Tool path correction algorithm for single-point ... - Springer Link

Int J Adv Manuf Technol (2013) 64:1239–1248 DOI 10.1007/s00170-012-4082-7

ORIGINAL ARTICLE

Tool path correction algorithm for single-point incremental forming of sheet metal Zemin Fu & Jianhua Mo & Fei Han & Pan Gong

Received: 8 November 2011 / Accepted: 19 March 2012 / Published online: 4 April 2012 # Springer-Verlag London Limited 2012

Abstract There exists some error between the manufactured part shape and the designed target shape due to springback of this part after forming. To reduce the error, an iterative algorithm of closed-loop control for correcting tool path of the single-point incremental forming, based on Fast Fourier and wavelet transforms, has been developed. Moreover, the data of the springback shapes, after unloading, of the sheet metal parts formed with the trial and corrected tool paths, used for iterative correction of tool path in the algorithm, are obtained with finite element model (FEM) simulation. Then, a truncated pyramid-shaped workpiece, whose average errors are +0.183/−0.175 mm, was made with the corrected tool path after three iterations solved by the above algorithm and simulation data. The results show that the tool path correction algorithm with Fourier and wavelet transforms is reasonable and the means with FEM simulation are effective. It can be taken as a new approach for single-point incremental forming of sheet metal and tool path design. Z. Fu (*) School of Mechanical Engineering, Shanghai Institute of Technology, 201408 Shanghai, People’s Republic of China e-mail: [email protected] J. Mo State Key Laboratory of Material Processing and Die & Mould Technology, Huazhong University of Science and Technology, Wuhan 430074, Hubei, People’s Republic of China F. Han College of Mechanical and Electrical Engineering, North China University of Technology, Beijing 100041, People’s Republic of China P. Gong Department of Mechanical Engineering, Tsinghua University, Beijing 100084, People’s Republic of China

Keywords Springback . Single-point incremental forming . Tool path correction . Close loop control . Wavelet transform . FEM simulation

1 Introduction Sheet metal single-point incremental forming (SPIF) is an innovative, flexible sheet metal-forming technology that uses principles of layered manufacturing. It transforms the complicated geometry information into a series of parameters of two-dimensional layers, and then the plastic deformation is carried out layer-by-layer through the computer numerically controlled movements of a spherical forming tool to get complex-shaped parts [1–5]. During the forming process (Fig. 1), a blank is put on the general mandrel and fastened at its edges by a clamping plate, which can move along the guide posts. The tool deforms the blank into the mandrel and moves along the contour lines until a sheet metal part is formed. However, there was some deviation between this formed part shape and the target part shape because of elastic deformation after unloading, that is commonly known springback. Due to the existence of springback, the precision of products and subsequent assembly operation were severely affected. How to effectively predict and compensate for springback has been the key to precision forming of sheet metal and programming of tool path. An approach to simply reduce springback lies in designing tool path in such a way to compensate for springback. That is, the springback may remain as large as before while the final part shape would closely approximate that of the target part shape. The first step in implementing such a strategy is the accurate prediction of the springback phenomenon. Assuming that springback can be predicted accurately, there still remains a

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Fig. 1 Sheet metal single-point incremental forming process

problem of how to use such results from predictions to solve a suitable tool path to produce a target part shape. That is, the springback predictions allow “forward” analysis of forming and springback, while a “backward” analysis is needed to work from these results back toward an optimized tool path. It is this second step of springback compensation that is mainly studied in this paper. Springback compensation has traditionally been carried out by simple trial-and-error. That is, the springback can gradually be compensated by experimental iterations. This practice would consume high labor cost and much time. Recently, much work has been investigated on how to compensate springback for improving profile accuracy. For example, many compensation algorithms for correcting die surface in stamping forming were developed by combination of finite element model (FEM) simulation and experimental method. Ambrogio [6] proposed a strategy, which is the design of modified trajectories by using a specific routine calculating the discrepancy between the actual coordinates and the ideal ones. In literature [7], focusing on the investigation of the influence of the process parameters on accuracy through a reliable statistical analysis is called response surface statistical model. Ghulam [8] developed a single empirical model for predicting and optimizing profile error in SPIF parts through parameters optimization. In literature [9], several new types of forming strategies are presented that aim to a more homogeneous distribution of material, among which a forming method suitable for computer-aided optimization was identified by finite element analyses. Gan [10] presented a displacement adjustment method (DAM). The concept of the method is to move the surface nodes defining the die surface in the direction opposite to the springback error. Weiher [11, 12] proposed the SDAM developed from DAM to the smoothed mould surfaces. Karafillis [13] reported a backlash compensation method. With this method, the compensation quantity of die surface can be solved by applying forces being contrary to the internal residual stresses of sheet metal parts after forming. However, all of the above methods do not reveal the relationship between the compensation and springback in

