Geometry-Based Process Planning For Multi-Axis Support-Free Additive Manufacturing

(1)

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Procedia CIRP 00 (2017) 000–000

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Peer-review under responsibility of the scientific committee of the 28th CIRP Design Conference 2018.

28th CIRP Design Conference, May 2018, Nantes, France

A new methodology to analyze the functional and physical architecture of

existing products for an assembly oriented product family identification

Paul Stief *, Jean-Yves Dantan, Alain Etienne, Ali Siadat

École Nationale Supérieure d’Arts et Métiers, Arts et Métiers ParisTech, LCFC EA 4495, 4 Rue Augustin Fresnel, Metz 57078, France * Corresponding author. Tel.: +33 3 87 37 54 30; E-mail address: paul.stief@ensam.eu

Abstract

In today’s business environment, the trend towards more product variety and customization is unbroken. Due to this development, the need of agile and reconfigurable production systems emerged to cope with various products and product families. To design and optimize production systems as well as to choose the optimal product matches, product analysis methods are needed. Indeed, most of the known methods aim to analyze a product or one product family on the physical level. Different product families, however, may differ largely in terms of the number and nature of components. This fact impedes an efficient comparison and choice of appropriate product family combinations for the production system. A new methodology is proposed to analyze existing products in view of their functional and physical architecture. The aim is to cluster these products in new assembly oriented product families for the optimization of existing assembly lines and the creation of future reconfigurable assembly systems. Based on Datum Flow Chain, the physical structure of the products is analyzed. Functional subassemblies are identified, and a functional analysis is performed. Moreover, a hybrid functional and physical architecture graph (HyFPAG) is the output which depicts the similarity between product families by providing design support to both, production system planners and product designers. An illustrative example of a nail-clipper is used to explain the proposed methodology. An industrial case study on two product families of steering columns of thyssenkrupp Presta France is then carried out to give a first industrial evaluation of the proposed approach.

Peer-review under responsibility of the scientific committee of the 28th CIRP Design Conference 2018.

Keywords: Assembly; Design method; Family identification

1. Introduction

Due to the fast development in the domain of communication and an ongoing trend of digitization and digitalization, manufacturing enterprises are facing important challenges in today’s market environments: a continuing tendency towards reduction of product development times and shortened product lifecycles. In addition, there is an increasing demand of customization, being at the same time in a global competition with competitors all over the world. This trend, which is inducing the development from macro to micro markets, results in diminished lot sizes due to augmenting product varieties (high-volume to low-volume production) [1]. To cope with this augmenting variety as well as to be able to identify possible optimization potentials in the existing production system, it is important to have a precise knowledge

of the product range and characteristics manufactured and/or assembled in this system. In this context, the main challenge in modelling and analysis is now not only to cope with single products, a limited product range or existing product families, but also to be able to analyze and to compare products to define new product families. It can be observed that classical existing product families are regrouped in function of clients or features. However, assembly oriented product families are hardly to find.

On the product family level, products differ mainly in two main characteristics: (i) the number of components and (ii) the type of components (e.g. mechanical, electrical, electronical).

Classical methodologies considering mainly single products or solitary, already existing product families analyze the product structure on a physical level (components level) which causes difficulties regarding an efficient definition and comparison of different product families. Addressing this

Procedia CIRP 78 (2018) 73–78

Selection and peer-review under responsibility of the scientific committee of the 6th CIRP Global Web Conference “Envisaging the future manufacturing, design, technologies and systems in innovation era”.

10.1016/j.procir.2018.08.175

ScienceDirect

Procedia CIRP 00 (2018) 000–000

2212-8271 © The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license

(https://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of the scientific committee of the 6th CIRP Global Web Conference “Envisaging the future manufacturing, design, technologies and systems in innovation era”.

doi:10.1016/j.procir.2017.04.009

6th CIRP Global Web Conference

“Envisaging the future manufacturing, design, technologies and systems

in innovation era”

Geometry-Based Process Planning for Multi-Axis Support-Free

Additive Manufacturing

Yavuz Murtezaoglu

a,b

_{, Denys Plakhotnik}

a,b,

_{*, Marc Stautner}

b

_{, Tom Vaneker}

a

_,

Fred J.A.M. van Houten

a

a_{University of Twente, Drienerlolaan 5, Enschede 7522 NB, the Netherlands} b_{ModuleWorks GmbH, Henricistr. 50, Aachen 52072, Germany} * Corresponding author. Tel.: +49-241-990004-32; fax: +49-241-990004-10. E-mail address: denys@moduleworks.com

