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DOI: 10.1007/s12541-014-0394-x
Design, Modeling, Control and Experiment for a 2-DOF Compliant Micro-Motion Stage Yangmin Li1,2,#, Shunli Xiao1, Longquan Xi2, and Zhigang Wu1 1 Department of Electromechanical Engineering, University of Macau, Taipa, Macao 2 School of Mechanical Engineering, Tianjin University of Technology, Tianjin, China # Corresponding Author / E-mail:
[email protected], TEL: +853-8397-4462, FAX: +853-2883-8314 KEYWORDS: Double four-bar prismatic joints, Compliance matrix method, Finite element analysis, Model reference adaptive PID
A new XY micro-motion stage is proposed with the double four-bar prismatic joints to transfer linear motions and mechanical displacement amplifier. The compliance models of the amplifier, the prismatic joints, and the whole stage are established based on the flexibility matrix method. The simulation is made by using finite element analysis ANSYS software. The output cross-talk is about 2% and the parasitic motions of the input points in the other limbs are less than 1.8%, which shows a good decoupling property. The mechanical prototype is fabricated, the experimental results show that the input/output of the stage has a very good linearity, the ratio of output displacement to input displacement is 5.06 and the working range is 156 µm × 156 µm. To tackle for the serious hysteresis nonlinear problems of the stage, model reference adaptive PID controller is designed, the final resolution can reach ±0.2 µm. Manuscript received: September 24, 2013 / Revised: February 18, 2014 / Accepted: February 26, 2014
1. Introduction Micro stage is usually constructed by flexure based mechanism, which is driven by piezoelectric actuators, therefore the micro/nano scale precision can be realized, so it has been paid much attention from academia and industry in recent years.1,2 The requirement of precision manufacture and assembly demands the microstage with a large workspace, but the stroke of piezoelectric actuator is within the range of a few to dozens of microns, an amplification device must be designed and connected with the actuators to enlarge the stroke.3 The shear stress actuated on the piezoelectric elements should be kept as small as possible in order to avoid any failures happened on the PZT stacks. Flexure hinge-based mechanism is a new kind of mechanism which is manufactured by cutting a piece of mental material into different shapes, the elastic deformation of the flexure hinges can help realizing rotational or translational motion replacing traditional joints, therefore the frictions between two motional parts will not exist, which can eliminate the flaws of creep and backlash during motion. It also has the features of no clearance and no lubrication needed, hence a high resolution can be realized.4 It is usually applied in micro-amplification to provide an amplified displacement of hundreds of microns. During the parallel flexure mechanism design, input and output decoupled design should be considered firstly, which may not only reduce the
© KSPE and Springer 2014
damages on piezoelectric actuators, but also try to eliminate parasitic motions in cross axes directions.5-8 Force sensors can be installed on micro stage to provide possible force feedback control information.9 It is found that the magnetic drive and piezoelectric drive can be integrated together to realize macro/micro control.10 Although the compliant flexure-based mechanism system driven by PZT actuators has many advantages, the nonlinear and hysteresis characteristics are main flaws.11 The nonlinear and hysteresis characteristics of the actuators make the controller design very complex.12 It is very hard to exploit a normal PID controller to control it with satisfied control results, therefore some advanced control strategies are proposed for controlling the system. If the mechanical system is decoupled, a single-input single-output (SISO) controller is enough for each direction of X and Y. For the decoupled system, the advantage is that the system can be modeled and controlled in each direction separately. Since model reference adaptive (MRAC) PID controller owns the ability of great robust characteristics for nonlinear, hysteresis and variable parameters systems,13 it is adopted in this research to control the stage. In this paper, the structural design of the micro-motion stage is improved through studying the working principle of the stage in detail, the compliance models of the components and the whole stage are obtained based on the flexibility matrix method,14 then the static and
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Fig. 1 A scheme of double four-bar prismatic joint
dynamic performances of the stage are analyzed via FEA. After controller is designed,15-18 extensive experiments are conducted to test the performance of the micro stage.
