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New Flexure Parallel-Kinematic Micropositioning System With Large Workspace Qingsong Xu, Member, IEEE

Abstract—Flexure-based micropositioning systems with a large workspace are attractive for a variety of precision engineering applications. In this paper, a new idea of multistage compound parallelogram flexure is proposed for the mechanism design of a novel parallel-kinematic XY micropositioning system, which has a motion range larger than 10 mm along with a compact structure. The established quantitative models and the stage performances are validated by conducting finite-element analysis (FEA) and experimental studies. Moreover, an enhanced model-predictive control (EMPC) is presented for positioning control of the system, which has a nonminimum-phase plant. It is shown that the EMPC is capable of producing a low magnitude of output tracking error by imposing an appropriate suppression on the control effort. Simulation and experimental studies reveal that the EMPC scheme outperforms the conventional proportional-integral-derivative (PID) and MPC methods in terms of transient response speed and steadystate accuracy. The idea that is presented in this paper is extendable to design and control of other micro-/nanopositioning systems with either minimum- or nonminimum-phase plants. Index Terms—Mechanism design, micro/nano robots, motion control, parallel robots, smart actuators.

I. INTRODUCTION FLEXURE-BASED compliant structure delivers motion by making use of elastic deformations of the material [1]–[3], which allows the generation of a smooth output motion by eliminating the adverse effects in terms of clearance and backlash [4]. Thus, micropositioning systems based on flexure mechanisms play an important role in precision applications, where a high-resolution motion over a microrange is required [5], [6]. For example, a flexure-based multiaxis scanner with a range of motion of less than 100 μm and the resolution of subnanometer level is commonly needed in a scanning probe microscope [7]. For such kind of applications, piezoelectric stack actuators (PSAs) are usually employed to build an XY micropositioning stage [8]–[10], since a PSA is capable of positioning with subnanometer resolution, rapid response, and large blocking force [11], [12]. Nevertheless, the PSA typically

A

Manuscript received May 5, 2011; revised September 9, 2011; accepted October 16, 2011. Date of publication November 24, 2011; date of current version April 9, 2012. This paper was recommended for publication by Associate Editor Y. Sun and Editor B. J. Nelson upon evaluation of the reviewers’ comments. This work was supported in part by the Macao Science and Technology Development Fund under Grant 024/2011/A and in part by the Research Committee of the University of Macau, Macao, China, under Grant SRG006-FST11-XQS. The author is with the Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Taipa, Macao, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TRO.2011.2173853

provides the stroke of less than 1 mm for a micropositioning system. For more applications, such as microassembly [13] or biological manipulation [14], [15], a large motion range greater than 10 mm is expected for each working axis [16]. In order to achieve such a large workspace, magnetic levitation motors [17], electromagnetic actuators [18], and voice coil motors [19] are popularly employed. These types of motors are lubrication free and vacuum compatible, which enable the fulfillment of ultraclean applications. To facilitate motion guiding of the stage, aerostatic bearings [20] and maglev bearings [21] are usually adopted. Some multiaxis positioning systems with large workspace are even commercially available (e.g., H2W Technologies, Inc., Santa Clarita, CA). It is observed that most of the commercial stages employ a serial-kinematic architecture, e.g., by stacking multiple single-axis positioning stages together. The serial connection of multiple stages allows the adoption of a relatively straightforward control strategy since each axis can be handled independently. However, it is at the cost of high inertia, low natural frequency, and cumulative errors. To conquer such drawbacks, the parallel-kinematic stage is a promising alternative because of its contribution to high load-carrying capacity, high accuracy, low inertia, and compact size [22], [23]. On the other hand, flexure bearings are more attractive in terms of no backlash, no friction, vacuum compatibility, and easy to manufacture [24]. Hence, flexures are widely applied in recent design and development of micropositioning systems. For instance, an XYZ flexure parallel mechanism is designed in [19], where two types of large-motion prismatic joints that are based on compound parallelogram flexures (CPFs) are used to achieve a motion range larger than 1 mm in each axis. Using double-compound linear spring flexures, a precision XY scanner is proposed in [25], which is capable of 2-mm working range in each axis by adopting voice coil motors (VCMs). The implementation of a compact flexure Xθ positioner is reported in [26] with a travel range of 1 mm in one working axis, which is produced by built-in electromagnetic actuators. In addition, the design of a large-range XY nanopositioning system with the workspace size of 10 × 10 mm2 is presented in [27], where a rotational-symmetric stage is constructed by using compound parallelogram leaf flexures [28]. A review of the previous works reveals that it is challenging to design an XY stage with both a large workspace and a compact physical dimension, simultaneously. It seems that the most compact stage is the one presented in [25], which has a dimension of 100 × 100 × 50 mm3 . However, it can only deliver a small workspace of 2 × 2 mm2 . In contrast, the XY stage designed in [27] produces a workspace of 10 × 10 mm2 , which is contributed by a large physical

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TABLE I AREA RATIO OF WORKSPACE TO THE PLANAR DIMENSION OF TYPICAL PARALLEL-KINEMATIC XY STAGES

Fig. 1.

dimension of 255 × 255 × 25 mm3 . A compact stage is desirable for the situations, where the micro-/nanomanipulation inside a limited space (e.g., inside a scanning electron microscope [29]) is required. In order to make a quantitative measure, the area ratio of workspace to the planar dimension of the stage is proposed in this research. A larger ratio implies that the stage can be fabricated as a smaller dimension for achieving a specified workspace size. Thus, the larger the ratio, the more compact the stage. The ratio values of some typical XY micropositioning stages are tabulated in Table I. It is evident that the stage that is presented in [27] is the most compact one, which has the potential to be scaled down to the smallest planar dimension if a particular workspace size (e.g., 1 × 1 mm2 ) is desired. The motivation of the current research is to design a new flexure XY micropositioning stage with a workspace range of 10 × 10 mm2 along with a compact dimension for micropositioning applications. The new concept of multistage CPF (MCPF) is proposed to achieve a large pure translational motion in one direction. Based on the idea of MCPF, a novel large-workspace XY stage is designed and fabricated with the unwanted buckling/bending phenomenon being eliminated. The system identification results show that the plant model of the micropositioning system is of nonminimum phase (NMP), which poses a challenge for the control system design. The NMP indicates that the system model possesses unstable zeros. Thus, many powerful control techniques that are based on high-gain feedback and inversion principle cannot be applied directly. Mathematically, the unstable zeros can be removed by the pole–zero cancellation approach. However, it is not feasible in practice since an arbitrarily small discrepancy between the zero and the pole results in instability [30]. On the contrary, an NMP system can be controlled by using a zero-phase-error tracking controller (ZPETC) or a zero-magnitude-error tracking controller (ZMETC) [31]. However, the ZPETC creates a zero-phase error at the cost of magnitude errors. Similarly, the ZMETC accomplishes a fine magnitude tracking as a sacrifice of phase delays. Hence, significant revisions are required to pursue a perfect tracking [32]. The output-feedback sliding-mode control algorithm with sufficient revisions [33] is also applicable to NMP systems. Nevertheless, the implementation of such kinds of controllers is a complicated and time-consuming procedure. In contrast, it has been shown [34] that model-predictive control (MPC) provides an efficient way for the control of an NMP system. As a robust control strategy, the MPC is well known for its ability to solve the problems with constraints, time delays, and disturbances by offering an optimal control effort [35]. Generally, an MPC controller is easy to implement since it only has one weight parameter to tune [36]. At the same time,

