Convergence Analysis of a Steerable Nip Mechanism for Full Sheet ...
Edgar Ergueta e-mail: [email protected]
Rene Sanchez e-mail: [email protected]
Roberto Horowitz e-mail: [email protected]
Masayoshi Tomizuka e-mail: [email protected] Department of Mechanical Engineering, University of California, Berkeley, CA 94720
1
Convergence Analysis of a Steerable Nip Mechanism for Full Sheet Control in Printing Devices Current approaches for high speed color printers require sheets be accurately positioned as they arrive to the image transfer station (ITS). This goal has been achieved by designing and building a steerable nip mechanism, which is located upstream from the ITS. This mechanism consists of two rollers that not only rotate to advance the paper along the track, but also steer the paper in the yaw direction. This paper briefly reviews the design and experimental setup of the system, and focuses on the design and analysis of a controller that precisely corrects the lateral, longitudinal, and angular positions of the sheet. The control strategy used is based on linearization by state feedback with the addition of internal loops for the local control of the actuators. This paper also provides a methodology to tune the controller parameters so that the desired performance specifications are met. The success of this mechatronic approach is corroborated through simulation and experimental results, which show that the system is able to correct sheet errors and meet all the performance specifications. 关DOI: 10.1115/1.3117195兴
Introduction
Current approaches for paper path control require sheets to be accurately positioned as they arrive to the image transfer station 共ITS兲. This is achieved by using a registration device, located between the paper path and the ITS, which not only corrects for longitudinal, lateral, and angular errors, but also delivers the sheet on time to the ITS. Prior work on sheet control in a printer paper path has been mainly focused in developing control techniques for coordinating multiple actuated sections of the paper path, in order to correct for longitudinal interspacing errors among sheets, and synchronize the arrival of sheets to the ITS with its corresponding image 关1–7兴. Other works, such as in Refs. 关8–12兴, do consider the sheet’s lateral and skew position error corrections, but they fail to do so at large speeds and without marking the page. In this paper we present the control architecture of a mechatronic solution that corrects for sheet position errors at high speeds without damaging the page. This is achieved by using the steerable nip mechanism 关13兴 depicted in Fig. 1, whose design is described in detail in Refs. 关14–16兴. The problem of controlling paper trajectories with steerable nips is similar to the control of two-wheel robots, such as the one studied in Ref. 关17兴. However, not only does the two-wheel robot have one less degree of freedom than the steerable nip system, but also the control law proposed by Yun and Sarkar 关17兴 fails to account for singularities that arise when the steering angle of the wheels approaches zero. Moreover, in the case of the two-wheel robot, three inputs are needed to follow a reference trajectory. This is not the case with steerable nips, where four inputs are needed due to the flexibility of the paper; two inputs rotate and steer roller 1, whereas the other two inputs do the same for roller 2 共see Fig. 1兲. Similar to the two-wheel robot, the steerable nip mechanism is a nonlinear system with nonholonomic constraints. For the steerable nip system, two nonholonomic constraints come from nonslip conditions on the rollers, and the other two come from local veContributed by the Dynamic Systems, Measurement, and Control Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received December 12, 2007; final manuscript received February 17, 2009; published online December 9, 2009. Assoc. Editor: Robert Gao.
locities 共of the paper兲 being zero in the direction perpendicular to the rotation of the rollers. Additional details on the constraints of this particular system can be found in Ref. 关14兴. The control objective of the steerable nip device consists of correcting the position of the sheet on a horizontal plane while the sheet is moving in the longitudinal direction at all times. Since the page should move without getting damaged, it is also necessary to control the sheet’s amount of buckling. The control strategy used to achieve these goals is based on state feedback linearization 关18兴 with inner loops for the control of the roller’s rotational angular velocity and steering angular position. Recently, Elliot and Gans 关19兴 presented a control strategy for an underactuated steerable nip mechanism for printer sheet registration devices similar to the device described in this paper. However, since this mechanism does not consider the sheet’s amount of buckling, it has one less degree of freedom than the mechanism presented in this paper. Thus, the control problem described in Ref. 关19兴 resembles more than that of the two-wheel robot in Ref. 关17兴, and the control strategy presented there is significantly different from the one we present. The remainder of this paper is organized as follows. Section 2 briefly describes the steerable nip section and the experimental setup. Section 3 presents the mathematical model of the system. Sections 4 and 5 describe the control strategy and convergence analysis for the closed-loop system, respectively. Section 6 presents the methodology proposed to tune the controller gains. Simulation and experimental results are shown in Sec. 7. Finally, conclusions are stated in Sec. 8.
2
Experimental Setup
The steerable nip mechanism has been designed so that it can correct for sheet lateral position errors without having to move the actuators and without inflicting any damage on the page. This has been achieved by steering two rollers, which are underneath two backer balls. As seen in Fig. 2, each roller is driven by a servomotor 共referred as process direction actuator兲 attached to a rotating table, which is in turn steered by another servomotor 共referred as steering actuator兲 through a coupling. The sheet moves along a flat surface, and passes between the two backer balls and rollers, as shown in Figs. 2 and 3; note that the page moves in the direction of the arrow labeled vគ . Figure 4 is a photograph of the experimental system we have
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Process Motor 1
Top View
Image Transfer Station
1 Roller 1
.θ
2b - δ
Optical Sensors
.θ
2
C
v−
2
1
2b
φ
1
φ
2
Process Motor 2
Fig. 3 Steerable nips with paper buckle
Roller 2 Laser Sensors
Backer Ball
Side View
Paper Roller RotatingTable
Steering Motor
In order to determine the position, orientation, and the amount of buckling of the sheet, it is necessary to detect the edges of the page. As seen in Fig. 1, two laser sensors are located on the right hand side of the page, which are used to measure the lateral and angular positions of the page. Furthermore, to measure the longitudinal position of the page, five single photodiode 共optical兲 sensors, spaced 52 mm apart, are located along the process direction. It should be noticed that whereas we are able to obtain continuous measurements for lateral and angular positions of the sheet, we need to estimate its longitudinal position when the leading edge of the page is between two consecutive photodiodes. In this paper we
Fig. 1 Schematic of steerable nip fixture
built. As shown in Fig. 4, the page is delivered to the steerable nip section by a feeder unit 共located at the back of the picture兲. Then, while the page moves on top of the horizontal plate, its position is being corrected by the steerable nip mechanism, which is located below the plate; a photograph of the dc motors driving the rollers is shown in Fig. 5. Finally, the sheet is removed through an exit roller 共located at the front of the picture兲. It is assumed that the location of the exit roller in the experimental setup is the location of the entrance to the ITS. Thus, the performance of the controller is determined by the position and orientation of the paper as it arrives to the exit roller.
sheet
Fig. 4 Experimental setup
-v Backer ball roller
Backer ball roller Process Direction Motors
Steering Motors
Fig. 2 Sheet moving through the steerable nip mechanism
Fig. 5 DC motors used for the steerable nip mechanism
011008-2 / Vol. 132, JANUARY 2010
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x-
1 s
-. φ1 +-
-. φ2 +-
CFB (s)
+-
1 s
.. φ1d 1
τ1
1 s
.. φ2d 1
τ2
1 s
u1
P (s) P1
CFF (s)
P (s)
. θ2
P (s)
φ1
P2
CFB (s)
+
u2
P2
P2
CFF (s) 1 s
. φ2d
+
P1
+-
. φ1d
. θ1
P1
. θ2d
.. θ2d CFBL
CFF (s)
S1
φ1d
CFB (s)
+-
+
u3
Kinematics
.. θ1d
x- d
y
. θ1d
S1
S1
CFF (s) S2
φ2d 1 s
CFB (s)
+-
S2
+
u4
φ2
P (s) S2
Fig. 6 System block diagram for implemented system
estimate longitudinal position through the use of an open-loop observer based on the kinematic relations described in Sec. 3. The same observer is used to estimate the amount of buckling of the sheet.
3
␦˙ = r2 sin 2˙ 2 − r1 sin 1˙ 1
where r1 and r2 are the radii of the two rollers. As mentioned in Refs. 关15,16兴, a simple model that adequately describes both the process direction and steering actuator dynamics is given by
System Model
As described in Fig. 3, the steerable nip mechanism has two independent steering rollers located at points 1 and 2, which are separated by a distance 2b. The space-fixed coordinates of the system 共x , y , , ␦兲 locate the leading right corner of the sheet, point C, which will be used to track the position of the page. Note that x, y, and are the lateral, longitudinal, and angular positions of the sheet, respectively. The amount of buckling, ␦, is defined as the difference between the distance separating points 1 and 2, as measured along the paper, 2b − ␦, and along the straight line, 2b 共see Fig. 3兲. Thus, a negative ␦ represents the amount of buckling on a sheet, whereas a positive ␦ occurs when the paper stretches; stretching needs to be avoided at all times. Also note that the origin 共0,0兲 of the space-fixed frame is located in the middle of points 1 and 2. Furthermore, ˙ i represents the angular velocity of the rollers in the direction parallel to the sheet, and i represents their angular position in the direction perpendicular to the sheet 共for i = 1 , 2兲. The kinematic model of the system is derived so that the four nonholonomic constraints mentioned in Sec. 1 are satisfied at all times. This model, whose complete derivation can be found in Refs. 关15,16兴, is represented by the following equations: x˙ = −
y˙ =
r1y y cos 1˙ 1 + r2 cos 2 + sin 2 ˙ 2 2b 2b
冉
冊
共1兲
r2共x + b兲 r1共x − b兲 cos 1˙ 1 − cos 2˙ 2 2b 2b
共2兲
1 ˙ = 共r1 cos 1˙ 1 − r2 cos 2˙ 2兲 2b
共3兲
Journal of Dynamic Systems, Measurement, and Control
共4兲
¨ i + ␣ pi˙ i =  piV pi
共i = 1,2兲
共5兲
¨ i + ␣si˙ i = siVsi
共i = 1,2兲
共6兲
where V ji is the voltage input to the motor, and ␣ ji and  ji are actuator coefficients that depend on the inertias and rotational viscous damping coefficients of the different components of the steerable nip mechanism; subindices p and s stand for process direction and steering actuators, respectively, and subindex i corresponds to each of the two rollers. The complete dynamic system model is composed of Eqs. 共1兲–共6兲. We can further define the state vector as xគ ˙1 ˙ 2兴T, the input vector as uគ = 关x y ␦ 1 2 ˙ 1 ˙ 2 T = 关u1 u2 u3 u4兴 = 关V p1 V p2 Vs1 Vs2兴T, and the output vector as yគ = 关x y ␦兴T.