Abstract

In contrast to standard layer based additive manufacturing methodologies, multi-axis material deposition can print structures without the need for support material. However, this method is jeopardized by potential collisions between a depositing unit (nozzle, wire, power and powder sources, etc.) and the already deposited material. The goal of this research is to initiate development of a methodology to check manufacturing feasibility of geometries and generate subsequent process planning strategies. The paper describes a geometry-based concept to decompose the product geometry into discrete volumes by using space partitioning with infinite planes and considering advantages and constraints of multi-axis additive manufacturing. The discrete volumes are used to generate process planning variants and to compute and generate boundary conditions for such process planning strategies. The algorithm generates multi-axis slices that require no support structures because of relative nozzle/workpiece orientation. In addition, the planning tackles more complex scenarios, in which overhangs, nozzle orientation, and gravity can be considered.

Keywords: additive manufacturing; process planning; multi-axis; slicing

1. Introduction

In the 1980s and 1990s, additive manufacturing (AM) was believed to replace subtractive machining. The new fabrication concepts promised easy production of very complex parts without complications that usually accompany conventional processes. Manufacturing of shapes with unlimited geometrical complexity was expected to become trivial and such that re-fixturing and multi-axis toolpaths will not be required. However, after the initial hype and obtaining broader and deeper experience, researchers and industrial

adopters realized that the classical 2.5-dimensional approach (horizontal slicing) has substantial limitations - need of support structures, the surface quality is likely to be insufficient on steep walls and overhangs, structural performance may be nonoptimal in the build direction, and other issues.

To improve quality of manufactured parts, new toolpath generation methods for AM have been developed. There exist two major aspects on how to define 3d spatial toolpaths. It is the same as in multi-axis milling. Toolpaths for milling are described with 3d points, which define a sequence of tool tip

ScienceDirect

Procedia CIRP 00 (2018) 000–000

2212-8271 © The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license

(https://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of the scientific committee of the 6th CIRP Global Web Conference “Envisaging the future manufacturing, design, technologies and systems in innovation era”.

doi:10.1016/j.procir.2017.04.009

6th CIRP Global Web Conference

“Envisaging the future manufacturing, design, technologies and systems

in innovation era”

Geometry-Based Process Planning for Multi-Axis Support-Free

Additive Manufacturing

Yavuz Murtezaoglu

a,b

_{, Denys Plakhotnik}

a,b,

_{*, Marc Stautner}

b

_{, Tom Vaneker}

a

_,

Fred J.A.M. van Houten

a

a_{University of Twente, Drienerlolaan 5, Enschede 7522 NB, the Netherlands} b_{ModuleWorks GmbH, Henricistr. 50, Aachen 52072, Germany} * Corresponding author. Tel.: +49-241-990004-32; fax: +49-241-990004-10. E-mail address: denys@moduleworks.com

Abstract

In contrast to standard layer based additive manufacturing methodologies, multi-axis material deposition can print structures without the need for support material. However, this method is jeopardized by potential collisions between a depositing unit (nozzle, wire, power and powder sources, etc.) and the already deposited material. The goal of this research is to initiate development of a methodology to check manufacturing feasibility of geometries and generate subsequent process planning strategies. The paper describes a geometry-based concept to decompose the product geometry into discrete volumes by using space partitioning with infinite planes and considering advantages and constraints of multi-axis additive manufacturing. The discrete volumes are used to generate process planning variants and to compute and generate boundary conditions for such process planning strategies. The algorithm generates multi-axis slices that require no support structures because of relative nozzle/workpiece orientation. In addition, the planning tackles more complex scenarios, in which overhangs, nozzle orientation, and gravity can be considered.

Keywords: additive manufacturing; process planning; multi-axis; slicing

1. Introduction

In the 1980s and 1990s, additive manufacturing (AM) was believed to replace subtractive machining. The new fabrication concepts promised easy production of very complex parts without complications that usually accompany conventional processes. Manufacturing of shapes with unlimited geometrical complexity was expected to become trivial and such that re-fixturing and multi-axis toolpaths will not be required. However, after the initial hype and obtaining broader and deeper experience, researchers and industrial

adopters realized that the classical 2.5-dimensional approach (horizontal slicing) has substantial limitations - need of support structures, the surface quality is likely to be insufficient on steep walls and overhangs, structural performance may be nonoptimal in the build direction, and other issues.

