process controller design for sheet metal forming

1、Proceedings of the American Control Conference San Diego, Cahfornla June 1999 Process Controller Design for Sheet Metal Forming Cheng-Wei Hsu and A. Galip Ulsoy Mechanical Engineering and Applied Mechanics Universi

2、ty of Michigan Ann Arbor, MI 48109-2125, USA Mahmoud Y. Demeri Ford Research Laboratories Ford Motor Company Dearbom, MI 48121, USA Abstract In-process adjustment of the blank holder force can lead to higher formabi

3、lity and accuracy, and better part consistency. There are many studies on the application of process con- trol to sheet metal forming. However, process controller de- sign has not been thoroughly addressed, and is stu

4、died in this paper. A constant gain proportional plus integral (PI) con- troller with approximate inverse dynamics will be presented to achieve small tracking error regardless of model uncer- tainty and disturbances.

5、 1 Introduction Sheet metal stamping is an important manufacturing process because of its high speed and low cost for mass production. Figure 1 shows a schematic of a simplified stamping process. Holder F i g u r

6、e1 :Schematic of a stamping process The basic components are a punch, and a set of blank holders which may include drawbeads. The punch draws the blank to form the shape while the blank holder holds the blank to cont

7、rol the flow of metal into the die cavity. Some process variables are also shown: Fp is the punch force, F bis the blank holder force, and F, is the restraining force within the blank. The good quality (i.e., no tear

8、ing, no wrinkling, and high di- mensional accuracy) of stamped parts is critical in avoiding problems in assembly and in the final product performance. Consistency (i.e., dimensional variations between parts) in t

9、he stamping process also significantly affects subsequent assem- bly in mass production. New challenges emerge from the use of new materials. For example, lightweight materials (e.g., aluminum) are essential for

10、 reduction of car weight to achieve high fuel economy. However, aluminum has reduced forma- bility and produces more springback [l, 21. The control of flow of material into the die cavity is crucial to good part qual

11、ity and consistency. Previous research showed that variable blank holder force during forming improves ma- terial formability [3, 41, reduces springback [l, 2, 51, and achieves part consistency [ 1,2]. One strateg

12、y (i.e.. process control) for the application of variable blank holder force is shown in Fig. 2 [l, 21. F i g u r e2 :Process control of sheet metal forming In this strategy, a measurable process variable (e.g., punch

13、force) is controlled by following a predetermined (e.g., punch force-displacement) trajectory through manipulating the blank holder force. A similar approach has also been re- ported [5,6,7]. Recent work on process co

14、ntrol in sheet metal forming led to the following conclusions [8]: 1. Consistency of part quality can be improved through 2. Better part quality can be achieved through selection of It is important to realize that a

15、 badly designed process con- troller cannot ensure good tracking performance, and, in t u r n ,cannot guarantee good part quality and consistency. Clearly, the process controller plays an important role in the feedb

16、ack control system and needs further investigation. Issues of process controller design for sheet metal forming have not been properly addressed, especially, from a control point of view. Modeling sheet metal forming

17、 for process con- troller design has been investigated [9]. Hsu et al. [lo] re- cently proposed a first-order non-linear dynamic model for u- channel forming which can capture the main characteristics of the proc

18、ess dynamics observed during experiments. Propor- tional plus integral (PI) control has been used for sheet metal forming and controller parameters were typically determined by trial and error [ll]. The disadvantage

19、 of PI control is that high controller gains can achieve good tracking performance but cannot maintain good stability robustness while low controller gains can main- tain good stability robustness but cannot achieve

20、 good track- ing performance. Since sheet metal forming is a highly non- linear process, it is difficult to tune a PI controller to stabilize the closed-loop system with good tracking performance. Hsu the tracking pr

21、operty of feedback control. the reference punch force trajectory. 0-7803-4990-6/99 $10.00 0 1999 AACC 192 then Eq. 8 becomes approximated by Equation 10 is negative definite because ‘r(Fb) > 0 [lo]. Ac- cord

22、ing to Lyapunov theorem, the tracking error dynamics is asymptotically stable if Eq. 9 is satisfied. Therefore, Eq. 9 is the inverse dynamics since F bcan be solved for a given Fpd. Solving F bfrom Eq. 9 depends on

23、because W. could be zero. Figure 5 shows contours of w.. The Bhkbolderhl’C%Fb (kN) Figure 5: Contours of a - .figure shows that ? *could be zero. This will cause nu- merical problems about solving Fb, because F bcoul

24、d be very large or become indeterminable, which implies that F bwill change abruptly or cannot be found. T oreconcile this prob- lem, that F b= 0 when I I < lo-* will be assumed in simulation. The inverse dynamics

25、 (i.e., Eq. 9) with this as- sumption will be called the approximate inverse dynamics. Later, Fm will denote the solution of Eq. 9. Generally, the in- verse dynamics or the feedforward control cannot sustain any

26、 disturbance or model uncertainty. A feedback control us- ing a constant gain PI controller is built to reject disturbance and improve robustness to model uncertainty. The constant gain PI controller is designed base

27、d on the perturbed process model. The perturbed process model is derived as follows. Assuming that the disturbance, Fd, comes at the input of the process model in Fig. 4, then 2.2.2 Constant Gain PI Control: The t

28、racking error, E, becomes E consists of e and the error due to F d .Assuming that F dis much smaller than Fm. then Fp(Fb,t) in Eq. 12 can be where 3 E $ is applied [lo]. Substituting Eq. 13 into Eq. 12 leads to Alt

29、hough e decays asymptotically, E is still influenced by F d .The perturbed process model is To maintain tracking performance (i.e., E approaches e), a feedback loop is designed to force to converge to zero. A consta

30、nt gain PI controller is investigated here. Figure 6 shows the block diagram. Kp is the proportional gain and Perturbed Process Model I I I I Figure 6 Block diagram for feedback loop design. Ki is the integral gain

31、. dFb is the output of the PI controller and also part of the calculated blank holder force through the proposed controller. Assuming that the perturbed process model is a constant gain, Go, then for the constant ga

32、in PI controller, the dynamic equa- tion of the closed-loop system becomes Its solution is where TO is the time constant and z is the dummy variable. For a given GO, choose Ki and Kp such that zo is as small a