3D printing path planning algorithm for thin walled and complex devices

2.2.1 Basic thought

The traditional potential field method only considers the optimization of the printing path but ignores the stress of special printing positions such as inflection point and thin-walled edge. As shown in Figure 2, a plane in the printing process is extracted, in which the black arrow represents the print path and the red arrow indicates the stress direction. At the edge of the device, due to the stress warping, the device is easy to deform, resulting in printing failure. Moreover, because only the potential field method is used to print step by step and line by line, the warping is more serious, even leading to the printing failure of the whole device.

Figure 2

Stress deformation diagram.

The main idea of the BPF method is to balance the sudden change of stress direction at the edge of the device and the stress superposition in the process of the device plane printing step by step. To eliminate this adverse effect, it is necessary to change the path direction in real-time according to the stress change trend in the printing process, as shown in Figure 3, to plan the circular path, cross path, and island path, so as to reduce the stress deformation effect.

Figure 3

Planning path to overcome stress deformation. (a) Circular path. (b) Cross path. (c) Island path.

2.2.2 Algorithm derivation

Before 3D printing, assuming that the global path of the printed device is a two-dimensional point set G ( X , Y ) , the current printing location is p now = ( x now , y now ) , the next step is to predict the printing path point as p next = ( x next , y next ) , the optimal trend point as p opt = ( x opt , y opt ) , the available gravitational function U att ( p next ) and repulsive function U rep ( p next ) are as follows:

where d = ( x now − x ) 2 + ( y now − y ) 2 , d next ⁎ = ( x now − x next ) 2 + ( y now − y next ) 2 and ( x , y ) , ( x now , y now ) , ( x next , y next ) ∈ G ( X , Y ) .

The resultant force of potential field function U ( p next ) is obtained as follows:

If there is U ( p next ) ≥ U ( p ) for any p = ( x , y ) ∈ G ( X , Y ) , then the optimal trend point of the next printing path is p opt = p next . Although the printing path can be drawn according to the potential field law, the problem of stress and deformation is not eliminated. Therefore, the region of balanced potential field B ( X , Y ) is introduced to balance the problem of global uneven stress G ( X , Y ) .

Define B ( X , Y ) as a partial region of G ( X , Y ) :

For any printing path L ( X , Y ) ∈ B ( X , Y ) , take continuous two points p 1 = ( x 1 , y 1 ) ∈ L and p 2 = ( x 2 , y 2 ) ∈ L given in the printing process on path L , and define the deformation stress F → p 2 → p 1 of p 2 to p 1 as:

where γ is the deformation force calculation coefficient, × is the cross multiplication symbol, the deformation force F → p 2 → p 1 obtained is the normal direction formed by p 2 p 1 → and L → , as shown in Figure 4.

Figure 4

Force deformation diagram.

It can be concluded that the superimposed deformation force F → L → on the path L is the superposition effect force of the sum of all continuous two points p i = ( x i , y i ) ∈ L and p j = ( x j , y j ) ∈ L :

If F → L → is greater than the threshold value, it may cause stress deformation of the device, even lead to the failure of printing in B ( X , Y ) . When the superposition effect force F → L → is close to the deformation threshold force, the path L → should be truncated to eliminate the superposition effect of stress, that is to balance the stress potential field. According to the different situations encountered in printing, ring path, cross path, and island path are planned respectively to balance the stress potential field.

1. Printing path based on balanced stress potential field

   For a printing path L ( X , Y ) , if the current superimposed effect force F → L → is greater than the set stress threshold, the cross path needs to balance the current stress and reduce stress deformation. The optimal point of the next printing path p opt is as follows:

   (14) p opt = p next , ∣ F → L → ∣ < F thresh L p balance , ∣ F → L → ∣ ≥ F thresh L

   where p balance is the direction point perpendicular to p next . When the algorithm feels that the path may produce stress deformation, it balances the stress deformation effect by changing the printing path direction to a normal direction perpendicular to the original direction.

2. Printing area based on balanced stress potential field

   For a partial area of the device B ( X , Y ) ∈ G ( X , Y ) , the path of the printing area is L 1 , L 2 , L 3 ,… L j , it can be considered that the region is equal to the superposition of multiple paths:

   (15) B ( X , Y ) ≈ { L 1 , L 2 , L 3 , … L j }

   Then the stress superposition of the region F → B is calculated as the superposition stress of each path:

   (16) F → B = F → L 1 → + F → L 2 → + F → L 3 → + ⋯ F → L k → = ∑ k F → L k → = = ξ ∑ k ∑ i , j p i p j → × 1 L k →

   When the superimposed stress of the region F → B is greater than the maximum deformation stress threshold F thresh B , the stress and deformation of a small part of the region may occur. Therefore, when the stress superposition approaches the maximum deformation stress threshold, the island path Θ = { l 1 , l 2 , l 3 , … l n } is re-planned in the region to balance the device deformation caused by regional stress superposition. The printing area B ( X , Y ) is:

   (17) B ( X , Y ) ≈ { L 1 , L 2 , L 3 , … L j } , ∣ F → B ∣ < F thresh B B − Θ , { l 1 , l 2 , l 3 , … l n } } , ∣ F → B ∣ ≥ F thresh B

3. Printing thin-walled edge based on balanced stress potential field

   For thin-walled complex devices, the stress balance method of edge is similar to print path. If F → C is greater than the maximum deformation stress threshold F thresh C of thin-walled edge, circular path is needed to balance the stress. It should be noted that the path C = { p 1 , p 2 , p 3 , … p m } of thin-walled edge needs to contain at least one connecting path to construct a circular path. As shown in Figure 5, when calculating the stress of the thin-walled edge (blue curve), there must be one connecting path at least (red line segment) to make C end to end.

Figure 5

Connecting path of balanced force potential field at the edge of thin wall.

In other words, the path C of the edge can be divided into two cases according to the relationship between F → C and the maximum deformation stress threshold F thresh C :

The BPF method can be used in 3D printing for thin-walled complex devices based on three situations. By calculating the relative situation of superimposed stress and stress threshold, the superimposed stress effect can be cut off, and the printing quality and success rate can be improved.