2. Mechanical Structural Design 2.1. A decoupled XY stage The XY stages can be designed with series or parallel form. Comparing with parallel stage, series structure is easier to design and control but its precision is not high. This is the reason why most micromotion stages with high precision and resolution will use parallel structure types. However, most proposed parallel stages have coupled motions, which make the kinematical model complex and the precise control difficult to be realized. An ideal flexure prismatic joint is adopted to eliminate the parasitic motions as shown in Fig. 1. A rigid double four-bar mechanism is not able to move, but in the compliant mechanism, the small axial elongation of the flexure hinge is permissible. Due to the symmetric property, when a horizontal force Fx is exerted at the motional part of the P joint, the motional part can move along X direction without parasitic motion in the Y direction. Due to the high stiffness in the vertical direction of the P joint, the motion part hardly moves in the vertical direction under the force Fy. As shown in Fig. 2, the mobile stage is connected to transverse P joints in the four limbs, so the output coupled motions are eliminated. For example, when the stage is actuated at the driving points within the limb B, the transverse P joint within limb A has the same displacement as the mobile stage, however, the vertical P joint within limb A hardly moves, so the amplifier connected to the vertical P joint in limb A hardly moves, i.e. the motion of one actuator dosen’t affect the motion of the other actuator, hence the XY stage has a good input decoupling property. 2.2 Displacement amplifier Comparing with other types of linear actuators, PZT has major advantages in terms of large blocking force, high stiffness, fast response, and compact size. The main drawback of PZT is its small travel stroke, in this paper, we adopt a proper amplification mechanism which has the merits including a compact size and a large amplification ratio. As shown in the Fig. 3(b), When the amplification mechanism is actuated, FF and FG are in the identical horizontal line, and FF = −FG. Due to the symmetrical structure, the output point E moves only in Y direction, the input points F and G move in X and Y directions and don’t
Fig. 2 A decoupled XY stage with actuators
Fig. 3 The amplification mechanism
rotate around Z axis. Because there is no rotation at the input points, so the actuators connected to the input points only bear compressive force and don’t bear shear stress. Those numbers have the following relationship: 2dFy = 2dGy = dEy, dFx = −dGx, dEy = λdFx. Here, λ is the amplification ratio, which will be elaborated in the section of the input compliance model.
3. Compliance Model of the Micro Stage 3.1 Flexibility matrix method The compliance matrix of a single right circular can intuitively express the lumped compliance model with the consideration of 6-D compliance in place. The stage adopts the right circular flexure hinges, so its compliance models including output stiffness and input stiffness can be obtained based on matrix method. The XY stage only moves in XY plane, so the lumped model of the stage only considers 3-D compliance in XY plane. As shown in Fig. 4, we will introduce the flexibility matrix of a single hingein XY plane. When an external force FOi in XY plane is exerted at the point Oi, the displacement DOi of the point Oi with respect to the ground can be calculated by
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Fig. 4 A single circular hinge with two different coordinates
dx c1 0 c2 fx DO = dy = 0 c3 0 fy = CO FO i i i θz
(1)
c2 0 c4 mz
where COi represents the compliance of the free-end Oj with respect to the other fixed end, C1 = dx/fx, c2 = −dx/Mz, c3 = dy/fy, c4 = θz/Mz, the values of ci (i = 1, 2, 3, 4) are mentioned in reference,14 which has the more details about them, we define COi =Ch. According to the transformation of force and displacement from the coordinate Oi to the coordinate Oj, the compliance of the free-end Oj with respect to the ground can be derived by Oj T
Oj
C Oj = T Oi C Oi ( T Oi )
Fig. 5 The P joints in limb A
amplCE
transCA
Oj TO i
r11 r12 0 1 0 –py Oj Oj Oj TO = r21 r22 0 0 1 p = ROi POi i x 0 0 r33 0 0 1 Oj
3.2 The compliance models of the amplifier and P joints As shown in Fig. 3(a), due to the symmetrical structure, we divide the amplifier into the left half part and the right half part, which connects the point E to the ground in parallel. In the left half part, the point E is connected to the fixed ground through hinges 2, 1, 5 and 6 in series and hinges 3, 4, 7, and 8 in series, the two branches are in parallel. The compliance of the left half part at point E can be derived by E
E
T
E
E
T
E
E
T
E
E
T
E
E
T
E
E
T
E
E
T
E
E
T
left1CE
= TO6 Ch ( TO6 ) + TO5 Ch ( TO5 ) + TO1 Ch ( TO1 ) + TO2 Ch ( TO2)
left2CE
