SHORT NOTES

cal sections of the performance surface or where suddenchanges caused by discontinuities, inequality con-

straints, or penalty functions occurred. With the abovemodifications, Rosenbrock's method, originally devel-oped for algebraic types of performance indices, was

found to be very suitable for parameter optimization offeedback-control problems, where the calculation of theperformance index involves the simultaneous solutionof an often complex set of ordinary differential equationstogether with an analysis of the corresponding solution.

It was concluded that termination of the searchshould not be based on the magnitude of the total steptaken during a given stage, but rather on no improve-ment of the performance index (within the precision ofthe computer) during a given number of consecutivestages. However, for practical purposes, an upper limiton the total number of stages should be given.Employing CSMP, especially the automatic sorting

applicable to the solution of simultaneous differentialequations (interconnected loops) and the direct simula-tion from a block diagram configuration, makes theapplication of the program very straightforward.

It is believed that a practical optimization algorithmfor the types of problems in question has been developedand that a corresponding predictable and flexible formfor a performance index has been found.

REFERENCES[1] T. Lange-Nielsen, "Parameter optimization of feedback control

systems using pattern search," Ph.D. dissertation, Univ. Iowa,Iowa City, May 1971.

[2] J. J. D'Azzo and C. H. Houpis, Feedback Control System Analysisand Synthesis. New York: McGraw-Hill, 1966.

[3] H. H. Rosenbrock, "An automatic method for finding the greatestor least value of a function," Comput. J., vol. 3, pp. 175-184, 1960.

[4] R. Hooke and T. A. Jeeves, "Direct search solution of numericaland statistical problems," J. Ass. Comput. Mach., vol. 8, pp. 212-229, 1961.

[51 R. Fletcher, Ed., Optimization. New York: Academic, 1969.[6] G. C. Newton, Jr., L. A. Gould, and J. F. Kaiser, Analytical De-

sign of Linear Feedback Controls. New York: Wiley, 1957.[7] D. T. McRuer and R. L. Stapleford, "Sensitivity and modal re-

sponse for single-loop and multiloop systems," Syst. Tech.,Inglewood, Calif., Tech. Doc. Rep. A SD-TDR-62-812, Jan. 1963.

[8] W. R. Perkins, "The sensitivity of feedback control systems toparameter variations," Stanford Electron. Lab., Stanford Univ.,Stanford, Calif., Tech. Rep. 2107-1, Sept. 1960.

On the Ordering of Connections forAutomatic Wire Routing

LUTHER C. ABEL

Abstract-Most wire-routing programs utilize a maze-runningtechnique to route one connection at a time. Once routed, a wirecannot be moved even if it is subsequently discovered to interferewith the successful completion of other connections. The order in

Manuscript received November 15, 1971; revised June 5, 1972.This work was supported in part by the Rome Air DevelopmentCenter under Contract USAF 30(602)-4144, in part by the JointServices Electronics Program under Contract DAAB-07-67-C-0199,in part by the Durham Army Research Office under Contract DAHC04-72-C-0001, and in part by NSF Grant GK-15459.

The author was with the Center for Advanced Computation, theCoordinated Science Laboratory, and the Department of ComputerScience, University of Illinois, Urbana, Ill. 61801. He is now withDigital Equipment Corporation, Maynard, Mass. 01754.

which the desired connections are presented to the routing algorithmhas therefore been thought to be of critical importance. Experimentalevidence is presented, however, to show that the performance of arouter, when measured in terms of the total of the minimum (orideal) lengths of the connections successfully completed, is, in fact,independent of the order in which connections are attempted.

Index Terms-Computer-aided design, computer design auto-mation, connection routing, interconnection, printed circuit, wiring.

INTRODUCTIONVirtually every automatic printed-circuit card wire-

routing program in existence today utilizes Lee's algo-rithm [1 ] or some other embodiment of Moore's maze-running technique [2]. The great advantage of thesemethods is their manner of systematically searching fora path between two points. But they also suffer from afundamental shortcoming in that they route preciselyone wire at a time, providing no feedback or anticipa-tion in the routing process to avoid conflict betweenwires or to assure that some early wire routing will notprevent successful completion of some later connection.

Because of this blindness, it has been claimed that theorder in which a set of connections is presented to awire-routing program (assuming the set is a priori com-pletely defined) is of crucial importance to the successfulrouting of a printed-circuit board. The topic of connec-tion ordering has been vigorously debated at designautomation conferences and workshops, but no mean-ingful data have been offered in support of particularviewpoints. The present work is a comparative study ofthe performance of various ordering methods over astatistically significant experimental base.

