Non-Planar Helical Path Generation Method for Laser Metal Deposition of Overhanging Thin-Walled Structures

Abstract

Laser metal deposition is a branch of additive manufacturing that offers advantages over traditional manufacturing techniques for forming overhanging thin-walled metal parts. Previously, helical paths that were suitable for manufacturing such parts were not only limited to stacking material on a flat surface but were also fixed to the model boundaries. In order to solve these two problems to meet more complex process requirements, a non-planar helical path generation method is proposed for laser metal deposition. The method is based on the characteristics of the additive manufacturing process planning flow, which first slices the model using curved surfaces, then offsets the contours on the sliced layering, and finally generates continuous helical paths according to the contracted or expanded contours. In order to verify the feasibility of the method, hollow blades are formed on cylindrical surfaces following the planned paths. The results show that the proposed method is not only capable of assisting the laser metal deposition process to fabricate thin-walled structures on non-planar surfaces but also capable of freely adjusting the contour dimension.

1. Introduction

Additive manufacturing (AM) is an important technology for building parts from scratch, which has the advantages of high design freedom, short build cycle time, and high material utilization over traditional manufacturing techniques [1]. When combined with metal materials, it has a wide range of industrial applications and mature material manufacturing processes. Laser metal deposition (LMD) is subordinate to the branch of metal AM, which is based on the principle of using high-energy lasers to melt a metallic material and then causing it to cool and solidify at a specified location. The process builds up the material layer by layer until the part is shaped [2]. Thin-walled structures have the advantages of being lightweight and compact in size and are increasingly used as critical parts in equipment and engineering for aerospace, marine navigation, and medicine [3]. The LMD process incorporating a multi-axis strategy does not require additional support structures for the fabrication of such structural parts compared to other processes, resulting in greater cost and time savings. Regardless of which AM technology is used, most require a similar planning process to be completed with the assistance of a computer. The process includes computer-aided design (CAD) modeling, slicing, path planning, and post-processing, which are usually integrated into a computer-aided manufacturing (CAM) system. The path data generated by this procedure in the computerized digital space are very critical; they not only determine the geometry of the forming but also affect the mechanical or metallurgical properties [4].

1.1. Slicing for Additive Manufacturing

The special geometrical characteristics of thin-walled structures mean that the focus is on the contour boundary region. Although it does not usually need to be filled internally, it still needs to go through a layered slicing phase due to the layer-by-layer buildup. In AM, the common format used for transferring and reading part models is STereoLithography (STL). Therefore, the input data for the slicing algorithm are usually in an STL file, which contains an unordered collection of triangular facets to fit the model surface, each with the attributes of vertex coordinates and normal vectors [5]. Initially, a uniform planar slicing algorithm emerged, which was based on the principle of using a set of equally spaced planes as cross sections to intersect the part model [6,7]. These types of methods suffer from a step effect that affects the surface quality when dealing with surfaces. This led to the emergence of adaptive slicing algorithms [8,9] dedicated to improving this problem, which use variables instead of constant layer thickness parameters. However, the adaptive algorithm not only fails to take into account the balanced transition of layer thicknesses but also affects the subsequent process flow, and thus, it has not been widely used and disseminated [10]. Later, non-planar slicing algorithms appeared, mainly including cylindrical slicing [11], spherical slicing [12], and free-form surface slicing [13]. Cylindrical and spherical slicing are designed to meet the process requirements of variable-curvature machining substrates, while free-form surface slicing is usually realized based on vector fields, which are difficult to control, and the high degree of uncertainty between neighboring layers does not apply to metal AM.

1.2. Path Planning

Once the contour is obtained by slicing, it needs to be further planned on its basis to generate a complete path. Maintaining the continuity of the path is important and can influence the forming process of the part [14,15]. Neighboring layers are usually transitioned with empty travel, and an unavoidable start–stop at the starting point not only leads to defects at the seam interface but also prolongs the build time. Yigit et al. [16] proposed a helical upward path generation method to improve the surface quality in such areas. The principle is to use two consecutive layered contours to construct direction vectors from the current layer to the next layer, and these vectors are used to articulate neighboring contours into a helical pattern in space. Zhao et al. [17], to improve the problem that helical paths are not capable of filling the area, broke through the limitation of the traditional 2.5-axis application and proposed a new helical filling path generation method with more desirable surface quality for the equipment that has extra rotary axes. Bhatt et al. [18] continued to utilize the potential of helical paths in conjunction with a multi-axis strategy by improving the method and applying it to a rotary table with two rotary axes, demonstrating the good suitability of helical paths for thin-walled structures by forming unsupported overhanging parts.

