Arrangement of multi-sensor for spatio-temporal systems: application to sheet-forming processes

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Chemical Engineering Science 56 (2001) 5709–5717

www.elsevier.com/locate/ces

Arrangement ofmulti-sensor for spatio-temporal systems:

application to sheet-forming processes

Der-Ming Chang

a

_{, Cheng-Ching Yu}

b;∗

_{, I-Lung Chien}

b

a_{Department of Food Engineering, Da-Yeh University, Chang-Hwa 515, Taiwan}

b_{Department of Chemical Engineering, National Taiwan University of Science and Technology, 43 Keelung Road, Sec 4,}

Taipei 106-07, Taiwan

Received 26 May 2000; received in revised form 9 February 2001; accepted 14 July 2001

Abstract

In sheet-forming process, we are dealing with a two-dimensional product, rolls of paper or metal, polymer 3lms etc. The control objective is to maintain product quality, e.g., thickness, uniformity during continuous production. Despite recent advances in measurement technology, it is di8cult to measure all points on the entire sheet. Therefore, we have a two-dimensional product with scattered measurements. The location ofmeasurement points not only decides the correctness ofthe estimation, it also a9ects the control performance. Previous study shows that, in the temporal mode, the optimal estimation implies equally spaced measurement updates. In this article, an image-based approach is proposed to evaluate the appropriateness ofmeasurement patterns for interactive spatio-temporal systems. The method is based on the concept of time to the next nearest neighbor(s) and the mean and standard deviation are utilized to discriminate among measurement patterns. It is useful in arranging multiple scanning sensors. The proposed approach is employed for the evaluation of multi-sensor arrangement for typical sheet-forming processes. Results show that the image-based analysis is e9ective for the design of sensor trajectories for manufacturing of two-dimensional products.

Keywords: Sheet-forming process; Sensor arrangement; Nearest neighbor; State estimation; Kalman 3lter; LQG

1. Introduction

Sheet forming is an important unit operation in many manufacturing processes. Common examples include: paper making, metal rolling, polymer 3lm extrusion and coating. Control objective ofthese processes is to main-tain the quality, e.g., thickness, uniformity of the end products, e.g., rolls ofpaper sheets or polymer 3lms. Therefore, we are controlling two-dimensional (2-D) products and the major concern is the quality variations occur in the direction ofAow, the machine direction (MD), and perpendicular to the direction ofAow, the cross direction (CD). The last decade has seen advances in CD control, such as Chen and Wilhelm Jr. (1986) proposed quadratic penalty function (QPF) techniques ∗_{Corresponding author. Present address: Department ofChemical}

Engineering, National Taiwan University, Taipei 106-17, Taiwan. E-mail address: [email protected] (C.-C. Yu).

that are more robust and easier to implement in real-time applications. Linear Quadratic Gaussian (LQG) con-trol theory, applied by Bergh and MacGregor (1987), was used to design multivariable controllers capable ofjointly controlling the machine and cross-directional property variations. The model predictive controller to maintain Aat pro3les ofcoating along the cross direction has been derived (Braatz, Tyler, Morari, Pranckh, & Sartor, 1992). The Linear programming based model predictive control was applied to solve large-scale control problems (Dave, Willig, Kudva, Penky, & Doyle, 1997). Robust decentralized con-trollers were proposed by Laughlin, Morari, & Braatz (1993).

