because of the complexty of the model. The expermental set was consdered adequate for demonstratng the valdty of the approach. We use dfferent resolutons to demonstrate accessblty computaton and mnmum tool sze for varyng granulartes of model approxmaton. A tessellated mode s, after all, an approxmaton of the NURBS model. The purpose of the experments are to show that the approach for computng tool accessblty (local and global) can be accomplshed for any sze tool and for varyng resoluton of tessellated models. We also demonstrate that the approach can be used to examne dfferent tool combnatons on a model. The approach was mplemented n C++ usng Standard Template Lbrares and the ACIS sold modelng kernel. All experments were conducted usng a Pentum IV machne wth a 2.2 GHz processng speed and 1 Gb of RAM. For each experment, three dfferent tool szes (0.25 nch, 0.50 nch, and 1.00 nch) were nvestgated to fnd the mnmum sze tool whch can ft onto the hghest curvature regons (whch means that ths tool can cut the entre part). Tab. 3-4 shows the results of the local accessblty analyses. Tab. 3 shows the results of the male face model and Tab. 4 shows the results of the bunny model. Local accessblty analyss s used to set the upper bound of the feasble tool radus (the mnmum sze tool). The expermentaton shows that there s a dfference n computng the mnmum tool sze based on the resolutons of the models. The hgher the resoluton, the more accurate the model. In practce, the resoluton of the tessellated model should be determned by the tolerance on the manufacture drawng. We found for these complex sculpture models that the tool that would cover the entre part was too small for machnng practce. From a machnng tme perspectve, t s better to use larger tools to machne parts. So n the local accessblty analyss we also look at three dfferent tool szes (0.25 nch, 0.50 nch, and 1.00 nch). The results of the experments show the percentage of area covered by each tool. Model # Model name Resoluton 1 Resoluton 2 Resoluton Male face Bunny Tab. 2: Three models used for expermentaton tessellated at dfferent resolutons (facets).

9 27 Resoluton No. (facets) Tool number Tool 1 Tool 2 Tool 3 Tool sze (nch) (Accessble area percentage) (Accessble area percentage) (Accessble area percentage) Tab. 3: Local accessblty analyss of the male face model wth three resolutons. Resoluton No. (facets) Tool number Tool 1 Tool 2 Tool 3 Tool sze (nch) (Accessble area percentage) Tab. 4: Local accessblty analyss of the bunny model (graphcs not shown). The red area on each model represents the hgh curvature regons that cannot be covered by the tool. The results n the tables are presented n groups. The pctures for each resoluton of the model form a group that occupes a row. In each group, the frst pcture s the result of usng all four tools where we frst cover all the facets of the model possble wth the largest tool, we then cover the porton of the model not covered by the largest tool wth the next largest tool and so on. The other pctures show the regon that would be covered by usng one tool sze and the hgh curvature regon uncovered s marked n red color. We also compute the percentage of accessble area covered by each tool. Tab. 5-6 shows the results of the expermentaton usng the global accessblty analyss. The mnmum tool sze calculated n the local accessblty analyss s used as the ntal feasble tool radus to fnd the mnmum tool sze n the global accessblty analyss. The results show the mnmum tool sze found n the global accessblty analyss may be smaller than that found n the local accessblty analyss. The mnmum tool sze found n the local accessblty analyss mght not have an accessble orentaton when we perform the global accessblty wth ths tool sze. In ths case, we wll shrnk the tool sze untl we fnd a tool sze that has at least one accessble orentaton when we perform the global accessblty process. In the global accessblty analyss, we looked at four tool szes. The accessble area percentage

10 28 and the machne tme were calculated. By comparng the machne tme, we demonstrate that usng a combnaton of tools s better machne plannng. Tab 7 shows the computng tmes for the BSM modfcaton approach for all model resolutons and tool szes. The results ndcate that the BSM modfcaton procedure takes consderably less processng tme than the patch proecton procedure (half lne proecton). Also, the larger the sze of the tool, the lower the processng tme. Resoluton No. (facets) Tool number Tool 1 Tool 2 Tool 3 Tool sze (nch) (Accessble area percentage (Cuttng tme n mnutes) Tab. 5: Global accessblty analyss of the bunny model (graphcs not shown) Resoluton No. (facets) Tool number Tool 1 Tool 2 Tool 3 Tool sze (nch) (Accessble area percentage) Cuttng tme (mnute) (Accessble area percentage) Cuttng tme (mnute) (Accessble area percentage) Cuttng tme (mnute) Tab. 6: Global accessblty analyss of the male face model wth three resolutons.

11 29 No. Model name Tool sze (nch) & computng tme (second) Tool sze Number of facets Mn tool sze 0* Mn Male face Bunny * half lne proecton Tab. 7: Computng tmes of the global accessblty analyss. 5. CONCLUSIONS An approach for global accessblty usng trangular tessellated sold models that consders tool sze (sphercal endmlls) was demonstrated. It ncludes usng local accessblty analyss to set an upper bound of the feasble tool radus, global accessblty analyss wth rays (no tool sze) to create a set of the BSMs, and modfyng the BSMs wth dfferent tool rad to analyze the global accessblty wth tool geometry. Kang s approach was used to create the ntal set of the BSMs by patches proecton. The approach s computatonally expensve. Three models at dfferent resolutons were used to demonstrate the approach. The experments ncluded fndng the smallest tool that would cover the part for the hgh curvature regons of the surface and also a global accessblty analyss wth dfferent tool rad. The results demonstrate that the approach can be used to analyze the global accessblty of dfferent tools and to compute the machnng tme wth the varous tools combnatons. The superor computaton effcency of the new approach was demonstrated and compared wth a prevous approach. Future work wll address other tool geometres and tool holder nteractons wth the workpece. Optmzaton of tool selecton s antoher area of future research. 6. REFERENCES [1] Balasubramanam, M.; Laxmprasad, P.; Sarma, S.; and Shakh, Z.: Generatng 5-axs NC roughng paths drectly from a tessellated representaton. Computer-Aded Desgn, 32, 2000, [2] Kang, J. K.; Suh, S. H.: Machnablty and set-up orentaton for 5-axs numercally controlled machnes of free surfaces. Internatonal Journal of Advanced Manufacturng Technology, 13, 1997, [3] Km, B. H.; Chu, C. N.: Effect of cutter mark on surface roughness and scallop heght n sculptured surface machnng. Computer-Aded Desgn, 26, 1994, [4] Rawat, R.: A Convex Hull Approach for Tool Accessblty Evaluaton. Thess, Arzona State Unversty, Tempe, AZ, [5] Sptz, S. N.; Requcha, A. A. G.: Accessblty analyss usng computer graphcs hardware. IEEE Transactons on Vsualzaton and Computer Graphcs, 6, 2000, [6] Vafaeesefa, A.; ElMaraghy, H. A.: Accessblty analyss n 5-axs machnng of sculptured surfaces. In Proceedngs of the 1998 IEEE Internatonal Conference on Robotcs and Automaton, Leuven, Belgum, 1998, (IEEE, New York). [7] Wang, J.: Local and Global Accesblty Analyss for Generatve Process Plannng. Dssertaton, Arzona State [8] Unversty, Tempe, AZ, May2006. Woo, T. C.; Gan, J. G.; and Tang, K.: Sphercal maps : Ther constructon, propertes, and approxmaton, Journal of Mechancal Desgn, 116, 1994,