design of a new parametric path plan for additive manufacturing of

Ibrahim T. OzbolatMechanical and Industrial Engineering,

The University of Iowa,

2130 Seamans Center,

Iowa City, IA 52242;

The Center for Computer-Aided Design,

The University of Iowa,

216 Engineering Research Facility,

Iowa City, IA 52242

A. K. M. B. KhodaIndustrial and Manufacturing

Engineering Department,

North Dakota State University,

202F CIE Building,

Fargo, ND 58102

Design of a New Parametric PathPlan for Additive Manufacturingof Hollow Porous StructuresWith FunctionallyGraded MaterialsIn this paper, a novel path planning approach is proposed to generate porous structureswith internal features. The interconnected and continuous deposition path is designed tocontrol the internal material composition in a functionally graded manner. The proposedlayer-based algorithmic solutions generate a bilayer pattern of zigzag and spiral toolpathconsecutively to construct heterogeneous three-dimensional (3D) objects. The proposedstrategy relies on constructing Voronoi diagrams for all bounding curves in each layer todecompose the geometric domain and discretizing the associated Voronoi regions withruling lines between the boundaries of the associated Voronoi regions. To avoid interfer-ence among ruling lines, reorientation and relaxation techniques are introduced to estab-lish matching for continuous zigzag path planning. In addition, arc fitting is used toreduce over-deposition, allowing nonstop deposition at sharp turns. Layer-by-layer depo-sition progresses through consecutive layers of a ruling-line-based zigzag pattern fol-lowed by a spiral path deposition. A biarc fitting technique is employed through isovaluesof ruling lines to generate G1 continuity along the spiral deposition path plan. Function-ally graded material properties are then mapped based on a parametric distance-basedweighting technique. The proposed approach enables elimination or minimization ofover-deposition of materials, nonuniformity on printed strands and discontinuities on thetoolpath, which are shortcomings of traditional zigzag-based toolpath plan in additivemanufacturing (AM). In addition, it provides a practical path for printing functionallygraded materials. [DOI: 10.1115/1.4028418]

Keywords: toolpath planning, porous structures, functionally graded materials, additivemanufacturing, heterogeneous objects

Introduction

Porous structures have attracted a great deal of interest in tissueengineering [1–3], drug delivery [4], fluid mechanics [5], andenergy [6]. This interest is largely driven by the wide applicationsin which porous structures manifest many appealing properties tosolid counterparts, such as larger internal surface areas and higherstrength-to-weight ratios. Interconnected and controlled porosityin a scaffold structure enables transport of media and biologicalcells to grow and regenerate new tissues [7]. Traditional porousstructure fabrication processes are mainly driven by chemicalprocesses, through which controllable uniformity, repeatabilityand/or distribution of material and internal architecture areextremely difficult to achieve in such techniques. AM aims tomake a physical prototype from its computer model, which willbe tessellated, sliced, and then built layer by layer. AM has a greatpotential to develop functionally graded materials with complexporous geometries.

In the literature, a plethora of work has been done to generateporous structure via AM applications, which mainly operates withCartesian coordinate based motion control system. Khoda et al.[1] proposed a toolpath methodology to build porous scaffoldswith controlled porous architecture. An offsetting technique wasapplied to generate functionally graded regions with different

porosity levels. Binary integer programming was then used toconnect filaments in different regions to maximize the connectiv-ity of porous channels. Although continuity was achieved for anonhollow interior geometry, the imposition of any hollowing fea-ture in the geometric domain brings jumps during the depositionprocess. Ozbolat and Koc [8] proposed a multimaterial toolpathplan, in which a 3D lofting process was developed to generatefunctionally graded regions for 3D wound dressings. The toolpathplan in the Cartesian coordinate system did not follow the materialblending direction precisely. Thus, a smaller raster size with asmaller gap between adjacent rasters is required to improve accu-racy in the composition of deposited material. Besides, introduc-ing any hollow features could result discontinuity in pathplanning. Such controlled and spatial deposition discontinuitydefined here as a jump (see Fig. 1) ensures the topology of hollowfeature but introduces complexity in path planning design. Themajority of proposed toolpath plans were constructed using theCartesian coordinate system [9], which triggers over depositionfor hollow object architecture [10]. Qui and Langrana [11]addressed this issue by filling void space with hollowing featuresusing an adaptive road width methodology, but continuity was notconsidered. That method was proposed to minimize gap forma-tion, which was independent of hollowing feature geometry; how-ever, imposed hollowing shapes brought a number ofdiscontinuities during the deposition process. Figure 1 highlightsthe drawbacks of the toolpath plan using the Cartesian coordinatesystem. In Fig. 1, material blending through multiple directions isindependent of the Cartesian coordinates. Thus, a smaller distance

Contributed by the Design Engineering Division of ASME for publication in theJOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscriptreceived August 1, 2013; final manuscript received August 17, 2014; publishedonline September 10, 2014. Assoc. Editor: Charlie C.L. Wang.

Journal of Computing and Information Science in Engineering DECEMBER 2014, Vol. 14 / 041005-1Copyright VC 2014 by ASME

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between adjacent rasters is necessary to obtain better approxima-tion when mapping material composition onto the geometric do-main. Besides, a number of jumps over hollowing features arecommon in toolpath planning in the Cartesian coordinate system.The jumps in Fig. 1 bring nonuniformity at each start and endpoint, which induces issues such as stress concentrations andover-deposition [1]. Xu and Shaw [12] introduced design of heter-ogeneous objects with internal holes for AM. They proposed theequal distance offset approach to generate gradients from thebounding curve to internal curves with identical material proper-ties in a 2D slice of a 3D object. Although material propertiesvary smoothly for a slice with a single internal curve, multiple in-ternal curves generate nonsmooth gradients when offset curvesapproach these curves due to the nature of offset curves. Kimet al. [13] addressed shortcomings of toolpath planning with an in-ternal hollow shape for ablation processes such as lasers and elec-tric power. The layer was decomposed into subregions byminimizing dwell time and energy requirement. Choi and Zhu[14] introduced virtual prototyping of multimaterial depositionthrough multiple nozzles. In that paper, a zigzag toolpath is usedto fill contours. One is embedded into the other according toparent–child relations, and a new methodology was developed toavoid nozzle collision. Zhu and Yu [15] proposed dexel-basedslicing of objects with assemblies with multimaterial properties,where an object is approximated using a number of dexels ordepth elements. The proposed method, however, created objectswith no more than single discrete entities. Chiu and Tan [16]designed and fabricated hollow objects in which voids were repre-sented as repeating Voxel elements, with a toolpath generatedusing ray representation. Ray representation of objects such ashoneycomb and square shape objects brings discontinuity numer-ous times during the fabrication process. During AM, materialdeposition may start and stop periodically depending on the inter-nal architecture. Starts and stops degenerate the uniformity of thedeposited material shape and need to be avoided [1]. Particularly,in precision engineering, over-deposition in micro-/nanofabrica-tion may trigger tremendous compromise in their functionality.

