Model-augmented methods for estimation of contact pressure distribution

Journal of Manufacturing Systems 30 (2011) 223–233

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Journal of Manufacturing Systems journal homepage: www.elsevier.com/locate/jmansys

Technical paper

Model-augmented methods for estimation of contact pressure distribution Sripati Sah, Robert X. Gao ∗ , Timothy Kurp Department of Mechanical Engineering, University of Connecticut, Storrs, CT, USA

a r t i c l e

i n f o

Article history: Received 27 June 2011 Accepted 7 July 2011 Available online 11 August 2011 Keywords: Contact pressure distribution (CPD) Spatial Blending Functions (SBF) Regularization Bayesian inference Kriging

a b s t r a c t On-line measurement of the contact pressure distribution (CPD) at the tool–workpiece interface during sheet metal stamping is critical to advancing the state-of-art in tool wear and product quality monitoring. Since the number of sensors that can be integrated into a tool structure is limited by concerns of structural integrity and cost, estimation of CPD through a small number of sparsely located sensors has created unique challenges in information acquisition and representation. Specifically, the problem of determining continuous CPD from discrete sensor measurements is under-constrained and thus ill-posed. A mathematical framework is needed for treating such a problem. This paper presents three mathematical approaches Regularization, Kriging, and Spatial Blending to address this problem and discusses their relative merits and limitations. © 2011 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.

1. Introduction The sheet metal stamping process is indispensible to the automobile, aviation, and ship building industry. Sheet metal products are also ubiquitous in modern life in the form of beverage cans, furnishings, bath tubs, and appliance body panels, to name but a few [1,2]. A majority of these products are formed in a stamping operation followed by finishing operations such as trimming, painting, and polishing. The principal geometric form of the finished product is imparted by the stamping process wherein the sheet metal undergoes plastic deformation caused by externally applied loads. These loads are applied to the sheet metal through stamping tools namely: the die, the binder, and the punch. The stamping tools are installed on single or multi action presses [3] which may be driven by hydraulic or mechanical means [4]. To better the sequence of events in the stamping process, each stamping operation can be thought to take place in four sequential stages. The stamping tool components and the four process stages (numbered 1 through 4) are illustrated in Fig. 1. In the first stage the sheet metal blank (optimal blank geometry is calculated prior based on geometry and load considerations [5,6]) is placed manually or by a robot-based automatic transfer system [7] on the die and positioned with the help of alignment pins. Once the blank is in geometric alignment with the tools and the press working volume is free of operators and robotic manipulators, the second stage commences. In this stage the binder clamps the edges of the sheet metal against the die flange.

∗ Corresponding author. E-mail address: [email protected] (R.X. Gao).

The clamping force exerted by the press is referred to as the binder force, and it plays a significant role in the stamping process beyond holding the blank in place during the process [8]. The frictional constraint created by the binder force allows membrane tensions to develop in the worksheet during the forming process. This is necessary to counteract compressive stresses that develop in the material during the forming process. In the absence of adequate tension the sheet metal warps leading to the formation of undesirable wrinkles [9]. For enhanced process control the magnitude and distribution of the binder force may be actively adjusted during the stamping process [10,11]. The major of sheet metal forming takes place during the third stage, in which the punch is pressed into the sheet metal to a desired depth. In this stage the punch works against the tension in the sheet metal, plastically deforming the material to form it to the design shape. Finally in the fourth and final stage, the punch and binder are withdrawn and the newly stamped part is ejected. The high speed of the operation and the additional stiffness imparted to the material by cold forming [12] makes stamping vital to mass manufacturing industries. The stamped product is formed as an outcome of the contact process between the sheet metal and the stamping tools. For this reason manufacturing stamped parts to tight geometric specification demands high repeatability and minimal variation in the tool–workpiece contact interaction [13]. Given the natural variability due to tool wear, shut height variation, fixture compliance, lubricant distribution, and other uncertainties in operating conditions, high repeatability in the contact process is difficult to achieve without direct measurements of the tool–workpiece interaction [14]. To meet this requirement recent research has addressed on-line measurement of contact pressure distribution

0278-6125/$ – see front matter © 2011 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jmsy.2011.07.003

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S. Sah et al. / Journal of Manufacturing Systems 30 (2011) 223–233

Fig. 1. Stages in a stamping process.