Int J Adv Manuf Technol (2013) 64:1239–1248

theory; therefore, when these methods were used in the correction of die surface, more iteration numbers of experiments were needed in order to get ideal results, which made the rate of iteration convergence be very slow. Cheng [14] summarized the above two methods and put forward an accelerated springback compensation algorithm. Although this algorithm can help make the convergence faster, the algorithm is prone to getting into local extreme and not achieving higher global accuracy. Therefore, the algorithm has certain problems that there exist errors in the accuracy of springback compensation. To solve this, the springback data obtained by experimental method are employed to correct die surface. For example, Webb [15] presented DTFM method. In this method, the transfer function between testing mould shapes and the stamping workpiece shapes was used to calculate the correction value of mould surface. Its Fourier transform is a global transform, with which the local information might be removed when processing the nonlinear signal such as springback. Therefore, there is an intrinsical error in this method. With this method, complex surfaces are described and the minor details might be rounded. But wavelet transfer, which has good time–frequency characters and suitable to nonlinear signal processing, could be utilized to solve this problem. Therefore, this study attempts to develop a closed-loop control algorithm. The algorithm, based on the combination of Fourier and wavelet transforms, is used to correct tool path of sheet metal singlepoint incremental forming, and the usefulness, convergence, and accuracy of this algorithm are tested. This paper presents four closed-loop control system models. Using these models, an algorithm based on Fourier and wavelet transforms is developed for correcting tool path. Then, the algorithm is verified by test results obtained with FEM simulation. Finally, a truncated pyramid-shaped workpiece with 140×140×50 mm is made with the third corrected tool path solved by this algorithm and further prove the validity and practicality of the algorithm.

2 Closed-loop control model 2.1 Modeling and processing Combined with the incremental forming process features, a closed-loop control system model for tool path correction is presented in Fig. 2. The tool path of incremental forming process in the model will be adjusted to get the desired final part shape. The adjusting procedure is as follows: shape of a formed part is measured after unloading, then the measured data as a feedback quantity are used to correct the tool path in the next forming process, and iterations will continue until the target part shape is attained within a specified tolerance. Where p* is the target workpiece shape, p is


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Fig. 2 The closed-loop control system model for tool path correction

measurement results with reverse engineering, e is the error between p and p*, d is the tool path envelope surface shape, p is the formed workpiece shape, and Γ is the forming parameters. It is clear that the process of single-point incremental forming is in essence a discrete process. This process can also be simplified with minor incremental linear. Thus, the closed-loop control system in Fig. 2 can be considered as a minor incremental linear discrete system. In addition, the input and output in closed-loop control system are all for discrete signal data of workpiece surface after sampling. Consequently, the model in Fig. 2 can be expressed with incremental model in Fig. 3. Where, p* is the target workpiece shape, di is the ith tool path envelope surface shape, pi is the ith formed workpiece shape, Δdi ¼ di di1 is the correction value for tool path envelope surface shape, Δpi ¼ pi pi1 is the variable quantity of the formed workpiece shape, ei ¼ p pi is the shape error, gp is the transfer function matrix for the incremental forming process, and gc is the control matrix. Due to the ith tool path envelope, surface shape di and formed workpiece shape pi are unknown before the ith forming process, which makes their feedback be only replaced by di−1 and pi−1 from the last results. Therefore, Fig. 3 Incremental model of closed-loop control system for tool path correction

the delay of z−1 needs to be added in the incremental model (Fig. 3). 2.2 Transfer function 2.2.1 Transfer function for incremental forming process The single-point incremental forming in Fig. 2 is related to the factors such as the envelope surface shapes of tool paths, processing parameters, and blank materials. After determining workpiece material, the formed workpiece shapes (p) is the function of the tool path envelope surface shapes (d) and process parameters (Γ). The function can be expressed as: p ¼ f ðxÞ

ð1Þ

where, x ¼ ðd; Γ Þis some state point of the forming process. It is assumed that x0 ¼ ðd0 ; Γ 0 Þ is the initial state. The envelope surface shape of the corrected tool path could be represented as d0 þ Δd . As the correction value of Δd is very small in springback compensation, if the forming process parameters Γ0 is a constant, a state point ðd0 þ Δd; Γ 0 Þ certainly exists around the original state point x0, which could be used just right to compensate for the dimensional

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error caused by springback. The formula (1) could be expressed as Taylor expansion at the initial state point x0.