To improve quality of manufactured parts, new toolpath generation methods for AM have been developed. There exist two major aspects on how to define 3d spatial toolpaths. It is the same as in multi-axis milling. Toolpaths for milling are described with 3d points, which define a sequence of tool tip

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positions. Additionally, tool orientations may change along the toolpath. Therefore, there are several possibilities on how to define multi-axis toolpaths, depending on tool orientation change and the tool tip trajectory shape.

Flat toolpaths may additionally include orientation of the nozzle. Some methods simultaneously vary the orientation of the nozzle, some methods preserve unit's orientation fixed for the entire process. Such 2D sliced toolpaths can be also called multi-axis or multi-directional if the nozzle alters its orientation between different slices, also called layers, or even within one slice. This approach is often called "Multi-Directional" or "Multi-Axis" AM.

In this paper, the authors suggest an algorithmic framework for multi-directional planar slicing and decomposing of volumes for 3+2 axis AM processes. The developed method is robust, and it is based on analysis of relationships between points, planes, and mesh surfaces. The method describes a computational geometry algorithm of splitting a volume with a plane considering some AM specific constraints. No derivation of surface curvature or other properties is needed. This paper focuses mostly on resolving geometric problems that arise in finding a plane position for volume decomposition for the sake of AM process planning (when gravity, upskin/downskip, and other factors may be taken into account). The algorithm generates multi-axis slices that require no support structures because of the relative nozzle/workpiece orientation.

Outline

This paper is organized as follows. After the introduction, a review on the existing developments in the field of process planning for multi-axis AM is presented. In this review, different steps of the process planning are identified and described to give an overview on the process chain algorithms. Then, the main content of the paper is presented.

2. Related work on multi-axis strategies

This paper focuses on process planning for multi-directional AM with fixed nozzle orientation (per slice). Previous studies addressed several important steps that are necessary for successful process planning.

2.1. Decomposition of the solid model

Several research papers highlight importance of decomposition of the part geometry as the starting point for multi-axis AM. As the result of decomposition, the part geometry can be represented as a set of several geometric features or primitives, which will be processed with different slicing directions. Decomposition strategies consider different geometric properties of solid bodies (feature detection and extraction [1,2], detection of the undercut surfaces [3], inscribing a cube inside the model [4], morphological skeleton of the solid [5,6], and topology and centroidal axis [7].

2.2. Build direction determination

For each decomposed volume, a build direction must be assigned. The build direction defines rules of how slicing will propagate, i.e. deriving orientations of the layer planes. Orientation of the layer may be either fixed for all layers (parallel slicing) or may vary and depend on some criteria, usually as a function of the part or feature geometry. Build directions may be defined as functions of various metrics, such as surface accuracy, time to build, or number of slices, as suggested in [8].

2.3. Sequencing of the build order

Sequencing of the build order is an order in which the decomposed volumes must be manufactured. As parts of one solid body, the decomposed volumes are to be produced in an order that assures physical existence of a base substrate on which to deposit (material of either the same volume or an adjacent one) [5,6,8]. Arranging the order of the layers in a suitable fabrication order faces not only the material adjacency problem but also problems of collision between the nozzle and already deposited material. Several solutions have been proposed, ranging from rescheduling the order of subdivided subvolumes to changing the nozzle orientation [1,4,5,6,8,9].

2.4. Slicing

Parallel slicing is well known from 3d printing. Every slice is parallel to each other with some extent. In adaptive parallel slicing, the distance between intersection planes may also vary.

In non-parallel slicing [1,2,7,8,10,11,12], build directions are usually described as curves, while the intersection planes slice the solid volumes orthogonally to the curve.

Deposition with non-uniform thickness resulted into development of several methods that can exploit advantages and reduce disadvantages. According to [13], by rearranging the direction of the non-uniform layers, variable slicing directions may produce surfaces of final parts with smaller cusp height. Similar approach was adapted for multi-axis slicing based on the centroidal axis [7], and then further extended in [9]. Also, slicing of each subregions of the CAD model by a set of parallel planes oriented normal to the morphological skeleton vector of the subregion was proposed in [6]. Besides, optimization of the segment thickness may produce smaller deviations, thus meeting allowable overhang length and cusp height [11].