at the tool–workpiece interface as a direct means of monitoring the stamping process [15]. The online measurements are achieved through tooling integrated sensing systems capable of measuring the normal interaction between the workpiece and die [16]. Inprocess knowledge about the tool–workpiece interaction enables real time control of the stamping process through multipoint cushion systems to achieve better repeatability [17,18]. The framework for a stamping tool-integrated sensing system for measuring contact pressure distribution is illustrated in Fig. 2. Piezoelectric force sensors are embedded into the stamping punch. Prior work has researched geometric techniques for optimizing sensor locations [19]. Structural integration into the tool surface is achieved through either top or bottom mount embedding techniques [20]. During the stamping process sensor measurements are acquired and preprocessed by dedicated hardware. The sensor measurements are filtered, time indexed, and converted to equiv-

alent pressure readings based on the area of the sensing element. Conventionally these measurements are spatially interpolated by means of CAD based numeric techniques such as Thin Plate Splines and Bézier surfaces to derive an estimate of the contact pressure distribution [21,22]. The objective of this paper is to provide an overview of different approaches that address the problem of estimating the contact pressure distribution (here on referred to as the pressure solution) from the spatially sparse sensor measurements. Mathematically speaking, such a type of inverse problem that seeks to determine the continuous contact pressure distribution from an observed set of sensor measurements, finite as well as sparse, is recognized as a under-constrained problem and ill-posed problem [23]. In this paper, three distinct mathematical approaches: Regularization, Kriging, and Model Augmentation, are investigated to achieve a solution to this ill-posed problem. The regularization technique introduces restrictions on the smoothness of the possible pressure solutions to compensate for the under-constrained nature of the problem [24]. It will be shown in this paper that the conventional CAD based numeric surface interpolations represent a subset of the larger set of possible pressure solutions that represent the regularization solutions. The second approach to pressure estimation explored in this paper, Kriging, is a form of regression analysis that has been adapted for making estimations of a spatial field based on partially observed measurements. In Kriging the stamping operation is modeled as a stochastic process with an assumption that the unknown pressure solution has an underlying spatial structure. Kriging is used widely in the field of geostatistics and geographical information systems (GIS) towards the estimation of parameters of geographic interest such as temperature, rainfall, soil nutrient concentration, land elevation, and mineral deposits [25,26]. The benefit of performing the analysis in a stochastic framework is that the pressure solution is accompanied by a calculated value of the estimation uncertainty, represented by the variance of the pressure estimate. It is of interest to note that both Regularization and Kriging for pressure estimation are a form of the Bayesian inference [27–31]. This topic is touched upon in this paper without going into too much detail. Prior work has established that accuracy of pressure estimation methods is directly dependent on the coverage density of the tool surface by the sensors [22]. Higher sensor density i.e. more sensors per unit tool area, predictably leads to better estimation accuracy. The accuracy of the pressure estimation can also be viewed from a spatial perspective. The estimated pressure is always 100% accurate at the locations where sensors are present. Once moving away from the sensor locations, the accuracy of the estimated pressure diminishes. The third approach presented in this study, referred to as Spatial Blending technique, improves the accuracy of contact pressure estimation at locations that are not close to any of the sensing

Fig. 2. Online sensing of contact pressure.

S. Sah et al. / Journal of Manufacturing Systems 30 (2011) 223–233

points. This technique uses a special class of functions called the Spatial Blending Functions (SBF) [32] to provide an improved estimate of the contact pressure distribution on the tool–workpiece interface. This is achieved by merging on-line measurements from tooling-embedded sensors with Finite Element (FE) simulationbased estimates of the contact pressure distribution. In this manner the contact pressure at locations that are far from sensors is estimated by factoring in FE-based knowledge models. The SBF method estimates the CPD as a spatially weighted sum of the experimentally measured pressure (by sensors) and simulated pressure (by FE). At points close to sensor locations, the SBF places a greater emphasis on sensor measurements, while points further are biased towards the FE results. The necessity of a blending scheme in the presence of an “accurate” FE model is better understood by examining the accuracy and resolution in the context of experimental and simulation techniques. The experimental technique consists of measuring the pressure at a limited number of sensor locations on the tool surface. While the accuracy of pressure measurement is high, limited only by the specifications of the sensor itself, the spatial resolution is low, since the pressure is known at only a few points on the tool. In contrast, FE models calculate contact pressure with high resolution, limited only by the mesh density. But they are unable to take into account the operational variations of the stamping process. The rest of the paper, which is based on the work presented in [33] is structured as follows: Section 2 discusses the experimental data source, based on the discrete pressure measurements made by the tool-embedded sensors, to provide a basis for demonstrating the proposed pressure estimation techniques. Section 3 presents an introduction to CAD-based interpolation techniques. In Section 4, the mathematical framework for Regularization-based pressure estimation is presented. It further discusses the link with CADbased techniques. The Kriging method and its mathematics are discussed in Section 5, where an example of Kriging-based pressure

