An optimization approach for path planning of high

Int J Adv Manuf Technol DOI 10.1007/s00170-017-0207-3

ORIGINAL ARTICLE

An optimization approach for path planning of high-quality and uniform additive manufacturing Yuan Jin 1 & Jianke Du 1 & Zhiyong Ma 1 & Anbang Liu 1 & Yong He 2

Received: 20 November 2016 / Accepted: 22 February 2017 # Springer-Verlag London 2017

Abstract Material extrusion-based additive manufacturing (AM) is an effective tool in producing prototypes and final parts without geometrical complexity limitations. Despite having widespread applications and enormous advantages over conventional manufacturing techniques, the proliferation of extrusion-based AM has been limited by the low deposition quality and the poor surface finish of printed parts. To address these issues, an optimized path planning technique is proposed in this paper. The sharp corners and the non-uniform spacing between adjacent path elements in the final planned path are two major causes of unevenness of the deposited surface. The proposed method tries to decrease the number of sharp corners by using an implicit algorithm derived from the level sets of the input contours. The curvature information is used to smooth the generated contour paths. Subsequently, to achieve uniform spacing, local optimization is applied on the smoothed path by adaptively adjusting the locations of points on the path. These optimizations lead to a smoother part surface, when compared to those of typical fill path techniques. The proposed method is validated using several examples of parts, many of which are then constructed using a 3D printer.

Keywords Extrusion-based additive manufacturing . Path planning . Level-set method . Local optimization

* Yuan Jin [email protected]

1

School of Mechanical Engineering and Mechanics, Ningbo University, Ningbo 315211, China

2

School of Mechanical Engineering, Zhejiang University, Hangzhou 310027, China

1 Introduction Additive manufacturing (AM), also termed as solid freeform fabrication (SFF), rapid prototyping (RP), or layered manufacturing (LM), refers to a family of technologies that integrates CAD/CAM, CNC, material science, and process control to build 3D objects by stacking layers of thin slices of material, such as plastic, metal, powder, and biocompatible hydrogel [1]. The biggest advantage of this fabricating process over conventional manufacturing techniques is the capability in producing arbitrary parts without geometric limitations, as well as the flexibility to achieve desirable part quality and fabrication efficiency. Besides, the merit in reducing the development period of a new product enables I ddts suitability for fabrication of customized components, e.g., biomedical prosthesis [2]. So far, a wide variety of AM processes have been developed and they are classified based on the employed materials and the manners of building and stacking layers [3]. One of the most universal and diffused AM technology is the material extrusion-based additive manufacturing, where the material is extruded and deposited with a successive layerby-layer process to build 3D objects [4]. Before undergoing practical manufacturing, several steps are required to apply on the models to be additively fabricated. These steps are collectively referred to as the process planning, by which the models transfer from the digital CAD files to the code that can be read and executed by AM machines. The CAD files are obtained from either the design models or scanned point cloud and then transformed into closed surfaces tessellated with a large number of triangular facets [5], which is commonly known as STL file. Next, the tessellated models are placed by choosing an appropriate orientation for the subsequent slicing and fabrication by considering many factors, such as the required support structures, the number of sliced layers, and the strength requirement [6]. Thereafter, the

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models are sliced into many layer contours by intersecting a set of parallel planes through the models of the design parts based on a critical parameter, layer thickness, which affects the fabrication quality and build time significantly [7]. At the same time, support structures to assist the fabrication of overhang features are generated before or after the slicing procedure with different methods for area identification. Afterwards, the areas on each layer (part and/or support) to be deposited are covered with a given filling path pattern and the generated paths are used to guide the fabrication. The above-mentioned steps are briefly illustrated in Fig.1. In the process planning of extrusion-based AM, generating the deposition path is one of crucial tasks that determine the trajectory for the deposition head to fill the cross section of specific layers [8]. Two major steps are involved in the path planning: interior filling and linking sequence [9]. The interior filling is about the method to fill up the internal area surrounded by contours continuously, while the linking sequence represents the deposition order of the generated interior sub-paths. Currently, two main filling path patterns used in the extrusion-based AM are direction parallel and contour parallel paths. They have their own pros and cons and have been used together as a hybrid strategy [10]. Based on the tasks of path planning for material extrusionbased AM, the objectives are summarized as follows: (1) obtain satisfied deposition quality; (2) improve fabrication efficiency; and (3) minimize the number of sub-paths to decrease the travel time spent on the linking paths [11]. To pursue these objectives, the factors affecting the part quality and build time are required to be clarified and different filling strategies are