ORDERING METHODSConnection lengths are typically measured in terms

of the rectilinear or "Manhattan" distance between theterminals to be joined (I=Ax+Ay). Two of the mostcommon methods for interconnection ordering are inascending order of lengtlh and descending order oflength. Proponents of the former argue that it is easierto route a long wire around a short one than vice versa;supporters of the latter counter that longer wires aremore difficult to lay out, and hence should be attemptedfirst. In the descriptions of a few reported systems [3],[4], the authors declare that shortest connections mustbe routed first, but give no justification; another author[5] concludes on the basic of just two observations that"sorting by net size [wire length] does not improvelayout. "When multilayer printed-circuit boards are used for

wiring, connections are frequently first assigned to indi-vidual layers by slope classes, that is, the sectors (say,octants) of a circle into which a line drawn between thetwo points to be joined would fall. For connectionswithin one slope class, another possible ordering is basedon the magnitude of the component of a connection'slength perpendicular to the sector axis; such an orderingis an attempt to measure the degree of interference awire presents to other wires within the sector, all ofwhich should be roughly parallel to the sector axis.

1227

IEEE TRANSACTIONS ON COMPUTERS, NOVEMBER 1972

DESCRIPTION oF EXPERIMENT

Wiring lists for 38 different types of printed-circuitboards from the Illiac IV processing element were usedas a data base for these experiments. Up to 20 16-pindual in-line integrated-circuit packages can be accom-modated on a board in 5 rows of 4 packages each, withthe packages parallel to the 100-pin connector locatedat one edge of the board. Integrated-circuit packageplacement data were included in the board wiring lists.For convenience in discussing board layouts, it will beassumed that a Cartesian coordinate system is superim-posed on the board, with the Y axis parallel to the edgewith the connector.Wiring was performed in an area approximately 4 in

square (10 cm by 10 cm) on a 50-mil (1.3-mm) grid. Itwas assumed that a pin connection pad could be formedentirely within one grid square. Signal sets were inter-connected using daisy-chain wiring (i.e., each signalconnection point had at most two wires emanatingfrom it), with the signal source constrained to be at oneend of the chain.'Two printed-circuit card layers were used for signal

interconnections in these experiments. Power andground were assumed to be supplied by other layers andare ignored in this work. A simple slope-class heuristicrationale was used to assign a connection to one of thetwo layers: all connections whose X-component oflength exceeded their Y-component (Ax >Ay) were as-signed to layer 1, all others were assigned to layer 2. Thenumber of connections assigned to each layer for routingwas approximately equal, but the connections on layer 1were on the average slightly longer, leading to a roughly3:2 ratio of total minimum length of the connectionsattempted. Each required pin-pair interconnection wasconsidered to be a separate wiring problem, divorcedfrom all reference to its electrically common fellow con-nections (i.e., signal networks were decomposed intosets of independent pin-pair interconnection problems).The wire-routing program for these experiments was

a straightforward implementation of Lee's algorithm[1 ]. More sophisticated routing techniques [7 ]- [9 ] pro-viding limited adaptive movement of wires and localoptimization of routing were not used in the experimentsboth to simplify the implementation of the experimen-tal router and because some of these techniques [7], [9]demand unrestricted interlayer transitions. Interlayertransitions by means of plated-through holes (vias)were forbidden in these experiments to preclude possiblecross-correlative effects of the routing of one layer on itsopposite. If a connection could not be completely routed

1 The size of the cards actually used in Illiac IV is slightly greaterand wiring is more complex because the emitter-coupled logic familyused in Illiac IV requires external pull-down and terminating re-sistors that were ignored for these experiments. Layouts of the actualcards were created manually; plated-through holes were permitted atfixed locations to provide interlayer stitching. A photograph of a typi-cal card is included in [6].

on the layer to which it was assigned, it was simplyabandoned and counted as a failure.

