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A. J . Lobbezoo et al., Robot control using adaptive transforma-tions, IEEE J . Robotics Autom at., vol. 4, pp. 104-108, 1988.P. Bondi, G. Casalino, and L. Grambardella, On the iterativelearning control theory for robotic manipulators, IEEE J . RoboticsAutomat. ,vol. 4, pp. 104-108, 1988.R. Horowitz and M. Tomizuka, An adaptive control scheme formechanical manipulators-Compensation of nonlinearity and decou-pling control, ASME J . Dynam. Syst. Measure. Control, vol.J . A. Berna rd, Use of a rule-based system for process control,IEEE Control Systems Mag., vol. 8, pp. 3-13, 1988.C. C. Lee, Fuzzy logic in control systems: Fuzzy logic controller-Part I, IEEE Trans. Syst. Man Cybern., vol. 20, pp. 404-418,1990.- Fuzzy logic in control systems: Fuzzy logic controller-Part11, IEEE Trans. Sys f. Man Cyb ern., vol. 20, pp. 419-435, 1990.

108, pp. 127-133, 1986.

Automated M achinability Checking for C A D / C A MChuan-Jun Su and Amitabha Mukerjee

Abstract-This short paper presents a method to determine whetheror not a part is mach inable on a three-axes milling machine by consider-ation of tool motion. The method focuses on determining sets ofcandidate machining directions, called the basis set, and testing if theprocess plan generated from any particular set of machining directionssucceeds in machining all of the part features. This constitutes amachinability test that can be used at design. The range of parts that aresuccessfully handled include all simple polyhedrons and simple three-dimensional curved faces suc h as cylinders, cones, and spheres.

I. INTRODUCTIONMachinability checking is an issue that surprisingly has gainedlittle attention in research in the integration of CAD and CAM. ACAD/CAM system should have the means to determine the limitsof its own capabilities so that a determination can be made whetheror not to proceed with more detailed process planning, NC codegeneration, and jig and fixture design. Otherwise, the system will

waste time generating an infeasible result, or worse yet, generate anincorrect solution that is not recognized as such. In order to achievethis level of introspection, the concept of machinability checking isintroduced. Thus, given a functional model of a machining process,and a solid model of a part, a determination is made concerning thecapability to machine the part from the raw material.

In the past, several researchers have suggested methods forgenerating an NC code for different types of objects [l], [2]. Thesemethods, however, do not address process-planning issues such asmultiple setups and process plan verification. Other attempts havelooked primarily at the process-planning aspects [3]-[5] but havenot investigated the issue of machinability itself.

The machinability checking module presented in this short paperacts as a filter that rejects improper part designs for a three-axesmilling machine before the NC code generation module is activated.Furthermore, it can be used as a design aid to provide informationfor modification if the given part is rejected.

The algorithm developed for machinability checking is based onconsideration of the tool motion. First, a convex enclosing object isManuscript received September 11, 1989; revised April 30, 1991.C.-J. Su is with the Knowledge Based Systems Laboratory, Departme nt ofIndustrial Engineering, Texas A&M University, College Station, TX 77843.A . Mukerjee is with the Department of Computer Science, Texas A&M

University, College Stati on, TX 77843.IEEE Log Number 9101704.

constructed about the part. The part faces are then orthogonallyprojected to the infinite planes containing the enclosing object faces.Next, the truncated right prisms that are bounded by the part facesand the projections are constructed. If for every part face thereexists at least one truncated right prism that does not intersect thepart itself, then the part satisfies the criterion for machinability, andthe part can be produced from the enclosing object. If there is someface for which such a truncated right prism cannot be constructed,then the enclosing object does not provide sufficient machiningdirections for producing the given part. In this case, one can eithertry an alternate enclosing object that might provide enough machin-ing directions, or generate new machining directions for these failedpart faces.Throughout this work, we restrict ourselves to polyhedral soIidsor polyhedral approximations of objects with curved surfaces. In thelatter case, the decision regarding the partitioning of the curvedfaces can be made based on the polyhedral model. However, ingenerating the final NC code, one can use the underlying curvedgeometry to obtain the exact results without compromising thesimplicity of the machinability test.

The organization of the paper is as follows. In Section 11,definitions are presented. The machinability test is discussed inSection III. The implementation of the machinability test is pre-sented in Section IV. Conclusions are discussed in Section V .