The identi3cation and estimation problems were stud-ied by Rawlings and Chien (1996) and by Rigopoulos, Arkun, & Kayihan (1997). These control strategies rely on the measurement ofthe property along both the CD and MD directions. The sensor ofinterest is the gauge sensor. It normally consists ofa radiation-emitting source

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Fig. 1. Schematic ofscanning sensor with zig-zag measurement pattern.

and a detector. The property (e.g., thickness) ofthe 3lm is sensed by the attenuation ofthe radiation signal. The sensor is often arranged as a scanning device mounted on a carriage which moves back and forth in the cross direc-tion. Since the 3lm is moving in the machine direction, this generates a zig-zag pattern on the 3lm (Fig. 1). This immediately leads to the following question: how does one reconstruct the full two-directional pro3le given only a zig-zag pattern on the 3lm. The Kalman 3lter is the natu-ral choice for the state estimation problem as discussed by several authors (Bergh & MacGregor, 1987; Chen, 1992; Wang, Dumont, & Davies, 1993; Tyler & Morari, 1995; Rawlings & Chien, 1996). Chang, Yu, and Chien (2000) discuss how the zig-zag pattern deteriorates the estima-tion and control performance. Recently, we have also seen rapid development in alternative sensor technologies. This includes full array wet or dry end sensor (Shapiro, 1998), additional dry end sensors (Tyler & Morari, 1995) and full sheet imaging system (Chen & Pfeifer, 1998).

Under current technology, a simple alternative is to use multiple scanning sensors for better estimation. The question ofthe sensor arrangement then arises. The objective ofthis work is to devise an image-based ap-proach to evaluate the appropriateness ofthe sensor arrangement provided with the digitized measurement patterns. The remainder ofthe paper is organized as fol-lows: A typical sheet-forming process is given in Section 2 and the estimation problem for the spatio-temporal system is discussed. The concept oftime to the near-est neighbors is developed and a procedure to evaluate the uniformity is proposed in Section 3. In Section 4, multi-sensor arrangements for sheet-forming processes are employed to test the correctness and e9ectiveness of the image-based approach. The conclusion is drawn in Section 5.

2. Process and estimation problem

Consider a typical sheet-forming process in Fig. 1 where the control objective is to maintain some prop-erty, e.g., thickness, uniformity of the 2-D product, e.g., rolls ofpaper sheets, by manipulating the actuators, e.g., opening ofthe slice lip which control the Aow out ofa headbox. Unlike typical control problem, only scattered measurements are available for control. Let us look at the measurement pattern as the scanning sensor moves along the cross directions (Fig. 1). Assume the sensor moves with a 3xed speed and the speed in the machine direction is also constant. Then, we have a linear mea-surement pro3le as the sensor sweeps forward and, on the return path, we obtain another linear trajectory. This is exactly the zig-zag trajectory mentioned in the literature. Rawlings and Chien (1996) point out the e9ectiveness ofestimation and control is a9ected by this particular measurement pattern. Therefore, the sheet-forming pro-cess is a distributed parameter propro-cess, a 2-D product evolved temporally, with speci3c measurement pattern for process control.

2.1. Process modeling

The complexity ofsuch systems is that derivation ofre-alistic theoretical models is either not feasible or too time consuming. Therefore, the goal of modeling is to provide the simplest mathematical model that can describe the essential features of the sampling mechanism. Following the approach ofBergh and MacGregor (1987), a linear time invariant dynamic model is employed.

x[k + 1] = Ax[k] + Bu[k] + w[k]; (1)

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D.-M. Chang et al. / Chemical Engineering Science 56 (2001) 5709–5717 5711

where x is a vector ofthe state variables, u is the process input and y is a process output. Many authors use Eqs. (1) and (2) to represent the process where A and B matrices are often assume to be constant and a time varying C[k] is employed. The C[k], matching the scanning pattern, is a vector with an entry 1 corresponding to the current measured state and 0 elsewhere (Bergh & MacGregor, 1987; Rawlings & Chien, 1996; Tyler & Morari, 1995; Chang, Yu, & Chien, 2000). Look at the measurement pattern as the scanning sensor moves along the cross di-rections (e.g., Fig. 1). Assume the sensor moves with a 3xed speed and the speed in the machine direction is also constant. This is exactly the zig-zag trajectory mentioned in the literature. For the scanning sensor with zig-zag pat-tern, the stacked C matrices becomes

                  C[1] C[2] ... C[n] C[n + 1] C[n + 2] ... C[2n]                   =                   1 0 0 · · · 0 0 0 0 1 0 · · · 0 0 0 ... 0 0 0 · · · 0 0 1 0 0 0 · · · 0 0 1 0 0 0 · · · 0 1 0 ... 1 0 0 · · · 0 0 0                   ; (3)

where n is the number oflanes in the cross direction. Notice that the C matrix can be constructed similarly if the measurements are di9erent from zig-zag pattern.