2.5. Other considerations

Research papers address mostly only the most significant aspects of the process planning, while other aspects, like considering the gravity, may become crucial. For instance, directed energy deposition (DED) with blown powder is very sensitive to the direction of the nozzle blow to the gravity force. Therefore, in some processes, adjusting the nozzle orientation may result in more efficient trade-off between overhang and process improvement via balancing between orthogonality to the surface and alignment to the gravity force

[14]. Additionally, volume subdivision can also be applied to optimize air-cut time [15] or rescheduling of the build sequence to avoid collisions.

3. Half-space decomposition method

Usually, process planning for multi-axis AM begins with model decomposition. Variations of feature-based decomposition are widely accepted by most of the researches. However, this approach faces at least two challenges. First, connecting geometries between geometric features may be not as trivial as the parent features and may require different planning, as highlighted in red on Fig. 1. Second, recent trends in Design for AM predict wide-spread usage of topology optimization, which is likely to result in free-form shapes with barely recognizable features.

Fig. 1. Connecting feature (red) in feature-based decomposition.

In this paper, a greedy algorithm for decomposition of the continuous free-form volumes is presented. A greedy strategy terminates at the largest possible increment of the search without considering smaller increments that may lead to better results. It starts scanning the part surface with a plane perpendicular to the Z direction, outwards from the table. While analyzing the surface geometry in the vicinity of the intersection between the scanning plane and the part surface, the scanning plane may adjust its orientation to search for its optimal position. Figure 2a depicts an example of the greedy decomposition, in which the black dash line is the first optimal border between two operations. Depending on optimization criteria, other results (blue or red dash lines) may be selected, as shown in Fig. 2b.

Fig. 2. (a) layers for decomposition with a vertical plane; (b) layers of alternative decomposition

Instead of defining more closely the details of the complex problem: constraints due to accessibility, collisions and the inevitable need for support structures in some remaining cases, we would like to define a limited problem as the following.

There is a given free-form geometry. This geometry is ensured to be manufacturable with the mentioned constraints, otherwise the algorithm should detect infeasible shapes. For the given free-form geometry located on a table, scan the part surface parallel to the table with plane E, starting with the scanning plane intersecting point , iterating forward in the direction of the table normal until it reaches point , in which encounters deposition conditions that must be avoided, like overhanging on the right side, as shown on Fig. 3. Then, a proper splitting plane (last feasible Z scanning plane) must be found. Since an infinite number of planes can be constructed, the method is to search (shift and rotate) a plane position that fits certain goals. Different objectives may result in different planes, like planes and shown in Fig. 3. The subvolume between the table and the next splitting plane, is the result of the greedy decomposition and it is considered for further slicing and toolpath planning. The subvolume is also removed from the part geometry, and the next iteration begins scanning the remaining part geometry from the splitting plane as if it is a table.

Fig. 3. Surface Z scanning with planes.

3.1. Stop-criteria for the parallel scanning

Given part is a triangle mesh, which is accepted across the AM industry as a standard part description. The scanning plane intersects triangles, and each plane-triangle intersection can be analyzed whether the staircase or overhanging is expected. Figure 4a and Fig. 4b depict several deposited layers that are enclosed by the triangles. Best printing results are achieved if printing direction and triangle normal are orthogonal to each other. Deviation from the ideal angle of 90° can be a relevant characteristic for defining the error function.

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positions. Additionally, tool orientations may change along the toolpath. Therefore, there are several possibilities on how to define multi-axis toolpaths, depending on tool orientation change and the tool tip trajectory shape.

Flat toolpaths may additionally include orientation of the nozzle. Some methods simultaneously vary the orientation of the nozzle, some methods preserve unit's orientation fixed for the entire process. Such 2D sliced toolpaths can be also called multi-axis or multi-directional if the nozzle alters its orientation between different slices, also called layers, or even within one slice. This approach is often called "Multi-Directional" or "Multi-Axis" AM.