4
Control Strategy
The block diagram of the control system is shown in Fig. 6. As will be explained in the next paragraphs, the control strategy designed uses feedback linearization to linearize only the kinematics, and uses internal loops to locally control the actuator’s positions and velocities. This technique is an extension to the dynamic feedback linearization controller presented in Refs. 关20–22兴, since we need to integrate the outputs from the nonlinear linearization law CFBL. However, robustness is gained through the use of the internal loops. In Fig. 6, the block Kinematics is represented by Eqs. 共1兲–共4兲, the process direction actuators, P p1 and P p2 by Eq. 共5兲, and the steering actuators, Ps1 and Ps2 by Eq. 共6兲. Furthermore, noticing that if we differentiate the output vector yគ twice, we obtain JANUARY 2010, Vol. 132 / 011008-3
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˙1 ˙ 2 兴T yគ¨ = m共xគ 兲 + N共xគ 兲关¨ 1 ¨ 2
共7兲
where m共xគ 兲 is a nonlinear 4 ⫻ 1 vector and N共xគ 兲 is a nonlinear 4 ⫻ 4 matrix; we design the following feedback linearization control law CFBL:
关¨ 1d vគ =
冤
T −1 ¨ 2d ˙ 1d ˙ 2d 兴 = N 共xគ 兲共vគ 共xគ 兲 − m共xគ 兲兲
x¨d + 共Kx + x兲共x˙d − x˙兲 + Kxx共xd − x兲 y¨ d + 共Ky + y兲共y˙ d − y˙ 兲 + Kyy共y d − y兲
¨ d + 共K + 兲共˙ d − ˙ 兲 + K共d − 兲
␦¨ d + 共K␦ + ␦兲共␦˙ d − ␦˙ 兲 + K␦␦共␦d − ␦兲
冥
y¨ = vគ 共xគ 兲 − N共xគ 兲关˙ p1 ˙ p2 共˙ s1 + ⑀1兲 共˙ s2 + ⑀2兲 兴T
␥ pi =
共8兲
共␣ pi +  pi pi − pi兲pi
si =
 pi 共␣si + ␥sisi − si兲si si
␥ pi ; s
CFBsi共s兲 = si + ␥ pis;
CFFpi共s兲 =
冉 冊 冉 冊
1 ␣ pi 1+  pi s
1 ␣si CFFsi共s兲 = 1+ si s
共i = 1,2兲 共9兲
Convergence Analysis of Closed-Loop System
In this section we will prove the local convergence of the control system described in Sec. 4 for a sheet of finite length by linearizing the system error dynamics. Thus, we will first set the error dynamics in terms of the paper and actuator errors as well as their corresponding surface errors. Then we will linearize the error dynamics around a predefined desired trajectory, and will show that, by proper selection of the controller gains, the error vector of the linearized control system converges asymptotically to zero. Let us first define paper coordinate errors and actuator errors by ˜x = xd − x, ˜ = d − , pi = ˙ id − ˙ i,
¯˙ i − ˙ id ⑀i =
˜␦ = ␦ − ␦ d
共10兲
and let us also define the following surface errors: sx = ˜x˙ + x˜x, ˜˙ + ˜, s =
˜y˙ = − y˜y + sy
˜␦˙ = − ˜␦ + s ␦ ␦
sy = ˜y˙ + y˜y
˙ p2 = − p2 p2 + sp2 ˙ s1 = − s1s1 + ss1 ˙ s2 = − s2s2 + ss2 s˙x = − Kxsx + n11˙ p1 + n12˙ p2 + n13共˙ s1 + ⑀1兲 + n14共˙ s2 + ⑀2兲 s˙y = − Kysy + n21˙ p1 + n22˙ p2 + n23共˙ s1 + ⑀1兲 + n24共˙ s2 + ⑀2兲 s˙ = − Ks + n31˙ p1 + n32˙ p2 + n33共˙ s1 + ⑀1兲 + n34共˙ s2 + ⑀2兲 s˙␦ = − K␦s␦ + n41˙ p1 + n42˙ p2 + n43共˙ s1 + ⑀1兲 + n44共˙ s2 + ⑀2兲 s˙p1 = − 共␣ p1 +  p1 p1 − p1兲sp1 s˙p2 = − 共␣ p2 +  p2 p2 − p2兲sp2 s˙s1 = − 共␣s1 + s1s1 − s1兲ss1 s˙s2 = − 共␣s2 + s2s2 − s2兲ss2
⑀˙ 1 = −
¯˙ 1 ¯˙ 1 1 ␦ ␦ ⑀1 + ⌿ គ˙ + 1 ␦⌿ គ ␦t
⑀˙ 2 = −
¯˙ 2 ¯˙ 2 1 ␦ ␦ ⑀2 + ⌿ គ˙ + 2 ␦⌿ គ ␦t
sp1 = ˙ p1 + p1 p1,
sp2 = ˙ p2 + p2 p2
ss1 = ˙ s1 + s1s1,
ss2 = ˙ s2 + s2s2
共15兲
គ is where nij is the 共i , j兲 element of matrix N共xគ 兲 in Eq. 共7兲, and ⌿ defined by ˜ s ˜␦ s␦ 兴T ⌿ គ = 关˜x sx ˜y sy
˙ s␦ = ˜␦ + ˜␦
共14兲
˙ p1 = − p1 p1 + sp1
共i = 1,2兲
共i = 1,2兲
共13兲
˜x˙ = − x˜x + sx
˜y = y d − y
si = id − i
共i = 1,2兲
˜˙ = − ˜ + s
共i = 1,2兲
where and ␥ are controller gains, and ␣ and  are the actuator coefficients defined in Eqs. 共5兲 and 共6兲. As shown in Fig. 6, in order to use feedforward for steering position control, we estimate ¨ id through the use of a first order filter the steering acceleration ˙ id with gain i 共i = 1 , 2兲. If i is sufficiently small, the value of ˙ ¯ i. This technique, which is somewill be very close to that of times referred as dynamics surface control 共DSC兲, has been proposed by Swaroop et al. 关23兴. Note that the success of the complete control strategy depends on the invertibility of matrix N共x兲. It is shown in Refs. 关15,16兴 that this matrix is invertible as long as the sheet always moves in the longitudinal direction 共˙ 1 , ˙ 2 ⫽ 0兲.
5
共i = 1,2兲
the time derivatives of the errors in Eqs. 共10兲 and 共11兲 can be expressed as
where K and are the feedback linearization controller gains. We then use feedback plus feedforward to locally control the actuator’s rotational velocities ˙ i and steering positions i. These local controllers are given by CFBpi共s兲 = pi +
共12兲
If we further express vគ 共xគ 兲 from Eq. 共8兲 in terms of paper and surface errors 共Eqs. 共10兲 and 共11兲兲, and let the actuator controller gains ␥ pi and si in Eq. 共9兲 be equal to
共16兲
If we now define the desired trajectory by ˙ d, ␦˙ d兲 = 共0, t,0,0,0, ,0,0兲 共xd,y d, d, ␦d,x˙d,y˙ d, 共11兲
Combining now Eqs. 共7兲, 共8兲, and 共10兲 we obtain the closed-loop expression
共17兲
where is the nominal longitudinal velocity of the sheet, and linearize the error dynamics in Eqs. 共14兲 and 共15兲 around ˜x = ˜y ˜ = ˜␦ = p1 = p2 = s1 = s2 = sx = sy = s = s␦ = s = s = s = s = p1 p2 s1 s2 = ⑀1 = ⑀2 = 0, we obtain an expression of the form
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e˙គ 共t兲 = G共t兲eគ 共t兲
共18兲
where eគ 共t兲 is the error vector, and it is defined by ¯ ¯␦ ¯ p1 ¯ p2 ¯s1 ¯s2 ¯⑀ 兴T eគ 共t兲 = 关¯x ¯y
共19兲
B␦⑀ =
whose elements are
冋册 冋册 冋册 ˜x
¯x =
¯␦ =
sx
˜y
¯y =
,
˜ ¯= s
,
sy
B⑀ 共t兲 =
冋册 冋 册 冋 册 冋 册 冋 册 冋册 ˜␦
s␦
s1
¯s1 =
p1
¯ p1 =
,
s2
¯s2 =
,
ss1
sp1
p2
¯ p2 =
,
B⑀ p1共t兲 =
sp2
⑀1 , ¯⑀ = ⑀2
ss2
共20兲
Similarly, the time-varying matrix G共t兲 in Eq. 共18兲 is given by
G共t兲 =
冤
0
Ax 0 0 Ay
0
0
0
A
0
0
0
B⑀x
0
0
0
0
0
0
s1
s2
Bx p1共t兲 Bx p2共t兲 Bx s1
0
By p1 Bp1
By p2 Bp2
0
0
0
A␦
0
0
B␦
B␦
B␦⑀
0
0
0
0
Ap1
0
0
0
0
0
0
0
0
0
Ap2
0
0
0
0
0
0
0
0
0
As1
0
0
0
0
0
0
0
0
0
As2
0
B⑀ 共t兲 B␦⑀ B⑀ p1共t兲 B⑀ p2共t兲 B⑀ s1 B⑀ s2 A⑀
Bx⑀ 0
冥
B⑀ s1 =
Ax =
A =
Ap1 =
As1 =
冋 冋
冋 冋
1
0
− Kx
−
1
0
− K
− p1
1
0
− g10,10
− s1
1
0
− g14,14 A⑀ =
Bx p1共t兲 =
冋
0
Bp1
=
冋 冋
冋
0
g2,9 g2,10
Bx s1 =
By p1 =
册 册
− x
册 册
Ay =
,
A␦ =
,
Ap2 =
,
As2 =
− ⑀1
册
g2,13 −
0
0
0
g4,9 g4,10 0
0
g6,9 g6,10
1
0
− Ky
− ␦
1
0
− K␦
冋 冋
0
册 册
0
− g12,12
− s2
1
0
− g16,16
册
,
By p2 =
,
Bp2
=
册 册
0
0
g17,9 g17,10
g18,9 g18,10
g17,13 g17,14
g18,13 g18,14
0
B␦⑀ =
,
g18,5 g18,6
g18,1 g18,2
0
g18,7 g18,8
,
B⑀ p2共t兲 =
g17,11 g17,12
,
B⑀ s2 =
0
g18,11 g18,12 0
g18,15 g18,16
共23兲
¯x˙ = Ax¯x + Bx p1共t兲 ¯ p1 + Bx p2共t兲 ¯ p2 + Bx s1¯s1 + B⑀x¯⑀ ¯y˙ = Ay¯y + By p1¯ p1 + By p2¯ p2 ¯˙ = A ¯ + Bp1¯ p1 + Bp2¯ p2 共24兲
共25兲
¯ + B␦¯␦ + Bp1共t兲 ¯⑀˙ = A⑀¯⑀ + Bx⑀¯x + B⑀ 共t兲 ¯ p1 + B⑀ p2共t兲 ¯ p2 + B⑀ s1¯s1 ⑀ ⑀
册
0
0
g4,11 g4,12 0
g17,5 g17,6
g17,1 g17,2
¯˙ s2 = A ¯s2 s2
− 0 0
Bx⑀ =
0
¯˙ s1 = A ¯s1 s1
冋 册 冋 册 冋 册
B⑀x =
,
0
g2,11 g2,12
0 0 , −
0
g8,15 −
¯˙ p2 = A ¯ p2 p2
共22兲
冋
B␦s2 =
,
¯˙ p1 = A ¯ p1 p1
1
册
0
¯␦˙ = A ¯␦ + Bs1¯ + Bs2¯ + B⑀¯⑀ ␦ ␦ ␦ s1 ␦ s2
− p2
Bx p2共t兲 =
,
册 册
− y
g18,17 − ⑀2
冋
0
冋 冋
,
0
g8,13
Elements gi,j, above, depend on system parameters and controller gains; elements g2,9, g2,10, g2,11, g2,12, g17,3, g17,9, g17,10, g17,11, g17,12, g18,5, g18,6, g18,9, g18,10, g18,11, and g18,12 depend explicitly on time, and they do so linearly. This dependence on time comes from the definition of the desired trajectory of the sheet shown in Eq. 共17兲. In order to show the convergence of the control system errors, we will look at the dynamics of each of the elements of the error vector in Eq. 共19兲, and will provide their algebraic solutions. Thus, combining Eqs. 共19兲–共23兲 we obtain the following expressions:
共21兲
whose matrix components are
冋 册 冋 册 冋 册 冋 册 冋 册 冋 册 冋 册 冋 册 册 册 冋 冋
B␦s1 =
0
g6,11 g6,12
Journal of Dynamic Systems, Measurement, and Control
+ B⑀ s2¯s2
共26兲
The solutions for pi, si, 共i = 1 , 2兲 can then be easily obtained from Eq. 共25兲 as follows: ¯ p1共t兲 = eAp1t¯ p1共0兲 ¯ p2共t兲 = eAp2t¯ p2共0兲 ¯s1共t兲 = eAs1t¯s1共0兲 ¯s2共t兲 = eAs2t¯s2共0兲
共27兲
¯ 共t兲, and ¯␦共t兲 can be Subsequently, the solutions for ¯x共t兲, ¯y 共t兲, obtained from Eqs. 共24兲 and 共27兲, and the solution for ¯⑀共t兲 from Eqs. 共26兲 and 共27兲 as JANUARY 2010, Vol. 132 / 011008-5
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¯x共t兲 = eAxt¯x共0兲 +
+
冋冕 冋冕 冋冕
冋冕
册
0
t
册 册 冕
t
t
+
t
eAx共t−兲B⑀x¯⑀共兲d
冋冕
冋冕
0
册
t
eAy共t−兲By p1eAp1d ¯ p1共0兲
册
0
t
eAy共t−兲By p2eAp2d ¯ p2共0兲
0
¯ 共t兲 = eAt ¯ 共0兲 +
冋冕 冋冕
冋冕
t
eA共t−兲Bp1eAp1d
册
0
t
+
iv.
