Research on trajectory planning of non-developable surface for automated tape placement processing of composite

1 Introduction

Advanced composite materials are famous for their light weight with high specific strength and high specific modulus. They also own the merits of integration for structure and function and easy formation of large components. With the growing use of composite materials in various kinds of large military transport planes and commercial aircrafts, automated tape placement (ATP) processing technology has been increasingly applied in a wider range [1, 2]. As a substitution for hand lay-up formation process of the prepreg tapes, ATP technology is used to lay the tape onto a molded surface precisely according to the designed structure using numerical control method [3, 4]. Moreover, it not only substantially improves the productivity of composite components and reduces the cost of production significantly but also raises the reliability and stability of the composite structure by an accurate measurement and precise control on the tape cutting quality and heating temperature, as well as the trajectory planning of the whole process [5, 6]. The tape placement process mainly serves for the fuselage and wing panel of the aero-planes characterized by small curvature and large size, and most of these components surfaces are non-developable. To ensure the tape lay-up quality, the ideal laying track should be in accord with the natural path (NP), which could lead the tape naturally stretched onto the mold, without defects as folding or tensile fracture [7, 8].

The trajectory planning problem is one of the key components for ATP technology. Developed countries, however, have different interpretation for its core algorithm and related software. Shinno and Shigematsu [9] proposed a directional control algorithm by dividing the curved surface into discrete points by a mesh grid in the first step. With the given initial point and direction, the iterative algorithm is established by solving the geodesic of the prepreg tape edge. Lewis and Romero [10] put forward a projection approximation method in which the initial point and laying angle is obtained in the 2-deimensional plane and projected into the curved surface. Then multiple iterative projection is applied to obtain the approximate solution. Meanwhile, Vistagy Company proposed FiberSIM software and Tape Laying Interface Module, which is developed under the CATIA environment [11, 12]. However, the details of this software are unknown, and its reliability is doubtable. The study on trajectory planning on automated tape laying is still in its initial stage domestically. Cui-ling et al. and Jian-feng et al. [13, 14] proved the NP is in conformity with the geodesic line in the developable surface. Sheng et al. [15] proposed the theoretical model and analytical equation of the geodesic line on non-developable surface. Hai-yan et al. [16] adopted the STL meshgrid method to turn the general surface into millions of triangle units and established core algorithm to solve the NP based on the method of Shinno and Shigematsu and Lewis and Romero. However, this algorithm was not designed on the basis of prepreg laying process and mechanical design, which could lead to the wrinkle and fracture of the tapes caused by stress concentration. Additionally, Li-Wei et al. [17] designed the tape-laying model applied to the elliptical cylinder, but it is only an analytical mathematical model, which is not usable for computer numerical calculations. Moreover, the actual mandrel provided in the paper is a pure cylinder, which surface is developable.

In this paper, the concept of UD is introduced to quantify the stretch and wrinkle extension of the prepreg tapes, and the existing algorithm is ameliorated in order to solve the stress defect problem during the laying process. To optimize the NP, the “variable step” and the “variable-angle” algorithms are presented to reduce and avoid the stress concentration problem of the tape.

2 Description of the NP problem

The theory of NP is proposed based upon the laying process of the prepreg tapes, while there was not a precise and quantified definition for it. The NP algorithm is designed just to meet the demand for “laying the tapes onto the surface naturally, with minimum deformation and without any stretch and fracture defect,” as it is shown in Figure 1.

Figure 1:

Analysis of the stress-strain for prepreg tape.