11. DEFINITIONSThe following definitions will be used throughout the paper.Minimum enclos ing box (MEB): The smallest rectangular boxConvex hull: The smallest convex set containing solid S.External plane: An infinite plane that contains an enclosing

object face or is constructed in orde r to provide access to a face of Sthat is not reachable from the enclosing object.Tmncated right prism: A prism is a polyhedron with two

congruent and parallel faces called bases. The other lateral faces areparallelograms formed by joining corresponding vertices of thebases. The intersections of lateral faces are lateral edges. A trun-cated prism is a portion of a prism lying between two nonparallelplanes that cut the prisms and have their lines of intersection outsidethe prism. A right prism is a prism whose lateral edges areperpendicular to the bases. If the lateral edges are orthogonal to atleast one base, and two bases of the prism can be either parallel ornonparallel, then this type of prism is defined as a truncated rightprism. This is illustrated in Fig. l(a).Point visibility: A point C n a solid S is said to be visible froma point D on an external plane if CD l in t (S) =0,here int isthe interior operator [6].

Point orthonormal visibility: A point C on a solid S is or -thonormally visible from an external plane G if andonly if there isa point D n G such that C is visible for D an d CD s orthogonalto G. For an illustration of this, see Fig. l(b).

containing a given solid S.

Face visibility: There ar e two possible cases.0 F is not orthogonal to G. In this case, F is said to be visiblefrom an external plane G, if and only if all the points of Fa r eorthonormally visible from G .0 F is orthogonal to G. F is visible from G if and only if all thepoints on F are orthonormally visible from G and there is noother face F of S such that Proj, (int (F )) n Proj, ( F ) )#0 nd n F . nF # 1, where Proj, ( F ) and Proj, ( F )are the

projections of F an d F on G , an d n F an d nF are the unitnormal vectors of F an d F. This is depicted in Fig. 2.1042-296X/91$01.00 01991 IEEE

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Lareral edge

Base

Rieht orism

Lateral fuce

LoferaledgeBase

Face H I /

Enclosing Object E(b)

Face G

D

Fig. 1 . (a) Illustration of oblique, right, and truncated right prisms. (b)Illustration of orthonormal visibility. The point C is orthonormally visiblefrom the face G and visible from the face F . G is not orthonormally visiblefrom F .

Fig. 2. Illustration of facewise visibility. (a) The intersection between theprojection of F and the projection F on G is nonempty, and the normalvectors of F and F have oppo site directions, and so by definition F is notvisible from G. (b) Both F and F are visible from G.

Facewise partial visibility: If only a subset of F is orthonor-mally visible from G , then F is partially visible from G .Solid VisibiIity: There are two types of solid visibility.0 Given a solid S , if a set of external planes r = {Gj i =

1, . . . ,n } can be found such that for each point p on theboundary of S , there exists an external plane G jE r such thatp is orthonormally visible from Gj, hen S is said to be visiblefrom r.

0 If the set of external planes r is equal to the set of planescontaining the faces of a convex enclosing object E , or th egeneration of r is based on E , then S is said to be visiblefrom E . The solid visibility criterion is depicted in Fig. 3.Residue: Le t H k , k = 1 , . . ., , be the faces of an enclosingobject E. Let Proj ( F , ) be the projection of solid face F , onexternal plane Gj . Le t D j be defined as U:= Proj ( f f k ) - *U:=Proj ( F , ) , where - is the regularized difference operator

[6]. If Dj is swept along both directions of the normal vector of Gj,an infinite sweeping solid U can be constructed. The residue R j isdefined as U n* where n* s the regularized intersection opera-to r [6]. For an illustration, see Fig. 4.

III. MACHINABILITYESTSince there are an infinite number of points on the boundary of asolid S , it is not possible to check the visibility of all boundarypoints. In practice, facewise visibility is checked for machinability

instead of pointwise visibility. In other words, if every solid face isvisible from some external plane, then the solid is accepted as avisible part. This method can be formalized mathematically asfollows:

1) Le t r be a set of external planes with respect to a solid S ,2 ) Le t F , , i = 1, . ..,n be the faces of S .3) Let Proj (e), = 1 , . . . , n , j = 1 , . . ,m be the orthogo-nal projection of F, on plane G j .4) If the area of Projj ( 4 ) s nonzero, then a truncated rightprism P i j with 6 and Projj(F,) as the two bases can be

constructed; otherwise, a finite polygon r i j can be generatedorthogonal to the plane G j . For an illustration, see Fig. 5.5) If the truncated right prism P j j constructed in 4) atisfies thecondition that Pj jn nt (S ) =0,hen Pi j is called a valid

prism. Similarly, if the xi j in 4) atisfies r j j l nt ( S ) =0and there is no other face Fk of S such that Proj (int (4)) lProj j ( F k )#0 nd nFk.nF k# 1 , where Projj ( F k ) s theprojection of Fk on G j , an d nF , an d nFk are the normalvectors of F,. an d Fk,he n r j j s called a valid polygon .