The characteristic equation of this system can be solved with formula (7), which is detailed in Section 2.3.

p ¼ f ðx0 Þ þ f ðx0 Þ0 ðx x0 Þ þ ½f ðx0 Þ0 0 =2!ðx x0 Þ2

2.3 Closed-loop stability

þ þ ½f ðx0 ÞðnÞ =n!ðx x0 Þn þ Rn ðxÞ

ð2Þ

Where Rn ðxÞ¼½ f ðsxÞðnþ1Þ=ðn þ 1Þ!ðx x0 Þðnþ1Þ 0 < s < 1 The over two-order polynomials in the formula (2) could be neglected for their values are close to 0. Then, the formula (2) can be simplified into formula (3). p ¼ p0 þ

@p j ðd d0 Þ @d d¼d0

i:e:; Δp ¼ gp Δd

ð3Þ ð4Þ

@p Where gp ¼ @d jd¼d0 . It means that the minor change of the tool path envelope surface shapes is linear to that of the workpiece shapes when the forming parameters are unchanged, which will be used in the solution of tool path correction algorithm in Section 3.2.

2.2.2 Transfer function for control system The overall gain (k) of the control system (Fig. 3) is defined as k ¼ gc gp :

ð5Þ

The open-loop transfer function of the control system in Fig. 3 is GðsÞ ¼

gc g p z pi kz ¼ : ¼ zðz 1Þ zðz 1Þ p

The stability limits of this system can be investigated by the use of the discrete-time root-locus [16]. The discrete-time root-locus of a system, also called as the locus of the roots of its characteristic equation, relies on the z-plane representation of the stable region of the system. And the stability of a system is determined by the location of the roots of its characteristic equation. Therefore, the closed-loop stability can be solved by the root-locus of a system in z-plane. According to formula (7) in Section 2.2.2, the characteristic equation of this closed-loop control system can be expressed as: DðzÞ ¼ z 1 þ k ¼ 0:

ð8Þ

When k changes from 0 to ∞, the locus of the roots of the characteristic equation in z-plane, namely the root-locus of the system in z-plane, can be obtained. It is shown in Fig. 5. As shown in Fig. 5, the root-locus is in the unit circle when k is in the range of 0–2, and the module of characteristic roots is smaller than 1. Under this condition, the linear discrete system is stable. The stable region in z-plane, along with constant damping and frequency lines, is shown in Fig. 5. When k01, the response of the system is the fastest, which can ensure stability, accuracy, and rapidity of the system. This makes the formed workpiece shape converge to the target workpiece shape at the fastest speed in the closed-loop control model.

ð6Þ 3 Tool path correction algorithm

Therefore, the control system in Fig. 3 can be simplified into a unit feedback control system in Fig. 4. In order to facilitate analysis of the closed-loop stability in Section 2.3, R(z), C(z), and E(z) are specified as the input, output, and error signal sequences of the Z-transform in Fig. 4, respectively. The closed-loop transfer function of the control system in Fig. 4 is described by

ΦðzÞ ¼

CðzÞ k ¼ : RðzÞ z 1 þ k

ð7Þ

Fig. 4 The simplified model of the closed-loop control system for tool path correction

3.1 Fourier and wavelet transforms The input and output signals in the closed-loop control system are all for 3D shape data of the tool path envelope surface and formed workpieces. Because the free curved surface can hardly be presented with analytic equation, Webb, etc. put forward a processing method using Fourier transform–inverse Fourier transform. The discrete data of curved surface could be processed with Fourier transfer, while the die surface could be solved with the inverse Fourier transform. But, the local information of processing signal was easily removed and the detailed feature of surface shape was easily neglected with Fourier transform method. It is because Fourier transfer is a global transform. This makes the optimum solution of the system be hardly solved with Fourier transform method. However, these disadvantages of Fourier transform method can be overcome with wavelet transform method for its good locality characteristic in time–frequency domain and suitability in nonlinear signal processing [17]. Because wavelet transform is developed from Fourier transform, it cannot replace


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Fig. 5 The root-locus of the closed-loop control system for tool path correction