eAx共t−兲Bx s2eAs2d ¯s2共0兲 +
¯y 共t兲 = eAyt¯y 共0兲 +
册
¯ p1共0兲
¯␦共t兲 = eA␦t¯␦共0兲 +
eA␦共t−兲B␦s1eAs1d ¯s1共0兲
册 冕
0
t
t
eA␦共t−兲B␦s2eAs2d ¯s2共0兲 +
0
eA␦共t−兲B␦⑀¯⑀共兲d
0
共28兲 ¯⑀共t兲 = eA⑀t¯e共0兲 +
冕 冋冕 冋冕 冋冕
冕
t
eA⑀共t−兲Bx⑀¯x共兲d +
0
t
+
eA⑀共t−兲B␦⑀¯␦共兲d +
0
+
t
t
¯ 共兲d eA⑀共t−兲B⑀ 共兲
册
eA⑀共t−兲B⑀ p1共兲eAp1d ¯ p1共0兲
册
eA⑀共t−兲B⑀ p2共兲eAp2d ¯ p2共0兲 t
册 册
t
0
eA⑀共t−兲B⑀ s2eAs2d ¯s2共0兲
m
共29兲
Finally, we can see from Eqs. 共22兲, 共23兲, and 共27兲–共29兲 that by proper selection of the controller gains, we can make matrices Ax, Ay, A, A␦, Ap1, Ap2, As1, As2, and A⑀ Hurwitz, and thus ¯x共t兲, ¯ 共t兲, ¯␦共t兲, ¯ p1共t兲, ¯ p2共t兲, ¯s1共t兲, ¯s2共t兲, and ¯⑀共t兲 will converge ¯y 共t兲, to zero asymptotically.
6
Given the initial and final maximum error bounds, ji共m兲共0兲 and ji共m兲共T兲, respectively, and their corresponding surface error bounds s ji共m兲共0兲 and s ji共m兲共T兲 for j = s , p
and i = 1 , 2, use Eq. 共27兲 to calculate the required controller gains 共pi , pi兲 and 共si , ␥si兲. ii. Given the initial maximum error bounds, ˜y 共0兲 and ˜ m共0兲, m ˜ m共T兲, their the final maximum error bounds, ˜y m共T兲 and corresponding surface error bounds, sy共m兲共0兲, s共m兲共0兲, sy共m兲共T兲 and s共m兲共T兲, and the controller gains obtained in step 共i兲, use the second and third expressions in Eq. 共28兲 to calculate the required controller gains 共y , Ky兲, and 共 , K兲, respectively. iii. Given the initial and final maximum error bounds, ⑀i共m兲共0兲 and ⑀i共m兲共T兲, respectively, the controller gains obtained in steps 共i兲 and 共ii兲, and an initial guess for the controller gains 共x , Kx兲 and 共␦ , K␦兲, use Eq. 共29兲 to calculate the controller gains i for i = 1 , 2. iv. Given the initial maximum error bounds, ˜x 共0兲 and ˜␦ 共0兲, m m the final maximum error bounds, ˜x 共T兲 and ˜␦ 共T兲, the
eA⑀共t−兲B⑀ s1eAs1d ¯s1共0兲
0
+
t
0
0
0
+
冋冕
冕
Now, by looking at the expressions in Eqs. 共27兲–共29兲, we can develop a procedure to calculate the required controller gains for a sheet of finite dimensions moving at a prespecified nominal longitudinal velocity with some given initial and final state errors. ¯ 共t兲, ¯ pi共t兲, and ¯si共t兲 Whereas the gains corresponding to ¯y 共t兲, 共i = 1 , 2兲 can be obtained directly, those corresponding to ¯x共t兲, ¯␦共t兲, and ¯⑀共t兲 cannot, since the expressions for ¯x共t兲 and ¯␦共t兲 depend on
i.
册
t
At time T, when the sheet exits the nip section, the maximum errors must be, respectively, smaller than or equal to ˜ m共T兲, and ˜␦m共T兲. ˜xm共T兲, ˜y m共T兲,
¯⑀共t兲 and vice versa. Thus, we need to use the following iterative procedure.
eA共t−兲Bp2eAp2d ¯ p2共0兲
0
冋冕
m
eAx共t−兲Bx s1eAs1d ¯s1共0兲
0
+
The sheet has finite dimensions and moves with a nominal longitudinal velocity, . ii. The distance between the first and last optical sensors in Fig. 1 is given by L. Thus, the leading edge of the sheet will exit the steerable nip section at time T = L / . iii. The sheet has maximum initial errors ˜x 共0兲, ˜y 共0兲, ˜ m共0兲, m m and ˜␦ 共0兲.
eAx共t−兲Bx p2共兲eAp2d ¯ p2共0兲
0
+
i.
eAx共t−兲Bx p1共兲eAp1d ¯ p1共0兲
0
+
册
t
Controller Design Methodology
Since the page moves at a constant nominal longitudinal velocity, , the leading edge of the sheet will enter the ITS at a prespecified time, T. Therefore, it is necessary to design a feedback system that will reduce all sheet positions and orientation errors to some prespecified level within the allowable control time T. We will now state these control specifications more precisely as follows.
v.
m
corresponding surface error bounds, sx共m兲共0兲, s␦共m兲共0兲, sx共m兲共T兲, and s␦共m兲共T兲, the controller gains initially guessed for 共x , Kx兲 and 共␦ , K␦兲 in step 共iii兲, and the controller gains obtained in steps 共i兲–共iii兲, use the first and fourth ˜ 共T兲兩 and expressions in Eq. 共28兲 to verify that the norms 兩x 兩˜␦共T兲兩 are smaller than ˜xm共T兲 and ˜␦m共T兲, respectively. Iterate between steps 共iii兲 and 共iv兲 if necessary.
As stated in step 共v兲, even though this procedure requires some iteration, it should be noted that the initial guesses mentioned in step 共iii兲 can be obtained from a simplified control system, which is described in Ref. 关24兴. The simplification consists in assuming that whereas we still control the velocity of the process direction motors through the internal loops shown in Fig. 6, inputs u3 and ˙ 1 = u3 ; ˙ 2 = u4兲, which are u4 are the steering angular velocities 共 obtained directly from the feedback linearization law in Eq. 共8兲. They are then integrated once and fed directly as inputs to the kinematics block in Fig. 6. In this way, we would have inner loops for the process direction actuators, but not for the steering motors. Using these assumptions, the control gain synthesis problem is simplified not only because of the reduction in the inner loops, but also because we no longer need the first order filters shown in Fig. 6. The simplified system only has controller gains 共x , Kx兲,
011008-6 / Vol. 132, JANUARY 2010
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Transactions of the ASME
Table 1 Maximum allowable initial and final state error bounds ˜, for simulation test. The unit for x˜, y˜, and ␦˜ is m; the unit for εs1, and εs2 is rad; and the unit for εp1, εp2, ⑀1, and ⑀2 is rad/s. Initial errors
gitudinal velocity of 0.5 m/s from the initial to the final state errors shown in Table 1, where T = 0.42 s represents the allowed time for control action. Based on the initial and final maximum errors shown in Table 1 and the methodology presented in Sec. 6, the following controller gains were obtained: 共x , Kx兲 = 共80, 4.5兲, 共y , Ky兲 = 共80, 8.0兲, 共 , K兲 = 共80, 4.9兲, 共␦ , K␦兲 = 共95, 4.0兲, 共p1 , p1兲 = 共100, 0.26兲, 共s1 , ␥s1兲 = 共100, 9.31兲, 共s2 , ␥s2兲 共p2 , p2兲 = 共100, 0.26兲, = 共100, 9.31兲, and 共1 , 2兲 = 共0.0042, 0.0013兲. Figure 7 shows the simulation results using these controller gains. In this figure, the horizontal dotted lines mark the final state errors at the allowable control time, T, which is in term marked by the vertical dotted lines. It should be noticed the efficacy of the methodology pre˜ , and ˜␦ are exactly reduced to the sented in Sec. 6, since ˜x, ˜y , prescribed limits shown in Table 1 within the allowed control time. Furthermore, we have also performed experimental tests in the system shown in Fig. 4. It should be noticed, however, that the actual experimental fixture cannot handle initial errors as large as those in Table 1 due to limitations in the setup design 共range of the laser sensors for lateral and angular measurements in Fig. 1兲. Thus, based on the initial and final errors shown in Table 2 and the methodology presented in Sec. 6, the following controller gains were obtained: 共x , Kx兲 = 共80, 12.5兲, 共y , Ky兲 = 共80, 13.2兲, 共 , K兲 = 共80, 11.2兲, 共␦ , K␦兲 = 共80, 7.9兲, 共p1 , p1兲 = 共100, 0.26兲, 共p2 , p2兲 = 共100, 0.26兲, 共s1 , ␥s1兲 = 共100, 9.31兲, 共s2 , ␥s2兲 = 共100, 9.31兲, and 共1 , 2兲 = 共0.0042, 0.0013兲. As shown in Fig. 8,
Final errors
˜ 共0兲兩 ⱕ˜xm共0兲 = 0.008 兩x ˜ 共0兲兩 ⱕ ˜y m共0兲 = 0.040 兩y ˜ 共0兲兩 ⱕ ˜ m共0兲 = 0.025 兩 ˜ ˜ 兩␦共0兲兩 ⱕ ␦m共0兲 = 0.0001 兩 p1共0兲兩 ⱕ p1共m兲共0兲 = 0.1 兩 p2共0兲兩 ⱕ p2共m兲共0兲 = 0.1 兩s1共0兲兩 ⱕ s1共m兲共0兲 = 0.1 兩s2共0兲兩 ⱕ s2共m兲共0兲 = 0.1 兩⑀1共0兲兩 ⱕ ⑀1共m兲共0兲 = 0.1 兩⑀2共0兲兩 ⱕ ⑀2共m兲共0兲 = 0.1
˜ 共T兲兩 ⱕ˜xm共T兲 = 0.0013 兩x ˜ 共T兲兩 ⱕ ˜y m共T兲 = 0.0016 兩y ˜ 共T兲兩 ⱕ ˜ m共T兲 = 0.0035 兩 ˜ ˜ 兩␦共T兲兩 ⱕ ␦m共T兲 = 0.00002 兩 p1共T兲兩 ⱕ p1共m兲共T兲 = 0.01 兩 p2共T兲兩 ⱕ p2共m兲共T兲 = 0.01 兩s1共T兲兩 ⱕ s1共m兲共T兲 = 0.01 兩s2共T兲兩 ⱕ s2共m兲共T兲 = 0.01 兩⑀1共T兲兩 ⱕ ⑀1共m兲共T兲 = 0.01 兩⑀2共T兲兩 ⱕ ⑀2共m兲共T兲 = 0.01
共y , Ky兲, 共 , K兲, 共␦ , K␦兲, 共p1 , p1兲, and 共p2 , p2兲, and they can be computed without any iteration. Furthermore, it should be noted that in most cases we only need one or two iterations between steps 共iii兲 and 共iv兲 before we arrive at the required controller gains for the desired specifications.