Since the routing of long wires immediately adjacentto a row of pins tended to sharply curtail the successfulrouting of any subsequent connections to those pins,two heuristics were incorporated into the router to avoidthis problem: wires that joined two points in the samerow of pins (i.e., connections with Ax=0) were forcedto swing away from the row of pins and, if possible, re-main at least two grid squares away. Wires joining twopoints not in the same row were (again, if possible)forced to follow a Z-shaped path after departing fromthe terminals in the X-direction, avoiding the L-shapedpath and concomitant blocking of a row of pins thatmight otherwise be chosen. (Note that the more sophisti-cated routing techniques intrinsically avoid such block-age problems.)Wire routings were attempted for each layer of each

board five times, using the following five connectionordering methods: 1) shortest connection first (abbrevi-ated by the letter S); 2) longest connection first L;3) random ordering R; 4) connections with the shortestX-component of length first, with shortest overalllength used for tie-breaking X; 5) Connections with theshortest Y-component of length first, again with short-est overall length used for tie-breaking Y.

In any of the ordering methods, if two connectionsranked equally in the selection process (e.g., were of thesame length for the S or L methods), the one to be at-tempted first was chosen arbitrarily.

Statistics recorded for each run include the number ofconnections attempted and the number successfullycompleted, the length of wire used in routing the suc-cessful connections, the sum of the-rectilinear distancesbetween points connected, and the overall sum for allattempted connections. These latter correspond to theminimum length of wire with which the connectionsmight have been wired (ignoring the increase due totopologically unavoidable detours).

EXPERIMENTAL RESULTS

In order to meaningfully compare ordering methods,there must be an unbiased means of presenting our ex-perimental results relating to the performance of therouter with each of the ordering methods.

Certainly some connections are easier to route thanothers. If the number of connections successfully com-pleted were simply counted and used as a performancecriterion, the measure of the quality of an orderingmethod would be biased in favor of those methods thatcaused the router to attempt the easiest connectionsfirst. Specifically, since geometric arguments and thephysical realities of board design indicate (at least to afirst-order approximation) that the ease of routing awire is inversely proportional to its length, tabulatingthe number of completed connections would undoubt-edly rank the shortest-first ordering method as best.Moreover, such a criterion would not give an adequate

1228

1229SHORT NOTES

measure of how difficult it would be to route the remain-ing uncompleted connections were extra layers or man-ual intervention in the task permitted.The relationship of the difficulty of routing to wire

length suggests another measure: the total length of thewiring successfully routed. With this criterion, success-ful completion of one long, difficult wire should rankequally with an equivalent total length of short, easywires. But evaluating ordering methods by the actuallength of wire used for the successfully routed connec-tions is still less than perfect, since it would bias the rat-ing in favor of some ordering that inherently causes alarge number of wires to be completed in greater thanminimum length.The real .task of a router is to specify the layout of

wires joining a set of terminals that, with ideal routing,would all be connected with wires of minimum length(actually, in minimum length plus some unavoidabledetours whose minimum length can be a priori deter-termined). Therefore, the most unbiased measure of theperformance of a router and/or the effect of a connec-tion-ordering method is the sum of the minimum orideal lengths of the connections that were successfullyrouted.

This measure may be used directly, or it may be nor-malized to the size of the board being routed by dividingthis length by the maximum length of wire that canpossibly be put on the board. Typically, both of theselengths are measured in terms of routing grid squares,one square being the fundamental quantum of length.The boards used in these experiments had approxi-mately 5800 available squares; wire lengths (wiredensities) are hereafter expressed as percentages of thisquantity.The statistical distributions of the total minimum

wire lengths for all attempted wire routings (i.e.,2(Ax+Ay) for all connections on a layer) for the 38card types are shown2 in Fig. 1. The total ideal lengthsof all connections ranged from 3 to 55 percent, with amean length of approximately 36 percent for layer 1 and24 percent for layer 2.

Fig. 2 shows the distribution of the sums of the mini-mum lengths of successfully completed connections forthe shortest-first, longest-first, and random orderingmethods. Fig. 3 shows these ideal length distributionsfor the minimum-X-component-first and minimum-Y-component-first orderings, with the shortest-first dis-tribution repeated for comparison. For these lattercurves, recall that the connections on layer 1 fall intothe octants of the circle adjacent to the X axis, whilethose on layer 2 occur in the octants nearest the Y axis.Hence a minimum- Y-component-first ordering for layer1 attempts earliest connections having minimum com-

2 All curves have been smoothed by introducing some statistical"fuzziness" into each data point. As the occurrence of each point wasplotted, specified fractional values were added to adjacent columnsof the histogram.

t>u

z

w

L)

O-Ae0

0

I..U

2

0iJA2

LA

0

WIRING DENSITY (PERCENT OF BOARD AREA)