(a), (b) Ordinary CPFs. (c) Basic module.

fewer tunable parameters implies less flexibility in design. In the current research, it is shown that the conventional MPC is not capable of producing a satisfactory positioning (in terms of transient response time and steady-state error) by adjusting the sole parameter. To overcome such limitation, an enhanced MPC (EMPC) is proposed by resorting to a popular proportionalintegral-derivative (PID) controller. It is demonstrated that the EMPC derives an optimal incremental control action by imposing suitable suppression on the control effort. Moreover, an output tracking error of the order O(T 2 ) (T is the sampling time) is achieved with the presented control scheme. The theoretical analysis and the effectiveness of the EMPC over the traditional MPC, as well as PID methods are verified by simulation and experimental studies that are performed on the micropositioning system with the NMP plant. The major contribution of this paper lies in the design of a novel flexure parallel-kinematic XY micropositioning stage for precise positioning applications. In the rest of the paper is organized as follows. The design procedure of the novel largeworkspace XY stage is outlined in Section II. The quantitative models that are established in the design process are verified with the finite-element analysis (FEA) and a prototype is developed in Section III. Then, in Section IV, we propose the EMPC strategy along with a state observer, where the tracking error bound is also analyzed. The designed controller is validated in Section V by both simulation and experimental studies. In Section VI, we conclude this paper. II. XY STAGE DESIGN In this section, the idea of MCPF is first presented to achieve a large pure translational motion in one direction. Based on the MCPF concept, a new parallel-kinematic XY stage is then proposed. A. Limitation of Existing Flexures In order to achieve a pure translational motion, the CPFs as shown in Fig. 1(a) and (b) have been widely used [19], [27]. It is observed that both ordinary CPFs are based on a module as depicted in Fig. 1(c). With an external force Fx that is applied at the primary stage of the CPF, the deformed shape of the first CPF is illustrated in Fig. 2(a). Each of the four flexures suffer from a combined force F and moment M . Considering the boundary conditions in terms of the rotation angle and translational motion, the following relationships can be derived: 0=

Ml F l2 − 2EI EI

(1)

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Fig. 2. (a) Deformation of a CPF. Dimensions of (b) one flexure and (c) cross section.

Fig. 3. (a) Proposed multistage CPF with N n secondary stages. (b) Its deformed shape.

B. Proposal of Multistage Compound Parallelogram Flexure M l2 F l3 − (2) δx = 3EI 2EI where δx represents the transverse displacement of one flexure [see Fig. 2(b)], E is Young’s modulus of the material, and I = bh3 /12 is the area moment of inertia of the cross section [see Fig. 2(c)] about the neutral axis. Solving (1) and (2) allows the generation: F =

2M l

(3)

F l3 δx = . (4) 12EI Since the four flexures are of the same length l, the relationship, i.e., δx = Δx/2, holds, where Δx denotes the one-sided translation of the CPF. Then, the stiffness of the CPF seen at the output direction can be computed: 3

2F Ebh Fx = = 3 . K = (5) Δx 2δx l If the maximum moment Mm ax is exerted by the flexures, the maximum stress σm ax (determined by the yield strength σy of the material) occurs at the outermost edge of the cross section, and it can be calculated by CPF

Mm ax h/2 I which gives the value of the moment σm ax =

σm ax bh2 . 6 In view of (3), (5), and (7), it can be derived that Mm ax =

(6)

δx =

Δx 2Nn

(9)

where Nn ≥ 1. It is noticeable that Nn = 1 represents the case of the traditional CPF. From Fig. 3(b) and noting (9), the stiffness of MCPF seen at the output direction can be computed as K M CPF =

2F Ebh3 Fx = = Δx 2Nn δx Nn l 3

(10)

which indicates that the Nn modules are connected in serial. Similarly, with the consideration of (3), (10), and (7), the maximum one-sided translation of the MCPF can be derived as CPF ΔxM = m ax

Fm ax Nn σm ax l2 . = M CPF K 3Eh

(11)

By comparison of (11) with (8), it can be deduced that, for the flexures with the same physical parameters, i.e., l and h, and material, the maximum translation of the proposed MCPF is enlarged by Nn times over the ordinary CPF.

(7)

Fm ax σm ax l2 (8) = K CPF 3Eh which indicates that the maximum one-sided translation Δx of the CPF is governed by the length l and thickness h of the leaf flexures for a given material. To obtain a larger Δx, the flexures are preferred to be designed with a larger length and smaller thickness. In practice, the physical parameter l is restricted by the compactness requirement of the device, and h is limited by the manufacturing process and the minimum stiffness restriction. In the following discussions, a new concept is proposed to achieve a large translation, while keeping the values of l and h unchanged. ΔxCPF m ax =

To realize a large motion range, the idea of MCPF with Nn modules is proposed as shown in Fig. 3(a). It is observed that the primary (output) stage is connected to the base through Nn modules with multiple secondary stages. All the flexures of MCPF are assumed to have the identical length l. Concerning an MCPF with Nn modules, if the onesided output translation Δx is given, the deformation of each flexure can be calculated as

C. XY Stage Design With Multistage Compound Parallelogram Flexure To illustrate the proposed idea of MCPF, Nn = 2 is selected to design an XY stage. That is, the MCPF is implemented with two modules as shown in Fig. 4(a) and (b). Although Nn = 3 or more may also be used, the development procedures are almost identical to the ones that are presented in the current research. Based on the concept of totally decoupling as presented in [37], one possible architecture of the XY stage with both input decoupling and output decoupling is proposed, as shown in Fig. 5(a). The motor decoupler is constructed by a double MCPF. It has large transverse stiffness at the output end, which guarantees that the motor does not suffer from transverse load. Thus, the two motors are isolated from each other and well are

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481

Fig. 4. Two configurations of the two-stage MCPF that is constructed with two modules.

Fig. 6. FEA results of deformations for an XY stage (a) without and (b) with the connecting bars.