7
Simulation and Experimental Results
In order to determine the efficacy of the controller developed, we first performed simulation tests for a letter-sized sheet moving along the steerable nip section length, 0.2 m, at the nominal lon-
Lateral Position Error
xe(mm)
8 6 4 2 0
0
0.1
0.2
0.3
0
0.1
0.2
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0.4
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0.6
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0.9
1
0.4
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Longitudinal Position Error
ye(mm)
40 30 20 10 0
Angular Position Error
e
φ (rad)
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1
Error in Buckling
0.05
δe(mm)
0 −0.05 −0.1 −0.15 −0.2
Time (sec) Fig. 7 Simulation results for large initial errors. The vertical dotted lines denote the allowable control time, T, and the horizontal lines denote the final errors at time T.
Journal of Dynamic Systems, Measurement, and Control
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Table 2 Maximum allowable initial and final state error bounds for the experimental test. The unit for x˜, y˜, and ␦˜ is m; the unit ˜ , εs1, and εs2 is rad; and the unit for εp1, εp2, ⑀1, and ⑀2 is for rad/s. Initial errors
Final errors
˜ 共0兲兩 ⱕ˜xm共0兲 = 0.004 兩x ˜ 共0兲兩 ⱕ ˜y m共0兲 = 0.0074 兩y ˜ 共0兲兩 ⱕ ˜ m共0兲 = 0.025 兩 兩˜␦共0兲兩 ⱕ ˜␦m共0兲 = 0.0001 兩 p1共0兲兩 ⱕ p1共m兲共0兲 = 0.1 兩 p2共0兲兩 ⱕ p2共m兲共0兲 = 0.1 兩s1共0兲兩 ⱕ s1共m兲共0兲 = 0.1 兩s2共0兲兩 ⱕ s2共m兲共0兲 = 0.1 兩⑀1共0兲兩 ⱕ ⑀1共m兲共0兲 = 0.1 兩⑀2共0兲兩 ⱕ ⑀2共m兲共0兲 = 0.1
˜ 共T兲兩 ⱕ˜xm共T兲 = 0.00026 兩x ˜ 共T兲兩 ⱕ ˜y m共T兲 = 0.0005 兩y ˜ 共T兲兩 ⱕ ˜ m共T兲 = 0.001 兩 兩˜␦共T兲兩 ⱕ ˜␦m共T兲 = 0.00002 兩 p1共T兲兩 ⱕ p1共m兲共T兲 = 0.01 兩 p2共T兲兩 ⱕ p2共m兲共T兲 = 0.01 兩s1共T兲兩 ⱕ s1共m兲共T兲 = 0.01 兩s2共T兲兩 ⱕ s2共m兲共T兲 = 0.01 兩⑀1共T兲兩 ⱕ ⑀1共m兲共T兲 = 0.01 兩⑀2共T兲兩 ⱕ ⑀2共m兲共T兲 = 0.01
by using these controller gains, the system was able to reduce the sheet’s initial position errors to prescribed levels in about 0.3 s. Note that the position increases constantly because the sheet moves in the longitudinal direction at all times. The discrepancies observed between simulation and experimental results can be attributed to sensor noise and model parameter uncertainties, such as friction coefficients. Particularly, Ergueta et al. 关25兴 discussed the possible existence of unmodeled dynamics in the actuators, and presented a proof of the robustness of this control strategy to such uncertainties.
8
Conclusion
In this paper we have presented an innovative design that permits a swifter correction of lateral, longitudinal, and angular position errors in a paper path control system for xerographic and printing devices. This mechanism accomplished this task by having steerable nips. In order to correct the sheet position errors we have used a controller based on dynamic feedback linearization 关18,20–22兴 with the addition of internal loops for the control of the process direction velocity and steering position of the rollers; for these inner loops we have used feedback plus feedforward control. In addition, not only have we provided a convergence analysis for the controller implemented, but also we have described a design methodology to determine the controller gains. Simulation and experimental results show that by using the controller gains obtained from the methodology previously mentioned, it is possible to drive a sheet from an initial state with nonzero longitudinal velocity to a final state, also with nonzero longitudinal velocity in a very short time.
Acknowledgment This work was supported by the National Science Foundation under Grant No. CMS 0301719, and by the financial support and collaboration with Xerox Corporation. In particular, the authors thank Dr. Martin Krucinski for his numerous critical remarks and suggestions during the development of this project.
Lateral Position Error
xe(mm)
4
Experimental Simulation
3 2 1 0 0
0.05
0.1
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0.3
Longitudinal Position 200
y(mm)
150 100
Experimental Simulation
50 0 −50
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Angular Position Error
φe(rad)
0 −0.02
Experimental Simulation
−0.04 0
0.05
0.1
0.15
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Error in Buckling
δe(mm)
1
Experimental Simulation
0.5 0 −0.5 −1
0
0.05
0.1
0.15
0.2
0.25
0.3
time(sec) Fig. 8 Experimental and simulation results for small initial errors
011008-8 / Vol. 132, JANUARY 2010
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Transactions of the ASME
References 关1兴 Krucin´ski, M., Cloet, C., Horowitz, R., and Tomizuka, M., 1998, “Interobject Spacing Control and Controllability of a Manufacturing Transportation System,” 1998 American Control Conference, Jun., Vol. 2, pp. 1259–1265, Session No. WA03-4. 关2兴 Cloet, C., Krucin´ski, M., Horowitz, R., and Tomizuka, M., 1999, “A Hybrid Control Scheme for a Copier Paperpath,” 1999 American Control Conference, San Diego, CA, Jun., Vol. 3, pp. 2114–2118. 关3兴 Cloet, C., Krucin´ski, M., Horowitz, R., and Tomizuka, M., 1998, “Intersheet Spacing Control and Controllability of a Copier Paperpath,” 1998 IEEE Conference on Control Applications, Trieste, Italy, Sept., Vol. 3, pp. 726–730. 关4兴 Krucin´ski, M., Cloet, C., Horowitz, R., and Tomizuka, M., 2000, “A Mechatronics Approach to Copier Paperpath Control,” First IFAC Conference on Mechatronic Systems, Darmstadt, Germany, Sept. 关5兴 Krucinski, M., 2000, “Feedback Control of Photocopying Machinery,” Ph.D. thesis, University of California, Berkeley, CA. 关6兴 Cloet, C., 2001, “A Mechatronics Approach to Copier Paperpath Design,” Ph.D. thesis, University of California, Berkeley, CA. 关7兴 de Best, J. J. T. H., Bukkems, B. H. M., van de Molengraft, M. J. G., Heemels, W. P. M. H., and Steinbuch, M., 2008, “Robust Control of Piecewise Linear Systems: A Case Study in Sheet Flow Control,” Control Eng. Pract., 16共8兲, pp. 991–1003. 关8兴 Williams, L. A., deJong, J. N. M., Dondiego, M., and Savino, M. J., 1997, “Sheet Registration and Deskewing Device,” U.S. Patent No. 5,678,159. 关9兴 Lofthus, R. M., 1986, “Apparatus and Method for Combined Deskewing and Side Registering,” U.S. Patent No. 4,971,304. 关10兴 Kamprath, D. R., and Malachowski, M. A., 1990, “Translating Electronic Registration Systems,” U.S. Patent No. 5,094,442. 关11兴 Castelli, V. R., deJong, J. N. M., Williams, L. A., and Wolf, B. M., 1986, “Agile Lateral and Skew Sheet Registration Apparatus and Method,” U.S. Patent No. 5,697,608. 关12兴 Tanaka, N., Fukumoto, H., Arimoto, K., and Iwashita, Y., 1997, “Skew Correction Mechanism for Thermal Transfer Type Color Printers,” International Conference on Micromechatronics for Information and Precision Equipment, Tokyo, Japan, Jul., pp. 635–638. 关13兴 Hwang, S. S., 2002, “Sheet Registration and Deskewing System With Independent Drives and Steering,” U.S. Patent No. 6,634,521.
Journal of Dynamic Systems, Measurement, and Control
关14兴 Sanchez, R., Horowitz, R., and Tomizuka, M., 2004, “Paper Sheet Control Using Steerable Nips,” 2004 American Control Conference Proceedings, Boston, MA, Jun. 30–Jul. 2, pp. 482–487. 关15兴 Sanchez, R., Ergueta, E., Fine, B., Horowitz, R., Tomizuka, M., and Krucinskic´, M., 2006, “A Mechatronic Approach to Full Sheet Control Using SteerAble Nips,” Fourth IFAC Symposium in Mechatronic Systems, Heidelberg, Germany, Sept. 12–15. 关16兴 Sanchez, R., Horowitz, R., and Tomizuka, M., “Full Sheet Control Using a Steerable Nips Mechanism,” IEEE/ASME Trans. Mechatron. 共to be published兲. 关17兴 Yun, X., and Sarkar, N., 1996, “Dynamic Feedback Control of Vehicles With Two Steerable Wheels,” 1996 IEEE International Conference on Robotics and Automation, pp. 3105–3110. 关18兴 Sastry, S. S., 1999, Nonlinear Systems: Analysis, Stability, and Control, Springer, New York. 关19兴 Elliot, J. G., and Gans, R. F., 2008, “Closed-Loop Control of an Underactuated Sheet Registration Device Using Feedback Linearization and Gain Scheduling,” IEEE Trans. Control Syst. Technol., 16, pp. 589–599. 关20兴 Isidori, A., 1995, Nonlinear Control Systems, 3rd ed., Springer, New York. 关21兴 Descusse, J., and Moog, C. H., 1985, “Decoupling With Dynamic Compensation for Strong Invertible Affine Non-Linear Systems,” Int. J. Control, 42共6兲, pp. 1287–1398. 关22兴 d’Andrea Novel, B., Campion, G., and Bastin, G., 1995, “Control of Noholonomic Wheeled Mobile Robots by State Feedback Linearization,” Int. J. Robot. Res., 14, pp. 543–559. 关23兴 Swaroop, D., Gerdes, J. C., Yip, P. P., and Hedrick, J. K., 2000, “Dynamic Surface Control for a Class of Nonlinear Systems,” IEEE Trans. Autom. Control, 45共10兲, pp. 1893–1899. 关24兴 Ergueta, E., Sanchez, R., Horowitz, R., and Tomizuka, M., 2007, “A Mechatronic Approach to Full Sheet Control Using Steer-Able Nips,” ASME International Mechanical Engineering Congress and Exposition, Seattle, WA, Nov. 11–15. 关25兴 Ergueta, E., Seifried, R., and Horowitz, R., 2008, “A Robust Approach to Dynamic Feedback Linearization for a Steerable Nips Mechanism,” 2008 ASME Dynamic Systems and Control Conference, Ann Arbor, MI, Oct. 20– 22.