In this paper, the authors suggest an algorithmic framework for multi-directional planar slicing and decomposing of volumes for 3+2 axis AM processes. The developed method is robust, and it is based on analysis of relationships between points, planes, and mesh surfaces. The method describes a computational geometry algorithm of splitting a volume with a plane considering some AM specific constraints. No derivation of surface curvature or other properties is needed. This paper focuses mostly on resolving geometric problems that arise in finding a plane position for volume decomposition for the sake of AM process planning (when gravity, upskin/downskip, and other factors may be taken into account). The algorithm generates multi-axis slices that require no support structures because of the relative nozzle/workpiece orientation.

Outline

This paper is organized as follows. After the introduction, a review on the existing developments in the field of process planning for multi-axis AM is presented. In this review, different steps of the process planning are identified and described to give an overview on the process chain algorithms. Then, the main content of the paper is presented.

2. Related work on multi-axis strategies

This paper focuses on process planning for multi-directional AM with fixed nozzle orientation (per slice). Previous studies addressed several important steps that are necessary for successful process planning.

2.1. Decomposition of the solid model

Several research papers highlight importance of decomposition of the part geometry as the starting point for multi-axis AM. As the result of decomposition, the part geometry can be represented as a set of several geometric features or primitives, which will be processed with different slicing directions. Decomposition strategies consider different geometric properties of solid bodies (feature detection and extraction [1,2], detection of the undercut surfaces [3], inscribing a cube inside the model [4], morphological skeleton of the solid [5,6], and topology and centroidal axis [7].

2.2. Build direction determination

For each decomposed volume, a build direction must be assigned. The build direction defines rules of how slicing will propagate, i.e. deriving orientations of the layer planes. Orientation of the layer may be either fixed for all layers (parallel slicing) or may vary and depend on some criteria, usually as a function of the part or feature geometry. Build directions may be defined as functions of various metrics, such as surface accuracy, time to build, or number of slices, as suggested in [8].

2.3. Sequencing of the build order

Sequencing of the build order is an order in which the decomposed volumes must be manufactured. As parts of one solid body, the decomposed volumes are to be produced in an order that assures physical existence of a base substrate on which to deposit (material of either the same volume or an adjacent one) [5,6,8]. Arranging the order of the layers in a suitable fabrication order faces not only the material adjacency problem but also problems of collision between the nozzle and already deposited material. Several solutions have been proposed, ranging from rescheduling the order of subdivided subvolumes to changing the nozzle orientation [1,4,5,6,8,9].

2.4. Slicing

Parallel slicing is well known from 3d printing. Every slice is parallel to each other with some extent. In adaptive parallel slicing, the distance between intersection planes may also vary.

In non-parallel slicing [1,2,7,8,10,11,12], build directions are usually described as curves, while the intersection planes slice the solid volumes orthogonally to the curve.

Deposition with non-uniform thickness resulted into development of several methods that can exploit advantages and reduce disadvantages. According to [13], by rearranging the direction of the non-uniform layers, variable slicing directions may produce surfaces of final parts with smaller cusp height. Similar approach was adapted for multi-axis slicing based on the centroidal axis [7], and then further extended in [9]. Also, slicing of each subregions of the CAD model by a set of parallel planes oriented normal to the morphological skeleton vector of the subregion was proposed in [6]. Besides, optimization of the segment thickness may produce smaller deviations, thus meeting allowable overhang length and cusp height [11].

2.5. Other considerations

Research papers address mostly only the most significant aspects of the process planning, while other aspects, like considering the gravity, may become crucial. For instance, directed energy deposition (DED) with blown powder is very sensitive to the direction of the nozzle blow to the gravity force. Therefore, in some processes, adjusting the nozzle orientation may result in more efficient trade-off between overhang and process improvement via balancing between orthogonality to the surface and alignment to the gravity force

[14]. Additionally, volume subdivision can also be applied to optimize air-cut time [15] or rescheduling of the build sequence to avoid collisions.

3. Half-space decomposition method

Usually, process planning for multi-axis AM begins with model decomposition. Variations of feature-based decomposition are widely accepted by most of the researches. However, this approach faces at least two challenges. First, connecting geometries between geometric features may be not as trivial as the parent features and may require different planning, as highlighted in red on Fig. 1. Second, recent trends in Design for AM predict wide-spread usage of topology optimization, which is likely to result in free-form shapes with barely recognizable features.