The critical axial load that causes elastic buckling of the flexure can be calculated by Pcr =

π 2 EI 2 lcr

(12)

where the critical length lcr is evaluated by using a specific coefficient k as follows: lcr = kl.

(13)

It is known that k takes values from 0.5 to 2, depending on the boundary conditions [24]. Assume that the MCPF deformation as shown in the FEA result [see Fig. 6(a)] is induced by the buckling effect. The buckling flexures are considered as fixed-free beams, which have a coefficient of k = 2. Thus, the ordinary MCPF has the critical load value π 2 EI . (14) 4l2 Equation (12) reveals that, in order to enlarge the critical load to avoid the occurrence of buckling, either the moment of inertia I can be increased or the critical length lcr may be decreased. For a flexure that is designed with specific dimensions, a method is proposed later to reduce the critical length lcr by reducing the coefficient k. By the investigation of the deformed shape of the two-stage MCPF [see Fig. 3(b)], it is found that the displacements of the two free ends are identical, i.e., x2 = x4 = Δx/2. Hence, these two ends can be connected together without influencing the motion output property. Once the two free ends of every MCPF are connected by a connecting bar as shown in Fig. 5(d), the concerned flexures become fixed–fixed columns. For a beam with the fixed–fixed boundary condition, the coefficient is reduced to k = 0.5 [24]. Thus, the critical load can be computed as PcrM CPF1 =

Fig. 5. Evolution of an XY stage using two-stage MCPFs. Blank circles are the mounting holes to fix the stage, and black holes in (d) denote the connecting holes to mount the connecting bars.

protected. In addition, the MCPF that is connected to the output platform ensures a decoupled translation of the XY stage. In order to generate a more compact XY stage with high space utilization rate, the decoupler is evolved as shown in Fig. 5(b). This way, a larger output platform is obtained and the planar space is utilized more efficiently. Furthermore, to improve the accuracy performance and reduce the temperaturegradient effect, a mirror-symmetric XY stage is implemented as described in Fig. 5(c). 1) Buckling/Bending Effect Consideration: Since the flexures are constructed by slender leaf springs with a relatively high ratio of length to thickness, they are prone to elastic (Euler) buckling/bending under compressive loads. As an adverse effect, the motion that can be transmitted to the output platform is greatly reduced. Thus, the buckling/bending should be avoided by the structure design.

4π 2 EI . (15) l2 It can be observed from (15) and (14) that the critical load of the improved design has been enhanced by 16 times in comparison with the original design. Even if the deformation as shown in Fig. 6(a) is merely because of the bending effect, the critical load will be improved furthermore since the bending load is lower than the critical buckling load (14). Therefore, the PcrM CPF2 =

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the motor can be expressed as Fm ax = Kin Dm ax .

(19)

When selecting motors, the VCM should be chosen with the maximum actuation force FmVCM ax satisfying Fm ax ≤ FmVCM ax ≤ Fcr

(20)

i.e.,

Fig. 7.

3Ebh3 Dm ax 2π 2 Ebh3 VCM ≤ F ≤ . (21) m ax l3 3l2 The aforementioned condition guarantees that the selected motor is powerful enough to drive the XY stage, yet not excites the buckling response of the structure at the same time. In addition, the given maximum one-sided displacement Dm ax should stay within the allowable maximum one-sided translation that is described by (11) for Nn = 2

Stiffness model of an XY stage seen at the input end.

buckling/bending phenomenon is alleviated by the improved XY stage as shown in Fig. 5(d). The FEA result [see Fig. 6(b)] shows that, with the same force that is applied to one input end of the XY stage, no buckling/bending occurs in the related flexures. As a result, a larger output displacement is generated as compared with the original XY stage [see Fig. 6(a)]. 2) Actuation Issues: As far as actuation is concerned, VCMs are selected to drive the XY stage in this research to generate a centimeter-level motion range. In comparison with other types of actuators that are based on smart materials, such as the PSA, the VCM provides a larger stroke. However, the VCM has a low blocking force as compared with the PSA. To facilitate the VCM actuation, the XY stage should be designed with low-enough stiffness seen at the actuator. At the same time, the actuation force should not induce the buckling phenomenon. Quantitatively, to guarantee the elimination of the buckling effect for the XY stage, the maximum actuation force should not exceed the critical force of actuation Fcr = 2PcrM CPF2 =

8π 2 EI . l2

(16)

It can be seen from Fig. 6(b) that, when an actuation force Fx is applied on the XY stage to generate a displacement x of the output platform, the deformations are primarily induced by the bending of six MCPFs. Taking into account the bending deformations only, the stiffness model of the XY stage is simplified as shown in Fig. 7. The stiffness K of each MCPF can be computed by substituting Nn = 2 into (10) Ebh3 . (17) 2l3 Then, the stiffness of the XY stage seen at the input end can be computed as K=

3Ebh3 . (18) l3 Given the predefined maximum one-sided displacement Dm ax of the stage in one axial direction, the driving force needed from Kin = 6K =

2σm ax l2 . (22) 3Eh Therefore, relationships (21) and (22) provide guidelines for the design of stage parameters to guarantee a well operation under the safety of the material. Dm ax ≤

III. MODEL VERIFICATION AND PROTOTYPE DEVELOPMENT In this research, an XY stage is developed to produce a motion range over ±5 mm in each working axis. Based on the design criteria (21) and (22), an XY stage is designed with the main parameters: a = 122.0 mm, b = 10.0 mm, h = 0.5 mm, and l = 25.0 mm [see Figs. 2 and 5(d)]. A. Model Verification and Performance Assessment With the Finite-Element Analysis The quantitative models are developed for ideal situations in the previous section. In order to test the performance of the designed stage and to verify the accuracy of the established models, FEA simulations are carried out with ANSYS software package. The main specifications of the adopted material (Al7075 alloy) are described as follows: Young’s modulus = 71.7 GPa, yield strength = 503 MPa, Poisson’s ratio = 0.33, and density = 2.81×103 kg/m3 . The analytical models reveal that the maximum allowable one-sided displacement of the stage is Δxm ax = 5.85 mm. Thus, the motion range in each axis is ±Δxm ax = ±5.85 mm. By assigning a motion range of ±5 mm, the maximum force that is required to drive the stage is Fm ax = 86.04 N. In addition, the critical force leading to buckling is calculated as Fcr = 943.53 N. To assess statics performance of the stage, the FEA simulation is carried out by applying an input displacement at the input end. With a 5-mm input displacement, the FEA result is illustrated in Fig. 8, which shows a safety factor of 1.7 of the material. It is found that the required actuation force is 68.83 N. By the consideration of the FEA result as “true” value, it is found that the analytical model overestimates the force capability by 25%. Additionally, the maximum allowable displacement

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Fig. 8.