Although continuity in toolpath plan is important, fabrication accu-racy also depends on consistency of toolpath plan points. TypicalAM motion control system provides motion between points via lin-ear or circular interpolation. A higher number of points provideshigher accuracy in geometry, but require longer motion executiontime. Moreover, motion profile within commands is nonuniform,which results in nonuniform deposition. As a result, there must havesome tradeoff between geometric accuracy and deposition accuracy.

In this paper, a new toolpath planning approach is developedfor hybrid hollow structures with multiple voids. A combinationof bilayer pattern with zigzag and spiral toolpaths is proposedwith arc and biarc fitting algorithm. Arc fitting is used to eliminatetwo consecutive sharp turns in the zigzag pattern while spiral tool-path is constructed with biarc fitting algorithm. This eliminatesthe abovementioned shortcomings of traditional and recentapproaches to hollow object prototyping. The proposed algorithmsare intended to generate toolpath for the multichannel single noz-zle AM process, which was developed in our past studies [4,8].The multichannel single nozzle process enables blending multiplematerials with dynamic or static mixer followed by extrusion ofthe material. The system can allow printing multiple materials ordifferent concentration of same material or encapsulated particlesto obtain functional gradients on the printed filaments spatially.

Geometric Modeling

To generate the continuous and interconnected toolpath for aporous structure with internal features as shown in Fig. 1, we pro-posed a new porous structure modeling for functionally gradedmaterials to direct the material deposition between the features. Inthis paper, a feature is defined as a generic shape of a structurewith which certain attributes can be associated for reasoning aboutthat structure [17]. For example, continuous material variationover a porous structure can be governed by its geometric features.Features are geometric entities such as points, curves, shells,surfaces, or solid primitives that define the geometry of a struc-ture. In such a scheme, the properties of geometric features can bechanged to obtain different designs where features dominate thestarting and ending points of variation (i.e., material compositionvariation) over the internal architecture. In this work, features areassigned to different shells on a model, which can be obtained viavarious means such as reverse engineering or medical image proc-essing or simply by using computer-aided design (CAD) as dis-cussed in Sec. 4.

For a single and concentric internal feature, connecting the cor-responding closest points between the two features (internal andexternal features) with a cyclic pattern would simply generate acontinuous zigzag toolpath. However, for an eccentric internalfeature, such a method could generate a twisted and interruptedtoolpath. Furthermore, in the case of multiple internal features,the connection between them is complex due to one-to-manyfeatures (one external feature needs to be connected to multipleinternal features), and arbitrary connection could generate inter-sections on the toolpath. To avoid this issue, a Voronoi diagram isintroduced as an intermediate feature (see Fig. 2(a)), and a one-to-one relationship (where one Voronoi cell needs to be connected toonly one internal feature) is established using Voronoi cell

Fig. 1 Shortcomings of traditional toolpath planning in Carte-sian coordinates such as jumps and independency of materialblending direction from the deposition direction in Cartesiancoordinates

Fig. 2 (a) Voronoi diagram generation. (b) Voronoi cell contour for corresponding internalfeature. (c) Voronoi cell contour for the external feature.

041005-2 / Vol. 14, DECEMBER 2014 Transactions of the ASME


contours as shown in Fig. 2. The Voronoi diagram in this paper isgenerated using offset curves as proposed by several researchers[18–20]. The Voronoi diagram was generated with a computa-tional accuracy of 10�3 using the codes developed in our earlierwork [20]. For each feature, a Voronoi cell is generated from theVoronoi diagram to introduce the one-to-one relationship. Theone-to-one relationship facilitates discretization of the geometricdomain into smaller regions, where a toolpath can be generatedwithout or minimum jumps overcoming discontinuity issue. AVoronoi cell contour can be defined as the segment of the Voronoidiagram that encloses a hollowing feature. Multiple features canalso share the same Voronoi segment in their Voronoi cell contouras shown in Fig. 2(b). Thus, the Voronoi diagram is generated,and a number of Voronoi cells VCcf gc¼1;:::;C are obtained for eachfeature Caf ga¼1;:::;A as illustrated in Fig. 2(a). Moreover, corre-sponding Voronoi cell contours for the external features areshown in Fig. 2(c). Next, each feature needs to be connected withits corresponding Voronoi cells VCcf gc¼1;:::;C via straight lines(ruling lines). Such lines will determine the toolpath for the po-rous structure, and the material composition will be definedthrough them. In this paper, the ruling line generation techniqueintroduced in our recent work [21] is implemented, which avoidsgenerating twisted and intersecting ruling lines between features.As shown in Fig. 3(a), both VCc and Ca have been parametricallydivided into the same independent number of equal cord lengthsections by sampled points from each curve, and those points areused to generate ruling lines. In order to find a correspondencebetween two points, both the distance between the points and theinner product of the unit normal vectors of the points are

considered. The optimization model is executed using a dynamic-programming-based algorithm presented in our earlier work [21].A ruling line in this work is feasible as long as it does not intersectwith other ruling lines and the bounding curves (except the con-nection points) satisfy the nonintersection constraint and the visi-bility constraint. A number of ruling lines between internalfeatures and their corresponding Voronoi cells are generated inde-pendently as shown in Fig. 3(b). The reader is referred to our ear-lier work for ruling line generation [21]. Ruling lines in this paperare proposed to be entities between two material features thatguide the toolpath. These entities are called “property changinglines (PCLs)” in this paper. The composition alters along thePCLs’ orientation direction gradually considering the neighboringfeatures. PCLs are inserted considering one-to-one feature interac-tion, which brings a mismatch or discontinuity among PCLsbetween Voronoi cells, as shown in Fig. 3(b). In other words,unconnected PCLs results in jumps on the bisector. In order toovercome this problem, a PCL matching technique is proposed toreduce over-deposition or stops and starts during AM.