Fourier transform. In addition, the frequency domain might be roughly divided with discrete wavelet transfer. After transfer, the frequency spectrum information cannot be obtained directly. Therefore, we should adopt wavelet transfer method on the basis of Fourier transform. Frequency domain wavelet transform, based on frequency domain Fourier transform, could be solved by their combination. With this method, the time domain expressions of Fourier for the shape data of the tool path envelope surfaces and the formed workpieces profile would be converted into the frequency domain expressions of wavelet for the same data to solve the transfer function gp which is detailed in Section 3.2.

be obtained from formula (5) in Section 2.2.2. k ¼ gc gp ¼ 1; or gc ¼ gp1

It is defined that tool path envelope surface shapes are d1 and d2 in two open-loop-forming experiments. Correspondingly, the springback shapes of the workpieces formed with d1 and d2 after unloading are p1 and p2. Then Δd ¼ d2 d1 and Δp ¼ p2 p1 . According to formula (4) in Section 2.2.1, when Δd is very small, the transfer function ^ matrix (gp) could be replaced approximately with gp which is obtained through two open-loop identification experiments. That is: ^

Δ p p2 p1 ¼ : Δ d d2 d1

3.2 Tool path correction algorithm with Fourier and wavelet transforms

gp ¼ gp ¼

According to analysis in Section 2.3, the overall gain of the closed-loop system (k) equals 1. Therefore, formula (9) can

Therefore; gc ¼ gp1 ¼

Fig. 6 The closed-loop control system model for tool path correction with wavelet transform method based on Fourier transfer

ð9Þ

d2 d1 : p2 p1

ð10Þ ð11Þ

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Table 1 Properties of blank for SPIF simulations Item

Value

Material Yield strength/MPa Ultimate tensile strength/MPa Ultimate elongation/% Elastic modulus/MPa Density/(g cm−2) Poisson ratio Strain-hardening exponent

08Al 175 303 44 2.1×105 7.83 0.30 0.227

Anisotropy coefficients, r

r0 01.71, r45 01.14, r90 01.83

Fig. 8 SPIF process parameters

Formula (13) can be arranged into

In order to solve transfer function gp, d1, d2, p1, and p2 in the function need to be converted into their parametric form in the spatial frequency domain with wavelet transfer method based on Fourier transform. It is assumed that the capital letters are taken as the frequency domain expressions of wavelet of their small letters. So, the wavelet transformation form of the formula (11) may be written as: ^

Gc ¼ Gp 1 ¼ Gp 1 ¼

ΔD D 2 D 1 ¼ : ΔP P2 P1

ð12Þ

Thus, the incremental model in Fig. 3 can be changed into the model in Fig. 6 by system identification and wavelet transform based on Fourier transform. As can be seen from Fig. 6, the relationship between the control matrix gc and transfer function matrix gp of the closed-loop system is established with the system identification. WF2 is the wavelet transfer based on Fourier transfer, and WF2−1 is the inverse wavelet transform. It is assigned that tool path envelope surface shape is d* in first closed-loop forming, namely first correction experiment. Correspondingly, the springback shape of the workpiece formed with d* after unloading is the target workpiece shape p*. Then, Δd 0 ¼ d d1 and Δp0 ¼ p p1 . According to formula (12), formula (13) can also be obtained. Gc ¼

ΔD ΔD0 D2 D1 D D1 ¼ ¼ ¼ ΔP ΔP 0 P2 P1 P P1

ð13Þ

D ¼ D1 þ

ðD2 D1 Þ ðP P1 Þ: ðP2 P1 Þ

ð14Þ

Formula (14) is the tool path correction algorithm. Inverse wavelet transform of D* in the algorithm is the discrete data point set of the tool path envelope surface. In the real production process, multi-iteration experiments are needed to correct the tool path for achieving the target part shape within a specified tolerance. It is assumed that the number of iteration is i. The (Gp)i could be replaced with (Gp)i−1 when ΔDi is very small. And the initial value of Gp could be derived from formula (12). Therefore, the iterative correction algorithm of tool path can be expressed as: ^

Diþ1 ¼ Di þ ðP Pi Þ ðGp Þ1 i

ð15Þ ^

That is : Diþ1 ¼ Di þ ðP Pi Þ Gp 1 :