= TO6 Ch ( TO6 ) + TO5 Ch ( TO5 ) + TO1 Ch ( TO1 ) + TO2 Ch ( TO2)
(4)
vertCA
–1
–1 –1
= ( ( left1CE ) + ( left2CE ) )
E –1
E –1 –1
E –1
E –1
E –1
E –1 –1
E –1
= [( C13 ) + ( C14 ) + ( C15 ) + ( C16 ) ]
limACO
In the same way, the compliance of the right half part at point E can be obtained. Because the point E is connected to the fixed ground through the left half part and the right part in parallel, the compliance of the amplifier at point E can be derived by
(9)
O
–1 – 1
–1
= [( tranCO + vertCO ) + ( amplCO ) ]
O T
O
O T –1
O T –1 – 1
O
(10)
= [( TA tranCA ( TA ) + TA vertCA ( TA ) ) + ( TE amplCE ( TE ) ) ]
Accordingly, the compliances of limb B, C and Dcan be derived. The output compliance of the whole stage can be calculated by
(5)
(6)
(8)
3.3 The output compliance model The output compliance is defined as the compliance at the point O, where the external force is exerted,is related to the ground. Because of the double symmetric property, we only select the limb A for the purpose of compliance model analysis. In the limb A, the output point O is connected to the ground through the branch consisting of transverse P joint and vertical P and the branch of amplifier in parallel, The compliance of the limb A at the point O can be derived by
stagCO
leftCE
E –1
= [( C9 ) + ( C10) + (C11 ) + ( C12 ) ]
In the same way, the compliance of the vertical P joint at point A can be derived by
(3)
where ROi is the rotation matrix for the rotation from coordinate Oj to coordinate Oi, rij is the entry in the i-th row and thej-th column of the Oj Oj rotation matrix ROi ; POi is the translation matrix, (px, py) is the vector of OiOj expressed in the coordinate Oi.
(7)
As shown in Fig. 5, when we calculate the compliance of the transverse P joint at point A, we consider that E is fixed. According to the parallel connection between the hinge 9, 10, 11 and 12, the compliance of them at coordinate A with respect to the E deemed to be fixed is transCA, which can be derived by
(2)
is the compliance transformation from Oi to Oj, which takes on the following form:
–1 –1
–1
= [( leftCE ) + ( righCE ) ]
–1
–1
–1
–1 –1
= [( limACO ) + ( limBCO ) + ( limCCO ) + ( limDCO ) ]
(11)
The stiffness and the compliance have such a relationship as: K = C-1. Similarly, stagKO represents the stiffness of the stage at the point O with respect to the ground.
3.4 The input compliance model The input compliance model of the stage is defined as the compliance
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Fig. 6 Stiffness model of the stage actuated in limb A
Fig. 7 The force distributing diagram of the amplifier
of the input point with respect to the ground. When the XY stage is actuated in the limb A, the stiffness mode of the stage is shown in the Fig. 6. The limb B, limb C and limb D are connected to the output point O in parallel, so the compliance of the part at the point O can be derived by limBCDCO
–1
–1 –1
–1
= [(limBCO ) + ( limCCO ) + ( limDCO ) ]
(12)
The part consisting of limb B, limb C and limb D is connected to the transverse P joint in series, then its compliance at point E can be derived by limBCD-tranCE
= limBCDCE + tranCE
(13)
The part consisting of transverse P joint, limb B, limb C, and limb D is connected to the vertical P joint in parallel, the compliance of the stage except for the amplifier in the limb A at the point E can be derived by –1
–1 –1
CE = [( limBCD-tranCE ) + ( vertCE) ]
Fig. 8 Displacements of the stage actuated in limb B (a)Vector displacement of the stage (b) Displacements of the output point and the input points within limb B (c) Displacements of the input points within limb A
d′Fx denotes the displacement of the point F with respect to E in X direction, d′Fy denotes the displacement of the point F with respect to E in Y direction. ⎛ d′ Fx⎞ ⎛ 0.5Fin ⎞ F⎜ ⎜ d′ ⎟ = ⎟ C ⎜ Fy⎟ quar E ⎜ 0.5FEy⎟ ⎝θ ′ ⎠ ⎝ M ⎠ Fz
(15)
Fz
(14)
As shown in Fig. 7(b), when considering the input compliance of the amplifier, we assume that the point E is fixed, so it has a double symmetrical structure. For illustration, one quarter is picked out. Based E on the matrix method, we can get quarCF which is the compliance of the quarter, with respect to the coordinate E. As shown in Fig. 7(b),
where θ ′Fz = 0, dEy = –2d′ Fy = −FEyd22, 2d′ Fx = dFx, d22 is a factor of CE. The input compliance of the stage can be derived by F Fin Cin = ------in- = ----------dFx 2d′ Fx
(16)
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Fig. 10 Mode shapes (a) The first order mode (b) The second or-der mode
Fig. 9 Displacements of the stage actuated in limb B and A (a) Vector displacement of the stage (b) The X-direction displacements of the output point (c) The Y-direction displacements of the output point
Accordingly, the relationship between the output displacement dEy and the input displacement dFx can be obtained by dEy = Aamp dFx
(17)
where Aamp represents the amplification ratio.