Fig. 1. Connecting feature (red) in feature-based decomposition.

In this paper, a greedy algorithm for decomposition of the continuous free-form volumes is presented. A greedy strategy terminates at the largest possible increment of the search without considering smaller increments that may lead to better results. It starts scanning the part surface with a plane perpendicular to the Z direction, outwards from the table. While analyzing the surface geometry in the vicinity of the intersection between the scanning plane and the part surface, the scanning plane may adjust its orientation to search for its optimal position. Figure 2a depicts an example of the greedy decomposition, in which the black dash line is the first optimal border between two operations. Depending on optimization criteria, other results (blue or red dash lines) may be selected, as shown in Fig. 2b.

Fig. 2. (a) layers for decomposition with a vertical plane; (b) layers of alternative decomposition

Instead of defining more closely the details of the complex problem: constraints due to accessibility, collisions and the inevitable need for support structures in some remaining cases, we would like to define a limited problem as the following.

There is a given free-form geometry. This geometry is ensured to be manufacturable with the mentioned constraints, otherwise the algorithm should detect infeasible shapes. For the given free-form geometry located on a table, scan the part surface parallel to the table with plane E, starting with the scanning plane intersecting point , iterating forward in the direction of the table normal until it reaches point , in which encounters deposition conditions that must be avoided, like overhanging on the right side, as shown on Fig. 3. Then, a proper splitting plane (last feasible Z scanning plane) must be found. Since an infinite number of planes can be constructed, the method is to search (shift and rotate) a plane position that fits certain goals. Different objectives may result in different planes, like planes and shown in Fig. 3. The subvolume between the table and the next splitting plane, is the result of the greedy decomposition and it is considered for further slicing and toolpath planning. The subvolume is also removed from the part geometry, and the next iteration begins scanning the remaining part geometry from the splitting plane as if it is a table.

Fig. 3. Surface Z scanning with planes.

3.1. Stop-criteria for the parallel scanning

Given part is a triangle mesh, which is accepted across the AM industry as a standard part description. The scanning plane intersects triangles, and each plane-triangle intersection can be analyzed whether the staircase or overhanging is expected. Figure 4a and Fig. 4b depict several deposited layers that are enclosed by the triangles. Best printing results are achieved if printing direction and triangle normal are orthogonal to each other. Deviation from the ideal angle of 90° can be a relevant characteristic for defining the error function.

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On the one hand, staircase is caused by surface normal when it is not orthogonal to printing direction. On the other hand, if is parallel to printing direction (top flat surface), the staircase effect is minimized as well. Although staircase effect may severely harm surface quality and the visual perception, overhangs should be considered as one of the main obstacles to realize AM parts. It is obvious that 3d printing cannot happen in the "air", so each succeeding printing layer can only slightly deviate from its predecessor, regarding generating overhangs without support. Defining a threshold angle α, computation can be constrained to secure overhang-relaxed slicing.

3.2. Search of the splitting plane

Parallel scanning begins from the table incrementally progressing upwards. Parallel scanning terminates as soon as either staircase measure or overhang threshold angle is exceed. Then, the splitting plane is to be found. In order to improve computational burden, the search of the splitting plane can be restricted to a smaller volume. The lower bound is the last feasible scanning plan . Another boundary can be an utmost plane defining the overhang threshold for the current printing direction, as shown in Fig. 5.

Fig. 5. Bounding box of the search space.

The goal is to iterate through search points scattered on the part surface within the Axis Aligned Bounding Box (AABB) such that all search points below the optimal splitting plane satisfy a function F defined as the number of search points below a plane E. All search points below E must tolerate upskin/downskin thresholds, otherwise F becomes invalid.

Since a plane in 3d space has three degrees of freedom. As shown in Fig. 6, plane orientation requires two rotational degrees (two Euler angles pair φ and θ), also the plane E can be described with the distance from origin d ( = | φ ∈ [0°, 360°] | θ ∈ [0°, 180°] | d ∈ [ , ], where and define AABB extent).

Since , ,

, , the search problem can be formulated as follows: to find such a plane that all sampling points below satisfy C, and the plane E does not intersect cross-sections between the part geometry on limiting planes (table and top).

Fig. 6. Plane definition with Euler angles.