(a)

06 to 15 20 25 30 35 40 45WIRING DENSITY (PERCENT OF BOARD AREA)

(b)

50 55 0

Fig. 1. Distribution of ideal total length of all connections.(a) Layer 1. (b) Layer 2.

ponent of length perpendicular to the sector axis, whilethe minimum-X-component-first ordering for the samelayer represents a particularly perverse ordering. ForLayer 2, the minimum-X-component-first ordering sim-ilarly routes earliest those wires with the minimum off-axis component. Principal characteristics of all five dis-tributions are summarized in Table I.These data represent our principal experimental re-

sult: when the summation of the distances between thepoints successfully connected by a router is used as a

IEEE TRANSACTIONS ON COMPUTERS, NOVEMBER 1972

Iat

b

05 10 15 20 25 30 35 40 45 50WIRING DENSITY (PERCENT OF BOARD AREA)

(a)

ID

00 05 10 15 20 25 30 35 40 45 50 55 so

WIRING DENSITY (PERCENT OF BOARD AREA)

(b)

Fig. 2. Distribution of ideal total lengths of successfully completedconnections for shortest-first, longest-first, and random orderings.(a) Layer 1. (b) Layer 2.

XI I ti' IIIco 05 lo 15 20 25 so 35 40 45 so 56 19

WIRING DENSITY (PERCENT Of BOARD AREA)

(a)-a~~~~~~~~~~a

;00 05 10 15 20 25 so 35 40 45 50 5s 60

WIRING DENSITY (PERCENT OF BOARD AREA)

(b)Fig. 3. Distribution of ideal total lengths of successfully completed

connections for minimum-X-component-first, minimum-Y-com-ponent-first, and shortest-first orderings. (a) Layer 1 (Ax>Ay forall connections on this layer). (b) Layer 2 (Ay> ax for all connec-tions on this layer).

1230

I

ahIC

3-Jw)

I

I11:

3.

TABLE IDISTRIBUTION OF IDEAL LENGTHS OF SUCCESSFULLY COMPLETED CONNECTIONS

Wiring Densities (Percent of Board Area)

Layer 1 Layer 2

Ordering Method Minimum Mean Maximum Minimum Mean Maximum

Shortest-first 7 17 26 3 16 24Longest-first 9 18 24 3 14 22Random. 9 17 26 3 14 22Min imu m-X-component-first 7 17 25 3 16 26Minimu m- Y-component-first 11 20 29 3 15 24

measure (and we contend this is the most unbiased com-

parison criterion), the order in which the connections are

attempted has little if any effect on router performance.True, there are some small differences between order-

ing methods; in these experiments ordering based on a

connection's component of length perpendicular to thesector axis for a layer cumulatively performed slightlybetter than the others, with shortest-first ordering rank-ing second. But no ordering proved conclusively bestfor all 38 cards. In fact, each of the methods performedbest for at least one card. Moreover, simply varying thearbitrary choices made when connections ranked equallyin the ordering process changed individual results bythe same percentages as the differences between thecumulative results. Also, if the set of layer 1 routingsand the set of layer 2 routings are regarded as separateexperiments (as essentially they are), different rankingsfor the methods are observed. We therefore feel that thelack of strong differences between the methods is farmore significant than the fact that small differences doexist.

Figs. 4 and 5 show the distributions of the actuallength of wire used for the successfully completed con-

nections, and Table II summarizes these characteristics.A strong correlation between these ideal-length distribu-tions is observed, with an average of 1.5 times as muchwire as the minimum actually required for the connec-

tions. But there are individual anomalies; for example,the ratio of actual-to-ideal wire lengths is consistentlyhigher for random ordering.The question of whether one ordering method might

demonstrate a substantial, consistent superiority forsome specific limited range of attempted wiring densi-ties was investigated by plotting the completion rate(measured in ideal length) versus the total length ofconnections attempted for the various methods; no suchprepotency was observed. Moreover, a large scatter wasobserved in the completion rates for sets of boards re-

quiring essentially the same amount of wire to be routed.This leads us to believe that complex geometric andwire-interrelationship factors considerably influence thesuccessful routing of a set of connections. Conversely,the completion rate was almost completely independentof the density of the attempted wiring.