4. Simulation Results via FEA The 3-D model of the stage is establishedby PRO-E software, then it is input into ANSYS where it is meshed automatically and solved. AL7075-T6 is selected as the manufacturing material for the XY stage, its material parameters are: Young’s modulus = 71.7 Gpa, Yield strength = 503 Mpa, Poisson’s ratio = 0.33, Density = 2810 kg/m3.
A set of vertical forces with 200, 400, 600, 800, and 1000 N are exerted at the input points H and I respectively in the limb B, while the limb A is not actuated.The displacements of the points H, I and O are listed in Fig. 8(b), and the displacements of the points F and G are listed in Fig. 8(c). According to the section of displacement amplifier, when the stage is actuated in the limb B, the Y-direction displacement of the point O is defined as the parasitic motion, which should be derived. The displacements of the point F and G determine the value of input coupling. The ratio between the X-direction displacement of the point O and the Y-direction displacements of the point H and I represents the amplification ratio. From the Fig. 8(b), we can see: the output displacement increases linearly with the increasing force, in other words, the input/output of the stage has a good linearity; the output parasitic motion d-Oy is less than 0.2 percent of d-Ox, showing a good output decoupling property; the amplification ratio is 5.23. From the Fig. 8(c), we can see that the displacements of the input points in the limb A are very small, the input cross-talk is less than 1.8%.While the limb B and A are actuated at the same time, different forces are exerted at the input-end in the limb B and A respectively to validate the output decoupling property. The relationship between the X, Y directions and the whole mobile stage displacements of the output-end and the driving force exerted is shown in the Fig. 9(b). From the Fig. 9(b), we can see that the proportional relationship between the X and Y directions displacements of the outputend are almost the same, but the X and Y directions displacements of
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Table 1 Structure parameters of the mechanical amplifier r (mm) 2.5
t (mm) 0.4
l (mm) 38
l1 (mm) 1.2
Fig. 11 Prototype of the positioning stage
Fig. 13 Open loop cross-talk study (a) Open loop cross-talk study when driving in X direction (b) Open loop cross-talk study when driving in Y direction
Fig. 12 Experimental system
the output-end are not the same to the sigle inputs. Since their output displacements errors in the two directions are shown in Fig. 9(c), the crosstalk is less than 2%. The first two mode shapes of the stage obtained by ANSYS software are shown in the Fig. 10(a) and (b), the natural frequency of the stage for the first two modes are 233.83 Hz and 234.01 Hz, respectively.
5. Prototype Fabrication and Preliminary Experimental Study After the compliance matrix model and the finite element model are constructed and tested, the parameters are carefully selected by considering the amplification ratio, fabrication accuracy and the safety factor of the material and structure. With the parameters listed in Table 1, the amplification is designed as 6 times. After the mechanical structure is designed and the parameters are optimized, the prototype can be fabricated via electro discharge machining as shown in Fig. 11. In the experimental study, two laser displacement sensors (Microtrak II head model: LTC-025-02 from MTI Instruments, Inc.) with a resolution of 0.12 µm and linearity better than 0.05% over a measurement range of 2 mm associated with a dSPACE DS1005 (from dSPACE GmbH) rapid
prototyping system equipped with DS2001 A/D and DS2102 D/A modular boards are employed to construct the experimental system. Two PZTs and the drivers from Physik Instrumente (PI) (model: P-840.2) with a resolution of 0.6 nm and working range of 30 µm driven by a 3channels PZT amplifier (model is E-503.00). Thewhole experimental system is shown in Fig. 12.