225

estimation is provided. The mathematical background of the Spatial Blending Function that combines the information from the experimental and simulation domains is presented in Section 6. Finally, conclusions from the presented study are drawn in Section 7. 2. Discrete pressure measurements The data set for evaluating the pressure estimation techniques was obtained from a prototype of the tooling integrated sensing system on a full scale stamping test setup. The stamping experiment forms a 432 mm × 508 mm aluminum panel from a 1.56 mm AA5182 blank on a 150 ton double action HPM hydraulic press. The tool setup consists of the punch, the die, and the binder. A total of 12 dynamic force sensors are integrated into the punch structure using the top mount sensor integration technique. The experimental setup is illustrated in Fig. 3. Contact pressure at the sensor locations is determined by dividing the sensor output by the area of the sensing element (2.85 cm2 ). The experiments were performed for 5 binder (edge clamping) forces and draw depths, with each test repeated 3 times. The sensor measurements were recorded at 100 Hz. Fig. 4 shows the pre-processed measurements for an experiment with 100 mm draw depth and 400 kN binder force, where good repeatability is observed. Furthermore, different locations on the punch have shown to have distinct pressure signatures. For example, sensors 1, 5, 10, and 12 are the first to sense the pressure increase, due to their position on the crest of the punch curvature. Of these four locations, 1 and 5 recorded larger pressures as they are close to the edges. Sensors 9, 10, 11, and 12 experienced much less pressure due to their internal locations on the punch face. The initial dip in the measured pressure can be attributed to the sliding of the sheet metal on the punch face. The late peak observed in most of the sensor measurements occurs when the punch is close to the bottom of the stroke. At this time the rounded corners and edges

Fig. 3. Experimental setup for tool-integrated sensing.

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Fig. 4. Pressure measured at the 12 sensor locations in three separate tests and their locations on the punch face [32].

are being formed, creating the high pressure. Sensors 7 and 3, being farthest away from the punch crest, are the last to come into contact and experience little pressure. This effect is more pronounced at location 3, as it is lower on the punch face.

3. Pressure estimation by CAD surfaces The conventional approach to estimation of contact pressure distribution on the tool–workpiece interface from instantaneous pressure measurements made by distributed sensors has been through the use of CAD-based surface techniques [21]. The sensor pressure measurements form the set of control points, which are used to generate the pressure estimate through interpolation techniques. Past research has shown that the Thin Plate Splines (TPS) interpolation technique achieves lower estimation errors as compared to other techniques while avoiding local instabilities and construction artifacts [22]. A TPS surface is mathematically defined as the unique function that minimizes the bending energy of the interpolation surface in 2D space. Thus a TPS surface is physically analogous to the shape that a planar, thin sheet of sheet metal would take when its position is constrained at certain locations. Under the assumption of small deformations, neglecting in-plane deformations and gravity, the bending energy for the thin sheet at any point (x, y) is proportional to the quantity G(x, y) defined in terms of the instantaneous shape of the sheet metal P(x, y) as follows [34]:

 G(x, y) =

∂2 P(x, y) ∂x2

2

 +2

∂2 P(x, y) ∂xy

2

 +

∂2 P(x, y) ∂y2

P(x, y) = a1 + ax x + ay y +

wi E(x, y)

(1)

(2)

i=1

Here, x and y are space coordinates, and a1 , ax , ay , and, wi are the n + 3 surface constants that define the TPS interpolated surface. These constants are determined from the instantaneous pressure measurements as described next. The symbol n refers to the total number of sensors, and the function E(x, y) is defined as follows: E(x, y) = U (||(Li − (x, y))||)

U(r) = r 2 log(r 2 )

(3)

(4)