Moreover, in the discrete time model (Eqs. (1) and (2)), v is the measurement noise with a covariance R; w is white noise with zero mean and has the strength which can be described by a covariance matrix

E{wwT_}_{= Q} =           1 s 2s · · · n−1s 1 s · · · n−2s ... ... ... s sym: 1           2_; ₍₄₎

where 2_{is the strength ofprocess noise and, physically,}

sdescribes the spatio-temporal interaction. Speci3cally,

s describes the propagation ofdisturbances to adjacent

lanes which takes a value between 0 and 1. When s= 0,

it means the disturbance will not a9ect adjacent lanes. On the other hand, if sis approaching 1, the disturbance

will a9ect all lanes in the CD with non-decaying strength. Generally, sfalls between 0 and 1 and its value depends

on the viscosity ofthe Auid. Moreover, for two spatial positions with|i−j|lanes apart, the degree ofinteraction can be described by |i−j|s .

2.2. E4ects of spatial interaction in estimation

The Kalman 3lter is e9ective in the state reconstruc-tion. The Kalman 3lter algorithm has been well docu-mented in the literature (Bryson & Ho, 1975; Grewal & Andrews, 1993) and the state estimation for CD=MD control is also studied (Bergh & MacGregor, 1987).

Consider the dynamic system in Eqs. (1) and (2) with equal dynamics on each lane (i.e., A = aI). In the Kalman 3lter algorithm, the element oferror covariance ( e[k−

1|k−1] = e[k−1|k−1]eT_[k₋₁_|_k₋_{1]) at the k}₋_{1 step}

is e2

o;ij. When the 3lter extrapolates a step forward, the

ijth entry ofthe error covariance e[k|k−1] becomes

e2

ij= a2e2o;ij+ 2qij; (5)

where qij is the ijth entry of Q. When the measurement

in the jkth lane is made, the output matrix becomes

C[k] = [0; : : : ; 0

jk−1

1 0; : : : ; 0]

n−jk

: (6)

The ith element ofthe Kalman 3lter gain can also be expressed as ki= a 2_e2 o;ijk + 2qijk a2_e2 o;jkjk + 2qjkjk + R : (7)

Substituting the 3lter gain into the Kalman 3lter algo-rithm, the iith entry ofthe error covariance after a mea-surement update at the jkth lane becomes

eii= a2(e2o;ii−kie2o;ijk) +

2₍₁₋_k

iqijk): (8) Certainly, ifnot updated, the error covariance remains the same as the extrapolated value.

eii= a2e2o;ii+ 2: (9)

Assume that the accumulated error is much smaller than process noise (i.e., the 3rst term in Eqs. (8) and (9) is nil). Hence, the ratio of error covariance after and before measurement update is

ri= 1−kiqijk: (10)

Furthermore, ifthe measurement noise is much less than the process noise, we have

ri= 1−q2ijk: (11)

Here qijk can be expressed explicitly in terms of s (i.e., qijk= |i−js k|). Therefore, we arrive at

ri= 1−2|i−js k|: (12)

Eq. (12) shows how a measurement update a9ects the es-timation error under certain assumptions. Let i = jk (i.e.,

making i the measured lane), the error will be reduced to 0 on this particular lane. Moreover, sis the

determi-nant factor in describing the error reduction in the cross direction.

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Fig. 2. E9ects ofspatio-temporal interaction parameter to estimation errors at di9erent spatial position.