6) If for every F , , i = 1 , . . . ,n , a valid prism or a validpolygon can be constructed, or a set of valid prisms P j j ca nbe constructed such that U j k = l ( P j j l 6)=F, , then S isvisible from r.7) If r is equal to the set of planes containing the faces of aconvex enclosing object E, or the generation of r is based onE , then S is visible from the enclosing object E.

r = {Gj I j = 1,. . . ,m } .

By the definition of solid visibility, if a solid S is visible from ase t of external planes r, hen for every solid face F , , there is a leastone valid prism Pij or valid polygon r i j hat can be constructedwith respect to some external plane G j . Suppose for the solid faceF, only the valid polygon x j j construction can be made. In otherwords, no valid prism Pij fo r F,. can be constructed with respect toany external planes in r. Then it is reasonable that r i j would beexpected to be on the boundary of some valid prism or on theboundary of the residue so that F,. can be machined while the valid

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693

M?B Co n vex h u l l part ME B Convexhull Part

Fig. 5.

(b) (C)Fig. 3. Illustration of visibility. (a) A solid that is visible from theminimum enclosing box (MEB). (b) A solid that is visible from the convexhull (CH) but not from the MEB. ( c) A solid that is not visible from eitherthe MEB or the CH , but is visible from the MEB with a generated externalplane.

R e s id ue s-U

Fig. 4. Illustration of the residue construction.prism or the residue is removed from the raw material. If T~ ~ s noton the boundary of any valid prism or the residue, the solid face F,can only be reached with an infinitely thin tool, which is notpossible in practice. Therefore, for the solid S to be machinable, allfaces F, for which only a valid polygon 7 r i j ca n be constructed mustlie either on the side face of some other valid prism, or on theexternal boundary of the part. The following theorems formally giveproof to the above.

Theorem 1: If x i j is a valid polygon and external planes areconstructed based on an enclosing object E , then r i j is on theboundary of a valid prism, a residue or the enclosing object E .Proof: Since riij is a valid polygon, this implies that thep ro je ct ion o f s z d f a ce F, on the exte rna lplane Gj is a linesegment, say AB . For each point q on AB , there are somecorresponding points p , , p 2 ,. . ,p m on the boundary for 5 such

prisms

Illustration of the truncated right prism and the polygon construc-tion.

/ I I 1 /( 1 1 1

li

Fig. 6. Illustration of the proof of Theorem 1 .-ha t pkq , k = 1, .. . ,m aie orthogonal to the external plane G j .The point p , , I E {1, . . . , m } that satisfies dist (p I , q) >dist (p k ,q), k =1, . .. ,m , k # I , is said to be a maximal depthpoint with respect to q, where dist is the function of distance. Theset of maximal depth points with respect to all the points on ABthen forms a set of maximal depth line segments, and the edges onthe boundary of F). can be classified into the maximal depth andnonmaximal depth segments. Let I, , I,, . . , , be the maximaldepth segments on 6 . he valid polygon r i j an then be segmentedinto regions r ,, z, . , r, by the maximal depth line segments. Foran illustration, see Fig. 6 .Each maximal depth line segment I, , p = 1, . . .,s, connects F,with some other part face F,. F, can either be in the positivehalf-space or in the negative half-space with respect to the normalvector of 4. (see Fig. 10 given below). If F, is in the positivehalf-space, then there is a valid prism that can be constructed byusing F,, or a subset of F,, and its projection on Gj as two bases,an d r, as a side face. Otherwise, 5 is not visible from G j . This is acontradiction. If F, is in the negative half-space, then if we extendI-,n a direction away from AB , we shall either (a ) reach someenclosing object face, which implies that r, would be on theboundary of the residue; or (b) meet som e other part face F, ,whichimplies that rp would be on the boundary of the valid prism formedby the part face F; . Each region r, , p = 1, . . ., of r j j s either