supposed to be analyzed and compared in terms of total path length, processing time, and surface quality [12, 13]. Similar to the scallop left as an uncut material between tool paths in traditional CNC machining, the surface roughness in the extrusion-based AM is mainly produced by the overfills and underfills between adjacent deposition paths out of many reasons, such as the speed change during the deposition process, the sharp corners, and the non-uniform space between adjacent paths [14]. Actually, the speed change is also dependent on the occurrence of sharp corners (or high curvature segments) in the planned path. Specifically, the moving speed of the deposition head has to decrease to ensure a smooth transition between two path segments at sharp corners, so an acceleration and deceleration process always appears near these locations. A zigzag path, as well as its associated deposition speed, is demonstrated in Fig. 2. The path and the speed are shown in x-y plane and z axis, respectively. On the other hand, the non-uniform spacing between paths would directly bring in unevenness of the surface profile. Therefore, the sharp corners and the non-uniform spacing are two major factors in affecting the deposition quality and would be addressed in this work. With respect to the fabrication efficiency, different strategies have been developed from a variety of perspectives, including the optimization of linking sequence [15], adaptive slicing [16], optimization of deposition angle [11], and speed optimization [10]. Meanwhile, to improve the continuity of the final generated deposition path, several novel filling patterns are developed, such as the connected Fermat spirals [17] and other continuous path patterns [18, 19]. The continuity of

Fig. 1 Illustration of process planning

Digital model

Two path patterns

Support generation

Paths on different layers

Slicing procedure

Sliced layers

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method in detail. Some example geometries are adopted in Section 4 to demonstrate the feasibility of the proposed approach. Conclusions are drawn in Section 5.

2 Modeling and analysis of deposition process in extrusion-based AM

Fig. 2 Zigzag path and its associated deposition speed variation

the path can not only improve the fabrication efficiency by reducing the number of linking paths but also enhance the deposition quality by avoid the frequent turning on/off of the deposition head. The objective of this work is to develop an optimization scheme for generation of contour parallel-based path to improve the deposition quality in material extrusion-based AM and to provide a new idea to build layers with acceptable deposition quality and desirable dimensional accuracy of fabricated parts. Several studies have been conducted in this field by some researchers. Chen et al. [7] introduced a new contour generation method utilizing the sectional projected data of a point cloud representation of the part surface to reduce the systematic part distortion during the additive manufacturing process. Ossino et al. [20] reported a comprehensive algorithm in contour parallel filling path generation for robot-assisted additive manufacturing, including finding intersection contour, grouping contours, generating support structure contours, and planning the contour parallel paths. The selfintersection issue was addressed by them [21] and then improved based on the BUFFERM function in MATLAB, which in turn uses the General Polygon Clipper (GPC) library [22] to perform Boolean operations between polygons. These works laid a solid foundation for further optimization of contour parallel-based path generation. In this paper, a level-set-based method is used to generate the contour parallel paths based on the input boundary, integrating the curvature information on the boundary to improve the path smoothness. Then, the uniformity of the smoothed path is achieved by a local optimization. The remainder of this paper is organized as follows: the model of deposited surface in material extrusion-based AM is established to lay a theoretical/geometrical foundation for the subsequent analysis in Section 2. Section 3 describes the proposed path-generation