Let us use a simple example to illustrate the e9ect of correlative degree in the correlation matrix Q. Consider a simple LTI system

˙x[k + 1] = 0:95I x[k] + 0:05I u[k] + w[k]; y[k] = C[k]x[k] + v[k];

where the strength ofthe process and measure-ment noises are 0:95 and 0:0095, respectively. Fig. 2 shows that the estimation errors at k = 1 with C[1] = [0 0 0 0 0 1 0 0 0 0 0] f or di9erent s. The results

clearly indicate that, for a spatially highly interactive system (i.e., s → 1), the placement ofmeasurements

in the CD is less critical. However, when s becomes

smaller, it is important to ensure the measurements are made across the entire CD in a certain period oftime. 3. Measurement patterns

Before getting into the e9ects of measurement pat-terns on the CD=MD estimation and control problems, we would like to explore the general measurement placement problem for a 2-D product evolved temporally. First, only the temporal model will be discussed followed by a more complex spatio-temporal mode. The concept ofthe near-est neighbor is employed to evaluate the appropriateness ofeach pattern and procedure will be given. The speci3c pattern as a result ofsweeping sensors will be discussed in Section 4.

3.1. Nearest neighbor—temporal mode

Considering a single variable with a given number ofmeasurements in period, Chang et al. (2000) show that optimal estimation can be achieved by making the

measurement update periods the same in the full pe-riod. In other words, when only the temporal mode is considered, the equal-spaced measurement update period provides the optimal state estimation. It can be shown an-alytically via the Kalman 3lter algorithm, provided with simple state-space equations.

To some degree, the approach ofequal-spaced mea-surement update period is equivalent to the concept of the nearest neighbor when analyzing the degree ofmix-ing usofmix-ing the digitized image ofa mixture (Wei, 1999). The idea behind the nearest neighbor is, 3rst, we take a snap shot ofa 2-D image and then analyze the uni-formity of the static image using di9erent sample sizes (Wei, 1999). The digitized image resulted from scanning sensors di9ers from the static image problem in that the image evolved dynamically, due to the movement ofthe sheet. For example, the speeds ofthe scanning sensor and=or sheet rolling may distort the image. Therefore, in-stead ofcounting the nearest neighbor contacts, we are interested in the time to the nearest neighbor. Since we are dealing with a periodical pattern, only a full period or the common denominator ofthe periods will be su8cient for the analysis. In the 1-D case (only the time horizon is considered), the mean time to the nearest neighbor is simply

QT =n

i=1

Ti=n; (13)

where Ti is the ith measurement update period in a full

cycle and n is the number ofmeasurements in that time period. The variance ofthe time to the nearest neighbor 2

T is a good measure ofthe uniformity in the temporal

mode. The variance is de3ned as 2 T= n i=1 (Ti− QT)2=n; (14)

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Fig. 3. Four di9erent measurement patterns with the same measure-ment update periods on the temporal mode.

where T is the standard deviation. An obvious way to

minimize the variance is to make Ti= QT(∀i). This is

ex-actly what we derived from the Kalman 3lter algorithm (Chang et al., 2000). Therefore, the concept of the mean time to the nearest neighbor o9ers a simple alternative to analyze measurement patterns.

3.2. Nearest neighbor—spatio-temporal mode

When both the temporal and spatial modes are con-sidered, it is di8cult to obtain an analytical solution from the Kalman 3lter algorithm especially when the spatio-temporal interaction factor (s) is involved.

Con-sider a system with 10 equal-spaced divisions (10 lanes) across the spatial direction (y-axis) and the correspond-ing temporal mode is expressed in the x-axis as shown in Fig. 3. The digitized 2-D images offour measurement patterns are also shown in Fig. 3 where the black square indicates the measurement. It should be emphasized here that the patterns shown here are utilized to establish the relationship between measurement patterns and the mean time to the nearest neighbor. Notice that, here, we do not restrict ourselves to patterns associated with any speci3c scanning device. The constraints will be enforced in the

Table 1

Standard deviations ofthe time to the nearest neighbor(s) T T1& T10 T2∼T8 TS Case A 0.0 7.07 5.77 6.03 Case B 0.0 5.66 4.04 4.36 Case C 0.0 4.24 2.31a _2.31 Case D 0.0 2.83 0.58 0.58 a_T5₌_T6_{= 1:54.}

next section once physical insight is gained. When only the temporal mode is considered (only looking at a par-ticular position in the y-axis), all four cases (A–D) give the same mean time to the nearest neighbor and all four variances are zero, i.e., QT = 10 and 2

T= 0.