1.3. Conclusions of the Literature Review

There is a growing need to stack materials around a fixed axis, such as the turbine parts necessary in engines. The hub surfaces of such parts are usually of variable curvature, and previous helical path methods can only be used to form parts with smaller overhang angles in a planar surface. In order to realize more potentials of helical path methods and promote the development of additive manufacturing in industry, this paper proposes a non-planar path generation method, that can not only fabricate overhanging structural parts on non-planar surfaces through the LMD process but also adjust the contour boundary dimensions as expected to flexibly meet the process requirements. The rest of this paper is organized as follows. In Section 2, the proposed method for generating non-planar helical paths is described in detail. In Section 3, solid parts are molded according to the paths to verify the feasibility and effectiveness of the method. In Section 4, the conclusion of the proposed method is given.

2. Method

The principle of the proposed method relies mainly on computational geometry algorithms, aiming to assist the LMD technique in forming thin-walled overhanging structures on non-planar surfaces through operations between geometries. The process planning flow of LMD is similar to that of the AM generalization as shown in Figure 1, which outlines how the method transforms the numerical model into a solid part according to the idea of input–output.

The computer represents the virtual digital space, and the machine tool represents the actual physical space. The input and output that pass between them consist of the STL model, the path data, and the physical workpiece. In the whole process, the most critical components are the path data, which are generated by the computer and guide the machine tool. Thus, path generation is the bridge between the digital space and the physical space. As shown in Figure 2, the path generation method mainly consists of three phases sequentially, namely, non-planar slicing, contour offset, and helical path generation. Each phase faces different geometric challenges and most of them are solved using intersection operations.

The inputs to the method are two STL models that represent the base surface to be deposited and the part to be formed. In the slicing stage, the surface model is offset by a given layer thickness and then intersected with the part model to obtain the deposition contour for each layer. In the contour offset stage, the curve contracts or expands on the surface by a given offset distance to affect the boundary dimensions of the part to be formed. In the helical path generation phase, the contours of each layer are connected in a helical upward trend. To introduce a multi-axis strategy to cope with non-planar and overhanging features, the coordinates of points on the path are matched to vectors that fit the surface of the model.

2.1. Non-Planar Slicing

Non-planar slicing is different from planar slicing in that the heights involved within the same layer of the former are variable, and the dimensionality of the operation cannot be reduced as in the latter. It can be seen that non-planar slices are more complex and need to involve more triangular facets to fit the surface. When forming cross sections, the parameters required for each layer are not just a height constant and a fixed Z-axis but a large number of mesh vertices and corresponding normal vectors. The variable-curvature base surface is offset as shown in Figure 3.

The triangular mesh of a fitted surface consists of vertices with topological relationships. Therefore, the effect of offsetting the surface can be achieved by moving the vertices. The direction of the vertex offset comes from the average sum of the normal vectors of the triangular faces in the neighborhood. Then, by inheriting the previous topology, this set of vertices forms a new triangular mesh for intersecting the part model. As shown in Figure 4, the essence of intersecting two meshes is the intersection computation between the triangular facets that belong to them. Triangular pairs can obtain a line segment in case of intersection; then, the line segments obtained between multiple triangular facets can form an open fold or a closed contour.

Since the triangular facets provided by the STL are unordered, it is also necessary to order the line segments on the contour to form an oriented contour. Based on the feature that neighboring line segments share common endpoints, a dictionary storing endpoints and line segments is created. As shown in Figure 5, the endpoint of a line segment is used as a keyword and the corresponding value stores the two line segments that contain it. Finally, a search is performed in the dictionary. For example, after reading the point $v0$, find its associated line segment $e0$ and compare it with its endpoint. If it is the same as its starting point, then the next point is the end point of $e0$; otherwise, the next point is the starting point. So, find the point $v3$ as the next point and take out the line segment that was stored when it was used as a keyword. Since $e0$ has already been processed, the edge $e3$ is used, and so on, repeating the above steps until all the edges have been used, and at this point, all the points have been read in an orderly sequence and form an oriented contour.