Matching of Property Changing Lines. If PCLs in a Voronoicell are inserted independent of PCLs in adjacent cells, this bringsdiscontinuities on bisectors, which is a crucial shortcoming intoolpath planning. Figure 4 highlights discontinuities betweenPCLs on adjacent Voronoi cells. During toolpath planning, the tool-path needs to be continuous, which necessitates the connection ofPCLs between two adjacent cells. Without matching of PCLs, thetoolpath follows the Voronoi diagram to travel through unconnectedPCLs. This, however, triggers over-deposition on the Voronoi dia-gram (bisectors). Thus, a PCL matching technique is proposed toconnect the dispatched PCLs on adjacent cells. For each bisectorbetween two cells, PCLs are ranked based on their priority. Two con-ditions can be used to express the priority analytically:

(i) Normal vectors on two points, where a PCL intersects ahollowing curve and a bisector, need to be collinear or par-allel for smooth material transition, which means that thePCL needs to be oriented in the direction of property com-position (i.e., material) change between the hollowingcurve and the bisector. Material composition over the ge-ometry needs to change smoothly without any sharpchanges, while this can result in material incompatibilitiesand stress concentrations.

(ii) In order to reduce fabrication time, the length of PCLsshould be minimized.

Fig. 3 Curve matching and ruling line generation between thefeatures and the Voronoi cells

Fig. 4 Matching between property changing lines based on an objective function

Journal of Computing and Information Science in Engineering DECEMBER 2014, Vol. 14 / 041005-3


Using the above conditions, an objective function can be intro-duced as

f ðpki ; q

amÞ ¼

~Nðpki Þ; ~Nðqa

mÞD Ewn

pki � qa

m

�� !wd(1)

where Pk ¼ pki

� �i¼1;:::;I

is a set of points sampled from bisectorBkðuÞ on the Voronoi diagram and N

!ðpki Þ is the corresponding

normal vector for 9pki : pk

i 2 Pk. In Eq. (1), qam is the mth point

sampled from the hollowing curve, Ca. Point sampling in eachcurve is performed over its parametric domain to even capturehighly nonlinear sections of the curve, such as

P ¼ pki

� �i¼1;:::;I

; where pki ¼ BkðuiÞ; ui 2 ulow; uhigh

� �;

ui < uiþ1; u0 < ulow; u1 < uhigh

pk0 ¼ Bkðu0Þ ¼ BkðulowÞ and pk

1 ¼ Bkðu1Þ ¼ BkðuhighÞ(2)

A greedy heuristic optimization approach developed in the litera-ture [22] is applied to perform PCL matching. The approachbegins with searching for a PCL with maximum objective func-tion, called max {PCL}. Then, the proximity of other PCLs fromthe adjacent cell is calculated on the bisector curve. The closestone is chosen, and the corresponding objective function is calcu-lated. A proximity upper bound is introduced as a user-inputtedvalue so that if the distance of the candidate PCL (CPCL) exceedsthe upper bound, then it is not considered feasible. This preventsconnection of a CPCL that is considerably apart from max {PCL}even though it is the closest among other CPCLs. In this case,max {PCL} is deleted when there is no feasible candidate formatching. Then, the connection point of the two PCLs on thebisector curve is determined by obtaining a weighted averagepoint between two points on the bisector curve based on the objec-tive function values of two PCLs. Next, the weighted averagepoint p�ðu�Þ is calculated as

p�ðu�Þ ¼pk

i ðuiÞf ðpki ;q

amÞ þ pk

j ðujÞf ðpkj ;q

bnÞ

f ðpki ;q

amÞ þ f ðpk

j ; qbnÞ

pki ¼ BkðuiÞ; pk

j ¼ BkðujÞ; p� ¼ Bkðu�Þ(3)

Two PCLs are then reconstructed, and their intersection pointswith the bisector curve are shifted to the weighted average pointp�ðu�Þ (see Fig. 4). This procedure is repeated until all PCLs onthe bisector curve are matched and connected with the PCLs ofthe adjacent cell. Those that do not have a matching pair areremoved, and the same technique is applied for each Bk. Figure 5illustrates the final result of the matched PCLs. As a result, thetoolpath can be constructed through each PCL continuously. Onceit approaches the bisector curve, it proceeds directly with thematched pair and passes to the other cell. This eliminates over-deposition and self-intersections on the bisector curve. AlgorithmI has been generated for PCL matching along each Bk (see theAppendix). The developed algorithm for PCL matching is appliedfor each bisector on the Voronoi diagram. Figure 5 highlights the

results of PCL matching, in which each PCL enables thecontinuous deposition path plan without over-deposition.

Relaxation of Property Changing Lines. Perfect alignment ofPCLs enables a smooth transition of material properties betweentwo features. PCLs that are close to branch points cannot bealigned perfectly; otherwise, they intersect each other. This enfor-ces density of PCLs over regions that are closed to branchingpoints (points where the Voronoi diagram branches into twobisectors). Although this can be acceptable for modeling purposes,it brings an over-deposition problem in the physical fabricationprocess. This will increase the severity of the problem in 3D witha cumulative increase in the height of deposited layers aroundbranching points. The nozzle may collide with the building struc-ture around these points during the deposition process and candamage the structure. Figure 6(a) shows the over-deposition prob-lem highlighted within a rectangle. Applying this procedure intoolpath planning, however, triggers over-deposition remarkablyover regions that are closed to branching points. This can be par-tially eliminated by increasing deposition speed or feed rate whileapproaching these regions. In order to decrease the effect of over-deposition considering the nonuniformity in these regions, a PCLrelaxation methodology has been developed. PCLs that are notaligned near the branching points are moved away from eachother by distributing them through the free space. This free spaceis created by reorienting two aligned PCLs (enclosing nonalignedPCLs) in the outward direction. This is followed by shifting thenonaligned PCLs through the shifted aligned lines with the sameparametric distance in between them. Hence, the gap betweeneach PCL is enlarged, and over-deposition is alleviated. Relaxa-tion can be applied iteratively until all denser regions are elimi-nated, depending on user input values. After relaxation, PCLs canbe used for toolpath planning. Algorithm II in the Appendix hasbeen developed for the relaxation of PCLs, Applying AlgorithmII, PCLs are distributed away from the branching points (seeFig. 6(b)), and the likelihood of over-deposition is alleviated. InAlgorithm II, the user can input the upper distance bound d todefine the closeness. This number depends on the diameter of thedeposited raster and the expected porosity. In this section, PCLsare preprocessed to enable a new toolpath plan without over-deposition. The next section introduces toolpath planning throughPCLs’ orientation direction.