Akers [10] and others have reported that routers per-form as if there existed within them a fundamental bar-rier to complete routing such that the ideal length ofconnections successfully routed simply cannot exceed afixed fraction of the available board area. Stated an-other way, any set of connections whose total minimumlength exceeds a given fraction of the board area is cer-tainly doomed to contain wires that will not be success-fully routed. The data of Figs. 2 and 3, with their ex-tremely sharp cutoff at 22-25 percent, strongly supportthis assertion.The value of the cutoff point seems dependent on

board geometry and the exact details of the routingalgorithm, such as the ordering method used and themethod of choosing among a multiplicity of shortestpaths. In addition to the small variations observed asthe connection-ordering method is changed, other ex-periments we have conducted indicate that the cutoffpoint is shifted by other geometric and algorithm varia-tions, but remains just as sharp.

CONCLUSIONS

Experimental evidence of a statistically meaningfulnature has been presented to show that the performanceof a maze-running-type router, when measured by thetotal ideal length of connections successfully routed, isindependent of the order in which connections are at-tempted. If the creator of a computer-aided design sys-tem feels compelled to include an ordering method,then ordering based on a connection's component oflength perpendicular to the sector axis for a layer wouldprobably perform over a long term in a marginallysuperior manner. The existence of a sharp fundamentallimit to the acceptable board wiring density for a rouiterhas also been demonstrated.Although the router used in these experiments did not

permit interlayer transitions, and the inclusion of sucha facility would undoubtedly increase the maximumwiring density that could be achieved, we feel the quali-tative result concerning uniformity of performance inde-pendent of connection ordering would not change.

Further research must be done on the relationship ofboard geometry and the geometric properties of packageplacement to successful routing. Why, for example, did

1231SHORT NOTES

IEEE TRANSACTIONS ON COMPUTERS, NOVEMBER 1972

tid~~~I

z

.1 -~~~~~~~~~~~31

>

00 05 10 15 20 25 30 35 40 45 50 55 60

WJIRING DENSITY (PERCENT OF BOARD AREA)

(a)0WI

Laye (b5 20ye 25.

w ~~ ~ ~ ~ ~ ~ ~

hi~~~ ~ ~ ~ ~ ~~~~Z

2.

>

0~~~~~~~~~~~~~~~a

20 15 1 5 2 5 0 3 0 4 0 5~ ~ ~ ~ IIGDNIY PRETO OR RA

0~~~~~~~~bFig. Dsrbto fata oa egh o ucsflyrue

wie o hretfrt ogstfrt.n admodrns

20y

(a$wcc0:

IL0

WIRING DENSITY (PERCENT OF BORD AREA)

(a)

I

0a

0

hi

lei

,ax

's1S 20 25 30 55 40 45 60

WIRING DENSITY (PERCENT OF BORD AREA)

(b)

Fig. 5. Distribution of actual total lengths successfully routed wiresfor minimum-X-component-first, minimum- Y-component-first,and shortest-first orderings. (a) Layer 1. (b) Layer 2.

1232

TABLE IIDISTRIBUTION OF ACTUAL WIRE LENGTHS FOR SUCCESSFULLY COMPLETEID CONNECTIONS

Wiring Densities (Percent of Board Area)

Layer 1 Layer 2

Ordering Method Minimum Mean Maximum Minimum Mean Maximum

Shortest-first 13 28 46 4 23 40Longest-first 17 29 44 3 21 36Random 18 30 44 3 23 42Minimum-X-component-first 14 28 49 4 24 45Minimum- Y-component-first 15 30 46 4 23 41

completion rates for boards of equal attempted wiringdensity vary by an alnmost 2:1 ratio in our experiments?Additional investigation is also needed into the algo-rithmic properties of routers to try to overcome theirone-wire-at-a-time blindness. Mlethods for choosing theoptimal path wxhen a choice of routings exist seem crucialto preventing early wire routings from blocking laterones, and the blocking properties of wire forced outsidetheir minimum-distance rectangles on the M\anhattangrid requires examination.Two additional factors may have a direct impact on

the successful routing of a board: the uniformity of thespatial distribution of connections (that is, the presenceor absence of "hot spots" through whiclh many wiresmust pass) and the total component of ideal wire lengthperpendicular to the major wiring axis for a layer. If acorrelation between these factors and the wireability ofa board does indeed exist, then they perlhaps can be usedas package placement criteria, either in addition to orinstead of the traditional placement objective of mini-mum wire length [1I].