483

FEA results of deformations for the designed XY stage.

evaluated by the FEA is 8.46 mm. Thus, the analytical model is more conservative since it underestimates the displacement by about 30.9%. It is seen that the simple analytical models predict the stage performance with relatively large deviations. In the future, nonlinear models will be established to evaluate the stage performances more accurately. Moreover, the modal analysis reveals that the first three natural frequencies are 48.3, 48.7, and 100.1 Hz, respectively. The modal shapes of the first two modes are the translations along the two working axes, and that of the third one is the rotation of the output platform in the working plane. The similar values of the first two natural frequencies imply that the XY stage has almost identical dynamics performance in the two working axes. The fact that the frequency of the rotational mode is two times higher than those of the translational modes confirms that the stage has two translational degrees of freedom (DOFs).

Fig. 9.

Prototype of the developed XY micropositioning stage.

B. Prototype Fabrication The FEA result suggests that the actuation force of the VCM should be chosen as 68.83 N ≤ FmVCM ax ≤ 754.82 N

(23)

to ensure a large output motion range (±5 mm), as well as the safety of the material. In view of the force capability and motion range requirements, the VCM (model: NCC05-18-060-2X, from H2W Techniques, Inc.) is selected to provide sufficient large output force of 27.8 N/A × 7 A = 194.6 N and a stroke of 12.7 mm. The fabricated prototype is graphically shown in Fig. 9. The stage output motion in each axis is measured by a laser displacement sensor with a resolution of 1.25 μm over a measuring range of 12.5 mm and a capacitive sensor with a resolution of 1.25 nm over the range of 125 μm. Similar to the measurement scheme as in [27], the two sensors in each working axis is connected in serial, and the sum of readings gives the displacement value in each axis. In addition, control algorithms are developed with MATLAB/Simulink software and downloaded to a dSPACE DS1005 PPC board via the ControlDesk interface to realize real-time control.

Fig. 10. Motion range and crosstalk test results with (a) VCM 1 and (b) VCM 2 driven for the XY micropositioning system.

C. Preliminary Experimental Results The motion range and crosstalk of the positioning system in the two working directions are tested by experimental studies. For a quasi-static test, a 1-Hz sine wave with 10-V amplitude is used as the input command. When the XY stage is driven by one VCM to translate along the x (or y) direction, the displacements in both axes are measured to determine the motion range and crosstalk. The experiments have been conducted using the open-loop control. The output displacements of the stage are measured as shown in Fig. 10. It is observed that XY stage has a workspace about 10.5 × 10.5 mm2 with the maximum crosstalk of 1.6% between the two working axes. Experimental

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results also exhibit that hysteresis effects exist in the positioning system. The hysteresis mainly arises from the nonlinearity of the employed VCM, which works based on the electromagnetic principle. As compared with an XY stage that was previously presented in [27], which has a dimension of 255 × 255 × 25 mm3 with 10-mm motion range, the current design is a cable of 10.5 × 10.5 mm2 motion while possessing a much smaller dimension of 214 × 214 × 10 mm3 . That is, the developed XY stage has an area ratio (workspace to planar dimension) of 0.2407%, which is 56.5% larger than the one (0.1538%) reported in [27]. Thus, the proposed parallel-kinematic XY stage owns a much compact size over the existing stages. It is noticeable that the compactness will be further enhanced by implementing an optimum design in the future research. The experimental results confirm the decoupling motion of the XY stage. In addition, the buckling phenomenon does not occur during the operation, which validates the effectiveness of the design procedure. It is found that the output decoupling of the XY stage is enabled by employing MCPFs, whereas the secondary stages increase the mass of moving components, which leads to a not-high natural frequency of the XY stage, as indicated by the FEA results. To suppress the nonlinearity and to achieve a precise positioning for the micropositioning system, a closed-loop control scheme is presented in the following section. IV. ENHANCED MODEL-PREDICTIVE CONTROL SCHEME DESIGN In this section, an EMPC scheme is proposed to realize a precise positioning of the micropositioning system. A. Problem Formulation The fine decoupling property of the stage allows the adoption of a single-input–single-output (SISO) controller for each working axis. The dynamics model of a VCM-driven micropositioning system for one working axis can be described by a linear-time-invariant model that is accompanied by disturbance: ˙ X(t) = Ac X(t) + Bc u(t) + f (t)

(24)

Y (t) = CX(t) + Du(t)

(25)

where the state vector X ∈ Rn (n ≥ 1), the output Y ∈ R, the control input u ∈ R, and the disturbance vector f ∈ Rn is supposed to be smooth and bounded. The system matrices Ac , Bc , C, and D are constant with approximate dimensions. In addition, the disturbance f describes the combined effects of unmodeled nonlinearity, higher order dynamics, and external perturbations. Using a sampling time T , the continuous-time system model (24) and (25) can be discretized as Xk +1 = AXk + Buk + fk Yk = CXk + Duk

(26) (27)

where the notation Xk = X(kT ), and  T A = eA c T , B= eA c τ dτ Bc 

(28)

0 T

fk =

eA c τ f ((k + 1)T − τ )dτ.

(29)

0

It is observed that both B and fk are of the order O(T ) with respect to the sampling time T . Moreover, the difference equation of the disturbance can be derived as    T (k +1)T −τ Ac T fk − fk +1 = e f˙ (θ) dθ dτ k T −τ

0 2

= O(T ).

(30)

Besides, a useful property as reported in [38] and proved in [39] is employed in the current research. Property 1 [38]: Assume that Ek +1 = ΞEk + δ k , where matrix Ξ is asymptotically stable (Ξ < 1), and the magnitude of δ k is of the order O(T r ) with r ≥ 1. Then, the ultimate bound of Ek is of the order O(T r −1 ) when k → ∞. It is assumed that the triplet (A, B, C) of the nominal system is both controllable and observable with the matrices B and C being of full rank. The concerned positioning control problem here is: How to make the output Yk precisely track a command reference input? Concerning a flexure-based micropositioning system in the current research, the result of system identification reveals that the plant is of the NMP. That is, the system has unstable zeros. The transfer function of the system model (24)–(25) can be expressed as G(s) = C(s I − Ac )−1 Bc + D. An NMP plant indicates that the inverse of the system model G−1 has unstable poles. Thus, some powerful control techniques that are based on the inversion principle cannot be applied directly. In the following section, an improved scheme that is based on the MPC technique is proposed for precise positioning control of the NMP micropositioning system. B. Enhanced Model-Predictive Control Scheme Design In this section, an EMPC strategy enhanced with the PID control is presented. 1) PID Controller Design: It is well known that PID is a model-free controller, which solves the control command by making use of control error only. A digital PID scheme can be expressed as follows: = K p Ek + K i upid k

k 

Ek + Kd (Ek − Ek −1 )

(31)

i=0

with the displacement error Ek = Rk − Yk

(32)

where Rk and Yk represent the desired and actual system outputs, and Kp , Ki , and Kd denote the proportional, integral, and derivative gains, respectively. The key issue in the PID control lies in the tuning of the three controller parameters. In the current research, the

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Ziegler–Nichols (Z-N) method is adopted because of its popularity. Based on the PID controller, an enhanced MPC strategy is presented later. 2) EMPC Controller Design: By using the conventional MPC strategy, the overall control effort can be expressed in the following different form: uk = uk −1 + Δuk .