The tape-laying technology is derived from the winding process, and the NP theory is the problem extension of the geodesic lines. The geodesic line is a curve on the surface, and its geodesic curvature is constantly equal to 0. It also equals the shortest path between 2 random points on the surface, and you can wind materials along this curve onto the model, avoiding skidding condition. Owing to the different processing procedures between tape-laying and filament winding, the 2 trajectory planning algorithms should be distinguishable. If the model surface is flat or developable, the laying and winding processes can both take on geodesic curves, and the NP equals geodesic ones. However, if the laying surface is a bit more complicated and non-developable, laying tapes on the geodesic lines could cause great deformation on the left and right edges. Consequently, this will lead to fold accumulations and tensile fractures of the tape edges and cause great trouble for the laying component due to the rigidity constraints of the tape width and curvature mutation. The NP theory is adopted to adjust the tape deformation properly based on a geodesic curve, in order to diminish the deformation and eliminate it partially [18, 19].

3 Introduction of UD

The geodesic can only solve the single curve trajectory problem, while NP is designed to fit the developed surface of the wide tape onto the model surface. The algorithm complexity of the 2 methods differs from each other. If you want to enhance the calculation precision of the geodesic, the iteration points during the process should be increased to a greater extent. But for the NP algorithm concerned on the precision control problem, not only the mechanism of longitudinal fibers and traverse tape width should be taken into consideration but the surface stretch and shear stress of the whole prepreg tapes should also be treated as important factors during the laying process [20, 21].

The existing NP algorithm gives fundemental solutions and implementation steps to solve stress concentration problem during tape-laying procedure and uses muti-point control and fitting methods to raise the computing accuracy, which ensures the feasiblity of the solution. Nevertheless, this algorithm has not analyzed deformation and stress quantitatively during the laying process, needless to say, to find the optimized trajectory points for the most NP. Considering the specificity of the tape-laying process, the UD definition is proposed to characterize stress and strain distribution property.

Suppose in a spatial curved surface ∑, the corresponding equation is r=r(u¹, u²), and a geodesic curve C₁ is on the surface ∑, which parametric equation is established as equation (1)

In equation (1) above, s is the arc length of C₁. Suppose a point P is on C₁ and v→ is the tangential vector of point P on curve C₁. A plane is created by the vector v→, and to form intersecting line L₁ with surface ∑, by intercepting points P₁ and P₂ on line L₁, it is ensured that PP1¯=PP2¯=w/2, in which PP1¯ and PP2¯ is the arc length of the intersecting line. Suppose another point P′ is on C₁ and also v→′ is the tangential vector, getting points P′1 and P′2 in the same way, and P′P′1¯=P′P′2¯=w/2. Based on the setting condition above, the arc length between the 2 points PP′¯=δ,P1P′1¯=δ,1P2P′2¯=δ2 along geodesic curves on surface ∑ can be figured out. Thereafter, the definition of UD could be given as follows:

Def. 1: The linearity of UD is defined as shown in Figure 2:

Figure 2:

Definitions of un-natural degree.

Given the angle α between the tangential vector of curve C₁ and P₁P₂ at point P, angle α₁ between P1P′1 and P₁P₂ at point P₁ and angle α₂ between P2P′2 and P₁P₂ at point P₂ could also be measured. The following definition could be provided below:

Def. 2: The angle of UD is defined as

Def. 3: The overall UD is defined as

In the equation above, λ₁ and λ₂ are the proportional weighting coefficients.

In Figure 2 above, the existing stress in the prepreg tape causes UD, while this pheonomenon could be characterized as stretching and compressing strain, as well as shear strain from the tape edge to the middle line. The stretching and compressing strain could be dipicted as |δ₁-δ₂| in the figure above; also, the shear strain could be represented as included angle difference |α₁-α₂| of the tape. The overall definition of UD includes UD linearity μ₁ and UD angle μ₂, which could quantify the deformation by initial step δ and initial angle α accurately.

4 Amelioration of NP algorithm

Based on the definition of UD, the quantified numerical value of the stress and strain created during the tape-laying process in general on a non-developable surface could be obtained. By adjusting the key parameter of the iteration step and laying angle in the NP algorithm, the optimized stress distribution of the tape on the laying surface will be achieved in order to figure out the best NP.