It should be pointed out here that we are dealing with a 2-D pro3le and the nearest neighbors evolve as we are moving along the time horizon. Therefore, at a speci3c lane, the nearest neighbors are the next measurements in the adjacent lanes as well as at this particular lane. In other words, we have three nearest neighbors to be met (instead ofone when the temporal mode is considered). Let us take case B in Fig. 3 as an example. Considering the 3rd lane, it takes one time unit to meet the 3rst neighbor (lane 2), another 8 time units to meet the 2nd neighbor (lane 4) and another 1 time unit to meet the 3rd neighbor (lane 3). Therefore for lane 3, the mean time to the next neighbor is

QT3= (1 + 8 + 1)=3 = 3:3:

This can be viewed as the averaged sampling time (time for a measurement update) in a local area. But, for control purpose, we are more interested in the di9erence among them. The variance thus becomes a good measure for the local (a particular lane) variation.

2

T3= (1−3:3)

2_{+ (8}₋_3:3)2_{+ (1}₋_3:3)2

3 = (4:04)2:

We can repeat the computing of Tjfor all 10 lanes. This

leads to the standard deviation from the mean time when both the temporal and spatial modes are considered. That is TS= m j=1 Tj=m; (15)

where m is the number oflanes. Here TS can be

inter-preted as the variation ofthe sampling time for the entire 2-D product, a measure of global variation. The 3rst col-umn ofTable 1 gives the standard deviations when only the temporal mode is considered. The result, T= 0,

in-dicates that all four cases are equally promising since the measurement update periods are the same on each lane. However, when the spatio-temporal mode is considered, case D turns out to be the best measurement pattern (i.e., having the smallest TSas shown in the last column). The

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Fig. 4. Estimation error covariance for four di9erent measurement patterns.

measurements are updated most uniformly to its near-est neighbors. The standard deviations on time to the nearest neighbors for each lane, Tj, are also given in

Table 1.

Simulation results, Fig. 4, also reveal that case D gives the optimal estimation over the entire range ofthe spatio-temporal interaction factor (s). It is

interesting to observe that when s approaches 1

(i.e., highly interactive system), cases B–D per-form equally well while case A gives poor esti-mation. The reason is the former three cases place measurement points on every time step along the time horizon (x-axis) and the interactive system en-sure good estimation quality across the spatial posi-tion (y-axis). To the other extreme, s= 0, all four

cases give the same estimation error since this is a non-interacting system and all one can do is to en-sure all the spatial positions are meaen-sured in a full period. This is exactly what these cases do as shown in Fig. 3.

3.3. Procedure

In reality, we are dealing with measurement patterns with some degree oftemporal and spatio-temporal inter-actions (di9erent s). Despite some special cases may

arise at extreme conditions, the pattern evaluation proce-dure using the concept oftime to the nearest neighbors o9ers an attractive alternative. The procedure has the fol-lowing steps:

1. Establish a digitized image ofthe measurement pattern (e.g., Fig. 3).

2. Select the nearest neighbors (For an evolving 2-D im-age, they are the next contact and the contacts above and below.)

3. Compute the mean time to the next nearest neighbor for each spatial position j.

QTj= n

i=1

Tji=n;

where n is the number ofthe nearest neighbors and Tji

is the time to the ith neighbor on jth spatial location after an immediate measurement update among the neighbors.