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on the boundary of residue or some valid prism. Therefore, 7 r i j iseither on the boundary of residue or some valid prism unless 7 r i j isW

Theorem 2: If a solid S is visible from an enclosing object E, oris visible from a set of external planes {Gj I j = 1, . . . ,m} asedon the enclosing object E (the set of external planes are formed bythe infinite planes that contain the enclosing object faces), then S ismachinable from E. That is , E - U;= U= P ij - U?= R j = S,where P i j are the valid prisms that can be constructed from thesolid face F, to G j , and R j is the residue with respect to theexternal plane Gj.Proof: Suppose E - U;= U= P j j- U?= I R = Q , Q # S.Then there exists a point set U that is not in S but in E such thatQ =s U U , U n U ~ = ~ U ; = ~ P ~ , ) 0,nd U n u ~ = ~ R ~ )0.If U flS =0,he n S is still machinable. Note that S is visibleso that all solid faces are reachable (visible). After all solid faces aremachined, the solid S would be separated from U , and hence U willnot be an obstacle in achieving the final solid S.If U an d S are not disjoint, that is they are contiguous, then thereis a nonempty subset w of the boundary of S such that w C U .Since U fl U y = I U ~ = l P j j ) 0 nd U n U, =,Rj )=0 his im-plies that w fl ( U y = l U j l P i j )=0 nd on U y = l R j ) =0.That is , w is not visible from any external plane G j . Since w is noton the boundary of S, his contradicts the assumption that solid S isvisible. Since E - U= lP j j- U,=_,Rj= S , then S is machinable.

W

part of the enclosing object face.

IV . IMPLEMENTATIONTo examine whether a given solid is machinable, w e need to

determine if the intersection of the truncated right prisms and theinterior of the solid is an empty set. To implement this machinabilitychecking concept, a methodology based upon the idea of a hiddenface is developed.

Le t Fl an d F, be two distinct faces of a solid. Let G be anexternal plane, and let Proj, ( F l ) nd Proj, (F,) be the orthogona lprojections of Fl an d F, on G , respectively. If Proj,(F,) flint (Proj, (F,)) #0,hen either Fl or F, is a hidden face withrespect to the external plane G. If the intersection between the twoprojections is nonempty, there is at least one point, say p , that canbe found in the intersection. Let q and r be the correspondingpoints ( from which the po i n t p was pro jected) on Fl an d F 2 ,respectively. If the length of p q is less than the length ofz,he nF2 is a hidden face with respect to G, an d F2 is said to be hiddenby F l . Otherwise, Fl is a hidden face, and Fl is hidden by F,. Fig.7 illustrates the hidden-face concept. This concept is similar to thez-buffer algorithm [7] in the context of computer graphics, exceptfor the orthogonal visibility restriction.

Le t P be a truncated right prism or a polygon that is constructedby orthogonal projection from a solid face F to an external plane G.If the truncated right prism P and the interior of the solid S has anonempty intersection, then there is at least one distinct solid faceF such that Proj, (F ) f l nt (Proj, (F)) is nonempty and F ishidden by F. This idea is used for the development of an algorithmfor the machinability implementation.A. Alg o r i th m

In this algorithm, each face is first tested to see if it is entirelyvisible from an external plane. If not, let p u j be the part of the facethat is visible from G j . The algorithm then determines if U= puj=5.Procedure machinability-checking(G,, G,, . , G,,,; F l , F2,

, F n )

Projection of F IProjection of Fz

Fig. 7. Illustration of the concept of hidden face. The intersection betweenthe projections of the part faces F , and FLon the enclosing object G isnonempty, and the length of pq is less that pr . Therefore, F2 is hidden byF,. P is an interior point in the intersection, and q and r are thecorresponding points on the part faces F, and F 2.Beginlet r be a set of external planes, r = {Gj I j = 1, . . .,m} ;let F,, i = 1 , . . .,n, e the faces of S;for every solid face Fkdobeginif there is an external plane Gj such that Fk s not a hidden face

thenelsebegin

with respect to GjFk s visible from G j

for j : = 1 to m dobeginfo r i := 1 to n doif Fk is hidden by F i, and i # k thenQij= (Proj j (Fk)- Proj, (4) )else

end;if U ~ = l p u j Fk thenelseFk s visible from a combination of external planesFk is not visible;end;end;End;