(33)

The purpose of MPC is to generate the incremental control effort Δuk in an optimal manner. However, the following simulation studies show that the conventional MPC produces a slow transient response with a large steady-state error. In order to overcome these limitations, an EMPC scheme is proposed by considering upid k as the previousstep control effort uk −1 . By this way, the controller performance will be significantly improved. Taking into account (26) and (33) together, a new state-space equation can be obtained as ¯ k + BΔu ¯ ¯ Zk +1 = AZ k + Ifk

(34)

¯ k Yk = CZ where





(35) 

Xk ¯ = A , A 0 uk −1   ¯I = I ¯ = [C 0]. C 0

Zk =



B , 1



¯ = B B 1

¯B ¯ C 0 ⎢ C ¯ ¯ ¯ ¯ ¯ AB CB ⎢ Φ=⎢ .. .. ⎢ ⎣ . . (N −1) ¯ (N −2) ¯ ¯ ¯ ¯ ¯ B CA B CA ⎡ ¯ ¯ CI 0 ⎢ C ¯ ¯ ¯ ¯ AI C¯I ⎢ Γ=⎢ .. .. ⎢ ⎣ . . (N −1) ¯ (N −2) ¯ ¯ ¯ ¯ ¯ CA I CA I

···

0

⎤

0 ⎥ ⎥ .. ⎥ ⎥ . ⎦ ¯B ¯ ··· C ⎤ ··· 0 ··· 0 ⎥ ⎥ .. ⎥ .. ⎥. . . ⎦ ¯ ¯I ··· C ··· .. .

(36)

¯ BΔu ¯ ¯ ¯ N −1 ¯Ifk + ··· + C k +N −1 + CA

Assume that the N -step preview, i.e., Rak = [Rk +1 , Rk +2 , · · · , Rk +N ]T , of the desired reference is available; then, a cost function for minimization can be expressed by T  J = (Rak − Yka )T (Rak − Yka ) + w Δuak −1 Δuak −1 (43) where w is the weighting to limit the partial control effort. It is noticeable that Δuak −1 approaches to zero as (Rak − Yka ) tends to zero. Thus, the cost function J vanishes at the steady state. Substituting (38) into the cost function (43) and applying the optimization criterion by setting ∂J/∂Δuak −1 = 0 yield

T T ˆf a = [f T f T k −1 k −1 k −1 · · · fk −1 ]

(37)

where the positive integer N is the prediction horizon. The N prediction equations can be stacked together into the form Yka = ΛZk + ΦΔuak −1 + Γfka−1

(38)

where the vectors for the future system output, incremental control effort, and disturbances are Yka = [Yk +1 Yk +2 · · · Yk +N ]T

(39) T

= [Δuk Δuk +1 · · · Δuk +N −1 ] T  = fkT fkT+1 · · · fkT+N −1 .

(44)

(40)

(45)

where fk −1 denotes the one-step delayed estimation of the disturbance, which can be obtained by noting (26). Since only the first value of the predicted control sequence is used, the optimal incremental control effort can be generated as Δuk = e(ΦT Φ + wI)−1 ΦT (Rak − ΛZk − Γˆfka−1 )

¯A ¯ N −2 ¯Ifk +1 + · · · + C ¯ ¯Ifk +N −1 +C

(42)

where the future disturbance values in fka−1 are unknown. Thus, they are estimated by

¯ ¯ ¯A ¯ N Zk + C ¯A ¯ N −1 BΔu ¯ ¯ N −2 BΔu Yk +N = C k + CA k +1

fka−1

⎡

Δuak −1 = (ΦT Φ + wI)−1 ΦT (Rak − ΛZk − Γfka−1 )



Based on (34) and (35), an N -step ahead prediction of the system output can be derived as follows:

Δuak −1

485

(46)

where the vector e = [1, 0, 0, · · · , 0]. Therefore, the total control action of the EMPC can be expressed as uk = upid k + Δuk = K p Ek + K i

k 

Ek + Kd (Ek − Ek −1 )

i=0

+ e(ΦT Φ + wI)−1 ΦT (Rak − ΛZk − Γˆfka−1 ). (47) A block diagram of the proposed EMPC controller is shown in Fig. 11. It is observed that the N -step preview Rak of the reference input is needed by the EMPC scheme.

(41)

Additionally, the three matrices take on the following forms: ⎡ ¯¯ ⎤ CA ⎢C ¯ ¯2 ⎥ ⎢ A ⎥ ⎢ Λ=⎢ . ⎥ ⎥ ⎣ .. ⎦ ¯A ¯N C

C. State Observer Design An insight into the controller (47) reveals that its implementation needs the full state Xk feedback of the system. However, only partial states are available in practice. Thus, a state observer is designed to estimate the full state by making use of the position information of the micropositioning system that is measured by displacement sensors.

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Fig. 11.

Block diagram of the proposed EMPC scheme.

Although various methods are available [40], a Luenbergerlike observer is adopted owing to its simplicity. The state observer takes on the form ˆ k + Buk + L(Yk − Yˆk ) + ˆfk ˆ k +1 = AX X ˆk Yˆk = CX

(48) (49)

ˆ k represents where L is the observer gain vector, the notation X the estimate of Xk , and the value of the disturbance fk in (26) is obtained by its one-step delayed estimation ˆfk = fk −1 = X ˆ k − AX ˆ k −1 − Buk −1 .

(50)

which indicates that the ultimate bound of the output tracking error is of the order O(T 2 ). The magnitude of the ultimate error is much lower than those produced in most of the previous works [38].