But just because there is a directional contour that does not mean that each layer is oriented in the same direction. The direction may be clockwise or counterclockwise. At the stage of intersection, the triangular facets of the model surface involved in each edge can be recorded. Each edge is shared by two triangular facets; then, their normal vectors are used as the normal vectors of the points. Thus, the result of the cross product of the tangent vector of the contour and the normal vector of the point is used to determine the direction of the contour. The normal vectors of the triangular facets that fit the surface of the model all face outward, and the normal vectors of the points at this stage also face outward. Then, the normal vectors of the points have the opposite result of the operation on the counterclockwise contour and the clockwise contour. For subsequent planning, the contours of each layer are standardized to be counterclockwise.

2.2. Contour Offset

After obtaining the contour, additional processing is required before generating the path to conform to the expectations of the boundary-shaping dimensions. Contour offsets are commonly found in planar surfaces and need to deal with the complexity of holes when used as filling paths [19]. Computational accuracy is lost and offset errors exist when the coordinates of the plane are applied directly to the surface through mapping transformations. To ensure that the curve representing the contour always fits the given surface when moving, a new algorithm for contour offset is proposed in this paper, but it is only applicable to surfaces that are fitted using triangle meshes. The constraints are obtained by using the intersection of the plane and the mesh as shown in Figure 6.

Here, each point can obtain the tangent vector of the contour at its location. Since the contour is ordered, each point only needs to form two vectors with two neighboring points, and their average sum is the tangent vector for that point. Then, based on the coordinates and tangent vectors of the current point, a plane that intersects the surface can be constructed so that each point has one such line of intersection, and this line of intersection subsequently restricts its movement on the surface.

When a tangent plane intersects a surface, it is essentially the intersection of the plane and the edges of the triangular facets. When the triangular facet is determined to transverse the plane based on the coordinates of the vertices, the problem is simplified to a directed line segment intersecting the plane. The method of finding the intersection of a plane with a line is shown in Figure 7.

Considering that the point of intersection is on both the line and the plane, it is reasonable to assume that the point of reference of the origin on the line is the point of intersection if it is moved a distance along that line. Knowing the distance $D$ from the plane to the origin $o$ and its normal vector $\overrightarrow{n_p}$, as well as the direction of the line $\overrightarrow{n_l}$, the distance $t$ can be found according to Equations (1) and (2).

$$o^{\prime }=o+o·\overrightarrow{n_l}$$

(1)

$$t=\left|D-o{o}^{″}\right|/\left(\overrightarrow{n_l}·\overrightarrow{n_p}\right)$$

(2)

Figure 7a shows the intersection of the plane and one edge of the triangular facets, assuming that the side view of the plane is in the direction of the Y-axis, which is then converted to the case of Figure 7b. The origin $o$ is projected onto the direction vector $\overrightarrow{n_l}$ of the line to obtain the point $o^{\prime }$; then, the reference point $o^{\prime }$ is projected onto the normal vector $\overrightarrow{n_p}$ of the plane to obtain the point $o^{″}$. The dot product of $\overrightarrow{n_l}$ and $\overrightarrow{n_p}$ yields $\cos \alpha$, while the difference between the distance of the point $o^{″}$ to the origin and $D$ and t is proportional to $\cos \alpha$. And so on, finding more intersections to form a curve. The line segments on the curve are ordered concerning the non-planar slicing, and the direction of the curve is arranged to be counterclockwise. Then, both contraction and expansion need to be discussed. The states at different positions and in different cases are shown in Figure 8.