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Rene Sanchez e-mail: [email protected]
Roberto Horowitz e-mail: [email protected]
Masayoshi Tomizuka e-mail: [email protected] Department of Mechanical Engineering, University of California, Berkeley, CA 94720
1
Convergence Analysis of a Steerable Nip Mechanism for Full Sheet Control in Printing Devices Current approaches for high speed color printers require sheets be accurately positioned as they arrive to the image transfer station (ITS). This goal has been achieved by designing and building a steerable nip mechanism, which is located upstream from the ITS. This mechanism consists of two rollers that not only rotate to advance the paper along the track, but also steer the paper in the yaw direction. This paper briefly reviews the design and experimental setup of the system, and focuses on the design and analysis of a controller that precisely corrects the lateral, longitudinal, and angular positions of the sheet. The control strategy used is based on linearization by state feedback with the addition of internal loops for the local control of the actuators. This paper also provides a methodology to tune the controller parameters so that the desired performance specifications are met. The success of this mechatronic approach is corroborated through simulation and experimental results, which show that the system is able to correct sheet errors and meet all the performance specifications. 关DOI: 10.1115/1.3117195兴
Introduction
Current approaches for paper path control require sheets to be accurately positioned as they arrive to the image transfer station 共ITS兲. This is achieved by using a registration device, located between the paper path and the ITS, which not only corrects for longitudinal, lateral, and angular errors, but also delivers the sheet on time to the ITS. Prior work on sheet control in a printer paper path has been mainly focused in developing control techniques for coordinating multiple actuated sections of the paper path, in order to correct for longitudinal interspacing errors among sheets, and synchronize the arrival of sheets to the ITS with its corresponding image 关1–7兴. Other works, such as in Refs. 关8–12兴, do consider the sheet’s lateral and skew position error corrections, but they fail to do so at large speeds and without marking the page. In this paper we present the control architecture of a mechatronic solution that corrects for sheet position errors at high speeds without damaging the page. This is achieved by using the steerable nip mechanism 关13兴 depicted in Fig. 1, whose design is described in detail in Refs. 关14–16兴. The problem of controlling paper trajectories with steerable nips is similar to the control of two-wheel robots, such as the one studied in Ref. 关17兴. However, not only does the two-wheel robot have one less degree of freedom than the steerable nip system, but also the control law proposed by Yun and Sarkar 关17兴 fails to account for singularities that arise when the steering angle of the wheels approaches zero. Moreover, in the case of the two-wheel robot, three inputs are needed to follow a reference trajectory. This is not the case with steerable nips, where four inputs are needed due to the flexibility of the paper; two inputs rotate and steer roller 1, whereas the other two inputs do the same for roller 2 共see Fig. 1兲. Similar to the two-wheel robot, the steerable nip mechanism is a nonlinear system with nonholonomic constraints. For the steerable nip system, two nonholonomic constraints come from nonslip conditions on the rollers, and the other two come from local veContributed by the Dynamic Systems, Measurement, and Control Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received December 12, 2007; final manuscript received February 17, 2009; published online December 9, 2009. Assoc. Editor: Robert Gao.
locities 共of the paper兲 being zero in the direction perpendicular to the rotation of the rollers. Additional details on the constraints of this particular system can be found in Ref. 关14兴. The control objective of the steerable nip device consists of correcting the position of the sheet on a horizontal plane while the sheet is moving in the longitudinal direction at all times. Since the page should move without getting damaged, it is also necessary to control the sheet’s amount of buckling. The control strategy used to achieve these goals is based on state feedback linearization 关18兴 with inner loops for the control of the roller’s rotational angular velocity and steering angular position. Recently, Elliot and Gans 关19兴 presented a control strategy for an underactuated steerable nip mechanism for printer sheet registration devices similar to the device described in this paper. However, since this mechanism does not consider the sheet’s amount of buckling, it has one less degree of freedom than the mechanism presented in this paper. Thus, the control problem described in Ref. 关19兴 resembles more than that of the two-wheel robot in Ref. 关17兴, and the control strategy presented there is significantly different from the one we present. The remainder of this paper is organized as follows. Section 2 briefly describes the steerable nip section and the experimental setup. Section 3 presents the mathematical model of the system. Sections 4 and 5 describe the control strategy and convergence analysis for the closed-loop system, respectively. Section 6 presents the methodology proposed to tune the controller gains. Simulation and experimental results are shown in Sec. 7. Finally, conclusions are stated in Sec. 8.
2
Experimental Setup
The steerable nip mechanism has been designed so that it can correct for sheet lateral position errors without having to move the actuators and without inflicting any damage on the page. This has been achieved by steering two rollers, which are underneath two backer balls. As seen in Fig. 2, each roller is driven by a servomotor 共referred as process direction actuator兲 attached to a rotating table, which is in turn steered by another servomotor 共referred as steering actuator兲 through a coupling. The sheet moves along a flat surface, and passes between the two backer balls and rollers, as shown in Figs. 2 and 3; note that the page moves in the direction of the arrow labeled vគ . Figure 4 is a photograph of the experimental system we have
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Process Motor 1
Top View
Image Transfer Station
1 Roller 1
.θ
2b - δ
Optical Sensors
.θ
2
C
v−
2
1
2b
φ
1
φ
2
Process Motor 2
Fig. 3 Steerable nips with paper buckle
Roller 2 Laser Sensors
Backer Ball
Side View
Paper Roller RotatingTable
Steering Motor
In order to determine the position, orientation, and the amount of buckling of the sheet, it is necessary to detect the edges of the page. As seen in Fig. 1, two laser sensors are located on the right hand side of the page, which are used to measure the lateral and angular positions of the page. Furthermore, to measure the longitudinal position of the page, five single photodiode 共optical兲 sensors, spaced 52 mm apart, are located along the process direction. It should be noticed that whereas we are able to obtain continuous measurements for lateral and angular positions of the sheet, we need to estimate its longitudinal position when the leading edge of the page is between two consecutive photodiodes. In this paper we
Fig. 1 Schematic of steerable nip fixture
built. As shown in Fig. 4, the page is delivered to the steerable nip section by a feeder unit 共located at the back of the picture兲. Then, while the page moves on top of the horizontal plate, its position is being corrected by the steerable nip mechanism, which is located below the plate; a photograph of the dc motors driving the rollers is shown in Fig. 5. Finally, the sheet is removed through an exit roller 共located at the front of the picture兲. It is assumed that the location of the exit roller in the experimental setup is the location of the entrance to the ITS. Thus, the performance of the controller is determined by the position and orientation of the paper as it arrives to the exit roller.
sheet
Fig. 4 Experimental setup
-v Backer ball roller
Backer ball roller Process Direction Motors
Steering Motors
Fig. 2 Sheet moving through the steerable nip mechanism
Fig. 5 DC motors used for the steerable nip mechanism
011008-2 / Vol. 132, JANUARY 2010
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x-
1 s
-. φ1 +-
-. φ2 +-
CFB (s)
+-
1 s
.. φ1d 1
τ1
1 s
.. φ2d 1
τ2
1 s
u1
P (s) P1
CFF (s)
P (s)
. θ2
P (s)
φ1
P2
CFB (s)
+
u2
P2
P2
CFF (s) 1 s
. φ2d
+
P1
+-
. φ1d
. θ1
P1
. θ2d
.. θ2d CFBL
CFF (s)
S1
φ1d
CFB (s)
+-
+
u3
Kinematics
.. θ1d
x- d
y
. θ1d
S1
S1
CFF (s) S2
φ2d 1 s
CFB (s)
+-
S2
+
u4
φ2
P (s) S2
Fig. 6 System block diagram for implemented system
estimate longitudinal position through the use of an open-loop observer based on the kinematic relations described in Sec. 3. The same observer is used to estimate the amount of buckling of the sheet.
3
␦˙ = r2 sin 2˙ 2 − r1 sin 1˙ 1
where r1 and r2 are the radii of the two rollers. As mentioned in Refs. 关15,16兴, a simple model that adequately describes both the process direction and steering actuator dynamics is given by
System Model
As described in Fig. 3, the steerable nip mechanism has two independent steering rollers located at points 1 and 2, which are separated by a distance 2b. The space-fixed coordinates of the system 共x , y , , ␦兲 locate the leading right corner of the sheet, point C, which will be used to track the position of the page. Note that x, y, and are the lateral, longitudinal, and angular positions of the sheet, respectively. The amount of buckling, ␦, is defined as the difference between the distance separating points 1 and 2, as measured along the paper, 2b − ␦, and along the straight line, 2b 共see Fig. 3兲. Thus, a negative ␦ represents the amount of buckling on a sheet, whereas a positive ␦ occurs when the paper stretches; stretching needs to be avoided at all times. Also note that the origin 共0,0兲 of the space-fixed frame is located in the middle of points 1 and 2. Furthermore, ˙ i represents the angular velocity of the rollers in the direction parallel to the sheet, and i represents their angular position in the direction perpendicular to the sheet 共for i = 1 , 2兲. The kinematic model of the system is derived so that the four nonholonomic constraints mentioned in Sec. 1 are satisfied at all times. This model, whose complete derivation can be found in Refs. 关15,16兴, is represented by the following equations: x˙ = −
y˙ =
r1y y cos 1˙ 1 + r2 cos 2 + sin 2 ˙ 2 2b 2b
冉
冊
共1兲
r2共x + b兲 r1共x − b兲 cos 1˙ 1 − cos 2˙ 2 2b 2b
共2兲
1 ˙ = 共r1 cos 1˙ 1 − r2 cos 2˙ 2兲 2b
共3兲
Journal of Dynamic Systems, Measurement, and Control
共4兲
¨ i + ␣ pi˙ i =  piV pi
共i = 1,2兲
共5兲
¨ i + ␣si˙ i = siVsi
共i = 1,2兲
共6兲
where V ji is the voltage input to the motor, and ␣ ji and  ji are actuator coefficients that depend on the inertias and rotational viscous damping coefficients of the different components of the steerable nip mechanism; subindices p and s stand for process direction and steering actuators, respectively, and subindex i corresponds to each of the two rollers. The complete dynamic system model is composed of Eqs. 共1兲–共6兲. We can further define the state vector as xគ ˙1 ˙ 2兴T, the input vector as uគ = 关x y ␦ 1 2 ˙ 1 ˙ 2 T = 关u1 u2 u3 u4兴 = 关V p1 V p2 Vs1 Vs2兴T, and the output vector as yគ = 关x y ␦兴T.