REFERENCES[1] C. Y. Lee, "An algorithm for path connections and its applica-

tions," IRE Trans. Electron. Comput., vol. EC-10, pp. 346-365'Sept. 1961.

[2] E. F. Moore, "Shortest path through a maze," in Annals of theComiputation Laboratory of Harvard University, vol. 30. Cam-bridge, Mass.: Harvard University Press, 1959, pp. 285-292.

[3] M. Freeman et al., "Multilayer printed wiring-Computer aideddesign," in Proc. 4th Share/A CM/IEEE Design AutomationWorkshop, 1967, pp. 16-1-16-28.

[4] S. Heiss, "A path cQnnection algorithm for m'ulti-layer boards,"in Proc. 5th Share/ACMI/IEEE Design Automation W'orkshop,1968, pp. 6-1-6-14.

[5] R. C. Moore, "Packaging flat pack intergrated circuits for earthsatellites," in 8th Int. Circuit Packaging Symp. Rec. (Advances inElectronic Circuit Packaging, vol. 8), sec. 4/4, 1967, pp. 1-9.

[6] D. L. Slotnick, "The fastest computer," Sci. Amer., vol. 224,pp. 76-87, Feb. 1971.

[7] A. Hashimoto and J. Stevens, "\Vire routing by optimizing chan-nel assignment within large apertures," in Proc. 8th Share/ACM/IEEE Design Automation lVorkshop, 1971, pp. 155-169.

[8] R. B. Hitchcock, "Cellular wiring and the cellular modelingtechnique,' in Proc. 6th Share/AC.M/IEEE Design AutomationWlXorkshop, 1969, pp. 25-41.

[9] S. E. Lass, "Automated printed circuit routing with a steppingaperture," Commun. Ass. Comput. Mach., vol. 12, pp. 262-265,May 1969.

[10] S. B. Akers, as reported in the minutes of the IEEE CircuitsStandards Committee meeting held at the Naval PostgraduateSchool, Monterey, Calif., J. Bordogna, Secretary, p. 19, May1969.

[11] M. Hanan and J. M. Kurtzberg, "A review of the placement andquadratic assignment problems," IBM Corp. Res. Rep. RC3046,Apr. 1970.

A Parallel Mechanism forDescribing Silhouettes

JACK SKLANSKY AND PAUL J. NAHIN

Abstract-We describe a parallel mechanism, based on an arrayof circularly nutating photodetectors, that computes an approxima-tion of the density of slopes of the boundary of any piecewise regularsilhouette. This density, when properly normalized, is invariant withrespect to the size, translation, and orientation of the given silhouette.

Among the results we derive are: 1) the fundamental Fourierharmonic of the slope density is zero, and 2) smoothing the slopedensity by a time-invariant linear filter multiplies the perimeter ofthe represented silhouette by the area under the filter's impulsiveresponse.

We present evidence indicating that the nutation process con-tributes usefully to the recognition of the "ideal" smoothed versionof the given silhouette.

Index Terms-Classification, pattern recognition, radius ofcurvature, silhouette, slope density.

I. INTRODUCTION

We define a silhouette to be a set of points consistingof a simple (non-self-intersecting) closed curve and itsinterior. We are particularly interested in silhouetteswhose boundaries are piecew-ise regular; i.e., the boun-dary of each silhouette consists of a finite number ofregular curves and a finite number of straight line seg-ments. We refer to suclh a silhouette as a piecewise regu-lar silhouette. (A "regular curve" is defined as the locusof points traced by the endpoint of a vector

r(t) = [rl(t), r2(t)] (1)

such that ri(t) (i= 1, 2) has a continuous second deriva-tive and the derivative of r1(t) (i= 1, 2) is nowhere zero.)We describe a parallel mechanism that converts any

piecewise regular silhouette into a periodic "signature,"i.e., a periodic function of time representing approxi-mately the density of slopes of the boundary of the sil-houette. The parallel mechanisml consists of a circularly

Manuscript received Jtuly 29, 1971; revised April 25, 1972.This work was supported by the National Science Foundation underGrant GK-4226. Part of this note was presented at the IFIP Con-gress 71, and appears in the Proceedings of that Congress.

J. Sklansky is with the School of Engineering, University of Cali-fornia, Irvine, Calif. 92664.

P. J. Nahin was with the Hughes Aircraft Company, Fullerton,Calif. He is now with Harvey Mudd College, Claremont, Calif. 91711.

1233SHORT NOTES