Let the ith sensor located at coordinates (xi , yi ) have the time varying pressure measurement vi (t). At any time instant T, the instantaneous sensor measurements v = [v1 (T), v2 (T), . . ., vn (T)] and the coordinates of the respective sensors form the interpolation boundary conditions for Eq. (2). The surface constants (a1 , ax , ay , and wi ) in Eq. (2) are determined as follows [35]: [ w1

w2

w3

···

wn | a1

ax

 ay ] = M −1 Y

(5)

In Eq. (5) the LHS contains the surface constants (a1 , ax , ay , and wi ) to be determined, and the RHS is the known boundary conditions. The vector Y contains the measurement results from the sensors at time T:

v2 (T ) v3 (T ) · · · vn (T ) | 0 0 0 ]

Y = [ v1 (T )

(6)

The square matrix M is composed of the sensor location matrix Q and functional matrix K, defined as follows:



M=



Kn

by n

Qn

by 3

Q3

by n

03

by 3

(7) (n+3) by (n+3)

The sensor location matrix Q, as the name suggests, is composed of the sensor locations Li = (xi , yi ):

⎡

2

The standard formulation for a TPS interpolated pressure surface P(x, y), which minimizes Eq. (1) for the entire surface, is known to be of the following form: n 

Here, Li = (xi , yi ) represents the location of the ith sensor on a 2D plane, the norm is the distance norm, and the scalar value function U(r) is defined as:

Qn

by 3

1 x1 ⎢ 1 x2 =⎢ . ⎣ .. ... 1 xn

⎤

y1 y2 ⎥ .. ⎥ ⎦ . yn

(8)

The functional matrix K is defined in terms of the distance norm of permutations of sensor location pairs as follows:

⎡

Kn by n

0 ⎢ U(r21 ) =⎣ ··· U(rn1 )

U(r12 ) 0 ··· U(rn2 )

··· ··· ··· ···

⎤

U(r1n ) U(r2n ) ⎥ ··· ⎦ 0

rij = ||Li − Lj ||

(9)

The mathematical framework outlined in Eqs. (1) through (9) was implemented in a MATLAB program to interpolate the pressure measurements presented in Fig. 4. The contact pressure distributions interpolated by TPS at 11 time instants are illustrated in Fig. 5. The red dots in the CPDs illustrated in Fig. 5 represent the fixed sensor locations. The estimated pressure distribution exhibits good

S. Sah et al. / Journal of Manufacturing Systems 30 (2011) 223–233

227

Fig. 5. TPS interpolated CPD (dimensions in m).

continuity and conforms to physical expectations. It is seen from the CPD that the pressure is initially distributed around sensors 1 and 5, and later towards 4 and 6. Furthermore, the central part with sensors 9 through 12 does not experience significant pressure at any time. The low pressure at 3 and 7 is clear at later stages. The CPD thus has shown to have captured the essence of the spatial component of the pressure distribution, and is of benefit to process designers in understanding the process mechanism. In the next section it is demonstrated that CAD based surface interpolation techniques may be viewed as a special case of Regularization-based pressure estimation techniques.

4. Pressure estimation by Regularization The problem of determining the actual contact pressure distribution P from the sensor measurements v is insufficiently constrained to achieve a unique solution. The selection of one solution from the multitude of possible solutions can be achieved by Regularization. Regularization is a mathematical approach to solving ill-posed problems through assumed restrictions on the smoothness of the final pressure solution [36]. The Regularization solution to the current problem is to find a surface P* that minimizes a total energy criteria ET (P) defined in Eq. (9) [37]. The particular

surface P* that minimizes ET (P) is referred to as the regularized solution: ET (P) = (1 − )εd (P, v) + εp (P)

(10)

Here, εd (P,v) is the component of the total energy function that measures the accuracy of the fit between the regularized solution and the measured data v. The smoothness constraint discussed before is embodied in the second energy term εd (P), also called the stabilizing function. The symbol  indicates the regularization parameter that controls the balance between having a regularized surface P* that perfectly satisfies all the input measurements v at the risk of having a noisy solution vs. a P* that may not satisfies all input data perfectly but has continuity and smoothness characteristic of most natural phenomena. The accuracy measure that quantifies the goodness of fit between a pressure solution P(x, y) and the measured data vi can be written as: 1  −2 i (P(xi , yi ) − vi )2 2 n

εd (P, v) =

(11)

i=1

Here, n is the number of data points or sensors, and the ith sensor located at coordinates (xi , yi ) shows a measurement variance of i2 .