An auxiliary vector is obtained by cross-multiplying the tangent vector of the contour at the point and the normal vector of the triangular facets where the point is located. This vector is used as the new value of the point’s normal vector, thus replacing the previous setting. The new normal vector of the point is used as the starting vector to determine whether the point needs to move forward or backward along the constraint curve. The direction of rotation is set to counterclockwise, the axis of rotation is the contour tangent vector, and the direction of the curve at the point is the target vector. Whether the point moves in a direction that obeys or resists the direction of the curve is a matter of whether or not to reverse the line segments on the curve. When the angle between the start vector and the target vector is obtuse, the curve needs to be reversed for the case of contraction. When the angle is acute, the curve is reversed for extended cases.

As shown in Figure 9, self-intersection and sawtooth problems also need to be solved when obtaining the biased contour. Figure 9a shows the redundant rings when the points on the contour are offset and then reconstituted into the contour in the original order. The adjacent points on the contour are formed into line segments, and it is judged whether there is intersection between the adjacent segments, including same-plane intersection and opposite-plane intersection. If it is the former, the intersection point $Q$ of the two line segments is directly retained and the points $P_{i+1}$ and $P_{i+2}$ on the anomalous ring are removed. If the latter, then, each line segment has a pseudo-intersection, taking the intermediate values of $Q^{\prime }$ and $Q^{″}$ that represent them. Figure 9b shows the phenomenon of sawtooth on a contour. When this occurs, there must be a conflict between the constraint curves of neighboring points. Therefore, when there is an intersection of the offset trajectories of two points, the neighboring point $P_{i+1}$ is removed.

2.3. Helical Path Generation

After obtaining the curve contour that undergoes the offset, a multi-axis helical path is generated on this basis. Each layer of the contour holds the coordinates of each point that makes it up and the normal vector corresponding to the point. The normal vectors of the points are the auxiliary vectors used in the previous section to determine whether the constrained curve needs to be inverted. Before articulating each layer as a path, it is necessary to find correspondences between the offset contour and the original contour and between neighboring offset contours. Since the smallest component unit of a contour is a point, it is finding the relationship between points. In the previous method, when finding the corresponding points, it is found according to the nearest distance. There are errors in such an approach, as shown in Figure 10.

To eliminate the existence of errors, an alternative way to find the corresponding points is needed, as shown in Figure 11. The plane formed by the point $P_i$ and the contour tangent vector cuts to the original contour, and the resulting intersection points $Q_j$ and $Q_k$ are used as candidates. The closest point $Q_j$ among the candidates is found as the corresponding point.

After finding the corresponding relationship between the contours, a parameter expressing the degree variable is also needed to satisfy the rule that the points are updated in an upward trend. As shown in Figure 12, the distance between the current point and the starting point is gradually increased until the distance reaches the total length of the contour when the starting point is retraced. Therefore, the ratio of the current length to the total length is between 0 and 1, which can be used as a reasonable ratio.

It is known that there are $n$ points on the current contour, and the coordinates of each point are ($x_{P_i}$, $y_{P_i}$,$z_{P_i}$). The length $L_k$ of the position of the $k^{th}$ point from the starting point is calculated according to Equation (3). And the total length $L$ can be calculated by replacing the parameter $k$ by the parameter $n$.

$$L_k=\sum _{i=0}^{k-1}\sqrt{{\left(x_{P_{i+1}}-x_{P_i}\right)}^2+{\left(y_{P_{i,j+1}}-y_{P_{i,j}}\right)}^2+{\left(z_{P_{i,j+1}}-z_{P_{i,j}}\right)}^2}$$

(3)

The length ratio is used not only to find the corresponding position of the current point on the neighboring contour but also to determine the distance the current point moves and the angle at which its normal vector is rotated. The principle of generating paths is further elaborated as shown in Figure 13.

First, let us discuss the situation within the current layer. When obtaining the normal vector $\overrightarrow{n_p}$ on a point $P_i^{\prime }$, if the corresponding point $P_c$ coincides with a point on the original contour, then it is straightforward to let the point $P_i^{\prime }$ inherit the normal vector. Otherwise, it will be on some line segment and then inherit the average sum of the normal vectors of the endpoints of that segment. Then, there is the case between neighboring layers. The corresponding point $Q_c$ found is the target point that the point $P_i^{\prime }$ wants to be close to, and the normal vector $\overrightarrow{n_Q}$ of $Q_c$ and the point $P_c$ are obtained in the same way. To generate the points $P_{helix}$ on the path with a helical regularity, the points and vectors on the offset contour need to be close to the next layer of the contour in different ratios depending on where they are located.