Fig. 5 Matching of property changing lines between Voronoicells

Fig. 6 Relaxation of property changing lines to alleviateover-deposition



Control of Material Deposition by the Orientation ofProperty Changing Lines. In this section, PCLs are used toorient zigzag pattern deposition. One of the most challengingproblems in the context of AM during zigzag motion is over-deposition during sharp turns [23]. Thus, the following methodol-ogy is proposed to overcome this problem.

Arc Fitting for Sharp Turns. While approaching sharp turns,deposition stops and then changes direction and acceleratesin the other direction. Although continuous deposition is

observable, there is considerable degeneration in the uniformityof deposited filament thickness [23]. This can be alleviatedby selecting a higher feed rate while approaching sharpturns; however, it is not completely avoidable. Similar issuesare also common in machining problems [24]. A generalapproach to such a problem is to generate a tangential continuityalong the toolpath. In this paper, arc fitting is used to eliminatetwo consecutive sharp turns. The fitted arc must be tangent totwo PCLs and connecting line simultaneously as shown inFig. 7(a). The radius of the curvature of the arc can be derived asin Eq. (4)

q ¼ qamþ1pk

iþ1

��!��

tan 0:5 qamþ1qa

m

��!; qa

mþ1pkiþ1

��!� �tan 0:5 qa

mpki

��!; qa

mqamþ1

��!E� �

tan 0:5 qamþ1qa

m

��!; qa

mþ1pkiþ1

��!� �þ tan 0:5 qa

mpki

��!;qa

mqamþ1

��!E� � (4)

Then, tangent points ðqamÞ0; ðqa

mþ1Þ0 �

in which the toolpath has G1

continuity through the arc can be derived as in Eq. (5)

ðqamþ1Þ

0

ðqamÞ0

( )¼

qamþ1

qam

� �

þq

pkiþ1�qa

mþ1

qamþ1pk

iþ1

��!��

1

tan 0:5 qamþ1qa

m

��!;qa

mþ1pkiþ1

��!� �pk

i �qam

qampk

i

��!��

1

tan 0:5 qamþ1qa

m

��!;qa

mpki

��!� �

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;

(5)

The toolpath should be planned through inserted PCLs with theminimum amount of over-deposition and the minimum number ofstarts and stops. In this work, the toolpath plan first crosshatchesthe region between the bounding curve and the hollowing curves.PCLs shown in Fig. 7(b) are used to generate the toolpath. Thetoolpath needs to start at a PCL just after one of the branchingpoints while the initialization of the toolpath on another PCLbrings jumps during the deposition process. In addition, if thenumber of PCLs on the bounding curve is odd, then the toolpathshould start from the bounding curve; otherwise, it should startfrom the hollowing curve to eliminate any possible jumps. Thisenables completion of crosshatching at the hollowing curve andtransition of the toolpath without any jumps through inner regions.As a proof of concept, we highlight the simulation of the deposi-tion plan along PCLs between the bounding curve and the hollow-ing curves developed using Rhinoscript in RHINOCHEROS 4.0 [25](see Fig. 8(a)). Once the external region is crosshatched, then thetoolpath proceeds with the internal region and fills the internal

region until it crosses a filled region. Figures 8(b) and 8(f) high-light the simulation of the deposition in the internal region: firstbetween C1 and C2, and then between C1 and C4. During cross-hatching over the internal or external regions, the toolpath selectsthe closest unoccupied PCL to move. The toolpath traces PCLsbetween C2 and C4, and then between C3–C4 and C2–C3. Once theclosest PCL is in a direction where a jump is needed, the toolpaththen proceeds with a jump to that PCL. These jumps are unavoid-able in this study during internal region deposition when the num-ber of hollowing features is greater than two; however, total jumpdistances can be minimized by selecting the shortest unoccupiedPCLs during jumps. Finally, the toolpath is completed as shownin Fig. 8(f). Algorithm III in the Appendix has been developed forpath planning in PCL orientation direction.

In this section, PCL-based zigzag toolpath planning has beendiscussed. The next section introduces a new deposition patternfor the consecutive layer in AM.

Fig. 7 (a) Arc fitting for smooth and uniform deposition and (b)arc fitted PCLs for smooth deposition

Fig. 8 Simulation of toolpath plan in RHINOCEROS 4.0 for thedeposition along PCLs



Spiral Pattern Toolpath. The lay-down pattern style, i.e., ingeneral 0–90 deg orientation, has been widely used in layer-by-layer manufacturing to build objects in the vertical direction, inwhich each layer supports the next one [26]. In order to supportthe zigzag pattern discussed in Sec. 2.3, a new spiral toolpath planis proposed for the next layer, in which spiral toolpath is closelyperpendicular to PCLs at the support point so that next layer canbe better supported structurally. The advantages of the spiral tool-path are as follows:

- The spiral toolpath enables a continuous deposition through ahollowing feature compared to the conventional zigzag-basedspace-filling technique.

- A smooth transition from one material composition propertyto another can easily be generated.

During the insertion of PCLs, imposing visibility and noninter-section constraints enables nonintersecting PCLs as shown inFig. 9. Our earlier study discussed the constraints extensively;extreme cases such as concavities were tested and zero degenerationwas achieved as well [19]. Thus, a continuous and nonintersectedspiral pattern can be obtained. The spiral curve can be constructed bygenerating knot points. Knot points are located on PCLs by a spiralfunction S (i.e., a polynomial function). A piecewise spiral curve hasbeen constructed for illustration purposes (see Fig. 9).