4. Compute the standard deviation on each spatial loca-tion j. Tj= _n i=1 (Tji− QTj)2=n ₁₌₂ ;

where Tjis the standard deviation oftime to the

near-est neighbors on jth spatial location.

5. Compute the averaged standard deviation (TS) over

all m spatial locations (Eq. (15)). 6. Select the pattern with the smallest TS.

4. Sensor arrangement

Let us use the arrangement ofthe scanning sensors in sheet-forming process to illustrate the appropriateness of the measurement patterns. The example in the previous section is studied. First a Kalman 3lter is used for state estimation. The implication ofestimation in control is also explored. In order to provide a fair comparison, a LQG (linear quadratic Gaussian) control is employed (Astrom & Wittenmark, 1990). The LQG control minimized the following objective function:

min

u J = x

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Fig. 5. Sweeping policies for two scanning sensors travelling in the same (pre3x s) and opposite (pre3x o) directions with the sweeping range of100% (su8x 1), 50% (su8x 1=2) and 75% (su8x 3=4).

where $ and % are the weightings for the states (x) and input (u), respectively. Here, we assume the weightings are $ = I and % = 0:1I.

4.1. Two scanning sensors

The zig-zag pattern from the scanning sensor (Fig. 1) leads to undesirable properties in estimation and con-trol, because it results in unequal-spaced measurement updates in the MD (large variance in time to the nearest neighbor). Approaches have been sought to improve the estimation (Chang et al., 2000). Under current sensor technology, an obvious choice is to put in another scan-ning sensor. That is using two scanscan-ning sensors for state estimation. The next question then becomes what is the best arrangement for these two sensors? What are the star-ing positions? Should they travel in the same direction or opposite direction, sweep the entire cross direction or only part ofit? The procedure from the concept ofthe nearest neighbors can be used to test the appropriateness ofthese arrangements.

Two factors, sweeping direction and range, are stud-ied. The two sensors can sweep in the same (pre3x s in Fig. 5) and opposite (pre3x o in Fig. 5) directions. They can sweep and full CD range (su8x 1 in Fig. 5), 3=4 of the entire range (su8x 3=4) or halfofthe entire range (su8x 1=2). This gives six possible sensor arrangements

Fig. 6. Measurement pattern for six sweeping policies in Fig. 5: (A) s-1, (B) o-1, (C) s-1=2, (D) o-1=2, (E) s-3=4 and (F) o-3=4. Table 2

Standard deviations ofthe time to the nearest neighbor(s) when considering only the temporal mode and spatio-temporal mode

Case T TS o-1 2.48 1.81 s-1 3.26 2.17 o-1=2 2.48 1.81 s-1=2 2.48 1.51 o-3=4 3.92 2.21 s-3=4 4.40 2.88

as shown in Fig. 5. For example, the sensors travel in the opposite direction while sweeping the entire CD is de-noted as the sweeping policy o-1. These six policies give di9erent measurement patterns as shown in the digitized images in Fig. 6.

Once the digitized images are available, we can pro-ceed with the proposed steps. When only the temporal mode is considered, the standard deviations oftime to the nearest neighbor (T) indicate that cases o-1, s-1=2 and

o-1=2 are better policies, but indistinguishable, because they give the same T (Table 2). Moreover, the

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Table 3

2-norm and ∞-norm oferrors for di9erent sensor arrangements

e x

2-norm ∞-norm 2-norm ∞-norm One sensor 0.01310 0.02060 0.01620 0.02100 o-1 0.00937 0.01474 0.01368 0.01596 s-1 0.00978 0.01809 0.01395 0.01874 o-1=2 0.00937 0.01474 0.01368 0.01596 s-1=2 0.00922 0.01474 0.01365 0.01596 o-3=4 0.00990 0.01841 0.01408 0.01919 s-3=4 0.01042 0.01877 0.01445 0.01932

as shown in Fig. 6. But when spatio-temporal mode (3 nearest neighbors) is taken into consideration, the results (Table 2) clearly indicate that s-1=2 is the best sweep-ing policy. Simulation results indicate that s-1=2, indeed, gives the least error in estimation ( e) and control ( x)