5.1. Open-loop crosstalk test After the positioning stage is fabricated and assembled, the first experiment is to validate the amplifier ratio of the mechanical bridge type amplifier. From the previous modeling study, the prototype is fabricated with parameters to make the positioning stage have amplifier ratio about 6 times. Since the working range of the PZT adopted in this research is 30 µm, the working range of the positioning stage can be estimated as 180 µm. When a sinusoidal reference signal is fed into the PZT driver, the reference signal and the measured response signal can be observed in Fig. 13(a). The maximum output displacement of the stage can be measured in average as 152 µm in both directions, compared with the PZT’s working range 30 µm, the amplifier ratio can be calculated as 5.07 times. It’s less than the previous model based design. The deviation about 3% mainly comes from the fabrication error, parameters error and the model simplification errors, etc.. Moreover, in this deisgn the PZTs will move along with the mechanism when they work, so the PZTs will affect the modeling accuracy in FEA analysis because the PZTs’ masses are ignored during simulations. The second experiment is to test its open-loop crosstalk. As shown
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Fig. 15 General structure of a typical MRAC scheme
Fig. 14 Open loop hysteresis study
in Fig. 13(b), the crosstalk is no more than 2.1% in both directions, which has well confirmed the decoupled design of the mechanical system. Since the crosstalk is caused mostly by either the machining tolerances or fabrication errors of the mechanical system, it is hard to establish an accurate model in the FEA. The experimental results demonstrate that the practical experiences are also very important in this engineering design.
5.2. Open-loop hysteresis study From the open-loop control experiment, the hysteresis and the nonlinear characteristics of the system can be observed clearly. The maximal hysteresis rate can reach 13.5%. The nonlinear characteristics of the system can be observed from Fig. 14, and at the beginning of the first 50 V, the mobile platform can just be moved about 50 µm. In the following 50 V voltages from 50 to 100 V, it can go about 110 µm displacement. Therefore, how to model the hysteresis PZT actuators will be a key issue for the control system. Since the PZT actuators have fast responses, when the input signal is within 10 Hz, the hysteresis loops look almost to have no differences. Therefore, a rate-independent inverse Preisach model can be adopted to compensate the hysteresis problems of the system.11 In this research, to obtain more robust response performance, some advanced control strategies are necessary.
6. MRAC PID Controller Design As indicated in preliminary study, we realize that the system is with typical nonlinear and hysteresis characteristics. Then, a more advanced controller should be adopted. Although in the model reference adaptive controller (MRAC), the desired index of performance is given by the reference model, and the controller can establish robustness with respect to bounded disturbances and unmodeled dynamics, it is hard to adjust the parameters to obtain a good performance, even it is very difficult to make the system stable. For conventional PID controller, it can only be used to control the nonlinear system with hysteresis in different working ranges with different PID parameters. That is to say, to obtain satisfy performance, the parameters of the PID controller should be adjusted online automatically. Many ways can be used to tune the PID parameters, aiming for combining the advantages of the MRAC and PID controller, the MRAC PID controller is adopted in this research. With the aid of MATLAB Simulink Toolbox and dSPACE system, the
adopted MRACPID controller is very easy to implement.13
6.1 System identification Although the system is with nonlinearity and typical hysteresis characteristics, due to the great robust characteristics and the self adaptive ability of the MRAC PID controller, an approximated model of the system is needed for the design of the control system.13 The approximated model can be obtained through many methods. In practice, the basic structure of a model reference adaptive control (MRAC) system is shown in the block diagram in Fig. 15. The reference model is chosen to generate the desired trajectory for the plant output to follow. The tracking error represents the derivation of the plant output from the desired trajectory. Accordingto the tracking error, output of the controller and the output of the plant, the adjustment mechanism automatically adjust controller parameters so that the behavior of the closed-loop control plant output closely