The rough workflow of the algorithm is presented in Fig. 7. At each search point, the algorithm places a plane and iterates over all possible plane orientations with a defined sampling step (1 degree is believed to be sufficiently small).

Search points are sampled on the part surfaces with any given precision. Successful search points must satisfy quality and threshold criteria C (mesh orientation at search points must tolerate upskin/downskin thresholds). We use points , where each Pi has an according normal , which is still the

triangle normal coming from of triangle, where is derived from. Accordingly, each search point in addition to its spatial position must be associated with a Boolean value. This approximation simplifies calculation of the function F.

Fig. 7. Pseudocode of the algorithm.

To determine whether a point is below the plane E, is tested against half-space , as shown in Fig. 8, with Eq. 1.

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Fig. 8. Half-space partitioning of a point cloud.

The plane E must not intersect intersection contours between the part geometry and AABB. The latter is important, overwise there will be two splitting planes, thus making impossible to proceed to the next decomposition. For instance, for a given continuous 3d shape, shown in Fig. 9, an AABB is calculated for 45° overhang as shown in Fig. 10.

Fig. 9. Twisted circle-square sweep shape.

Fig. 10. AABB for the twisted-sweep shape.

The optimal splitting plane position is sought at search points, shown in Fig. 11, via brute-force computation with fixed sampling of angles φ and θ defining a plane. If a splitting plane intersects cross-section contours on the faces of AABB (vertical contour and slice contour in Fig. 11), the connection between two subvolumes will be represented by

two different planes, thus making parallel slicing for the next subvolume impossible. For this AABB, there are 5059 search points along with 9280 triangles to be checked at utmost 90*360 = 32,400 plane orientations.

Fig. 11. AABB search space.

The best plane, according to the greedy strategy, accounts for the maximal number of sampling points in the half-space aligned to the table plane. The search space is intentionally restricted to a prismatic primitive (AABB), while ideally the search space should be a truncated pyramid with the base comprised of the part profile polygon at the utmost horizontal scanning plane and the face slope attributing to the maximal allowed overhang. On the one hand, the use of AABBs limit a variety of geometries that can be handled by the algorithm, on another hand, splitting planes may not have a negative slope, thus easing collision-free process planning.

Figure 12 shows three different splitting planes. Two planes are found with restricted plane orientation (45° and 90°). Third result included an additional condition not to include upskin points with surface orientation exceeding 60°.

Fig. 12. Different search results, depending on the constraints.

Besides, gravity can be taken into account as well. Due to tilting, the local printing direction will not be Z-axis aligned. Therefore, the gravity force vector must be transformed to the local coordinate system, and the angle between the vector of the printing direction and the gravity force vector can be penalized. The splitting plane restricted to 45° is an example of considering gravity in planning. This restriction may severely limit a search space, while assuring better process conditions.

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On the one hand, staircase is caused by surface normal when it is not orthogonal to printing direction. On the other hand, if is parallel to printing direction (top flat surface), the staircase effect is minimized as well. Although staircase effect may severely harm surface quality and the visual perception, overhangs should be considered as one of the main obstacles to realize AM parts. It is obvious that 3d printing cannot happen in the "air", so each succeeding printing layer can only slightly deviate from its predecessor, regarding generating overhangs without support. Defining a threshold angle α, computation can be constrained to secure overhang-relaxed slicing.

3.2. Search of the splitting plane

Parallel scanning begins from the table incrementally progressing upwards. Parallel scanning terminates as soon as either staircase measure or overhang threshold angle is exceed. Then, the splitting plane is to be found. In order to improve computational burden, the search of the splitting plane can be restricted to a smaller volume. The lower bound is the last feasible scanning plan . Another boundary can be an utmost plane defining the overhang threshold for the current printing direction, as shown in Fig. 5.

Fig. 5. Bounding box of the search space.

The goal is to iterate through search points scattered on the part surface within the Axis Aligned Bounding Box (AABB) such that all search points below the optimal splitting plane satisfy a function F defined as the number of search points below a plane E. All search points below E must tolerate upskin/downskin thresholds, otherwise F becomes invalid.

Since a plane in 3d space has three degrees of freedom. As shown in Fig. 6, plane orientation requires two rotational degrees (two Euler angles pair φ and θ), also the plane E can be described with the distance from origin d ( = | φ ∈ [0°, 360°] | θ ∈ [0°, 180°] | d ∈ [ , ], where and define AABB extent).