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Fig. 6. CPD by Regularization (dimensions in m).

The stabilizing energy function εd (P) can have many forms with the only criteria being that lower energy leads to a more smoother surface. Commonly used forms of smoothness functions are based on physical principles, such as energy of a membrane, Eq. (12), under the assumption of small in-plane deformations, and the energy of a surface, Eq. (13), under small out-of-plane deformations [38]: 1 εMembrane (P) = P 2 1 Surface εP (P) = 2





∂P ∂x

∂2 P ∂x2

2

2



+

 +2

∂P ∂y

2 

∂2 P ∂xy

dx dy

2

 +

∂2 P ∂y2

(12)

2  dx dy

(13)

Eq. (13) is the area integral of Eq. (1), thus TPS based surface interpolations (and CAD splines in general) are recognized as a special case of Regularization-based pressure estimation, where putting the regularization parameter  = 1 in Eq. (10) gives: ET (P) = εp (P)

(14)

Eq. (14) is a mathematical statement of the definition for TPS surfaces. The mathematical framework outlined in Eqs. (10) through

 1 εp (P) = 2

(x, y)(1 − (x, y))



∂P ∂x

2

 +

∂P ∂y

(13) was coded in a MATLAB program to estimate the pressure measurements presented in Fig. 4. The regularized surfaces generated by minimization of the total energy criteria of Eq. (10) (taking the value of the regularization parameter to be  = 0.75) at 11 time instants are illustrated in Fig. 6. It is instructive to compare the regularized pressure surfaces ( = 0.75) in Fig. 6 with the TPS interpolated pressure in Fig. 5. It is readily recognized that the peak pressure in regularized surfaces are about 20–30% lower than the peak pressure in TPS interpolation. This is because the strict interpolation condition is relaxed when  < 1, leading to regularized pressure surfaces that are smoother at the expense of not satisfying all the data points. This property of Regularization can be very useful when dealing with noisy data. In the presence of noise, pure interpolation by TPS (or any other technique) leads to a condition called over fitting, wherein the interpolated surface describes the noise rather than the underlying trends. Regularization methods avoid over fitting by discounting calculated error in favor of underlying smoothness [39]. Another advantage of Regularization based pressure estimation is that the smoothness functions of Eqs. (12) and (13) can be combined to provide a means of constructing pressure estimates with depth and orientation discontinuities [37]:

2 

 + (x, y)

∂2 P ∂x2

2

 +2

∂2 P ∂xy

2

 +

∂2 P ∂y2

2  dx dy

(15)

S. Sah et al. / Journal of Manufacturing Systems 30 (2011) 223–233

Here, setting (x, y) = 0 allows orientation discontinuities, and (x, y) = 0 allows depth discontinuities to be built into the regularized pressure solution. A more in depth discussion of these properties can be found in [37]. The Regularization-based methods address the ill-posed pressure estimation problem by assuming smoothness in the pressure field. The next section takes an alternative, stochastic approach to solving the pressure estimation problem.

229

Here, L = (x, y) is a location on the tool at which the pressure needs to be estimated, Li = (xi , yi ) is the known location of the ith sensor, P(x, y) is the actual pressure distribution on the surface that needs to be estimated, and i are the weights of the linear combination subject to an unbiased condition: n 

i = 1

(17)

i=1

5. Pressure estimation by Kriging In Kriging, the stamping operation is modeled as a stochastic process with an assumption that the unknown pressure solution has an underlying spatial structure that can be quantified by either a variogram or as a stochastic model with expectation and covariance values. The benefit of performing the analysis in a stochastic framework is that the pressure solution is accompanied by a calculated value of the estimation uncertainty, represented by the variance of the pressure estimate. The Kriging operation calculates the best linear unbiased estiˆ mator (BLUE) P(L) based on the spatial dependency quantified by a semi-variogram [25,26,40]: ˆ P(L) =

n  i=1

i · P(Li )

The weights i are chosen to minimize the estimation variance criterion given in Eq. (18), leading to a pressure estimate with the least error: 2

ˆ − P(L)) ) E2 = E((P(L)

(18)

In Eq. (18), E stands for the expected value and E2 is the estimation variance. In order to minimize the estimation variance, it is needed to introduce information about the spatial dependency using a semi-variogram. The semi-variogram (x, y) is a measure of spatial dependency in distributed data values. Spatial dependency is the assumption that observations of a physical parameter taken close together in space are likely to be more alike than observations taken at locations farther apart from each other. If the spatial separation between two sensor locations L1 = (x1 , y1 ) and L2 = (x2 , y2 ) were described by the lag variable (h), defined as:

(16) h = ||L1 − L2|| =

Fig. 7. CPD by Kriging (dimensions in m).