4
Control Strategy
The block diagram of the control system is shown in Fig. 6. As will be explained in the next paragraphs, the control strategy designed uses feedback linearization to linearize only the kinematics, and uses internal loops to locally control the actuator’s positions and velocities. This technique is an extension to the dynamic feedback linearization controller presented in Refs. 关20–22兴, since we need to integrate the outputs from the nonlinear linearization law CFBL. However, robustness is gained through the use of the internal loops. In Fig. 6, the block Kinematics is represented by Eqs. 共1兲–共4兲, the process direction actuators, P p1 and P p2 by Eq. 共5兲, and the steering actuators, Ps1 and Ps2 by Eq. 共6兲. Furthermore, noticing that if we differentiate the output vector yគ twice, we obtain JANUARY 2010, Vol. 132 / 011008-3
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˙1 ˙ 2 兴T yគ¨ = m共xគ 兲 + N共xគ 兲关¨ 1 ¨ 2
共7兲
where m共xគ 兲 is a nonlinear 4 ⫻ 1 vector and N共xគ 兲 is a nonlinear 4 ⫻ 4 matrix; we design the following feedback linearization control law CFBL:
关¨ 1d vគ =
冤
T −1 ¨ 2d ˙ 1d ˙ 2d 兴 = N 共xគ 兲共vគ 共xគ 兲 − m共xគ 兲兲
x¨d + 共Kx + x兲共x˙d − x˙兲 + Kxx共xd − x兲 y¨ d + 共Ky + y兲共y˙ d − y˙ 兲 + Kyy共y d − y兲
¨ d + 共K + 兲共˙ d − ˙ 兲 + K共d − 兲
␦¨ d + 共K␦ + ␦兲共␦˙ d − ␦˙ 兲 + K␦␦共␦d − ␦兲
冥
y¨ = vគ 共xគ 兲 − N共xគ 兲关˙ p1 ˙ p2 共˙ s1 + ⑀1兲 共˙ s2 + ⑀2兲 兴T
␥ pi =
共8兲
共␣ pi +  pi pi − pi兲pi
si =
 pi 共␣si + ␥sisi − si兲si si
␥ pi ; s
CFBsi共s兲 = si + ␥ pis;
CFFpi共s兲 =
冉 冊 冉 冊
1 ␣ pi 1+  pi s
1 ␣si CFFsi共s兲 = 1+ si s
共i = 1,2兲 共9兲
Convergence Analysis of Closed-Loop System
In this section we will prove the local convergence of the control system described in Sec. 4 for a sheet of finite length by linearizing the system error dynamics. Thus, we will first set the error dynamics in terms of the paper and actuator errors as well as their corresponding surface errors. Then we will linearize the error dynamics around a predefined desired trajectory, and will show that, by proper selection of the controller gains, the error vector of the linearized control system converges asymptotically to zero. Let us first define paper coordinate errors and actuator errors by ˜x = xd − x, ˜ = d − , pi = ˙ id − ˙ i,
¯˙ i − ˙ id ⑀i =
˜␦ = ␦ − ␦ d
共10兲
and let us also define the following surface errors: sx = ˜x˙ + x˜x, ˜˙ + ˜, s =
˜y˙ = − y˜y + sy
˜␦˙ = − ˜␦ + s ␦ ␦
sy = ˜y˙ + y˜y
˙ p2 = − p2 p2 + sp2 ˙ s1 = − s1s1 + ss1 ˙ s2 = − s2s2 + ss2 s˙x = − Kxsx + n11˙ p1 + n12˙ p2 + n13共˙ s1 + ⑀1兲 + n14共˙ s2 + ⑀2兲 s˙y = − Kysy + n21˙ p1 + n22˙ p2 + n23共˙ s1 + ⑀1兲 + n24共˙ s2 + ⑀2兲 s˙ = − Ks + n31˙ p1 + n32˙ p2 + n33共˙ s1 + ⑀1兲 + n34共˙ s2 + ⑀2兲 s˙␦ = − K␦s␦ + n41˙ p1 + n42˙ p2 + n43共˙ s1 + ⑀1兲 + n44共˙ s2 + ⑀2兲 s˙p1 = − 共␣ p1 +  p1 p1 − p1兲sp1 s˙p2 = − 共␣ p2 +  p2 p2 − p2兲sp2 s˙s1 = − 共␣s1 + s1s1 − s1兲ss1 s˙s2 = − 共␣s2 + s2s2 − s2兲ss2
⑀˙ 1 = −
¯˙ 1 ¯˙ 1 1 ␦ ␦ ⑀1 + ⌿ គ˙ + 1 ␦⌿ គ ␦t
⑀˙ 2 = −
¯˙ 2 ¯˙ 2 1 ␦ ␦ ⑀2 + ⌿ គ˙ + 2 ␦⌿ គ ␦t
sp1 = ˙ p1 + p1 p1,
sp2 = ˙ p2 + p2 p2
ss1 = ˙ s1 + s1s1,
ss2 = ˙ s2 + s2s2
共15兲
គ is where nij is the 共i , j兲 element of matrix N共xគ 兲 in Eq. 共7兲, and ⌿ defined by ˜ s ˜␦ s␦ 兴T ⌿ គ = 关˜x sx ˜y sy
˙ s␦ = ˜␦ + ˜␦
共14兲
˙ p1 = − p1 p1 + sp1
共i = 1,2兲
共i = 1,2兲
共13兲
˜x˙ = − x˜x + sx
˜y = y d − y
si = id − i
共i = 1,2兲
˜˙ = − ˜ + s
共i = 1,2兲
where and ␥ are controller gains, and ␣ and  are the actuator coefficients defined in Eqs. 共5兲 and 共6兲. As shown in Fig. 6, in order to use feedforward for steering position control, we estimate ¨ id through the use of a first order filter the steering acceleration ˙ id with gain i 共i = 1 , 2兲. If i is sufficiently small, the value of ˙ ¯ i. This technique, which is somewill be very close to that of times referred as dynamics surface control 共DSC兲, has been proposed by Swaroop et al. 关23兴. Note that the success of the complete control strategy depends on the invertibility of matrix N共x兲. It is shown in Refs. 关15,16兴 that this matrix is invertible as long as the sheet always moves in the longitudinal direction 共˙ 1 , ˙ 2 ⫽ 0兲.
5
共i = 1,2兲
the time derivatives of the errors in Eqs. 共10兲 and 共11兲 can be expressed as
where K and are the feedback linearization controller gains. We then use feedback plus feedforward to locally control the actuator’s rotational velocities ˙ i and steering positions i. These local controllers are given by CFBpi共s兲 = pi +
共12兲
If we further express vគ 共xគ 兲 from Eq. 共8兲 in terms of paper and surface errors 共Eqs. 共10兲 and 共11兲兲, and let the actuator controller gains ␥ pi and si in Eq. 共9兲 be equal to
共16兲
If we now define the desired trajectory by ˙ d, ␦˙ d兲 = 共0, t,0,0,0, ,0,0兲 共xd,y d, d, ␦d,x˙d,y˙ d, 共11兲
Combining now Eqs. 共7兲, 共8兲, and 共10兲 we obtain the closed-loop expression
共17兲
where is the nominal longitudinal velocity of the sheet, and linearize the error dynamics in Eqs. 共14兲 and 共15兲 around ˜x = ˜y ˜ = ˜␦ = p1 = p2 = s1 = s2 = sx = sy = s = s␦ = s = s = s = s = p1 p2 s1 s2 = ⑀1 = ⑀2 = 0, we obtain an expression of the form
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e˙គ 共t兲 = G共t兲eគ 共t兲
共18兲
where eគ 共t兲 is the error vector, and it is defined by ¯ ¯␦ ¯ p1 ¯ p2 ¯s1 ¯s2 ¯⑀ 兴T eគ 共t兲 = 关¯x ¯y
共19兲
B␦⑀ =
whose elements are
冋册 冋册 冋册 ˜x
¯x =
¯␦ =
sx
˜y
¯y =
,
˜ ¯= s
,
sy
B⑀ 共t兲 =
冋册 冋 册 冋 册 冋 册 冋 册 冋册 ˜␦
s␦
s1
¯s1 =
p1
¯ p1 =
,
s2
¯s2 =
,
ss1
sp1
p2
¯ p2 =
,
B⑀ p1共t兲 =
sp2
⑀1 , ¯⑀ = ⑀2
ss2
共20兲
Similarly, the time-varying matrix G共t兲 in Eq. 共18兲 is given by
G共t兲 =
冤
0
Ax 0 0 Ay
0
0
0
A
0
0
0
B⑀x
0
0
0
0
0
0
s1
s2
Bx p1共t兲 Bx p2共t兲 Bx s1
0
By p1 Bp1
By p2 Bp2
0
0
0
A␦
0
0
B␦
B␦
B␦⑀
0
0
0
0
Ap1
0
0
0
0
0
0
0
0
0
Ap2
0
0
0
0
0
0
0
0
0
As1
0
0
0
0
0
0
0
0
0
As2
0
B⑀ 共t兲 B␦⑀ B⑀ p1共t兲 B⑀ p2共t兲 B⑀ s1 B⑀ s2 A⑀
Bx⑀ 0
冥
B⑀ s1 =
Ax =
A =
Ap1 =
As1 =
冋 冋
冋 冋
1
0
− Kx
−
1
0
− K
− p1
1
0
− g10,10
− s1
1
0
− g14,14 A⑀ =
Bx p1共t兲 =
冋
0
Bp1
=
冋 冋
冋
0
g2,9 g2,10
Bx s1 =
By p1 =
册 册
− x
册 册
Ay =
,
A␦ =
,
Ap2 =
,
As2 =
− ⑀1
册
g2,13 −
0
0
0
g4,9 g4,10 0
0
g6,9 g6,10
1
0
− Ky
− ␦
1
0
− K␦
冋 冋
0
册 册
0
− g12,12
− s2
1
0
− g16,16
册
,
By p2 =
,
Bp2
=
册 册
0
0
g17,9 g17,10
g18,9 g18,10
g17,13 g17,14
g18,13 g18,14
0
B␦⑀ =
,
g18,5 g18,6
g18,1 g18,2
0
g18,7 g18,8
,
B⑀ p2共t兲 =
g17,11 g17,12
,
B⑀ s2 =
0
g18,11 g18,12 0
g18,15 g18,16
共23兲
¯x˙ = Ax¯x + Bx p1共t兲 ¯ p1 + Bx p2共t兲 ¯ p2 + Bx s1¯s1 + B⑀x¯⑀ ¯y˙ = Ay¯y + By p1¯ p1 + By p2¯ p2 ¯˙ = A ¯ + Bp1¯ p1 + Bp2¯ p2 共24兲
共25兲
¯ + B␦¯␦ + Bp1共t兲 ¯⑀˙ = A⑀¯⑀ + Bx⑀¯x + B⑀ 共t兲 ¯ p1 + B⑀ p2共t兲 ¯ p2 + B⑀ s1¯s1 ⑀ ⑀
册
0
0
g4,11 g4,12 0
g17,5 g17,6
g17,1 g17,2
¯˙ s2 = A ¯s2 s2
− 0 0
Bx⑀ =
0
¯˙ s1 = A ¯s1 s1
冋 册 冋 册 冋 册
B⑀x =
,
0
g2,11 g2,12
0 0 , −
0
g8,15 −
¯˙ p2 = A ¯ p2 p2
共22兲
冋
B␦s2 =
,
¯˙ p1 = A ¯ p1 p1
1
册
0
¯␦˙ = A ¯␦ + Bs1¯ + Bs2¯ + B⑀¯⑀ ␦ ␦ ␦ s1 ␦ s2
− p2
Bx p2共t兲 =
,
册 册
− y
g18,17 − ⑀2
冋
0
冋 冋
,
0
g8,13
Elements gi,j, above, depend on system parameters and controller gains; elements g2,9, g2,10, g2,11, g2,12, g17,3, g17,9, g17,10, g17,11, g17,12, g18,5, g18,6, g18,9, g18,10, g18,11, and g18,12 depend explicitly on time, and they do so linearly. This dependence on time comes from the definition of the desired trajectory of the sheet shown in Eq. 共17兲. In order to show the convergence of the control system errors, we will look at the dynamics of each of the elements of the error vector in Eq. 共19兲, and will provide their algebraic solutions. Thus, combining Eqs. 共19兲–共23兲 we obtain the following expressions:
共21兲
whose matrix components are
冋 册 冋 册 冋 册 冋 册 冋 册 冋 册 冋 册 冋 册 册 册 冋 冋
B␦s1 =
0
g6,11 g6,12
Journal of Dynamic Systems, Measurement, and Control
+ B⑀ s2¯s2
共26兲
The solutions for pi, si, 共i = 1 , 2兲 can then be easily obtained from Eq. 共25兲 as follows: ¯ p1共t兲 = eAp1t¯ p1共0兲 ¯ p2共t兲 = eAp2t¯ p2共0兲 ¯s1共t兲 = eAs1t¯s1共0兲 ¯s2共t兲 = eAs2t¯s2共0兲
共27兲
¯ 共t兲, and ¯␦共t兲 can be Subsequently, the solutions for ¯x共t兲, ¯y 共t兲, obtained from Eqs. 共24兲 and 共27兲, and the solution for ¯⑀共t兲 from Eqs. 共26兲 and 共27兲 as JANUARY 2010, Vol. 132 / 011008-5
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¯x共t兲 = eAxt¯x共0兲 +
+
冋冕 冋冕 冋冕
冋冕
册
0
t
册 册 冕
t
t
+
t
eAx共t−兲B⑀x¯⑀共兲d
冋冕
冋冕
0
册
t
eAy共t−兲By p1eAp1d ¯ p1共0兲
册
0
t
eAy共t−兲By p2eAp2d ¯ p2共0兲
0
¯ 共t兲 = eAt ¯ 共0兲 +
冋冕 冋冕
冋冕
t
eA共t−兲Bp1eAp1d
册
0
t
+
iv.