After the preparation work is completed, the generation of the path starts. The first step is to calculate the coordinates on the path. The point $P_i^{\prime }$ on the offset contour, its direction of travel $\overrightarrow{P_i^{\prime }{Q}_c}$ and length ratio $L_i/L$ are known. The point $P_{helix}$ on the helical path is calculated according to Equation (4).

$$P_{helix}=P_i^{\prime }+{\displaystyle \frac{L_i}{L}}\overrightarrow{P_i^{\prime }{Q}_c}$$

(4)

In generating the tool orientations $\overrightarrow{n_{helix}}$ on the helical path, the angle $\delta$ of rotation of the vector $\overrightarrow{n_p}$ toward $\overrightarrow{n_Q}$ is first calculated according to Equation (5). Then, a new vector is obtained by rotating the rotation of $\overrightarrow{n_p}$ according to this angle. The operation used for vector rotation is the quaternion definition [20], which gives $\overrightarrow{n_{helix}}$ according to Equations (6) and (7).

$$\delta =acrcos\left(\overrightarrow{n_p}·\overrightarrow{n_Q}\right)\times {\displaystyle \frac{L_i}{L}}$$

(5)

$$q=\mathit{cos}\left({\displaystyle \frac{\delta }{2}}\right)+Asin\left({\displaystyle \frac{\delta }{2}}\right)$$

(6)

$$\overrightarrow{n_{helix}}=q\overrightarrow{n_p}{q}^{-1}$$

(7)

where $q$ is the unit quaternion, $A$ is the unit vector perpendicular to the vector $\overrightarrow{n_p}$, and $\delta$ is the angle of rotation of $\overrightarrow{n_p}$ around $A$. And the perpendicular vector here is is not an arbitrary line in the plane with $\overrightarrow{n_p}$ as the normal vector; it is the tangent vector of the contour at this position.

3. Results

To validate the feasibility and effectiveness of the proposed method, the algorithms were developed in a Windows 11 (64-bit) operating system using the C# code language and run on a laptop configured with an i9-12900H 2.50 GHz CPU and 16.0 GB of RAM. The experiment was divided into two phases, the first was testing on different models to generate paths, and the second one was to form the solid parts from the paths using the LMD process.

3.1. Path Generation and Visualization

Most of the demand for metal parts deposited on non-planar surfaces is reflected in the growth of overhanging structures on a given component. Therefore, CAD models of this type of structure were chosen as the input to the algorithm. The four models used for testing are shown in Figure 14. These were modeled as parts with overhanging structures grown on non-planar surfaces, with triangle meshes fitted to their surfaces.

Since different models have different geometrical features and dimensions, different parameters were set for them. The layer thickness parameter was uniformly set to 1.0 mm to display the results. The offset parameters were labeled in detail to verify the effect of combining helical paths and contour offsets. As shown in Figure 15, several twisted rectangular sheets were suspended from the ring body, which had its offset parameter set to $\pm$1.0 mm. Positive values represent outward expansion, and negative values represent inward contraction.

As shown in Figure 15a, the path was a continuous helical shape curve, and the short lines growing above it were tool orientations. Since the offset parameter is negative, the path shrinks inside the model. The only way to see the model’s path is to make it more transparent. As shown in Figure 15b, the path was positive due to the offset parameter, and it hovered outside the model. As shown in Figure 16, turbine A with thicker blades was selected for testing and it had an offset parameter of $\pm$1.5 mm.

As shown in Figure 16a, the path was similarly constricted inside the blade. The path hovered around the outside of the blade as shown in Figure 16b. The effect of the offset was also more pronounced due to the increased offset distance. In addition to the cylindrical surface, paths were also generated on the spherical surface. As shown in Figure 17, some overhanging cylinders were distributed on the surface of the sphere. The offset parameter of the radiation sphere was set to $\pm$2.0 mm.

As shown in Figure 17a, the helical path was contracted inside the cylinder. As shown in Figure 17b, the helical path surrounded the surface of the cylinder. The method remained correct on the sphere where the surface was more curved. Finally, a freer variation in hub surface curvature and a greater degree of blade twist in the overhang were chosen on a complex model. As shown in Figure 18, the path was generated on turbine B, which had an offset parameter of $\pm$2.5 mm.