The deposition process of extruded filament along a specified path is empirically simulated for the subsequent analysis of the deposited surface as illustrated in Fig. 3. The materials during the extrusion and deposition process are supposed to obey the law of conservation of volume, which can be used to analyze the cross-sectional shape of the deposited bead. The initial shape of the extruded filament is determined by the nozzle tip, which is commonly round in shape. Under the influence of the gravity and interaction between edge of nozzle tip and former deposited layers (or the base plate), the cross section may be approximated with simple quadratic and superelliptic primitives with dimensions determined by the bead width w and layer height h [23]. The flowrate of the extruding material is assumed as Q, while the moving speed of the deposition head is considered as F, then the area of the cross section of deposited bead can be expressed as s¼

Q F

ð1Þ

For an easy computation, the approximation and expression in our previous work are used to describe the relationship between the area and two major parameters (w and h) in Eq. (2) [24].
2 h þ ðw−hÞ*h ð2Þ s¼π 2 As the layer height is set in advance as the layer thickness in the slicing procedure, the bead width becomes the only variable in the deposition process. Based on Eqs. (1) and (2), the bead width w can be expressed as
2  Q π −1 h2 − 4 F w¼ ð3Þ h

Fig. 3 Illustration of extrusion and deposition process

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Fig. 4 Moving speed of the deposition head vs bead width

So, the bead width w is always changing when the proportion between the flowrate Q and moving speed of head F changes. It should be noted that the bead width cannot go larger than that of the outer diameter of the nozzle tip since the surface profile of the deposited filament would be irregular under this circumstance and should be hence avoided. Meanwhile, the bead width is impossible to be smaller than that of the inner diameter of the nozzle tip. So, the bead width w is confined to [Di, Do]. In order to demonstrate the influence on the bead width imposed by the moving speed, Fig. 4 shows the variation of bead width and the corresponding moving speed with the assumption of a constant flowrate of the extruding material. It can be observed that the bead width can be adjusted within a certain range depending on the moving speed of the deposition head, which is achievable in the practical deposition process. It is understandable that the deposited surface is affected by the bonding effect between adjacent deposited filaments within one layer and the joining effect between filaments on adjacent layers. These two effects are both dependent on the filament width when the layer thickness is determined, while the filament width is affected by the flowrate and the moving speed. Hence, the bonding effect between adjacent deposited filaments can be enhanced by adjusting the flowrate with a given moving speed F. An illustration of the bonding between adjacent filaments is shown in Fig. 5 with increasing flowrate and constant speed and same path space. Besides, the deposition at corners is another factor affecting the surface quality. Overfills usually appear at the inside of the corner, while the underfills appear at the outside of the corner. This phenomenon can be analyzed geometrically from Fig. 6 and it can be concluded that the issue becomes more serious when the corner is sharper. The same situation occurs at Fig. 5 Bonding effect under different flowrates

Fig. 6 Deposition at sharp corners

locations with high curvature (curves with high curvature are comprised of line segments with sharp corners in between essentially). Therefore, two fundamental requirements for the path planning of extrusion-based AM can be obtained based on the above-mentioned analysis. The first is the path spacing should be kept within a certain range and the second is the sharp corners (or high-curvature segments) are supposed to be avoided. The path-generation method developed in this paper tries to meet these requirements.

3 Path planning for high-quality and uniform deposition 3.1 Level-set method The level-set method was initially developed for capturing moving interfaces with intricate topological features by embedding the moving interface into a higher dimensional levelset function and it has become an effective numerical tool in a variety of fields, such as image processing [25], topology optimization [26], and computational fluid dynamics [27]. Assume the input contour of one 2D layer is γ, ω is the filling region, and the boundary γ can be expressed by ∂ω. With an implicit representation of the contour, the area to be filled is embedded in a level-set function ϕ, and then the contour is the intersection of the level-set function and the zero plane. Figure 7 is an example of the level-set function of an input geometry. Based on the above-mentioned demonstration, the contour of one region can be defined by the zero level-set function expressed as [28]