eAx共t−兲Bx s2eAs2d ¯s2共0兲 +
¯y 共t兲 = eAyt¯y 共0兲 +
册
¯ p1共0兲
¯␦共t兲 = eA␦t¯␦共0兲 +
eA␦共t−兲B␦s1eAs1d ¯s1共0兲
册 冕
0
t
t
eA␦共t−兲B␦s2eAs2d ¯s2共0兲 +
0
eA␦共t−兲B␦⑀¯⑀共兲d
0
共28兲 ¯⑀共t兲 = eA⑀t¯e共0兲 +
冕 冋冕 冋冕 冋冕
冕
t
eA⑀共t−兲Bx⑀¯x共兲d +
0
t
+
eA⑀共t−兲B␦⑀¯␦共兲d +
0
+
t
t
¯ 共兲d eA⑀共t−兲B⑀ 共兲
册
eA⑀共t−兲B⑀ p1共兲eAp1d ¯ p1共0兲
册
eA⑀共t−兲B⑀ p2共兲eAp2d ¯ p2共0兲 t
册 册
t
0
eA⑀共t−兲B⑀ s2eAs2d ¯s2共0兲
m
共29兲
Finally, we can see from Eqs. 共22兲, 共23兲, and 共27兲–共29兲 that by proper selection of the controller gains, we can make matrices Ax, Ay, A, A␦, Ap1, Ap2, As1, As2, and A⑀ Hurwitz, and thus ¯x共t兲, ¯ 共t兲, ¯␦共t兲, ¯ p1共t兲, ¯ p2共t兲, ¯s1共t兲, ¯s2共t兲, and ¯⑀共t兲 will converge ¯y 共t兲, to zero asymptotically.
6
Given the initial and final maximum error bounds, ji共m兲共0兲 and ji共m兲共T兲, respectively, and their corresponding surface error bounds s ji共m兲共0兲 and s ji共m兲共T兲 for j = s , p
and i = 1 , 2, use Eq. 共27兲 to calculate the required controller gains 共pi , pi兲 and 共si , ␥si兲. ii. Given the initial maximum error bounds, ˜y 共0兲 and ˜ m共0兲, m ˜ m共T兲, their the final maximum error bounds, ˜y m共T兲 and corresponding surface error bounds, sy共m兲共0兲, s共m兲共0兲, sy共m兲共T兲 and s共m兲共T兲, and the controller gains obtained in step 共i兲, use the second and third expressions in Eq. 共28兲 to calculate the required controller gains 共y , Ky兲, and 共 , K兲, respectively. iii. Given the initial and final maximum error bounds, ⑀i共m兲共0兲 and ⑀i共m兲共T兲, respectively, the controller gains obtained in steps 共i兲 and 共ii兲, and an initial guess for the controller gains 共x , Kx兲 and 共␦ , K␦兲, use Eq. 共29兲 to calculate the controller gains i for i = 1 , 2. iv. Given the initial maximum error bounds, ˜x 共0兲 and ˜␦ 共0兲, m m the final maximum error bounds, ˜x 共T兲 and ˜␦ 共T兲, the
eA⑀共t−兲B⑀ s1eAs1d ¯s1共0兲
0
+
t
0
0
0
+
冋冕
冕
Now, by looking at the expressions in Eqs. 共27兲–共29兲, we can develop a procedure to calculate the required controller gains for a sheet of finite dimensions moving at a prespecified nominal longitudinal velocity with some given initial and final state errors. ¯ 共t兲, ¯ pi共t兲, and ¯si共t兲 Whereas the gains corresponding to ¯y 共t兲, 共i = 1 , 2兲 can be obtained directly, those corresponding to ¯x共t兲, ¯␦共t兲, and ¯⑀共t兲 cannot, since the expressions for ¯x共t兲 and ¯␦共t兲 depend on
i.
册
t
At time T, when the sheet exits the nip section, the maximum errors must be, respectively, smaller than or equal to ˜ m共T兲, and ˜␦m共T兲. ˜xm共T兲, ˜y m共T兲,
¯⑀共t兲 and vice versa. Thus, we need to use the following iterative procedure.
eA共t−兲Bp2eAp2d ¯ p2共0兲
0
冋冕
m
eAx共t−兲Bx s1eAs1d ¯s1共0兲
0
+
The sheet has finite dimensions and moves with a nominal longitudinal velocity, . ii. The distance between the first and last optical sensors in Fig. 1 is given by L. Thus, the leading edge of the sheet will exit the steerable nip section at time T = L / . iii. The sheet has maximum initial errors ˜x 共0兲, ˜y 共0兲, ˜ m共0兲, m m and ˜␦ 共0兲.
eAx共t−兲Bx p2共兲eAp2d ¯ p2共0兲
0
+
i.
eAx共t−兲Bx p1共兲eAp1d ¯ p1共0兲
0
+
册
t
Controller Design Methodology
Since the page moves at a constant nominal longitudinal velocity, , the leading edge of the sheet will enter the ITS at a prespecified time, T. Therefore, it is necessary to design a feedback system that will reduce all sheet positions and orientation errors to some prespecified level within the allowable control time T. We will now state these control specifications more precisely as follows.
v.
m
corresponding surface error bounds, sx共m兲共0兲, s␦共m兲共0兲, sx共m兲共T兲, and s␦共m兲共T兲, the controller gains initially guessed for 共x , Kx兲 and 共␦ , K␦兲 in step 共iii兲, and the controller gains obtained in steps 共i兲–共iii兲, use the first and fourth ˜ 共T兲兩 and expressions in Eq. 共28兲 to verify that the norms 兩x 兩˜␦共T兲兩 are smaller than ˜xm共T兲 and ˜␦m共T兲, respectively. Iterate between steps 共iii兲 and 共iv兲 if necessary.
As stated in step 共v兲, even though this procedure requires some iteration, it should be noted that the initial guesses mentioned in step 共iii兲 can be obtained from a simplified control system, which is described in Ref. 关24兴. The simplification consists in assuming that whereas we still control the velocity of the process direction motors through the internal loops shown in Fig. 6, inputs u3 and ˙ 1 = u3 ; ˙ 2 = u4兲, which are u4 are the steering angular velocities 共 obtained directly from the feedback linearization law in Eq. 共8兲. They are then integrated once and fed directly as inputs to the kinematics block in Fig. 6. In this way, we would have inner loops for the process direction actuators, but not for the steering motors. Using these assumptions, the control gain synthesis problem is simplified not only because of the reduction in the inner loops, but also because we no longer need the first order filters shown in Fig. 6. The simplified system only has controller gains 共x , Kx兲,
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Table 1 Maximum allowable initial and final state error bounds ˜, for simulation test. The unit for x˜, y˜, and ␦˜ is m; the unit for εs1, and εs2 is rad; and the unit for εp1, εp2, ⑀1, and ⑀2 is rad/s. Initial errors
gitudinal velocity of 0.5 m/s from the initial to the final state errors shown in Table 1, where T = 0.42 s represents the allowed time for control action. Based on the initial and final maximum errors shown in Table 1 and the methodology presented in Sec. 6, the following controller gains were obtained: 共x , Kx兲 = 共80, 4.5兲, 共y , Ky兲 = 共80, 8.0兲, 共 , K兲 = 共80, 4.9兲, 共␦ , K␦兲 = 共95, 4.0兲, 共p1 , p1兲 = 共100, 0.26兲, 共s1 , ␥s1兲 = 共100, 9.31兲, 共s2 , ␥s2兲 共p2 , p2兲 = 共100, 0.26兲, = 共100, 9.31兲, and 共1 , 2兲 = 共0.0042, 0.0013兲. Figure 7 shows the simulation results using these controller gains. In this figure, the horizontal dotted lines mark the final state errors at the allowable control time, T, which is in term marked by the vertical dotted lines. It should be noticed the efficacy of the methodology pre˜ , and ˜␦ are exactly reduced to the sented in Sec. 6, since ˜x, ˜y , prescribed limits shown in Table 1 within the allowed control time. Furthermore, we have also performed experimental tests in the system shown in Fig. 4. It should be noticed, however, that the actual experimental fixture cannot handle initial errors as large as those in Table 1 due to limitations in the setup design 共range of the laser sensors for lateral and angular measurements in Fig. 1兲. Thus, based on the initial and final errors shown in Table 2 and the methodology presented in Sec. 6, the following controller gains were obtained: 共x , Kx兲 = 共80, 12.5兲, 共y , Ky兲 = 共80, 13.2兲, 共 , K兲 = 共80, 11.2兲, 共␦ , K␦兲 = 共80, 7.9兲, 共p1 , p1兲 = 共100, 0.26兲, 共p2 , p2兲 = 共100, 0.26兲, 共s1 , ␥s1兲 = 共100, 9.31兲, 共s2 , ␥s2兲 = 共100, 9.31兲, and 共1 , 2兲 = 共0.0042, 0.0013兲. As shown in Fig. 8,
Final errors
˜ 共0兲兩 ⱕ˜xm共0兲 = 0.008 兩x ˜ 共0兲兩 ⱕ ˜y m共0兲 = 0.040 兩y ˜ 共0兲兩 ⱕ ˜ m共0兲 = 0.025 兩 ˜ ˜ 兩␦共0兲兩 ⱕ ␦m共0兲 = 0.0001 兩 p1共0兲兩 ⱕ p1共m兲共0兲 = 0.1 兩 p2共0兲兩 ⱕ p2共m兲共0兲 = 0.1 兩s1共0兲兩 ⱕ s1共m兲共0兲 = 0.1 兩s2共0兲兩 ⱕ s2共m兲共0兲 = 0.1 兩⑀1共0兲兩 ⱕ ⑀1共m兲共0兲 = 0.1 兩⑀2共0兲兩 ⱕ ⑀2共m兲共0兲 = 0.1
˜ 共T兲兩 ⱕ˜xm共T兲 = 0.0013 兩x ˜ 共T兲兩 ⱕ ˜y m共T兲 = 0.0016 兩y ˜ 共T兲兩 ⱕ ˜ m共T兲 = 0.0035 兩 ˜ ˜ 兩␦共T兲兩 ⱕ ␦m共T兲 = 0.00002 兩 p1共T兲兩 ⱕ p1共m兲共T兲 = 0.01 兩 p2共T兲兩 ⱕ p2共m兲共T兲 = 0.01 兩s1共T兲兩 ⱕ s1共m兲共T兲 = 0.01 兩s2共T兲兩 ⱕ s2共m兲共T兲 = 0.01 兩⑀1共T兲兩 ⱕ ⑀1共m兲共T兲 = 0.01 兩⑀2共T兲兩 ⱕ ⑀2共m兲共T兲 = 0.01
共y , Ky兲, 共 , K兲, 共␦ , K␦兲, 共p1 , p1兲, and 共p2 , p2兲, and they can be computed without any iteration. Furthermore, it should be noted that in most cases we only need one or two iterations between steps 共iii兲 and 共iv兲 before we arrive at the required controller gains for the desired specifications.