Biarc Fitting. The spiral pattern curve can be constructed as aset of piecewise linear curves through the inserted knot points dueto its simple implementation and computational efficiency (seeFig. 9); however, this exaggerates the deposited filament uniform-ity and needs to be smooth for uniform deposition. In addition, thenumber of knot points on the spiral profile necessitates a plethoraof commands in the toolpath. Thus, a curve fitting methodologyneeds to be implemented along with some degree of continuity toensure a smooth and precise toolpath. A spiral with G1 continuity

could provide the required smoothness in toolpath design. Due tothe nature of the freeform geometry, the knot points may not besuitable for monotonically increasing/decreasing the curvature ofthe spiral [27]. Instead, biarc fitting among those knots might bemore appropriate in both C and S type shapes while precisely fol-lowing the global spiral pattern. A biarc curve can be defined astwo consecutive arcs with identical tangents at the junction pointJ that preserves G1 continuity while maintaining a given accuracy.When applied to a series of points, it determines a piecewise cir-cular arc interpolation of given points. The following informationis required to construct a biarc [28]:

- the number of points n through which it must pass- the coordinate ðxi; yiÞ of points kpið2 � i � nÞ- the tangent at the first and the last point.

A set of discrete knot points KP ¼ fkpngn¼0;1…N has beenachieved on the set of property changing lines PCL¼ PCLtf gt¼1;…;T by a spiral function S. To Introduce abiarc between two end knot points kps and kpe that consists oftwo segments of circular arcs A1 and A2 (see Fig. 10), the knotpoint set needs to match Hermite data [29] (i.e., both coordinates)and the unit tangents ts and te. Here the biarc can be denoted asB ¼ fkps; ts; kpe; teg for notational convenience. Figure 10(a)illustrates a C-shape, and Fig. 10(b) illustrates a S-shape biarc,where kps and kpe are two knot points, ts and te are correspondingtangents, J is the junction points of two arcs, l is the cord lengthkpskpej j, r1 and r2 are the radius of two arcs, and O1 and O2 are

the corresponding centers. Some conventions used:

- A1 must pass through the knot point kps, and A2 must passthrough the knot point kpe with unit tangents ts and te,respectively.

- The junction point J is determined by minimizing the differ-ence in curvatures.

Fig. 9 Feasible PCLs for spiral path planning resulting in generation of knot points andpiecewise spiral curve



- A positive angle is defined as counterclockwise from thevector kpskpe

��!to the corresponding tangent vector.

- For a ¼ b ¼ 0, the associated arc is straight line. The biarc isC-shaped if a and b have the same sign; otherwise, it isS-shaped.

- Minimization of the Hausdorff distance [30] technique isused for error control.

The tangent vector tn at each knot point is approximated byinterpolating the three consecutive neighboring knot points intro-duced in the literature [31] as in the following equation:

tn¼ðkpn�kpn�1Þkpnþ1�kpnj jkpn�kpn�1j jþðkpnþ1� kpnÞ

kpn� kpn�1j jkpnþ1� kpnj j 8 n

(6)

Referring to the literature [32], the optimum location is obtainedby minimizing the difference in curvatures as

r1

r2

" #¼

l 4 sin3aþ b

4

� �cos

a� b4

� �� 1

�l 4 sinaþ 3b

4

� �cos

a� b4

� �� 1

266664

377775 (7)

O1ðx1; y1Þ ¼ �r1 sin a; r1 cos a½ �T (8)

O2ðx2; y2Þ ¼ 1� r21 sin b; r2 cos b½ �T (9)

Jðx; yÞ ¼ l

2;

l

2tan

a� b4

� �� T

(10)

The spiral toolpath pattern is generated by initializing the toolpathat the first PCL. Then, knot points on the next two PCLs are calcu-lated and a biarc is fitted. The knot point is determined on the nextPCL, and the biarc is refitted again. The fitting accuracy of a biarcis determined based on one-sided Hausdorff distance [33]. For twocompact nonempty, bounded subsets A ¼ fah1

gh1¼ 0; 1…H1 and

B ¼ fbh2gh2¼ 0; 1…H2 with element a and b of the d dimensional

space Rd , the Hausdorff distance simply assigns to each point ofone set the distance to its closest point on the other and takes themaximum of all these values. Mathematically, it is expressed as

dðah1;BÞ ¼ minH2

h2¼0ðdðah1; bh2ÞÞ ¼ eh1

(11)

Here, dðah1; bh2Þ is the Euclidean distance. A represents the knot

point set KP, and B represents the point set defining the biarc.The Hausdorff distance from A and B can be represented asfollows:

hðA;BÞ ¼ maxH1

h1¼0ðdðah1;BÞÞ ¼ maxðe1;…eh1;…eH1

Þ ¼ emax

(12)

If emax stays within the user input tolerance range f, the new pointis included in biarc fitting; otherwise, the previously generatedbiarc is kept and the new point is considered for the new biarc.Subsequent points are checked and included in biarc fitting untilthe maximum error exceeds the tolerance (see Fig. 10(c)). Thesame procedure is applied for subsequent biarcs. Thus, biarc fit-ting is implemented to generate a C1 continuous and smooth tool-path with significantly reduced knot points. In order to clarify theprocess, a computational algorithm (Algorithm IV) is developed andpresented in the Appendix. Increasing the number of PCLs improvesthe accuracy of the spiral curve; however, toolpath planning throughPCL orientation direction generates over-deposition due to theincreasing density. This technique is applied for each Voronoi cell togenerate a spiral toolpath for a structure with multiple hollowing fea-tures. As shown in Fig. 8, jumps during the shift from one Voronoicell to another are observable in some circumstances, and developinga toolpath with zero starts and stops might be a future direction forwork. In this section, geometric modeling has been discussed. Thenext section presents material composition modeling.