The main idea of the “variable step-angle” algorithm is to weaken the stress and strain distribution condition of the laid tapes by adjusting the searching steps and laying angle along geodesic curves. During the NP searching procedure, UD linearity μ₁ and UD angle μ₂ would be changed to ensure μ≤μ₀ to meet the technological requirement of the tape-laying process. The algorithm flow chart is shown in Figure 3 as follows:

1. To mesh, generate surface ∑ into a plurality of discrete plane unit and extract planar vertex matrix information.

2. Set the starting point P and searching angle α, and expand along the normal value of α by the distance of ω/2 on the surface ∑ to acquire initial points P₁ and P₂ with corresponding angles α₁ and α₂.

3. Start from the initial points P₁ and P₂ and searching angles α₁ and α₂, and use the geodesic initial value algorithm to explore the iterative points P′1 and P′2 by step length δ along the left and right edges.

4. Figure out the geodesic curve C₂ between the points P′1 and P′2 according to geodesic boundary value algorithm, and get the middle point P′ on curve C₂. The corresponding tangential angle α′ is obtained by linking points P and P′.

5. Create the plane Π by normal vector α′, which intersects with surface ∑ by line L′ Points P′1 and P′2 could be intercepted from point P′ by the distance of ω/2 on the intersecting curve L′. Calculate the shortest path δ₁ and δ₂ of P1P′1 and P2P′2 on the surface and angle deformation valueα′1 and α′2 simultaneously.

6. Figure out UD linearity μ₁ and UD angle μ₂ by using equations (2) and (3), and overall UD μ could be settled, in which λ₁ and λ₂ are determined according to the laying condition.

7. If μ≥μ₀, compare the UD linearity μ₁ and UD angle μ₂. If μ₁≥μ₂, adjust step length δ and make δ′=εδ while ε∈(0, 1), whereas adjust the laying angle α, make α′=ε1α′1+ε2α′2 with ε₁/ε₂=(α₂-α₁)/(α-α₁), in which ε₁+ε₂=1 and ε₁ε₂∈(0, 1) and jump to step (3). If the condition μ≤μ₀ is reached, execute sequential step (8).

8. Repeat the steps from (3) to (7), and a series of center track points for tape-laying coordinate could be obtained, such as P, P′, P″ … P′⁽ⁿ⁾, which is optimized by the “variable step-angle” algorithm and fitting for the tape-laying process.

Figure 3:

The flowchart of angle shifting algorithm.

5 Simulation of NP algorithm

On the basis of NP algorithm calculation steps as shown in the above figure, a mathematical model of the hyperbolic function is created. Equation function (5) is described below:

In equation (5), we set a=30, b=25, c=20. In the Matlab environment, this surface is created and the meshgrid function is applied to subdivide it into small planes, as shown in Figure 4.

Figure 4:

Non-developed surface and trajectory planning of NP.

5.1 Simulation of “variable step-angle” algorithm

Given the intial point P=P₀(x₀, y₀) and initial angle V₀=θ₀, one NP track could be figured out with the step length of 3. Then according to the definition of UD in the optimized NP algorithm, verify the unnatural degree (UD) of the control point in the track. By setting μ=μ₀ as the error convergence condition and adjusting the step length and the laying angle, the following comparison data could be reached, as shown in Figure 5.

Figure 5:

The optimization effect of “variable step-angle” algorithm.

In Figure 5 above, adjusting the step length is more effectable to reduce the UD linearity. In the global area, the maximum value of UD linearity amounts to 0.24% as the step length is set to 3 units, but when the step length is shortened to 1.5 units, the UD linearity shrinks severly to 0.1%. However, it is not obvious for adjusting the UD angle. Although the maximum value of UD angle could be reduced from 0.03% to 0.015% correspondingly, it causes great fluctuation and the system is unstable. Thus, the “variable-step” method is more effectable than the “variable-anglegly, it causes great fluhe NP track in a smooth curvature surface. Compared with the UD angle, the UD linearity accounts for a greater effect and takes the main role in the algorithm. Generally, the “variable-step” method is used to optimize the global area while the “variable-angle” method is used to adjust the part area of the whole laying surface. Therefore, the “variable-step” method could be adopted to adjust UD to the proper range to meet the fundamental demand for NP trajectory planning, and the abscissa is the X-axial coordinate of the iteration points on the laying track.