(x1 − x2 )2 + (y1 − y2 )2

(19)

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then the semi-variogram can be calculated in terms of the lag variable h by the following equation [41]: (h) =

1 2 E[(P(L) − P(L + h)) ] 2

(20)

Eq. (20) can be written in terms of the covariance function C as: (h) = C(0) − C(h)

(21)

Based on Eqs. (19) through (21) the estimation variance criterion of Eq. (18) that is to be minimized can be written as follows [42]:



2

ˆ E2 = E((P(L) − P(L)) ) = C(0) − 2 +

 i

i C(||Li − L||) Fig. 8. Variance of Kriging based CPD estimate.

i

i j C(||Li − Lj ||)

(22)

j

The minimization of Eq. (22) subject to the unbiased condition given in Eq. (17) provides the value of , which is the Lagrangian coefficient used in minimization analysis, and the weights i . This allows the calculation of the minimized Kriging variance as: 2

ˆ E2 = E((P(L) − P(L)) ) = +

n 

i (||Li − L||) − (0)

(23)

i=1

Prior research indicates similarities between the Kriging stochastic analysis and Bayesian inference methods [27–31]. The prior model which is the starting point in Bayesian inference methods is represented in Kriging by the semi-variogram which specifies a spatial dependency model for the observations. When a set of new measurements are observed a revised estimate of the pressure distribution is generated. This is achieved by the combination of the Gaussian prior with a likelihood function for the observed values, resulting in a revised estimate of the pressure distribution which is also a Gaussian distribution with a calculated mean and variance. This is similar to the posterior model in Bayesian inference technique. The Kriging method has been investigated for estimation of contact pressure distribution, using a MATLAB toolbox [42,43]. The Kriging based pressure estimations generated by minimization of the estimation variance criterion given in Eq. (18) at 11 time instants are illustrated in Fig. 7. Kriging estimates are distinct from both the TPS based interpolations (Fig. 5) and Regularization based estimates (Fig. 6) in that they have very sharp features

that persist through each time instant. This is because the semivariogram based spatial model, which is the underpinning of the Kriging based pressure estimation, remains the same for all time instants. An advantage of the Kriging analysis over Regularization techniques is that it provides a measure of the uncertainty in the calculated pressure. The uncertainty is described by the pressure variance (square of standard deviation). It is noted that this variance measure addresses the inherent stochastic uncertainty in the true value of P(x, y) while overlooking the uncertainty in the spatial dependency model described by the semi-variogram. This limitation can be addressed by treating the spatial dependency model as an unknown in a Bayesian framework [44]. The variance for the pressure distributions in Fig. 7 is shown in Fig. 8. As expected, the uncertainty in the pressure estimate is higher at locations far from sensors and is minimal at locations close to the sensor. Such observation has motivated research into the Spatially Blended Functions, as described next. 6. Pressure estimation by Spatial Blending It is seen in Fig. 8 that the accuracy of estimated CPD at a surface location lessens as its distance from sensors increases. It is hypothesized that the pressure estimation accuracy can be improved if TPS pressure estimates based on sensor measurements are merged with FE model results. The merging of TPS and FE estimates is realized through the Spatial Blending Functions (SBF). The mathematical expression for a SBF is given in Eq. (24). Essentially, the SBF based pressure estimate at a location (x, y) is a weighted sum of the TPS-

Fig. 9. Spatial Blending Function in two views.