The flow chart shown in Fig. 8 reveals the structure of thisalgorithm.If every solid face Fk , k E {1, . . ,m} is either visible fromsome external plane or can be visible from a combination of externalplanes, then the given solid S is visible from the set of externalplanes I. Otherwise, S is not visible from r. If r is equal to theset of planes containing the faces of a convex enclosing object E, orthe generation of r is based on E, then S is visible from . theenclosing object E. By Theorem 2, therefore, S is machinable fromE.B . Testing f o r S imp le C u r v ed Face

The previous analysis of machinability focuses on polygonalparts . Ho wever, curved surfaces can also be incorporated via polyg-

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(a) (b) (C)Fig. 11 . Illustration of the clearance requirements of the hole-making pro cess. I , 5 ensures that the hole can be made (a). 1, 2 1ensures that no collision will occur between the spind le and the wall of the hole (b). These two criteria still cannot detect the overallspindle-part collision as shown in (c).

NA (The normal o fA )NL

V . isible fromN .Not Visible fromH . idden by part faCe(S)

External Plane Fd

Enclosing Object (ME B )(b)

y n g bject Face

(C)Fig. 12. (a) A sample part design which is machinable. (b) The enclosing object of the sample part in (a). (c) Illustration of theconcept of hidden face. (d) The summary of the status of each part f ace after the visible-hidden face classification process.

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IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 7, NO. 5 , OCTOBER 1991 697

2) Tolerance: Multiple tolerancing issues are not explicitly ad-dressed in the current implementation. Following Kimura et al . [9],one can search through the space of tolerance-constrained dimen-sions and determine a unique component that is within all thetolerance constraints.3) Clearance: Another practical issue that needs to be consid-ered is the clearance of the part [8]. The part clearance is theminimal tool length required in machining to prevent spindle-partcollision. Th is problem is usually solved by modeling the movementof the tool and spindle along the tool p ath. T he collision can then bedetected by performing a solid intersection test. As an example,consider a hole-making process. For this process, the depth of thehole can be obtained by computing the distance between the bottomface and the top face of the hole, while the cutting length and totaldrill length can be obtained from the tool specification. With thisinformation, two criteria can be constructed for this hole makingprocess:

1) I , 2 , where I, is cutting length of the drill and I is the hole2) I , L I , where I , is the total length of the drill.The satisfaction of criterion 1 ensures that the hole can be madeby the drill while criterion 2 ensures that no collision would occurbetween the spindle and the wall of the hole. This is illustrated inFig. ll(a) and (b). The two criteria, however, are still not sufficientfor detecting the overall spindle-part interference, as shown belowin Fig. 14(c). In order to detect such interference, a simulation ofthe whole machining process is needed.

length.

D. esultsThe machinability algorithm has been incorporated into twoevolved enclosing object-based automatic NC code-generation sys-tems. ZEN0 [lo] is a prototype coded in Pascal and implementedon a VAX/VMS platform. The Intelligent Process Assistant (IPA)system, which is funded by the Navy, is implemented in Lisp on aSymbolics Lisp workstation. IPA makes extensive use of theobject-oriented facilities of the Lisp environment. The systemsprocess parts is designed on the solid modeler I-DEAS, whichchecks the machinability and generates NC code for producing thepart. B oth systems also automatically determine raw stock suitabilityand perform optimization of the machining directions with respect tominimizing the number of setups. The systems use a boundary

representation as input and have G code as output.E. Exam ples

In this section, we provide three examples to illustrate the test ofmachinability. Further, the implementation of the concept of hiddenface described earlier for NC machining is presented.Example 1: In the machinability checking algorithm, each partface is checked for visibility along the normal vectors of theenclosing object faces by using the hidden-face concept. Consider

the part shown in Fig. 12(a). To begin, the system generates anenclosing object such as the MEB (see Fig. 12(b)). The normalvectors of the enclosing object thus provide an initial set of check-ing directions. The enclosing object itself can be used as areference for raw material generation. In the second step, thesystem orthogonally projects part faces to the external planes de-fined by the enclosing object faces (see Fig. 12(c)). In Fig. 12(c),part faces number 10 an d 12 are projected to the external plane Fdefined by the front face of the enclosing object. Since the regular-ized intersection between projections (the shaded region in thefigure) is nonempty, a point P in the interior of the intersection can

i(b)

sample part in (a). This view reveals the inaccessibility of the hole.Fig. 13 . (a) A sample part that is not machinable. (b) The top view of