5.2 Comparison of amelioration effect for reduced surface curvature

To take further validation on the optimization results for the “variable step-angle” algorithm, the surface curvature is reduced. In equation (5), we reset c=10 and c=15, and the corresponding track could be calculated, as shown in Figure 6.

Figure 6:

Trajectory planning of NP on smaller curvature surface. (A) c=10, (B) c=15.

By following the “variable step-angle” algorithm orders, re-adjust the step length and laying angle; the adjusted results of the UD could be achieved as shown in Figure 7.

Figure 7:

Optimization effect of “variable step-angle” algorithm on smaller curvature surface. (A) Comparison of UD linearity (c=10). (B) Comparison of UD angle (c=10). (C) Comparison of UD linearity (c=15). (D) Comparison of UD angle (c=15).

From Figure 7, we can see that with smaller surface curvature, changing step length is still effectible to control the UD linearity. With the step length shortened from 3 to 2 units, the UD linearity is reduced by 30%~40% approximately, which still raises steadily with the increasing curvature. However, changing the laying angle is not effected obviously to control the UD angle, and the error is randomly distributed.

5.3 Comparison of amelioration effect for enlarged surface curvature

Change the curvature parameter again to enlarge the surface curvature, and we set c=25 and c=30, respectively. The following result can be reached in Figure 8.

Figure 8:

NP trajectory planning on larger curvature surface. (A) c=25, (B) c=30.

By taking the “variable step-angle” calculation steps, modulate the step length and laying angle, and the similar results of UD on larger curvature surface can be reached, as shown in Figure 9.

Figure 9:

Optimization effect of “variable step-angle” algorithm on larger curvature surface. (A) Comparison of UD linearity (c=25). (B) Comparison of UD angle (c=25). (C) Comparison of UD linearity (c=30). (D) Comparison of UD angle (c=30).

Owing to the larger variation of surface curvature, the mutation of the surface is more obvious, which leads to a different effect for the “variable step-angle” algorithm. The overall distribution of UD on the surface causes greater fluctuation. The maximum value of UD linearity reaches 0.6% by the step length of 3 units, but this number still holds to 0.55% while the step length is shortened to 1.5 units. This is because of the high elastic modulus along the longitudinal direction of the fibers in the tape, which seriously limited the tensile strain capacity in the fiber axial direction. Consequently, the main form for the tape deformation is wrinkling and buckling created in the compression of fiber along the longitudinal direction, which is also limited by the tape width. Thereafter, for the surface of larger curvature, usually, it is to diminish the laying tape width to weaken the UD and meet the basic requirement for NP algorithm.

6 Conclusion

In order to solve the problems of trajectory planning on non-developable surface for ATP process, on the basis of existing NP algorithm, 3 points can be concluded:

1. The theoretical model of the UD is presented, and the UD linearity and the UD angle are characterized to quantify tensile and shear strain during the tape-laying process.

2. The UD can be controlled and reduced by adjusting the iteration steps and fine tuning the laying angle of the NP algorithm in order to weaken the partial stress and to achieve the goal of trajectory optimization.

3. The simulation model on non-developable surface is established to verify the adjusting effect of the “variable step-angle” algorithm. The results show that on the surface of smaller curvature, the “variable-step” method is more effectible than the “variable-angle” method to optimize the NP tracks, while on the surface of larger curvature, the overall distribution of the UD on the surface causes greater fluctuation. In general, decreasing the laying tape width can weaken the UD and meet the basic requirement for NP algorithm.