Since , ,

, , the search problem can be formulated as follows: to find such a plane that all sampling points below satisfy C, and the plane E does not intersect cross-sections between the part geometry on limiting planes (table and top).

Fig. 6. Plane definition with Euler angles.

The rough workflow of the algorithm is presented in Fig. 7. At each search point, the algorithm places a plane and iterates over all possible plane orientations with a defined sampling step (1 degree is believed to be sufficiently small).

Search points are sampled on the part surfaces with any given precision. Successful search points must satisfy quality and threshold criteria C (mesh orientation at search points must tolerate upskin/downskin thresholds). We use points , where each Pi has an according normal , which is still the

triangle normal coming from of triangle, where is derived from. Accordingly, each search point in addition to its spatial position must be associated with a Boolean value. This approximation simplifies calculation of the function F.

Fig. 7. Pseudocode of the algorithm.

To determine whether a point is below the plane E, is tested against half-space , as shown in Fig. 8, with Eq. 1.

(1)

Fig. 8. Half-space partitioning of a point cloud.

The plane E must not intersect intersection contours between the part geometry and AABB. The latter is important, overwise there will be two splitting planes, thus making impossible to proceed to the next decomposition. For instance, for a given continuous 3d shape, shown in Fig. 9, an AABB is calculated for 45° overhang as shown in Fig. 10.

Fig. 9. Twisted circle-square sweep shape.

Fig. 10. AABB for the twisted-sweep shape.

The optimal splitting plane position is sought at search points, shown in Fig. 11, via brute-force computation with fixed sampling of angles φ and θ defining a plane. If a splitting plane intersects cross-section contours on the faces of AABB (vertical contour and slice contour in Fig. 11), the connection between two subvolumes will be represented by

two different planes, thus making parallel slicing for the next subvolume impossible. For this AABB, there are 5059 search points along with 9280 triangles to be checked at utmost 90*360 = 32,400 plane orientations.

Fig. 11. AABB search space.

The best plane, according to the greedy strategy, accounts for the maximal number of sampling points in the half-space aligned to the table plane. The search space is intentionally restricted to a prismatic primitive (AABB), while ideally the search space should be a truncated pyramid with the base comprised of the part profile polygon at the utmost horizontal scanning plane and the face slope attributing to the maximal allowed overhang. On the one hand, the use of AABBs limit a variety of geometries that can be handled by the algorithm, on another hand, splitting planes may not have a negative slope, thus easing collision-free process planning.

Figure 12 shows three different splitting planes. Two planes are found with restricted plane orientation (45° and 90°). Third result included an additional condition not to include upskin points with surface orientation exceeding 60°.

Fig. 12. Different search results, depending on the constraints.

Besides, gravity can be taken into account as well. Due to tilting, the local printing direction will not be Z-axis aligned. Therefore, the gravity force vector must be transformed to the local coordinate system, and the angle between the vector of the printing direction and the gravity force vector can be penalized. The splitting plane restricted to 45° is an example of considering gravity in planning. This restriction may severely limit a search space, while assuring better process conditions.

(6)

Different weights for the overhang and gravity converge to different decomposition and process planning, as shown in Fig. 13, Fig. 14, and Fig. 15.

Fig. 13. Slicing planning for a 45° splitting plane.

Fig. 14. Slicing planning for a splitting plane excluding 60° upskin.

Fig. 15. Slicing planning for a 90° splitting plane.

4. Conclusions

This work involves searching for a second 3d printing direction using multi-axis (5-axis) kinematics. The above described algorithm is limited to one half-space E, we believe it has the potential to be extended to multiple subsequent half-space calculations for multi-axis 3d printing.

The user needs to define the relative penalty for allowing overhangs and depositing against the gravity force. Different input value generated different volume decomposition with parallel slicing. Since this method is based on the greedy approach, it inherits its strengths and weaknesses. The

algorithm is robust and computationally efficient, but it may end up into sub-optimal solutions.

The algorithm can be improved by adopting adaptive slicing (layer thickness) to reduce the staircase effect on curved surface. The algorithm has some limitations that are planned to elaborate in the future development. The missing functionality may be extended by including collision check and avoidance in subvolume decomposition and slicing, implementation of non-uniform and curved slicing.

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