Material Composition Modeling

A porous structure can be defined in the geometric domainXG � E3, where E3 is 3D Euclidean space. In addition to the ge-ometry that is discussed extensively in Sec. 2, material composi-tion needs to be defined over the porous domain. The design spacecan be defined as E3 � EK , where Euclidean space E3 is the basespace, EK is the material space, and K > 1 is the number of pri-mary materials. Thus, a set of materials Mkf g in material domainXM (where Mkf gk¼1;…K2 XM � EK) constitutes the heterogene-ous structure (see Fig. 11). Material distribution of the structurecan be defined by mapping from the material domain to the geo-metric domain by a function g : XG ! XM (see Fig. 11). Materialcomposition mapping of the heterogeneous structure is initializedwith by assigning boundary conditions. Material composition of aset of boundaries in geometric domain XG does not necessarilyshow homogeneity on their respective domain. Material composi-tion of a sampled point P ¼ CmðvkÞ on boundary curve CmðvÞ canbe expressed using the following equation:

gðCmðvkÞÞ¼X

j

wjðdjÞgðPjÞ; 0�wjðdjÞ� 18jX

j

wjðdjÞ¼ 1;

(13)

where Pj is the jth constructive point on curve Cm(v). Constructivepoints of a boundary curve can be sampled points that govern thematerial composition. In Eq. (13), wj stands for the materialblending weight or the contribution of Pj, where the total weightof constructive points on each boundary curve is equal to 1. Inthis paper, we use one of the most popular methods, the inversedistance based technique [34], for designing weight function

Fig. 10 Biarc fitting: (a) C-shape biarc, (b) S-shape biarc, and (c) determining number of points with error control



wjðdjÞ, which is based on the parametric distance dj from thesampled point vk to the constructive point Pj.

wj¼d�1

jXi

d�1i

� where dj¼minðPj�vk;vhighþPj�vkÞ (14)

For example, in Fig. 11, if the parametric distance dj from the con-structive point Pj increases, then it may be desirable to expect thatwjðdjÞ increases. In other words, the weight wjðdjÞ is inverselyproportional to parametric distance. While there exist two para-metric distances between vk and Pj on a closed curve, one shouldensure the minimum one for the parametric distance formulation.Thus, wj is the parametric inverse distance-based weight that isobtained by considering the minimum parametric distance fromthe constructive point, where entire parametric domain of CmðvÞcan be defined as v 2 ½vlow; vhigh�. As presented in Eq. (14), theshortest parametric distance between two arbitrary sampled pointsvj and vk thus can be obtained as the minimum ofðvj � vkÞ; ðvhigh þ vj � vkÞ �

in R3. A similar methodology can beused for determining the composition of a PCL between twoboundary curves. Suppose PCLt is defined as a parametric curvebetween hollowing curves CmðvÞ and CnðuÞ, wherePCLtðsÞ 2 CmðvkÞ;CnðulÞ½ � and m 6¼ n (see Fig. 11). Then, thematerial composition of PCLt can be modeled using theparametric inverse distance based technique as

gðPCLtðsÞÞ ¼ gðCmðvkÞÞsþ gðCnðulÞÞð1� sÞ 0 � s � 1 (15)

Equation (15) can be applied to all PCLs in the geometric domainto map the material composition properties. While the materialcomposition is known along each PCL, material composition of abiarc can also be easily interpolated. In the next section, computa-tional implementation and examples are presented.

Implementation and Examples

Porous structures have been used as one of major tissue engi-neering strategies for tissue reconstruction/repair [1]. Considering

their spatial and temporal multifunctionality requirements, con-trollable variational porosity as well as functionally graded mate-rial composition is highly anticipated into their designarchitecture. We implement our proposed methodology for suchporous biostructures as well as a free-form arbitrary structure inthis section. The proposed methodologies are coded in VisualBasic-based Rhinoscript, and examples are generated in the RHI-

NOCEROS 4.0 software [25]. Material properties are represented inRGB color space, and examples are generated on a computer withan Intel (R) Core(TM) i7-2.8 GHz processor and 8GB of RAM. Inthis work, we used the source codes developed in our earlier work[20] to generate the Voronoi Diagram.

Case I: Heterogeneous Porous Structure. In this case study, ahuman femur bone is considered for demonstration purposes. Thefemur bone model shown in Fig. 12(b) is obtained through medi-cal imaging using the public free software ITK-SNAP (seeFig. 12(a)). Then, a generated mesh file is exported to the NURBSmodeling software RHINOCEROS 4.0. As shown in Figs. 12(c) and12(d), two different materials are imposed on the contours (theexternal boundary and the internal femoral artery boundary)following slicing. Materials can be assigned spatially based onload-bearing capacity and ease of cell attachment and prolifera-tion. While the load-bearing capacity of external regions is higherthan that of internal regions, and internal regions necessitate bio-activity due to femoral artery in a natural femur bone, two differ-ent materials are preferred for the design objective. Thus, aheterogeneous composition is preferred with 100% of M1 at C1

and 100% of M2 at C2 to meet multiple local needs. The heteroge-neity of the material composition needs to be altered gradually tominimize sharp changes, which may result in worsened materialproperties such as material incompatibilities and stress concentra-tions. The heterogeneous femur bone scaffold illustrated inFig. 12(d) is designed in two consecutive steps: (i) a PCL-basedzigzag pattern, and (ii) a biarc spiral path. The material composi-tion inside the scaffold geometric domain changes smoothly andcontinuously through the toolpath planning direction. In thispaper, the porosity is not intended to control; however, algorithms

Fig. 11 Mapping of the material domain to the geometric domain by function g to obtainmaterial composition



developed in our recent papers [10] can be easily applied tocontrol the porosity or the air gap between rasters, where thesparser or the denser regions in Fig. 12(d) can be easily altered asshown in Fig. 12(e). The porosity is reduced and sparser sectionsare diminished, where uniform porosity is obtained through thegeometry by controlling the space between spiral and zigzagcurves in addition to the number of PCLs and the spiral function.The porosity of sparser regions (with large space between zigzagcurves) can also be reduced by reducing the distance betweenparallel spiral curves in these regions as shown in Fig. 12(e). In thiscase, the designer will have the freedom to control the porosity spa-tially, which mediates the mechanical and structural properties infact.

Similar to the femur scaffold design methodology, the CADmodel of an aorta in Fig. 13(b) is generated through medicalimaging using ITK-SNAP software, and a set of consecutive slicesare extracted in RHINOCEROS. Based on the material compositionrequirement shown in Fig. 13(d), a continuous toolpath is designedas demonstrated in Fig. 13(e) with 39 PCLs. Material selection onthe boundaries can be as follows: a material (M3) that enablessmooth flow of blood over the internal boundary after the implanta-tion, and a material (M1) with high elasticity on the external bound-ary that provides flexibility during the motion exerted by pumping of

blood. As shown in Figs. 13(c) and 13(d), an intermediate materialrequirement (M2) brings additional functionality in design where thecomposition of M2 increases through the inner section of the aortawall. M2 may have intermediate properties enabling a smooth transi-tion from M1 to M3. Simulation of the deposition process takes 5 sand 20 s for the zigzag layer and the biarc layer, respectively.