(51)

V. CONTROLLER VERIFICATION AND DISCUSSIONS

Subtracting (48) from (26) allows the derivation  k + (fk − fk −1 )  k +1 = (A − LC)X X

 k = Xk − X ˆ k denotes the estimation error of the state where X Xk . It is seen that the state observer is independent of the controller. To guarantee the stability of the observation unit, the observer gain L should be designed to make all the eigenvalues of the matrix (A − LC) locate inside the unit circle. The convergence speed of the observer relies on the location of the poles of the matrix (A − LC). The poles can be arbitrarily placed by the gain vector L. Generally, small pole value will give rapid convergence, and the convergence speed of the observer should be faster than the system response so that the observer dynamics is insignificant as compared with the system dynamics. D. Tracking Error Analysis In view of Property 1 and considering that (A − LC) is  k = O(T ). stable, it can be deduced from (51) and (30) that X That is, the ultimate state error is of the order O(T ). With the proposed controller, the tracking error can be derived as Ek +1 = Rk +1 − Yk +1 = Rk +1 − CXk +1 = Rk +1 − C[AXk + B(upid k + Δuk ) + fk ] = Rk +1 − CAXk − CBupid k − Cfk − CBΔuk . (52) Substituting (47) into (52) along with a necessary algebraic operation results in Ek +1 = C(fk − fk +1 ) = O(T 2 )

Fig. 12. Magnitudes of frequency responses of the two-input two-output micropositioning system. G x 1 describes the x-axis displacement response when the stage is driven by a voltage applied to VCM 1.

(53)

In this section, the designed controller is verified by a series of simulation and experimental studies that are conducted on the micropositioning system. A. Plant Model Identification The micropositioning system is viewed as a linear system with bounded disturbances as described in (24) and (25). The linear plant model is identified by the swept-sine approach via experiments. Specifically, swept-sine waves with the amplitude of 0.05 V and frequency range of 1–300 Hz are applied to the power amplifier to drive one VCM. The position responses of the XY stage in the two working directions are simultaneously recorded using a sampling rate of 2 kHz. With the two VCM actuated individually, the frequency responses (see Fig. 12) in the two axes are obtained by the fast Fourier transform. The frequency response of the i-axis motion induced by driving the VCM k is denoted by Gik (jω) = di (jω)/uk (jω), where the axis index i = x and y, and actuator index k = 1 and 2, respectively. It is observed that, with one VCM driven, the response in the passive axis is about 30 dB lower than that in the primary axis in the low-frequency range. This confirms that the two axial motions of the XY stage are well decoupled. Thus, an SISO controller can be designed for each axis. For the purpose of verifying the proposed control scheme, only the x-axis motion control is handled in this research. The input/output datasets are used to identify the plant transfer function by estimating the model from the frequency response data. The frequency responses that are obtained by experiment and estimated by the identified model shown later are compared in Fig. 13. The first resonant mode occurs around 23 Hz, and

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487

Fig. 13. Plant frequency responses in the x-axis that are obtained by experiments and the identified second-order model, which indicate that the model accurately captures the system dynamics up to 100 Hz.

the identified second-order model matches the system dynamics well in the frequencies below 100 Hz. In order to capture highfrequency dynamics accurately, a much higher order model is required to be identified. Here, a simple second-order model is employed to demonstrate the effectiveness of the proposed control scheme. The transfer function (54) shows that the two zeros (1587.4 ± 1341.7i) locate in the right-hand side half-plane. Thus, the micropositioning system has an NMP plant 5.7656s2 − 1.8305 × 104 s + 2.4907 × 107 . (54) s2 + 30.8676s + 2.0772 × 104 It is observed that the resonant mode, i.e., 23 Hz, is much lower than the FEA result that is obtained in Section III-A, i.e., 48.3 Hz. The discrepancy mainly comes from the mass of the moving coils of the VCM, capacitive sensors, as well as sensor targets, which are not considered in the FEA simulation. Gx1 (s) =

B. Controller Parameters Design Comparing the state-space model (24) and (25) with (54) yields the system parameters, which allow the calculation of the discrete-time model (26) and (27) with a sampling time T . In order to capture the system behavior precisely, a sampling frequency that is larger than 15 times of the first resonant mode, i.e., 23 Hz of the system is preferred. In the current research, a sampling rate of 500 Hz is adopted. With T = 0.002 s, the discrete-time system is represented by     0.9005 0.0019 12.8832 A= , B= −39.7318 0.9596 49 648.7140 C = [1 0],

D = 5.7656.

Based on the generated linear model, the state observer is constructed. The observer gain vector L is designed by the pole-placement technique. The trial simulations suggest that small pole values produce rapid convergence since a high bandwidth is achieved by the observer. However, experiments re-

Fig. 14. Simulation results of set-point positioning using (a) the conventional MPC controller and (b) the proposed EMPC controller. (c) Components of control action for EMPC with w = 4 × 105 .

veal that a pole that is too small leads to instability of the system, which arises from the sensor noises and spill-over effects of the neglected dynamics. To make a tradeoff between the two performances, the two poles are assigned as (0.85, 0.85), which are chosen to guarantee a relatively quicker convergence, as well as robustness of the observer in the presence of unmodeled dynamics. The corresponding gains are calculated as L = [0.1601, 0.0016]T . In addition, the PID control gains are tuned by means of the Z-N method, which gives Kp = 0.0010, Ki = 0.0809, and Kd = 3.1612 × 10−6 . Moreover, a prediction horizon N = 10 is selected for an illustration. C. Simulation Studies and Discussions Before experimental investigations, the performance of the designed controller is tested by several simulation studies conducted later. For demonstration, a 1-mm set-point positioning is carried out to discover the influence of the weight parameter w on the positioning performance. First, the conventional MPC without the PID control is tested. Increasing the weight w, the corresponding positioning results are depicted in Fig. 14(a). It is observed that with a smaller weight, i.e., w = 2 × 107 , a quicker response is obtained.

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Fig. 15. N.

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Settling time and RMS steady-state error versus the prediction horizon

However, the control action oscillates as a result, which leads to an oscillation in the positioning result. On the contrary, a larger weight, i.e., w = 8 × 107 , results in less oscillation in positioning. Nevertheless, it is at the cost of a large steady-state error. Hence, the conventional MPC is not capable of producing a satisfactory positioning. That is the major reason why EMPC scheme is proposed in this paper. Anyway, for a comparative study, w = 4 × 107 is selected to make a tradeoff between response speed and positioning accuracy. It produces a 5% settling time of 0.059 s with a root-mean-square (RMS) steady-state error of 5.209 μm. Second, the performance of the proposed EMPC scheme is verified. For different weight values, the positioning results are compared in Fig. 14(b). As the w increases, the control effort Δuk is more constrained. Thus, it is observed that the EMPC positioning result is more like the stand-alone PID result. The weight value, i.e., w = 4 × 105 , gives better result in terms of a short settling time of 0.046 s and a low RMS steady-state error of 5.574 × 10−5 μm. As compared with the conventional MPC method, the EMPC scheme has reduced the settling time by 22% and the steady-state positioning error by over 105 times. The effectiveness of the proposed controller is evident from the positioning results. Besides, for the EMPC with w = 4 × 105 , the components of control effort are depicted in Fig. 14(c). It is observed that the incremental action Δuk takes effect initially and then vanishes gradually as the time elapses. As a result, the positioning error is gradually reduced to zero. This process is enabled by minimizing the cost function (43) using the MPC technique. The results intuitively illustrate the efficiency of the EMPC strategy. Third, the effect of the prediction horizon selection is investigated. As the variation of the prediction horizon N , the settling time and steady-state RMS positioning errors are depicted in Fig. 15. It is observed that the larger the prediction horizon, the more rapid the system response. Concerning the RMS positioning error, it appears that N = 4 leads to the lowest error. For a specific application, a trial-and-error approach is needed to select a suitable prediction horizon. The performed simulation investigations validate the effectiveness of the proposed controller. The control scheme is further verified by experimental studies that are conducted later.