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Fig. 8 Demonstration of offsetting process with level-set method

Fig. 7 The level-set function and zero plane

γ ðt Þ ¼ fxðt Þjϕ½xðt Þ; t  ¼ 0g;

ð4Þ

where ϕ[x(t),t] is the level-set function, x(t) is the real variables, and t is the time. From the level-set function, the following level-set equation can be obtained:[28]. 0

ϕt þ ∇ϕðxðt Þ; t Þ⋅x ðt Þ ¼ 0

ð5Þ

If F is defined as the speed along outward normal direction, namely, 0

F ¼ x ðt Þ⋅n n¼

∇ϕ j∇ϕj

ð6Þ ð7Þ

The level-set equation can be simplified as ϕt þ F j∇ϕj ¼ 0

ð8Þ

Hence, with a given contour and the moving speed F of the contour, the moving zero level-set boundaries can be computed from Eq. (8). That is to say, the offset contours from the initial boundary can be obtained with this method with the assumption that the offset distance is controlled by the speed function and the time. This process can be illustrated in Fig. 8.

At first, a level-set function ϕ is required to be constructed to embed the input geometry of one sliced layer. Normally, the signed distance function in Eq. (9) is selected as the initial level-set function. The |x−x γ | in the equation is the Euclidean distance between one point on the plane and the contour. The plus and the minus signs represent the location of the point. Specifically, the level-set function of the points inside the filling area is larger than zero, while the area outside the contour has a value less than zero. So the area to be filled can be identified based on the sign of the level-set function.   8 x is inside of ω < minx−xγ  ð9Þ ϕðxÞ ¼ 0 x is on γ  : x is outside of ω −minx−xγ  During the offsetting process, the signed distance level-set function also requires preserving the signed characteristic by being periodically reinitialized. With a given offset distance, the offset contours are computed several times by ensuring the product of the speed function and the time equals to the offset distance. The offsetting process is finished once the termination condition is satisfied. The termination condition here is that the number of newly offset contours is zero after one offsetting operation. Through the above-mentioned steps, the offset contours are generated with an example showing in Fig.9. It can be observed that some sharp corners appear along the offset

3.2 Contour path generation Based on the aforementioned knowledge about the level-set method, the contour parallel paths can be obtained by offsetting the original contours with a given offset distance, which is controlled by the speed function and the time according to Eq. (8). The generation of offset contours is realized with the following steps:

Fig. 9 Offset contours with level-set method

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(a)

(b)

(c)

(d)

Fig. 10 Paths with different smoothness via revised level-set equation

contours that are detrimental for the deposition quality. This issue would be solved in the next section.

3.3 Path smoothness In order to improve the smoothness of the high-curvature points along the offset contours, the curvature information of the contour is integrated in a revised speed function described as [29] F n ðxÞ ¼ 1−εk;

ð10Þ

3.4 Continuity and uniformity The following step is to connect the generated offset contours sequentially into a spiral path to achieve a desirable continuity, which is beneficial for the deposition quality. The related algorithm for the generation of spiral path is not described here because some effective algorithms have been reported [30]. Meanwhile, the generated offset paths may suffer from nonuniform path spacing after the smoothing process. The nonuniform spacing is very common in the narrow thin-walled area, and it should be considered and avoided.

where ε is a smaller valued constant and k is the curvature information of the moving contour. Combining Eqs. (8) and (10), the level-set equation is expressed by ϕt þ ð1−εk Þj∇ϕj ¼ 0

ð11Þ

By solving Eq. (11) with appropriate ε, the smoothed offset contours can be obtained as shown in Fig. 10. Different values of ε would lead to different smoothing effects. Specifically, a larger ε can enhance the smoothing effect and the generated path tends to be much smoother with larger curvature.