ð16Þ

4 Verification of the tool path correction algorithm In order to test the usefulness, convergence, and accuracy of the tool path correction algorithm, three-dimensional shape of one truncated pyramid workpiece (Fig. 11a) was subjected to tool path correction design using this algorithm. First, two tool path envelope surface shapes (d1 and d2), which have the similar profile with minor changes, were designed for two open-loop-forming experiments. The springback shapes of two workpieces (p1 and p2) formed with d1 and d2 after unloading were obtained by FEM simulation. Then, the tool path envelope surface shape for forming the target workpiece was exactly solved with the iterative correction algorithm after just three iterations. In Table 2 Process parameter values utilized for simulations

Fig. 7 Stress–plastic strain curve of 08Al blank

Process parameter

Value

Tool depth step (△z)/mm (layer distance) Tool diameter (D)/mm Tool speed (mm/s)

0.75 10 200


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4.1.1 Simulation algorithm

Fig. 9 FEM model for SPIF process

iterative correction process of tool path, the springback shapes, after unloading, of sheet metal workpieces formed with the corrected tool paths, compared with the target workpiece shape every time, were also obtained by simulation. Finally, a truncated pyramid workpiece was manufactured with the final corrected tool path at the WMSC-800× 500/500 single-point incremental forming machine developed at Huazhong University of Science and Technology. The formed workpiece shape was then measured using Atos-II three-dimensional laser measuring system. And the results showed that shape error of the workpiece was within a specified tolerance. 4.1 FEM simulation for the forming and springback of the test workpieces

The sheet metal SPIF is an extremely complicated physical process. It is a high nonlinear problem containing geometric nonlinear, material nonlinear, and contact nonlinear. To solve this, the explicit algorithm ABAQUS/Explicit module fitted for dynamic and nonlinear analysis could be used to simulate the sheet metal-forming process, and the implicit algorithm ABAQUS/standard module adapted to static and steady analysis to simulate the springback process. Due to the forming result of any time during the ABAQUS/Explicit module running could be treated as initial condition run in the ABAQUS/Standard module for further calculating and analyzing springback result, and the springback result of any time during the ABAQUS/Standard module running could be treated as initial condition run in the ABAQUS/Explicit module for further calculating and analyzing the next step forming result [18]. Therefore, ABAQUS software function is very suitable for multiple-working-step singlepoint incremental forming process. We integrated ABAQUS/ Explicit and ABAQUS/Standard to carry out the mixed operations. This method can make the nonlinear behavior of each working step in multiple-working-step incremental forming be solved with higher precision and faster convergence, compared with algorithms of other FEM software. 4.1.2 Simulation setup

To reduce cost and save time, the forming and springback of test workpieces were replaced with numerical simulation. Therefore, the traditional experimental iteration method for compensating springback could be replaced with the improved numerical simulation iteration method to rapidly optimize tool path of workpiece-forming. Fig. 10 Truncated pyramid workpiece formed with the third corrected tool path by simulation

Material and process parameters The sheet metal with 300× 300×1-mm dimensions are made from 08AL (Table 1) whose property curve of the stress–plastic strain is shown in Fig. 7. The SPIF process parameters can be observed in Fig. 8, and the process parameter values are listed in Table 2.

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Finite element model A three-dimensional finite element model (Fig. 9) was established for the single-point incremental forming process. The sheet metal was meshed with 14,400 elements (ABAQUS type CPS4R, namely the fournode bilinear plane stress quadrilateral, reduced integration, and hourglass control shell elements) with five integration points through the thickness. Its material was assumed to be planar anisotropic following Hill's 1948 yield criterion with kinematic hardening. The tool, supporting die, clamping plate, and backing plate were modeled as rigid surfaces. Coulomb friction principle was applied with a friction coefficient of 0.05 between the blank and the tool and of 0.15 between the blank and the supporting die. The contact condition was implemented through a pure master–slave contact-searching algorithm and penalty contact force algorithm. 4.2 Tool path iterative corrections based on the simulation results Firstly, the registration for the element nodes of the models of the truncated pyramid workpieces simulated using finite elements and their corresponding models of the tool path envelope surfaces as well as the target workpiece model was obtained with Geomagic Qualify in the same coordinate system. Through the discretization of these models, their respective points cloud could be gained in registration condition. All of the points cloud data and element nodes data were then loaded into the Matlab program. The griddata function in the program could be used to interpolate to their respective data, so that regular meshes and points cloud data of these models can be obtained. Secondly, the tool path correction algorithm, based on the frequency domain expressions of wavelet for the 3D shapes data of all these models, could be used to solve the meshes and points cloud of the tool path envelope surface after correction. Here, all procedures were performed in Matlab program. Thirdly, the tool path envelope surface after correction could be obtained by surface reconstruction from its discrete points cloud. This surface, processed with low pass filter, was loaded into UG software, and a tool path first corrected by first closed-loop experiment was solved. Finally, the workpiece formed with the tool path was measured, and then the errors between the formed workpiece shape and target workpiece shape were evaluated. When the errors are within specification, the tool path is desired. Otherwise, tool path needs to be modified again using closed-loop experiment until the springback errors meet the need of precision. In iterative correction process of the tool path, each time their correction process will need to go through all four steps explained above. Figure 10 shows the truncated pyramid workpiece formed with the third corrected tool path by simulation. The workpiece (Fig. 11a) is symmetric with respect to the