Fig. 16.

Histograms of (a) open-loop and (b) closed-loop positioning noises.

D. Experimental Results and Discussions In experimental studies, a spectrum analysis reveals that the power source introduces noises with a fundamental frequency of 50 Hz. Thus, a fourth-order Butterworth low-pass filter with the cutoff frequency of 50 Hz and a fourth-order Butterworth bandstop filter with stopbands of 42 and 60 Hz are employed to filter the sensor output. Correspondingly, the PID control parameters are tuned as Kp = 0.000 18, Ki = 0.0084, and Kd = 9.6429 × 10−7 by using the Z-N method. In addition, to achieve better experimental results, the weights, i.e., w = 9 × 108 and w = 8 × 107 , are selected for the conventional MPC and proposed EMPC, respectively. The prediction horizon is assigned as N = 10. First, the noise level of the micropositioning system is tested. The histograms of steady-state sensor readings under open- and closed-loop conditions are shown in Fig. 16. It is observed that the standard deviation of sensor output in open loop (RMS = 20.4 μm) is much larger than that in closed loop (RMS = 7.8 μm). This means that about 68.3% open-loop readings are less than 20.4 μm, whereas the value for the closed-loop readings is only 7.8 μm. The results clearly demonstrate the 61.8% improvement of the positioning resolution under the EMPC action over the open-loop method. Second, a consecutive-step positioning with a step size of 25 μm is carried out with the EMPC scheme, and the results are shown in Fig. 17. The steps can be clearly identified, which indicates that the positioning resolution of the system is better than 25 μm. Third, the set-point positioning capability of the micropositioning system is examined. By commanding the output platform from the home position (0, 0) to the position (1 mm, 0) in the workspace, the positioning results of PID, MPC, and EMPC are depicted in Fig. 18. For the purpose of quantitative comparisons, the results are tabulated in Table II. It is observed that the EMPC produces the quickest response since it reduces the settling time by 19% and 10%, as compared with PID and MPC schemes, respectively. Concerning the overshoot, the MPC gives the smallest value of 3.01%, which is slightly lower than

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TABLE II SET-POINT POSITIONING RESULTS OF DIFFERENT CONTROLLERS

Fig. 17. Experimental results of the 25-μm consecutive steps positioning, which indicate that the positioning resolution is better than 25 μm.

Fig. 18. Experimental results of set-point positioning using (a) PID control, (b) conventional MPC, and (c) the proposed EMPC scheme.

the EMPC result of 3.26%. In addition, both MPC and EMPC alleviate the overshoot by more than 65% in comparison with the PID control. The steady-state RMS positioning errors are also summarized in Table II, which shows that the three methods yield similar results at the steady state. Specifically, the EMPC delivers a positioning accuracy, which is about 1% better than the PID result, while it is 8% worse than the MPC output. The

reason why the MPC produces a smaller steady-state error than the EMPC lies in that the MPC is implemented with a larger weight value, which imposes more constrains on the control effort. As the cost, it gives a lower response speed in comparison with the EMPC as mentioned earlier. Moreover, the observer output as shown in Fig. 18 indicates that the designed state observer is capable of estimating the position information from the noisy sensor readings. By comparing the experimental results with the simulated ones as shown in Fig. 15, it is observed that a much slower transient response is obtained by experiments. The reason mainly arises from the low-pass filter (50-Hz cutoff frequency) that is employed in the experimental studies. The filter is adopted for the conditioning of sensor output signals; however, it causes a phase delay and moderated response at the same time. Another reason of discrepancy between the simulation and experimental results aries from the sensor noises. By employing a flexure-based mechanism, the micropositioning system is sensitive to environmental conditions. The sources of sensor noise mainly come from floor vibration, table vibration, and even air-flow variations. In the future, vibration isolation measures will be taken to improve the positioning resolution of the system. It is noticeable that the controller parameters are not optimally designed, and the positioning performance of the designed controller is limited by the resolution of the displacement sensors. The sensors are the bottleneck in ultraprecision positioning applications. Using sensors with a higher resolution along with proper signal conditioners, more accurate positioning can be achieved with the controller that is proposed in this paper. Although there is plenty of room for performance improvement of the proposed controller, the enhancement of positioning accuracy for the micropositioning system over the traditional PID and MPC elaborated by the conducted investigations validates the effectiveness of the proposed EMPC scheme and displays great potential for the future research. As a model predictive controller, the major restriction of the proposed control scheme lies in that the preview of the reference input is required to implement the control algorithm. This is not feasible when the desired input is unknown beforehand. How to eliminate this limitation will be explored in future research. Moreover, a dual-actuation and dual-sensing strategy may be explored to improve the positioning accuracy of the system. Experimental testing for biological micromanipulation will be conducted in the future. VI. CONCLUSION The concept of MCPF that has been proposed in this paper is useful in designing a multiaxis micropositioning system with both a compact size and a large workspace. A new XY

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micropositioning stage has been implemented for an illustration, which produces a workspace of 10.5 × 10.5 mm2 . Moreover, it owns a compact structure as reflected by the large area ratio (workspace to planar physical dimension) of 0.2407%, which is much larger than the existing parallel-kinematic XY stages driven by smart actuators. It has been found that the presented EMPC scheme is capable of improving the positioning performance in terms of settling time and steady-state error, as compared with conventional PID and MPC methods. Since the realization of the controller does not require the inverse of the plant model, it can be easily extended to both minimumand nonminimum-phase systems. Although only the leaf spring has been used in this research, the concept can be extended to the design of micropositioning systems with any other types of flexure hinges for achieving more types of motions.