7
Simulation and Experimental Results
In order to determine the efficacy of the controller developed, we first performed simulation tests for a letter-sized sheet moving along the steerable nip section length, 0.2 m, at the nominal lon-
Lateral Position Error
xe(mm)
8 6 4 2 0
0
0.1
0.2
0.3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.4
0.5
0.6
0.7
0.8
0.9
1
Longitudinal Position Error
ye(mm)
40 30 20 10 0
Angular Position Error
e
φ (rad)
0.02
0.01
0
0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.5
0.6
0.7
0.8
0.9
1
Error in Buckling
0.05
δe(mm)
0 −0.05 −0.1 −0.15 −0.2
Time (sec) Fig. 7 Simulation results for large initial errors. The vertical dotted lines denote the allowable control time, T, and the horizontal lines denote the final errors at time T.
Journal of Dynamic Systems, Measurement, and Control
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Table 2 Maximum allowable initial and final state error bounds for the experimental test. The unit for x˜, y˜, and ␦˜ is m; the unit ˜ , εs1, and εs2 is rad; and the unit for εp1, εp2, ⑀1, and ⑀2 is for rad/s. Initial errors
Final errors
˜ 共0兲兩 ⱕ˜xm共0兲 = 0.004 兩x ˜ 共0兲兩 ⱕ ˜y m共0兲 = 0.0074 兩y ˜ 共0兲兩 ⱕ ˜ m共0兲 = 0.025 兩 兩˜␦共0兲兩 ⱕ ˜␦m共0兲 = 0.0001 兩 p1共0兲兩 ⱕ p1共m兲共0兲 = 0.1 兩 p2共0兲兩 ⱕ p2共m兲共0兲 = 0.1 兩s1共0兲兩 ⱕ s1共m兲共0兲 = 0.1 兩s2共0兲兩 ⱕ s2共m兲共0兲 = 0.1 兩⑀1共0兲兩 ⱕ ⑀1共m兲共0兲 = 0.1 兩⑀2共0兲兩 ⱕ ⑀2共m兲共0兲 = 0.1
˜ 共T兲兩 ⱕ˜xm共T兲 = 0.00026 兩x ˜ 共T兲兩 ⱕ ˜y m共T兲 = 0.0005 兩y ˜ 共T兲兩 ⱕ ˜ m共T兲 = 0.001 兩 兩˜␦共T兲兩 ⱕ ˜␦m共T兲 = 0.00002 兩 p1共T兲兩 ⱕ p1共m兲共T兲 = 0.01 兩 p2共T兲兩 ⱕ p2共m兲共T兲 = 0.01 兩s1共T兲兩 ⱕ s1共m兲共T兲 = 0.01 兩s2共T兲兩 ⱕ s2共m兲共T兲 = 0.01 兩⑀1共T兲兩 ⱕ ⑀1共m兲共T兲 = 0.01 兩⑀2共T兲兩 ⱕ ⑀2共m兲共T兲 = 0.01
by using these controller gains, the system was able to reduce the sheet’s initial position errors to prescribed levels in about 0.3 s. Note that the position increases constantly because the sheet moves in the longitudinal direction at all times. The discrepancies observed between simulation and experimental results can be attributed to sensor noise and model parameter uncertainties, such as friction coefficients. Particularly, Ergueta et al. 关25兴 discussed the possible existence of unmodeled dynamics in the actuators, and presented a proof of the robustness of this control strategy to such uncertainties.
8
Conclusion
In this paper we have presented an innovative design that permits a swifter correction of lateral, longitudinal, and angular position errors in a paper path control system for xerographic and printing devices. This mechanism accomplished this task by having steerable nips. In order to correct the sheet position errors we have used a controller based on dynamic feedback linearization 关18,20–22兴 with the addition of internal loops for the control of the process direction velocity and steering position of the rollers; for these inner loops we have used feedback plus feedforward control. In addition, not only have we provided a convergence analysis for the controller implemented, but also we have described a design methodology to determine the controller gains. Simulation and experimental results show that by using the controller gains obtained from the methodology previously mentioned, it is possible to drive a sheet from an initial state with nonzero longitudinal velocity to a final state, also with nonzero longitudinal velocity in a very short time.
Acknowledgment This work was supported by the National Science Foundation under Grant No. CMS 0301719, and by the financial support and collaboration with Xerox Corporation. In particular, the authors thank Dr. Martin Krucinski for his numerous critical remarks and suggestions during the development of this project.
Lateral Position Error
xe(mm)
4
Experimental Simulation
3 2 1 0 0
0.05
0.1
0.15
0.2
0.25
0.3
Longitudinal Position 200
y(mm)
150 100
Experimental Simulation
50 0 −50
0
0.05
0.1
0.15
0.2
0.25
0.3
Angular Position Error
φe(rad)
0 −0.02
Experimental Simulation
−0.04 0
0.05
0.1
0.15
0.2
0.25
0.3
Error in Buckling
δe(mm)
1
Experimental Simulation
0.5 0 −0.5 −1
0
0.05
0.1
0.15
0.2
0.25
0.3
time(sec) Fig. 8 Experimental and simulation results for small initial errors
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References 关1兴 Krucin´ski, M., Cloet, C., Horowitz, R., and Tomizuka, M., 1998, “Interobject Spacing Control and Controllability of a Manufacturing Transportation System,” 1998 American Control Conference, Jun., Vol. 2, pp. 1259–1265, Session No. WA03-4. 关2兴 Cloet, C., Krucin´ski, M., Horowitz, R., and Tomizuka, M., 1999, “A Hybrid Control Scheme for a Copier Paperpath,” 1999 American Control Conference, San Diego, CA, Jun., Vol. 3, pp. 2114–2118. 关3兴 Cloet, C., Krucin´ski, M., Horowitz, R., and Tomizuka, M., 1998, “Intersheet Spacing Control and Controllability of a Copier Paperpath,” 1998 IEEE Conference on Control Applications, Trieste, Italy, Sept., Vol. 3, pp. 726–730. 关4兴 Krucin´ski, M., Cloet, C., Horowitz, R., and Tomizuka, M., 2000, “A Mechatronics Approach to Copier Paperpath Control,” First IFAC Conference on Mechatronic Systems, Darmstadt, Germany, Sept. 关5兴 Krucinski, M., 2000, “Feedback Control of Photocopying Machinery,” Ph.D. thesis, University of California, Berkeley, CA. 关6兴 Cloet, C., 2001, “A Mechatronics Approach to Copier Paperpath Design,” Ph.D. thesis, University of California, Berkeley, CA. 关7兴 de Best, J. J. T. H., Bukkems, B. H. M., van de Molengraft, M. J. G., Heemels, W. P. M. H., and Steinbuch, M., 2008, “Robust Control of Piecewise Linear Systems: A Case Study in Sheet Flow Control,” Control Eng. Pract., 16共8兲, pp. 991–1003. 关8兴 Williams, L. A., deJong, J. N. M., Dondiego, M., and Savino, M. J., 1997, “Sheet Registration and Deskewing Device,” U.S. Patent No. 5,678,159. 关9兴 Lofthus, R. M., 1986, “Apparatus and Method for Combined Deskewing and Side Registering,” U.S. Patent No. 4,971,304. 关10兴 Kamprath, D. R., and Malachowski, M. A., 1990, “Translating Electronic Registration Systems,” U.S. Patent No. 5,094,442. 关11兴 Castelli, V. R., deJong, J. N. M., Williams, L. A., and Wolf, B. M., 1986, “Agile Lateral and Skew Sheet Registration Apparatus and Method,” U.S. Patent No. 5,697,608. 关12兴 Tanaka, N., Fukumoto, H., Arimoto, K., and Iwashita, Y., 1997, “Skew Correction Mechanism for Thermal Transfer Type Color Printers,” International Conference on Micromechatronics for Information and Precision Equipment, Tokyo, Japan, Jul., pp. 635–638. 关13兴 Hwang, S. S., 2002, “Sheet Registration and Deskewing System With Independent Drives and Steering,” U.S. Patent No. 6,634,521.
Journal of Dynamic Systems, Measurement, and Control
关14兴 Sanchez, R., Horowitz, R., and Tomizuka, M., 2004, “Paper Sheet Control Using Steerable Nips,” 2004 American Control Conference Proceedings, Boston, MA, Jun. 30–Jul. 2, pp. 482–487. 关15兴 Sanchez, R., Ergueta, E., Fine, B., Horowitz, R., Tomizuka, M., and Krucinskic´, M., 2006, “A Mechatronic Approach to Full Sheet Control Using SteerAble Nips,” Fourth IFAC Symposium in Mechatronic Systems, Heidelberg, Germany, Sept. 12–15. 关16兴 Sanchez, R., Horowitz, R., and Tomizuka, M., “Full Sheet Control Using a Steerable Nips Mechanism,” IEEE/ASME Trans. Mechatron. 共to be published兲. 关17兴 Yun, X., and Sarkar, N., 1996, “Dynamic Feedback Control of Vehicles With Two Steerable Wheels,” 1996 IEEE International Conference on Robotics and Automation, pp. 3105–3110. 关18兴 Sastry, S. S., 1999, Nonlinear Systems: Analysis, Stability, and Control, Springer, New York. 关19兴 Elliot, J. G., and Gans, R. F., 2008, “Closed-Loop Control of an Underactuated Sheet Registration Device Using Feedback Linearization and Gain Scheduling,” IEEE Trans. Control Syst. Technol., 16, pp. 589–599. 关20兴 Isidori, A., 1995, Nonlinear Control Systems, 3rd ed., Springer, New York. 关21兴 Descusse, J., and Moog, C. H., 1985, “Decoupling With Dynamic Compensation for Strong Invertible Affine Non-Linear Systems,” Int. J. Control, 42共6兲, pp. 1287–1398. 关22兴 d’Andrea Novel, B., Campion, G., and Bastin, G., 1995, “Control of Noholonomic Wheeled Mobile Robots by State Feedback Linearization,” Int. J. Robot. Res., 14, pp. 543–559. 关23兴 Swaroop, D., Gerdes, J. C., Yip, P. P., and Hedrick, J. K., 2000, “Dynamic Surface Control for a Class of Nonlinear Systems,” IEEE Trans. Autom. Control, 45共10兲, pp. 1893–1899. 关24兴 Ergueta, E., Sanchez, R., Horowitz, R., and Tomizuka, M., 2007, “A Mechatronic Approach to Full Sheet Control Using Steer-Able Nips,” ASME International Mechanical Engineering Congress and Exposition, Seattle, WA, Nov. 11–15. 关25兴 Ergueta, E., Seifried, R., and Horowitz, R., 2008, “A Robust Approach to Dynamic Feedback Linearization for a Steerable Nips Mechanism,” 2008 ASME Dynamic Systems and Control Conference, Ann Arbor, MI, Oct. 20– 22.
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