Case II: Heterogeneous Porous Structure With MultipleHollowing Features. In the second case study, relatively complexgeometry is imposed to generate a toolpath plan. For implementa-tion purposes, the right and left ventricle sections of an anony-mous human heart are selected as illustrated in Fig. 14(a). A heartscaffold with continuous material deposition is developed for thecomplex section with two hollowing features according to thegiven material composition requirement for each material as afunction of parametric distances between each feature couple (seeFig. 14(b)). For 100 PCLs, simulation of the deposition processtakes 29 s and 40 s for the zigzag layer and the biarc layer, respec-tively. A 16-layer scaffold is also illustrated with a simulation runtime of approximately 9 min. It should be noted that the computa-tion time is relatively short; however, the rendering took most oftime during the simulation. Figures 15(a) and 15(b) demonstratesanother multiple hollowing features case, where a continuoustoolpath plan is generated for 16 layers (including zigzag andbiarc layers) in 6.5 min.

Fig. 12 Femur bone CAD model is generated: (a) using medi-cal imaging and then (b) sliced into layers. (c) The proposedmethodology is applied based on material needs (d) to generatecontinuous toolpath for femoral artery section of the femur. (e)Uniform porosity, where the porous space is controlled by con-trolling the distance between zigzag and spiral curves by apply-ing the algorithms developed in our earlier work [10].

Fig. 13 (a) Medical imaging is used to generate (b) a 3D CADmodel of an aorta, which (c) is then sliced into layers for (d) and(e) toolpath generation in two consecutive steps based on ma-terial requirements defined by a function. (f) A 32-layer scaffoldis also developed for demonstration purposes.



Case III: Complex-Shaped Structure. Figures 15(c) and15(d) illustrate a complex-shaped structure with four hollowingcurves, each with a different material requirement. While thenumber of hollowing curves is greater than two, the structure isgenerated with two jumps in Fig. 15(c); however, jump distancesare minimized using Algorithm III. For a total of 137 PCLs, simu-lation of the deposition process takes 35 s and 85 s for the zigzaglayer and the biarc layer, respectively. A 16-layer scaffold is alsodemonstrated in Fig. 15(d). Figures 15(e) and 15(f) demonstrateanother complex-shaped structure, where the internal hollowingfeature has a concave shape. Despite its concave shape, the pro-posed algorithms are robust enough to handle the continuous tool-path generation.

Discussion and Conclusion

In this paper, a new continuous path planning approach is pro-posed for functionally graded material printing in porous hollowstructures. In contrast to earlier toolpath planning studies in Carte-sian coordinates, this study introduces a new parametric path plan-ning strategy for complex geometries, in which the toolpathfollows material blending directions. A novel PCL based zigzagalgorithm is developed to trace the heterogeneous domain fol-lowed by a spiral toolpath. Arc and biarc fitting is applied to gen-erate a C1 continuity path for uniform material deposition. Several

examples are illustrated to test the proposed algorithms. The algo-rithm can also work for more complex cases such as internal fea-tures with very complex concave cavities or sharp edges withadditional work, where similar type of cases were already demon-strated in our earlier work [20]. For more complex concave cav-ities in internal features, the Voronoi was extended toward thecavities that enabled ruling line generation successfully. Althoughit is not the intent of this paper, the porosity of the structure canbe easily controlled by controlling parameterization process whengenerating PCLs as demonstrated in our recent work [10]. Thisallows the designer to alleviate nonuniformity of porosity bychanging the density of denser or sparser regions to mediate themechanical and structural properties of the porous artifacts.

After designing the geometric domain, heterogeneous materialproperties are then mapped onto the geometric domain accordingto the parametric distance-based weighting method. The method-ologies developed in this work enable development of heterogene-ous structures with functionally graded material composition.Although a feature-based approach is developed to assign materialproperties, other methods can be easily applicable such as materialproperties can be mapped onto the geometric domain based on ananalysis results, which gives more freedom to the designer. Thispaper has the potential to enrich theoretical systems in toolpathplanning as well as heterogeneous object modeling.

Despite its great advantages, the proposed toolpath has somelimitations compared to traditional zigzag raster approach in theliterature. First of all, the traditional zigzag approach is more com-putational efficient compared to the proposed method, which doesnot need biarc fitting, line matching, curve parameterization andVoronoi generation. Although 2D Voronoi generation was appliedin this paper, 3D Voronoi generation might be considered forfuture work, which can improve the computational efficiency oftoolpath planning considerably. Second, fabrication techniques forthe proposed multimaterial toolpath plan is limited compared tothe conventional single material AM techniques. In addition, theproposed arcs do not capture boundaries closely, however this donot bring any major issues related to mechanical and structural in-tegrity of the constructs.

The proposed methodologies will enable fabrication of a heter-ogeneous structure while the continuous material variation overthe structure is known. Print-on-demand biomanufacturing techni-ques such as multichamber single nozzle systems in our earlierstudies [8,35] and a recent study from Cormier and his coworkers[34] can be used to fabricate the designed structures with func-tionally graded material composition. The developed AM systemwith the proposed toolpath algorithms can have great potential inmany different applications such as but not limited to medicine(i.e., tissue scaffolding, protein or growth factor deposition forguided cell response [8]) and energy (solid oxide fuel cell printingfor porous spatially controlled chemical composition [34]). More-over, the proposed toolpath algorithms can be used not only in thecontext of functionally graded materials but also to allow develop-ment of a single material structure with appealing properties suchas controlled porosity and enhanced permeability.

Fig. 14 A CAD model of an anonymous human heart is (a) sliced, and the new methodology is applied to generate continuoustoolpath for a section enclosing right and left ventricle (b) based on the material requirements as functions of parametric dis-tances from features. Toolpaths with (c) a double layer and (d) multiple layers are developed for demonstration purposes.

Fig. 15 A porous structure: (a) a continuous toolpath withtwo-hollowing features, (b) the structure is designed in 3D; acomplex-shaped structure: (c) a continuous toolpath with fourinternal hollowing features, (d) the structure is designed in 3D;another complex-shaped structure: (e) a continuous toolpathwith a concave hollowing feature, and (f) the structure isdesigned in 3D.