be selected. Let q and r be the corresponding points (from whichthe point p was projected) on part faces 12 and L0 , respectively.As the last step, the lengths of 2 and p r are computed todetermine the status of the p s aces21n this exam ple, part face 10is hidden by face 1 2 since lpq l < l p r l . That is, part face 1 0 is notaccessible from the external plane F , which defines a m illing plane.On the other hand, part face 12 is not hidden by any part face, so itis accessible from the external plane F . Fig. 12(d) summarizes thestatus of each part face after such a process. Based on the statusgenerated, the system responds that the part in machinable sinceeach part face is visible from at least one external plane.Example 2: Fig. 13(a) shows a part with a slot and a bridge andthe two holes in the slot that is not machinable. T he part seems to bemachinable from perspective viewing although it is actually not. Th ebottom face of the smaller hole is hidden by the faces of the bridge.In other words, the smaller hole is not accessible from the enclosingobject due to the blocking of the bridge. The top view of the part asshown in Fig. 13@) reveals the inaccessibility of the hole. In thiscase, the system will suggest that either the hole or bridge or bothshould be modified (repositioned, rescaled, etc.).Example 3: This example illustrates how the proposed machin-ability checking technique can be applied in N C machining. If a partis machinable, the system generates valid prisms and residues fromeach external plane. The volumes of raw material that need beremoved from each external plane are then constructed. Before thetool path and NC code generation are performed, the systemdetermines the number of setups required for machining. Since thetool is always orthogonal to the external plane (milling plane), thetask of de te rmining machining se tups i s to de te rminethe sequence of external planes that the part is to be machined from .

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NEEElG7EG9EG94NEEE2G5EX11.3848Y1.13EE25.EEEENE883518EEF15.EEEEII3NEBE4GlX2.3945Y-1.875E24.EEEENEEE5G122.5000NEBEfiGlX2.8125..... ~NEEE7GlY-3.8125NEBE8GlX2.52E8NEEE9GlX2.6875Y-2.3125NEElEGlX2.3945Y-2.4268NEBllGlY-1.8758NEE12GlX2.207EY-1.6875NEE1361 3 . BEBENEE14GlY-4.EEEENEE15GlX2.5EBBNE016GlX2.6875Y-2.3125NEEl7GlX2.2878Y-2.5EEENEE18GlY-1.6875NEE19GlX2.394SY-1.875ENEE2EGlX2.6125NEEZlGlY-3.8125NE022GlX2.52E8N8823GlX2.6875Y-2.3125NEE24GlX2.3945Y-2.4266NEE25GlY-1.8758NEE26GlX2.287EY-1.6875NEE27GlX3.EBEENEE2861Y-4.EEEENEE29GlX2.5EEENEE3EGlX2.6875Y-2.3125NBE32GlY-1.6875NEE33GlX2.3945Y-1.8758N E ~ ~ ~ G ~ X Z . Z E ~ B Y - ~ . ~ E E ~

hE034GlX2.6125hEE35Gll-3.8125hEE36ClX2 5288hE837Glx2.6875~-2 125

(C)Fig. 14. (a) A sample part fo r an illustration of the relationship between the proposed approach and N . C . machining. @)Removal Volume construction based on the MEB for machining the sample part in (a). (c) The tool path generated by theIntelligent Planning Assistant (IPA) system.

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IEEE TRANSACTIONS ON ROBOTICS AN D AUTOMATION, VOL. 7 , NO. 5 , OCTOBER 1991Because of the number of setups directly affects the fixturing,tolerancing, and machining time, an algorithm has been developedto determine the minimum number of setups required for producingthe part. The algorithm first attempts to fathom the possibility ofachieving the target part from a single setup. This step involvesexploring the possibility of achieving the target part from a singlemachining direction in the set of candidate machining directionsdefined by the enclosing object faces. If multiple setups is inevitablyrequired, then the algorithm searches from the minimum number ofsetups to achieve the target part by enumeration.

As an illustration, consider the part shown in Fig. 14(a). This partis machinable from the enclosing object MEB since all the part facesare visible. The system then determines that at least two machiningsetups are required to achieve the finished part requirement. Therem oval volume forme d by the union of valid prisms and residuesfrom each setup is subsequently constructed as shown in Fig. 14(b).The prisms provide depth information for material removal from thepart in the form of z =f ( x , y ) where z is the depth from theenclosing object (or raw stock) face, and (x, ) are the coordinatesof a point on an enclosing object face. Using this depth informationwith linear interpolation techniques, cutting strategies and a toolpath can be derived (see Fig. 14(c)). The generated enclosing objectand the orientation of the enclosing object face can be used as areference for the raw stock and fixture positioning, respectively.