S. Sah et al. / Journal of Manufacturing Systems 30 (2011) 223–233

estimated pressure, TPS(x, y), and the FE-simulated pressure, FEA(x, y): PSBF (x, y) = ˛(x, y) · FEA(x, y) + (1 − ˛(x, y)) · TPS(x, y)

(24)

Here, ˛(x, y) is the spatial weighting function, which controls the merging balance. At locations close to sensors, ˛ limits to zero such that PSBF limits to TPS(x, y). At elements located away from sensors, ˛ is large and controls the blending-in of FEA(x, y). Additionally, ˛ is differentiable everywhere to preserve smoothness in the SBF pressure estimate. Under these constraints, the weighting function ˛ is devised as [32]:



˛(x, y) = 0.5c · 1 − cos



2 d1 (x, y) d1 (x, y) + d2 (x, y)



(25)

Here, d1 and d2 are the distances between location (x, y) and its two nearest sensors, respectively, ˛(x, y) is differentiable everywhere, allowing for smooth blending. The value of c is chosen between 0 and 1, limiting the influence of FEA(x, y) on the CPD estimate. This constant can be calibrated by minimizing the error in pressure estimation with respect to c at some test point. The weighting

231

function ˛(x, y) for 9 sensor locations from the experimental setup and limiting factor c = 0.5 is plotted in Fig. 9. Having laid out the mathematical framework for generating SBF-based contact pressure distribution (CPD) estimates, experimental sensor measurements and FE simulation results are used to generate SBF pressure estimates at equally spaced time intervals in the stamping process. The SBF based CPD estimates are illustrated in Fig. 10, alongside the original TPS interpolated estimate and the FE results. On the left column are TPS surfaces calculated from 9 sensor measurements based on the formulation in Section 3. On the right are the FE-simulated contact pressure distributions from the model presented in [32]. In the center column is the SBF estimate. Examining the SBF pressure estimate in the context of the sensor positions, it is evident that the pressure estimates near sensor locations are biased towards the TPS estimate. Away from the sensors, features of the FE simulation are recognizable. In the CPD corresponding to 12.0 and 13.8 s, the advantage of the SBF scheme can be seen. At these times the sheet metal is not yet in full contact with the tool. This is reflected at sensors near the top and bottom of the tool,

Fig. 10. SBF, TPS, and FE estimates of CPD.

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Bayesian method that integrates these approaches into a stochastic framework. Acknowledgements This work is supported by the National Science Foundation under CMMI-0936075. The authors thank Prof. J. Cao for providing experimental facility at Northwestern University. References

Fig. 11. Quantitative assessment of accuracy.

where the pressure measurements are near zero. However, due to the relatively low density of sensors the TPS interpolation indicates areas of nonzero, somewhat high pressure in these regions. This is noticed in the upper right region of the TPS pressure at time 12.0 s. Examining the CPD estimate using the Spatial Blending Function at this time instant, it is evident that this unwanted effect is reduced. In this region, which is further away from the sensors, FE results are heavily blended in. It is realized that a direct weighted average of the FE simulated results and experimental measurements may not be optimal in scenarios where there is a large disagreement in the two models. This disagreement could stem from the need for fine tuning of simulation parameters or due to unavoidable modeling limitations and can be quantified along the lines of the model inadequacy index (MI) [27]. In order to quantitatively assess the accuracy with which the different methods estimate the contact pressure distribution over the entire surface, a redundant sensor test has been conducted. In this test, a set of 11 sensors was used to estimate the pressure distribution on the contact interface. The sensor measured pressure at the location of the 12th sensor (location 10 in Fig. 4) is compared against the pressure estimated by the four model augmented techniques. In this analysis, Kriging output the best result, with a 12% estimation error, followed by Regularization (16%), TPS (17%), and SBF (28%). The results exemplify the expected errors for CPD estimation (Fig. 11). 7. Conclusions An analysis of different methods for estimating continuous contact pressure distribution from spatially sparse sensor measurements is presented. Three distinct mathematical techniques were discussed: Regularization, Kriging, and Spatial Blending. These techniques take different approaches in addressing an illposed problem: Regularization assumes an inherent smoothness in the pressure solution; Kriging assumes that there exists an underlying stochastic spatial dependency model to the pressure solution, and Spatial Blending provides a framework for merging FE results into the pressure solution as a means of improving accuracy at locations far from sensors. Evaluation of these methods based on experimental data has revealed the specific strengths of these methods. Specifically, Regularization is good for estimating pressure from noisy data and can represent spatial discontinuities. Kriging, being a stochastic analysis, has the advantage of reporting the variance of the estimated pressure, which serves as a measure of uncertainty. Spatial Blending directly improves the spatial resolution by selective merging of experimental and simulated results. Future research will investigate

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