Fig. 11 Several simple geometries with contour parallel paths under a certain path space

Fig. 12 Demonstration of medial axis of shapes with a single contour and b multi-contours

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The path spacing is supposed to be adaptively and locally adjusted to enable the path elements equally distributed to improve the evenness of the deposited surface. As shown in Fig. 11, the contour parallel paths are generated according to a certain path space for the shapes of letters “A”, “B”, and “C”. It is easy to find out that some overfills or underfills would

appear if the areas are filled with the obtained paths directly. In fact, these overfills or underfills are located at areas marked as red dot lines in the figures. Geometrically, the red lines belong to the skeleton (or medial axis) of the input geometry, so the issues of overfill and underfill in the contour parallel paths can be addressed based on the skeleton of the geometry.

(a)

(c)

(b)

(d)

(e) Fig. 13 Adjusting process of the generated offset contours. a Initial offset contours b Medial axis. c Sampling points. d Adjustment procedure. e Closedup of adjusted points

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Skeleton of a 2D shape can be used to concisely represent the domain shape by geometrically bisecting the domain and becomes the shape skeleton [31]. The skeleton is bounded by the planar curve and is the set of the locus of centers of locally maximal circles that are tangent to the curve in two or more points as illustrated in Fig.12. The skeleton of a simple polygon can be visually described as a tree whose leaves are the vertices of the polygon, while the edges are either straight segments or arcs of parabolas [32]. Based on the definition of the skeleton, the detailed algorithm is described as follows and demonstrated in Fig. 13: Step 1: Obtain the skeleton of the outmost offset contour as shown in Fig. 13b; Step 2: The skeleton curve is adaptively sampled based on the curvature distribution; specifically, more samples are selected near locations with higher curvature, while less samples are picked up at locations with smooth segments; Step 3: Generate the maximal inscribed circles at the selected samples, and the intersections between path elements and the line connecting the center of the inscribed circle and two associated tangent points are obtained as shown in Fig. 13c; Step 4: Adjust the location of the intersections achieved from the last step to make them equally spaced as shown in Fig. 13d; Step 5: Connect and fit all the adjusted points into a set of curves as shown in Fig. 13e;

(a)

(c) Fig. 14 Fabricated parts based on the parts with different smoothness

Step 6: Connect associated curves with smooth curve to ensure smooth transition between them to get the optimized spiral offset path.

With the local adjustment, the obtained path in a thinwalled area has a relatively uniform path spacing and the issue of overfills and underfills out of the non-uniform distance between paths becomes acceptable. It should be noted that one of inherent disadvantages of the contour parallel path is the tendency for fill paths in adjacent layers to be very similar, which can lead to poor bonding between the filament paths. So, it is not recommended to adopt contour parallel path between layers, while the direction parallel is used to fill the interior part to strengthen bonds between layers and increase isotropy by alternate orientation of continuous filaments within the part. The contour parallel path is mainly used to fill the top surface to guarantee the deposition quality.

4 Implementation and discussions Firstly, the feasibility of the proposed level-set-based contour parallel path-generation method is verified using the illustrative geometry mentioned in section 3. The C++ programming language is used to generate the Gcode file for the 3D printer according to the planned paths. The layer thickness is set to be 0.2 mm, and the whole model is comprised of 20 layers. A luzblot TAZ 3 FDM printer is adopted to fabricate the parts

(b)

(d)