As shown in Figure 18a, the test was performed on a blade grown on a free-form surface. Even though the inward contraction was more complex and less fault-tolerant compared to the expansion, the path still exhibited contraction according to the given parameter distance. As shown in Figure 18b, the path hovered around the outside of the blade with more complex surface curvature, showing an outwardly expanding pattern.

3.2. Machine Tool Simulation and Manufacturing

To validate the proposed method, two expectations need to be met. First, the ability to deposit parts on non-planar surfaces. Secondly, to form parts with different dimensions of the contour according to given parameters. Thus, a machine tool was used to form three overhanging hollow blades on a cylindrical hub. Regarding the parameters of the generated paths, they had a layer thickness of 0.4 mm and offset parameters of −1.5 mm, 0.0 mm, and 1.5 mm as shown in Figure 19.

Before the path data are applied to the actual equipment, they also need to be post-processed according to the positive and negative solution motions [21], not only to set the process parameters but also to be converted into program instructions according to the characteristics of the equipment.

In order not to negatively affect the forming stage, a multi-axis machine with an increased rotary table is more suitable for the LMD process with metal powders. Therefore, although the Rotation Tool Center Point (RTCP) subtracts the updating of coordinate data, the tool orientation needs to be decomposed from vectors to rotational angle combinations. The data were converted and confirmed in the machine simulation as shown in Figure 20. They were the coordinates needed for the three moving axes and the angles needed for the two rotating axes.

The practical application is a coaxial powder-feeding device, which is a five-axis CNC machine equipped with the LMD process module, as shown in Figure 21a. The machine mainly consists of a laser deposition head, a rotary table, and a CNC system. The deposition head is composed of a powder feeder and a laser device that forms a molten pool at a specified location to absorb the metal powder. Figure 21b shows the process of forming a blade with a laser deposition head. The deposition head has only a moving axis and is not capable of rotating, so it is necessary to flip the cylindrical base to avoid collapse and light leakage when forming overhanging structures.

Regarding the process parameters, the laser power was set at 600 W, the scanning speed was set at 600 mm/min, and the spot diameter was 3 mm. As shown in Figure 22, three blades with different dimensions were formed in different directions of the cylinder. In Figure 22a, the paths were generated with the model boundary offset $-$1.5 mm. In Figure 22b, the path was generated along the model boundary without offset. In Figure 22c, the path was generated with a model boundary offset of 1.5 mm. Based on the measurements at the same location, it can be seen that the difference between their contour dimensions was around 3 mm, which is consistent with the setting of the offset parameter. The experimental result shows that the paths generated by the proposed method allow the LMD process to form overhanging thin-walled blades on the non-planar surface, as well as to freely offset the paths to form contour boundaries that match the expected dimensions of any machining allowance.

4. Conclusions

This paper presents a helical path generation method that aims to assist the LMD process in fabricating thin-walled structures with overhangs of adjustable contour dimensions on a non-planar surface. The method is divided into three main stages, each of which has its own geometrical operations to deal with. In the slicing stage, the curved layering is obtained by intersection computation between triangular meshes to conform to the discrete-stacking feature realized on a non-planar surface. In the contour offset stage, orientation curves are created to constrain the contour to be offset on the surface, and self-intersections and jaggedness are eliminated. In the path generation stage, point coordinates and tool orientations are updated to convert the path from a multilayer discrete contour to a single continuous curve. Finally, three hollow blades with different path parameters are grown on the variable-curvature surface of the cylinder. It can be observed from the experimental results that the paths generated by the method are not only capable of forming unsupported overhanging structures on non-planar surfaces but also capable of meeting the process requirements of different profile sizes according to the given offset parameters.

The essence of the proposed method consists of computational geometry algorithms that need to focus on computational cost and time while solving the problem. Therefore, we will continue to improve the efficiency of the method in the future in order to apply it to more complex and larger part models. In addition, we will study and analyze more thin-walled structures with different geometrical characteristics and improve the method to realize its possibilities.