V . CONCLUSIONSWe have presented a machinability test that determines whether agiven part can be machined from a certain set of machining direc-tions, such as those obtained from the convex hull, the minimum

enclosing box, or by the augmentation of these surfaces. Theanalysis of machinability focuses only on polyhedral parts. How-ever, curved surfaces can also be incorporated via polygonalfaceting. This provides an automatic method for segmenting thesurface into portions that need to be accessed from different direc-tions.

ACKNOWLEDGMENTThe authors would like to thank Dr. G. L . Hogg and Dr. R. J.Mayer of Texas A&M University for their support of this research

an d A . A . Keen for his helpful discussions.REFERENCES

G. T. Armstrong, G . C. Carey, and A. DePennington, Numericalcode generation from a geometric modeling system, in Solid Model-ing from Computers, M. S. Pickett and J. Boyse, Eds. New York:Plenum, 1984.A. Parkinson, The use of solid models in BUILD as a database forNC machining, in Software for Discrete Manufacturing, J. P.Crestin and J. F. M cWaters, Eds.D. S. Nau and T. C. Chang, Prospects for process selection usingartificial intelligence, Computers Industry, vol. 4, pp. 253-263,1983.A. A. G. Requicha and J. Vandenbrande, Automated systems forprocess planning and part programming, in Artificial IntelligenceImplications o r CIM , A. Kusiak, Ed. New York: Springer-Verlag,1988.T . C. Chang and R. Wysk, An integrated CAD/automated processplanning system, AZZE Trans.,pp. 223-233, 1981.A. A. Requicha and H. B. Voelker, Constructive solid geometry,Tech. Mem o. 25, Production Automation Project, Univ. of Rochester,Rochester, NY, 1977.J. D. Foley and A. Van Dam, Fundamentals of Interactive Com-puter Graphics.S . C. Chan and H. B. Voelker, An introduction to MPL: A new

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A Retraction Method for Learned Navigation inUnknown Terrains for a Circular RobotNageswara S . V. Rao, N . Stoltzfus, and S . S. Iyengar

Abstract-We consider the problem of learned navigation of a circu-lar robot R , of radius 6 ( B 0), through a terrain whose model is not apriori known. W consider two-dimensional finite-sized terrains popu-lated by an unknown (but, finite) number of simple polygonal obstacles.The number and locations o f the vertices of each obstacle are unknownto R ; R is equipped with a sensor system that detects all vertices andedges that are visible from its present location. In this context, we dealwith to problems: the visit problem and the terrain model acquisitionproblem. In the visit problem, the robot is required to visit a sequenceof destination points, and in the terrain model acquisition problem, therobot is required to acquire the complete model of the terrain. Wepresent an algorithmic framework for solving these tw o problems basedon a retraction of the free space onto the Voronoi diagram of theterrain. We then present algorithms to solve the visit problem and theterrain model acquisition problem.

Keywords-Incidental learning, polygons, retraction, unknown ter-rains, Voronoi diagrams.NOMENCLATURE

Number of obstacles.Number of vertices of outer face of D(0).Expansion factor in obtaining E( 0) ro m C(0 ).Number of concave vertices.Set of obstacles or terrain.ith obstacle. polygon.Number of positions in the visit problem.Total number of obstacle vertices.Robot.Set of visited nodes by NA V .Set of nodes with neighbors visited by N A V .Radius of the robot.Free-space.Maximal connected component of set of free positions ofR .Su m of * an d R .Convex hull of vertices of 0.Dual graph of Vor ( 0 ) .Expanded convex hull of 0.

Manuscript received April 24, 1990; revised May 7, 1991. Preliminaryresults of the terrain model acquisition portion of this work were presented atthe IEEE International Conference on Robotics and Automation, April25-29, 1988, Philadelphia, PA.N. S . V. Rao is with the Department of Computer Science , Old DominionUniversity, Norfolk, VA 23529-0162.N . Stoltzfus is with the Departm ent of Mathematics, Louisiana StateUniversity, Baton Rouge, LA 70903.S . S . Iyengar is with the Department of Computer Science, LouisianaState University, Baton Rouge, LA 70803.IEEE Log Number 9102214.1042-296X/91$01.00 01 99 1 IEEE