follows that of reference model. Structures of the reference model and the adaptive gains are chosen based on the requirements of control performance. The adjustment mechanism of MRAC system constructs by a popular method for computing the approximated sensitivity functions as so-called MIT rule. The algorithms for MIT rules can be derived byapproximated model of the system, which can be obtained via system identification conveniently. Even if the system is with great nonlinearity and hysteresis feature, we can still use the system identification toolbox to obtain an approximated linearity auto-regression exogenous (ARX) model of the system through least-square (LS) method. In system model identification process, the most used model structure is the simple linear difference equation which relates the current output y(t) to a finite number of past outputs y(t−k) and inputs u(t−k). The structure is thus entirely defined by the three integers na, nb, and nk. na is equal to the number of poles and nb is the number of zeros, while nk is the pure time-delay (the deadtime) in the system. For a system under sampled-data control, typically nk is equal to 1 if there is no dead-time. The orders na, nb, and nk can either be directly entered into the edit box orders in the parametric models window, or selected using the popup order menus in the editor. When selecting “Estimate”, models corresponding to all of these structures are computed, a special plot window will then be opened which shows the fit of these models to validation data. There are two methods to estimate the coefficients a and b in the ARX model structure: least squares and instrumental variables methods. As the most known method, least square method is selected in this research, it can minimize the sum of squares of the right hand side minus the left-hand side of the system differential expression, with respect to a and b. After the system is approximately identified, the system model can be used for
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the design of the MRAC PID controller. The reference model is chosen to generate the desired trajectory for the plant output to follow. The tracking error represents the derivation of the plant output from the desired trajectory. According to the tracking error, output of the controller and the output of the plant, the adjustment mechanism automatically adjust controller parameters so that the behavior of the closed-loop control plant output closely follows that of reference model. Structures of the reference model and the adaptive gains are chosen based on the requirements of control performance. The algorithms for MIT rules can be derived. Tracking error is e = yp – ym
(18)
1 2 J (θ ) = --- e (θ ) 2
(19)
Fig. 16 MRAC PID controller block diagram
Cost function is where ∂J/∂ε = ε , ∂ε /∂y = 1 , ∇ = d/dt .
From the MIT rule, the change rate of θ is proportional to negative gradient of J: dθ ∂J ∂e ------ = –γ ----- = –γ e -----dt ∂θ ∂θ
(20)
where e denotes the model error and θ is the controller parameter vector. The components of ∂e/∂θ are sensitivity derivatives of the error with respect to θ. The parameter γ is known as the adaption gain. The MIT rule is a gradient scheme that aims to minimize the squared model cost function. According to the design process of the mechanism system of the positioning stage, the system can be described by second order model b/(s2 + a1s + a2). A block diagram of control system can be shown in Fig. 16. Then, the closed-loop transfer function can be derived as:
∂y b∇ --------p- = --------------------------------------------------------------------------------------- ⋅ [Uc – yp] ∂Kp ∇3 + (a + bK )∇2 + ( a + bK )∇ + bK
(25)
∂y b -------p- = --------------------------------------------------------------------------------------- ⋅ [Uc – yp ] ∂Ki ∇3 + ( a + bK )∇2 + ( a + bK )∇ + bK
(26)
2 ∂y b∇ --------p- = --------------------------------------------------------------------------------------- ⋅ [Uc – yp] 3 2 ∂Kd ∇ + (a + bK )∇ + ( a + bK )∇ + bK
(27)
1
1
1
d
d
d
2
2
2
p
p
p
i
i
i
Then, dKp/dt, dKi/dt, dKd/dt can be derived by dKp b∇ ∂J --------- = –γ -------- = –γp ε --------------------------------------------------------------------------------------- ⋅ [Uc – yp ] 3 2 dt ∂Kp ∇ + ( a1 + bKd )∇ + ( a2 + bKp )∇ + bKi (28) dKi b ∂J -------- = –γ i ------- = –γi ε --------------------------------------------------------------------------------------- ⋅ [Uc – yp ] 3 2 dt ∂Ki ∇ + ( a1 + bKd )∇ + ( a2 + bKp )∇ + bKi (29)
2
b ( Kd s + Kp s + Ki ) yp ( s ) ------------- = -------------------------------------------------------------------------------UC ( s ) s ( s2 + a s + a ) + b ( K s2 + K s + K ) 1
2
d
p
(21)
i