as shown in Table 3. However, it is only slightly bet-ter than the other two policies (o-1 and o-1=2) as also shown in Table 3. The s-1=2 policy di9ers from the o-1 and o-1=2 policies in that, at most, 2 adjacent points in the spatio-temporal mode are measured for s-1=2 instead of4 for the other two cases (e.g., lanes 5 and 6 in Fig. 6D and Figs. 6A and C). It should also be noted that, from engineering point of view, the policy o-1 is a bet-ter choice over o-1=2, despite giving exactly the same measurement pattern. The reason is that the full range sweeping, o-1, requires much less frequent acceleration and deceleration and this prevents wear in the motors. Therefore, the sweeping policy o-1 is recommended for its performance as well as reliability.

The results presented here indicate the e9ectiveness ofthe image-based analysis for sensor arrangement. It clearly identi3es the better measurement patterns for an obvious reason: they place the measurement uniformly over the entire product. More importantly, it provides quantitative evaluation ofdi9erent sweeping policies. It should be noticed that, alternatively, one can always run the Kalman 3lter equations for all possible con3gurations and obtain covariance matrices.

4.2. Extension to multi-sensor

From on-going previous analysis, a heuristic can be es-tablished. That is, for a system with l sweeping sensors, sweep on the same direction with each sensor covering 1=l ofthe entire range. Consider a case with 5 scanning sensors. Ifeach sensor sweep 1=5 ofthe entire cross di-rection, Fig. 7A shows the scanning pattern when the sensors are travelling in the same direction (case A) and the case for moving in the opposite direction (case B) is shown in Fig. 7B. The digitized images for the example of5 scanning sensors are given in Fig. 8. Without quanti-tative computing, it immediately becomes clear that case

Fig. 7. Arrangement of3ve scanning sensors with the sensor travelling in the same (A and C) and opposite (B and D) direction while sweeping 20% (A and B) and 100% ofthe entire range.

Table 4

2-norm and∞-norm oferrors for the example of3ve scanning sensors

e x

2-norm ∞-norm 2-norm ∞-norm Case A 0.005879 0.008483 0.01155 0.01212 Case B & D 0.005809 0.008525 0.01166 0.01215

A is a better arrangement since better uniformity in the spatio-temporal mode for case A is rather obvious. Sim-ulation results (Table 4) also reveal that case A is a lit-tle better than case B. Unfortunately, neither case A nor case B can be implemented in practice since they only sweep 1=5 ofthe entire cross direction (e.g., 10 m) which can lead to substantial wear in the motors. Again, sim-ilar to the two scanning sensor example (e.g., o-1=2 vs. o-1 in Fig. 5), the sub-optimal measurement pattern (case B) can be reproduced by arranging the scanning sensors sweeping the entire CD while moving in the opposite di-rection (case B vs. case D in Fig. 7). This is exactly the case D as shown in Fig. 7. Also in Fig. 7, while sweeping

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Fig. 8. Measurement patterns for four di9erent sweeping policies shown in Fig. 7.

the entire range, the sweeping policy ofcase C is a poor choice which can be ruled out immediately by inspection the measurement pattern (Fig. 8). Again, the concept of the nearest neighbor o9ers an e9ective way to discrim-inate di9erent sensor arrangement while considering its maintainability.

5. Conclusion

In this article, an image-based analysis is proposed to evaluate the appropriateness ofmeasurement patterns for interactive spatio-temporal systems. The method is based on the concept oftime to the nearest neighbor(s) and the mean and standard deviation are utilized to discriminate among measurement patterns. It is useful in arranging multiple scanning sensors while considering its practi-cality. A sheet-forming example is used to illustrate the e9ectiveness ofthe proposed procedure.

Acknowledgements

Preliminary version ofthis work was presented Au-tomation 2000 Taipei. This work was supported by the National Science Council ofTaiwan.

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