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(a) Example shape 1

(b) Example shape 2

(c) Example shape 3

(d) Example shape 4

Fig. 15 Example shapes for offset contours generation

Deposition path

Fill

Underfill

(a) Original deposition path

(b) Estimated deposition result I

(d) Optimized deposition path

(e) Estimated deposition resultI I

Fig. 16 Analysis of deposition quality with paths before and after optimization

(c) Actual deposition result I

(f) Actual deposition resultI I

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based to the generated Gcode. It is observed from the fabricated parts in Fig. 14 that a same input geometry can be achieved with various paths of different smoothness, which can be adjusted by revising the parameter in the level-set function. This property can be used to handle geometries whose curvature varies along the contour, and thus address some quality issues in the material extrusion-based AM process. To further evaluate the effectiveness of the proposed levelset method for the generation of contour parallel paths, several simple example shapes as shown in Fig. 15 are used and processed. The offset distance is 0.5 mm. The offset contours of the input geometry are generated step-by-step as demonstrated in section 3.2. Firstly, the signed distance function of each sliced layer is constructed based on the coordinates of points on the boundary as the level-set function. In computing the level-set function, the product of moving speed at each point and propagating time equals to the offset distance. In the smoothing process, ε and k are selected to smooth the sharp corners in the original offset curves with high curvature. The obtained offset contours and the associated fabricated parts are shown in Fig. 15, respectively. We can see from Fig. 15d that some terrible voids appear in the interior without any adjustment and local optimization. The obvious advantage of the generated contour parallel paths using level-set method is the reduction of highcurvature segments that is conducive to the deposition quality.

The issue of overfills and underfills from the sharp turns can be relieved based on the analysis of deposition quality in section 2. On the other hand, the deposition with obtained offset contour paths (Fig. 16a) may also bring in some overfills and underfills as illustrated in Fig. 16b, c. Overfills are mainly from two aspects: the turns and the non-uniform path spacing. Some path segments after the smoothing procedure still have some features with high curvature that cannot be deposited justly with smooth filaments and overfills are inevitable under these situations. Meanwhile, the path spacing that is not considered in the generation of offset contours is not constant and fair. So, besides overfills, some underfills appear when the distance between adjacent path segments does not suit to the width of the deposited bead. The local optimization is applied on the generated offset contours to adaptively adjust the locations of points on the original offset contours. The optimized deposition path is shown in Fig. 16d, the estimation of overfills and underfills based on the modeling of deposition process is shown in Fig. 16e, and the 3D printed part is shown in Fig.16f. It can be observed that overfills due to the large path spacing can be completely avoided and other underfills are scattered into several smaller void from larger voids. This would be beneficial for the deposition via improving the surface evenness and smoothness. At the same time, the example shape in Fig. 13 is also demonstrated in Fig. 17, the original planned paths using the level-set-based method are shown in Fig. 17a, and

Fig. 17 Illustration of deposition quality with paths before and after optimization

(a) Original path without local optimization

(b) Deposition result using path in (a)

(c) Deposition path after local optimization

(d) Deposition result using path in (c)

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there are some big voids due to the overfill issue occurring on the additively fabricated part as shown in Fig. 17b. However, after some local adjustment on the generated paths to improve its uniformity in Fig. 17c, those evident voids are eliminated that is conducive to enhance the deposition quality as shown in Fig. 17d.

4.

5.

6.

7.

5 Conclusions In this work, a level-set-based optimization method to generate contour parallel deposition path for the material extrusionbased additive manufacturing is proposed. As the issue in terms of overfills and underfills is mainly resulted from two causes, non-uniform path spacing and sharp corners, the levelset method is adopted in our work to implement the offsetting process to avoid self-intersection and other issues. Instead of explicit expression, the level-set method uses an implicit paradigm to represent the input contour to enable the offsetting process to be convenient and simple. More importantly, the smoothness of the generated offset contours could be easily handled by optimizing the speed function in the level-set equation. At last, the non-uniform spacing between adjacent path segments is solved by a local optimization strategy. The deposition using the generated paths could be effectively improved in terms of fabrication quality, which has been verified by several examples. Although the proposed path-generation method could generally reduce the number of sharp corners or high-curvature segments along the deposition path, there are still more details required to be investigated to extend this method to arbitrary shapes. Considering the relatively high computational cost, future research will aim towards minimizing the computational cost of path generation by developing computational process and avoiding redundant iteration. Therefore, further studies are required to integrate the proposed approach into the general path-generation scheme to improve its universality and applicability. Acknowledgements This paper is sponsored by the National Natural Science Foundation of China (No. 51375440), Initial Foundation of Ningbo Univerity (No.421610040), and K.C. Wong Magna Fund in Ningbo University.

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