2 dKd b∇ ∂J --------- = –γd -------- = –γd ε --------------------------------------------------------------------------------------- ⋅ [Uc – yp ] 3 2 dt ∂Kd ∇ + ( a1 + bKd )∇ + ( a2 + bK p )∇ + bKi (30)
Furthermore 2
yp ( s ) b (Kd s + Kp s + Ki ) ------------- = --------------------------------------------------------------------------------3 UC ( s ) s + ( a + bK )s2 + ( a + bK )s + bK 1
2
d
p
(22)
i
Consequently, the reference model can be chosen as the following form: 2
b m1 s + b m2 s + bm3 Ym ( s ) ------------- = ------------------------------------------------U m ( s ) s3 + a s2 + a s + a m1
m2
(23)
m3
According to the MIT rules, the parameters of PID controller Kp, Ki, Kd can be determined by dK ∂J- = –γ ⎛ ∂J ⎞ ⎛ ∂ε ⎞ ⎛ ∂yp ⎞ --------p- = –γ p -------p ⎝ -----⎠ ⎝ --------⎠ ⎝ ---------⎠ dt ∂Kp ∂ε ∂yp ∂Kp dK ∂J ∂J ∂ε ∂y --------i = –γ i ------- = –γ i ⎛ -----⎞ ⎛ --------⎞ ⎛ -------p-⎞ ⎝ ∂ε ⎠ ⎝ ∂yp⎠ ⎝ ∂Ki ⎠ dt ∂Ki dKd ∂J- = –γ ⎛ ∂J ⎞ ⎛ ∂ε ⎞ ⎛ ∂yp ⎞ --------- = –γd -------d ⎝ -----⎠ ⎝ --------⎠ ⎝ ---------⎠ dt ∂Kd ∂ε ∂yp ∂Kd
(24)
A block diagram is shown in Fig. 17, the model reference based adaptive PID controller in Simulink will be compiled and downloaded into dSPACE for real time running.
6.2 Experiment After the approximate model of the system is identified and the MRAC PID controller is built in the MATLAB Simulink real-time workshop (RTW) environment, the Simulink model can be complied and downloaded into the dSPACE to run in realtime with the hardware in loop. The software Control Desk is used to supervise the experimental process and upload the experimental data instantaneously through optical fiber communication tool set. In the experiment, two sinusoidal waves in two directions are used to drive the mobile stage to follow a circle, the learning and self adaptive time can be seen clearly form Figs. 18 to 20. It needs about 50 s for the controller to learn and adapt the plant well. After the adaptive time, it can be seen that the mobile stage can track the given reference path exactly. It can be seen from Figs. 19 and 20, after 100 s, the hysteresis problems are almost canceled, the response can track the reference signals well in both directions. The whole
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Fig. 17 The Simulink model for running under real-time work environment in dSPACE
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Fig. 20 Hysteresis performance in y-direction after applying the MRACPID controller on the stage
Fig. 21 Reference and response of x- and y-directions for the circle contouring Fig. 18 Performance of the system when tracking a circle
7. Conclusion
Fig. 19 Hysteresis performance in x-direction after MRACPID controller on the stage
applying the
tracking performance can be seen in Fig. 21. After the learning and adaptive period, the response follows the reference path exactly. Compared with the previous simple PID control method, the MRACPID controller can tune the PID parameters automatically and make the system stable in the whole working range. But such a long learning and adaptive time may limit the application field of the designed positioning stage. This long time adaptive time may come from the approximated system model and the setting of adaptive gain. If more accurate model is used, or even associated with inverse hysteresis feed forward compensator, and proper adaptive gain is selected, much better performance will be obtained.
A novel compliant flexures-based totally decoupled XY micropositioning stage is designed and manufactured. An amplification mechanism is designed with the merits of a compact size and a large amplification ratio. The compliance model of the stage is obtained based on the flexibility matrix method. FEA simulation results demonstrate that the stage has a high linearity relationship between the input force and the output displacement, the cross-talk is less than 2%, the input parasitic motion is less than 1.8%. After the prototype system is built, preliminary experiments are carried out. It is found that the system is with very typical nonlinear and hysteresis features. A model reference adaptive PID controller is adopted in this research to cope with the nonlinear characteristics of the system. The controller is built in realtime work environment and down loaded into dSPACE for running. Experimental results verify the performance of the micro stage and demonstrate that the designed controller fulfills the expected control results.
ACKNOWLEDGEMENT This work was supported by Macao Science and Technology Development Fund (108/2012/A3), Research Committee of University of Macau (MYRG183(Y1-L3)-FST11-LYM, MYRG203 (Y1-L4)FST11-LYM).
744 / APRIL 2014
INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 15, No. 4
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