Fig. 11 Experimental validation of the tool path correction algorithm. a Target workpiece shape. b Tool path envelope surface and workpiece profiles along the X–Z symmetry plane. c Convergence behavior of tool path correction


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plane can be used to reflect forming results of the entire workpiece. The tool path envelope surface after three tool path correction iterations and the simulated workpiece (Fig. 10) and target workpiece profiles along X–Z symmetry plane are shown in Fig. 11b. Obviously, the springback profile along the symmetry plane is nearly identical to its target after just three iterations. Figure 11c shows the normalized springback error for the entire part of tool path correction iteration. It is defined as the root mean square (RMS) shape error for the ith tool path correction cycle over that of the initial cycle, where RMS shape error is calculated as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN 2 k¼1 Δzk ; ð17Þ N in which Δzk is the shape difference in the z direction for the kth node and N is the number of nodes. As shown in Fig. 11b, it takes only three cycles of tool path correction iteration to find an appropriate tool path envelope surface shape to minimize springback error. As can be seen from Fig. 11c, the convergence rate is very fast. After just three cycles, the algorithm converges to an accurate solution and reduces springback error to an acceptable value. 4.3 Experimental verification Fig. 12 The experimental workpiece. a Truncated pyramid workpiece formed at the WMSC-800×500/500 single-point incremental forming machine. b Truncated pyramid workpiece after processing at electric spark linear cutting machine

X–Z and Y–Z planes, and profiles along the two symmetry planes are identical, so only one half along X–Z symmetry Fig. 13 Matching results of measured points cloud from the manufactured workpiece and CAD model

A truncated pyramid workpiece (Fig. 12a) was formed at the WMSC-800 × 500/500 single-point incremental forming machine with the third corrected tool path. The workpiece after processing at electric spark linear cutting machine is shown in Fig. 12b.

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The surface profile of the workpiece was measured using Atos-II three-dimensional laser measuring system, and 3D point cloud model could be obtained. The 3D point cloud model was matched with CAD model with Geomagic Qualify software, and the results showed that the average dimensional errors of the workpiece were +0.183/−0.175 mm (Fig. 13), which can completely meet the requirements of workpiece form and position accuracy.

5 Conclusions An algorithm for the tool path correction of single-point incremental forming of sheet metal to produce a specified part shape, taking springback into account, has been developed. The tool path correction algorithm, based on wavelet and Fast Fourier transforms, is an iterative technique based on comparing a target workpiece shape with a formed-andunloaded workpiece shape simulated using finite elements. A truncated pyramid workpiece with high accuracy was manufactured with the algorithm. From the foregoing discussion, the following conclusions can be made: 1. The closed-loop control model for the incremental forming makes the tool path correction algorithm converge faster. 2. The tool path correction algorithm, based on Fast Fourier and wavelet transforms, provides the optimum parameters for tooling design as well as tool path planning in singlepoint incremental forming, which speeds up the progress of mould design and improves production efficiency. 3. The tool path correction algorithm has been confirmed by experiment to produce accurate tool path envelope surface shape and final part shape with few iterations, which demonstrates that the finite element method exhibits higher accuracy and faster convergence in the simulation of sheet metal single-point incremental forming process. 4. The manufacture results show that the methods of closed-loop control, tool path correction, and FEM simulation for single-point incremental forming of sheet metal are reasonable and effective. Acknowledgments This work was supported by Shanghai Institute of Technology Scientific Research Project of Talent Introduction, under YJ 2011-20; Shanghai Leading Academic Discipline Project, under J51501; and the National Nature Science Fund of People's Republic of China, under 51105256/E050301.


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