Appendix

Algorithm I: PCL MatchingInput: A set of available property changing lines PCL ¼ MCLtf gt¼1;…;T , upper distance bound d, hollowing curves Caf ga¼1;…A,bisectors Bkf gk¼1;…;KOutput: A set of new property changing lines nPCL ¼ nPCLvf gv¼1;:…;V1. 8PCLt

2. While (PCL 6¼ null) {***/Do matching until the set is empty/***3. CPCL! ;; i! 1; ***/Initialization/***4. For t ¼ 1 to T ***/Determination and objective function calculation of candidate PCLs/***5. If PCLt \ BkðuÞð Þ 62 ; then CPCLif g ! CPCLif g [ PCLt;6. Calculate objective function fi using Eq. (1);7. i! iþ 1;8. End If9. I ! i;10. End For ***/End of first For-loop/***11. max! 0; ***/Initialization/***12. For i ¼ 1 to I ***/Finding candidate with maximum objective function/***13. If fi > max then max! fi; temp! i;14. End If15. End For ***/End of second For-loop/***

16. Get dist ¼ min pktemp; p

kj

�� n o***/Get the closest PCL from the adjacent cell (with shortest distance)/***

17. If dist < d then calculate p� using Eq. (3) And generate nPCLv ¼ qam;p

� [ p�;qbn and Delete PCL : pk

tempqam [ pk

j qbn

� �; ***/if

the distance is smaller than the upper bound, then get the new PCL(nPCL) by connecting matching pairs and subsequentlydelete matching pairs/***

18. Else Delete PCLtemp; ***/Otherwise delete the PCL while there is no matching pair/***19. End If20. End While ***/End of While-loop/***

Algorithm II: Relaxation of Property Changing Lines.Input: A set of available property changing lines PCL ¼ PCLtf gt¼1;…;T , upper distance bound d, small incremental number D, shift-ing distance ds, bounding curve C0

Output: A set of new property changing lines after relaxation nPCL ¼ nPCLvf gv¼1;…;V1. S¼ 0; CRL! ;;/*** Initialization***/2. For t ¼ 1 to T/*** Loop to detect PCLs on the bounding curve***/3. If PCLt \ C0ðuÞð Þ 62 ; then CRLf g ! CRLf g [ PCLt

4. End If5. End For/*** End of first for-loop***/6. For u ¼ 0 to 1 Step D/*** Loop to sort PCLs based on their parametric distance from the start point of the bounding curve***/7. For s ¼ 1 to S8. P1 ! PCLs \ C0ð Þ; P2 ! C0ðuÞ;9. If P1P2

�� D then F CRLsð Þ ! u;/*** store the parametric distance u from the start of the bounding curve using functionF, if point at u is close enough to a sampled point***/

10. End If11. End For/*** End of third for-loop***/12. End For/*** End of second for-loop***/13. For s ¼ 1 to S� 1/*** Search PCLs for relaxation***/14. temp¼ 1;15. If FðCRLsþ1Þ � FðCRLsÞ � d then/*** If the condition for relaxation is satisfied***/16. lower_bound¼ s; ulower_bound¼FðCRLsÞ;17. temp¼ 0;18. End If19. If temp¼ 0 then upper_bound¼ sþ 1; uupper_bound¼FðCRLsþ1Þ;20. Switch PCLs toward the opposite side of the branching points by a distance of 2 �ds =(upper bound - lower bound);21. s¼ 1;/*** Restart searching unless the distance between each PCL is greater than the upper distance bound***/22. nPCL ¼ nPCL [ CRLs/*** Update the set of new PCLs***/23. End If24. End For/*** End of fourth for-loop***/

Algorithm III: Zigzag Toolpath Planning Through PCL DirectionInput: Set of curves C ¼ Cif gi¼0;…;I; set of property changing lines PCL ¼ PCLtf gt¼1;…;T

Output: Zigzag toolpath plan in PCL orientation direction1. 8MCLt

2. Zi;j ! fg;/*** Initialization ***/3. For i ¼ 0 To I{/*** Grouping each PCL between curve pairs***/4. For j ¼ iþ 1 To I{5. If ArraySize PCLt \ Cið Þ ¼ 1 And ArraySize PCLt \ Cj

�¼ 1 Then

6. Zi;j ¼ Zi;j [ PCLt;/*** Update the set ***/



7. End If8. }9. End For/*** End of second For-loop ***/10. End For/*** End of first For-loop ***/11. If

Pj nðZ0;jÞ � 1ðMod2Þ Then/*** check if the number of PCL in the outer boundary is odd***/

12. Start tracing from the bounding curve;13. Else Start tracing from the hollowing curve;14. End If15. If (# of nontraced PCLt> 0) Then/*** Check if there is still nontraced PCL***/16. Search the closest nontraced PCLt;/*** Proceed with the closest PCL for the deposition***/17. For i ¼ 1 To I{18. For j ¼ iþ 1 To I{19. If PCLt 2 Zi;j Then20. Continue zigzag for Zi;j;21. End If22. End For/*** End of fourth For-loop ***/23. End For/*** End of third For-loop ***/24. Else Exit;25. End If26. Go to Step 15 until all PCLs are traced;

Algorithm IV: Spiral Toolpath GenerationInput: A set of available property changing lines PCL ¼ PCLtf gt¼1;…;T , spiral function S, tolerance range fOutput: Spiral toolpath ST1. t 1; ST ¼ fg;2. While t � T Then/*** Initialize the first three points ***/3. For i ¼ t to tþ24. kpi ¼ PCLi Sði=TÞð Þ;5. Next6. While emax � f Then/*** Check if the next point is in the tolerance range ***/7. m ¼ mþ 1;8. kpm ¼ PCLm Sðm=TÞð Þ;9. Biarcðkpt; kpmÞ10. End While/*** End of Second While loop ***/11. t m;12. ST ¼ ST [ Biarcðkpt; kpmÞ;13. End While/*** End of First While loop ***/14. Biarcðkpt; kpmÞf/*** Biarc Subfunction ***/15. Use Eqs. (7)–(10) to proceed with the toolpath by generating Biarcðkpt; kpmÞ;16. }

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