REFERENCES [1] C. M. DiBiasio, M. L. Culpepper, R. Panas, L. L. Howell, and S. P. Magleby, “Comparison of molecular simulation and pseudo-rigid-body model predictions for a carbon nanotube-based compliant parallel-guiding mechanism,” J. Mech. Des., vol. 130, no. 4, pp. 042308-1–042308-7, 2008. [2] X. Ding and J. S. Dai, “Compliance analysis of mechanisms with spatial continuous compliance in the context of screw theory and Lie groups,” Proc. Inst. Mech. Eng. C, Mech. Eng. Sci., vol. 224, no. 11, pp. 2493–2504, Nov. 2010. [3] J. Choi, S. Hong, W. Lee, S. Kang, and M. Kim, “A robot joint with variable stiffness using leaf springs,” IEEE Trans. Robot., vol. 27, no. 2, pp. 229–238, Apr. 2011. [4] T. J. Teo, I.-M. Chen, G. Yang, and W. Lin, “A flexure-based electromagnetic linear actuator,” Nanotechnology, vol. 19, no. 31, pp. 315501-1– 315501-10, Aug. 2008. [5] H. C. Liaw, B. Shirinzadeh, and J. Smith, “Sliding-mode enhanced adaptive motion tracking control of piezoelectric actuation systems for micro/nano manipulation,” IEEE Trans. Control Syst. Technol., vol. 16, no. 4, pp. 826–833, Jul. 2008. [6] Y. Li and Q. Xu, “A novel piezoactuated XY stage with parallel, decoupled, and stacked flexure structure for micro-/nanopositioning,” IEEE Trans. Ind. Electron., vol. 58, no. 8, pp. 3601–3615, Aug. 2011. [7] F. Krohs, C. Onal, M. Sitti, and S. Fatikow, “Towards automated nanoassembly with the atomic force microscope: A versatile drift compensation procedure,” J. Dyn. Syst., Meas., Control, vol. 131, no. 6, pp. 061106-1–061106-8, 2009. [8] K. K. Leang, Q. Zou, and S. Devasia, “Feedforward control of piezoactuators in atomic force microscope systems: Inversion-based compensation for dynamics and hysteresis,” IEEE Control Syst. Mag., vol. 29, no. 1, pp. 70–82, Feb. 2009. [9] Y. K. Yong, S. Aphale, and S. O. R. Moheimani, “Design, identification, and control of a flexure-based XY stage for fast nanoscale positioning,” IEEE Trans. Nanotechnol., vol. 8, no. 1, pp. 46–54, Jan. 2009. [10] S. Polit and J. Dong, “Development of a high-bandwidth XY nanopositioning stage for high-rate micro-/nanomanufacturing,” IEEE/ASME Trans. Mechatronics, vol. 16, no. 4, pp. 724–733, Aug. 2011. [11] J. Minase, T.-F. Lu, B. Cazzolato, and S. Grainger, “A review, supported by experimental results, of voltage, charge and capacitor insertion method for driving piezoelectric actuators,” Precis. Eng., vol. 34, no. 4, pp. 692–700, Oct. 2010. [12] T. W. Secord and H. H. Asada, “A variable stiffness PZT actuator having tunable resonant frequencies,” IEEE Trans. Robot., vol. 26, no. 6, pp. 993– 1005, Dec. 2010. [13] M. Probst, M. Fluckiger, S. Pane, O. Ergeneman, Z. Nagy, and B. J. Nelson, “Manufacturing of a hybrid acoustic transmitter using an advanced microassembly system,” IEEE Trans. Ind. Electron., vol. 56, no. 7, pp. 2657– 2666, Jul. 2009. [14] H. B. Huang, D. Sun, J. K. Mills, and S. H. Cheng, “Robotic cell injection system with position and force control: Toward automatic batch biomanipulation,” IEEE Trans. Robot., vol. 25, no. 3, pp. 727–737, Jun. 2009.

[15] C. Leung, Z. Lu, N. Esfandiari, R. F. Casper, and Y. Sun, “Automated sperm immobilization for intracytoplasmic sperm injection,” IEEE Trans. Biomed. Eng., vol. 58, no. 4, pp. 935–942, Apr. 2011. [16] Y.-J. Choi and S. V. Sreenivasan, B. J. Choi, “Kinematic design of large displacement precision XY positioning stage by using cross strip flexure joints and over-constrained mechanism,” Mech. Mach. Theory, vol. 43, no. 6, pp. 724–737, Jun. 2008. [17] S. Verma, W.-J. Kim, and J. Gu, “Six-axis nanopositioning device with precision magnetic levitation technology,” IEEE/ASME Trans. Mechatronics, vol. 9, no. 2, pp. 384–391, Jun. 2004. [18] D.-H. Kim, M. G. Lee, B. Kim, and Y. Sun, “A superelastic alloy microgripper with embedded electromagnetic actuators and piezoelectric force sensors: A numerical and experimental study,” Smart Mater. Struct., vol. 14, no. 6, pp. 1265–1272, 2005. [19] X. Tang and I.-M. Chen, “A large-displacement and decoupled XYZ flexure parallel mechanism for micromanipulation,” in Proc. IEEE Int. Conf. Autom. Sci. Eng., Shanghai, China, Oct. 2006, pp. 75–80. [20] W. Dong, J. Tang, and Y. ElDeeb, “Design of a linear-motion dual-stage actuation system for precision control,” Smart Mater. Struct., vol. 18, no. 9, pp. 095035-1–095035-11, Sep. 2009. [21] M.-Y. Chen, C.-C. Wang, and L.-C. Fu, “Adaptive sliding mode controller design of a dual-axis maglev positioning system,” in Proc. Amer. Control Conf., Arlington, VA, Jun. 2001, pp. 3731–3736. [22] B. H. Kang, J. T.-Y. Wen, N. G. Dagalakis, and J. J. Gorman, “Analysis and design of parallel mechanisms with flexure joints,” IEEE Trans. Robot., vol. 21, no. 6, pp. 1179–1185, Dec. 2005. [23] J. B. Hopkins and M. L. Culpepper, “Synthesis of multi-degree of freedom, parallel flexure system concepts via freedom and constraint topology (FACT)—Part I: Principles,” Precis. Eng., vol. 34, no. 2, pp. 259–270, Apr. 2010. [24] N. Lobontiu, Compliant Mechanisms: Design of Flexure Hinges. Boca Raton, FL: CRC, 2002. [25] D. Kang, K. Kim, D. Kim, J. Shim, D.-G. Gweon, and J. Jeong, “Optimal design of high precision XY-scanner with nanometer-level resolution and millimeter-level working range,” Mechatronics, vol. 19, no. 4, pp. 562– 570, Jun. 2009. [26] M.-Y. Chen and H.-W. Tzeng, “Implementation of a novel large moving range submicrometer positioner,” Mechatronics, vol. 19, no. 7, pp. 1143– 1151, Oct. 2009. [27] S. Awtar and G. Parmar, “Design of a large range XY nanopositioning system,” in Proc. ASME Int. Design Eng. Tech. Conf., Montreal, QC, Canada, 2010, pp. 387–399. [28] S. Awtar and A. H. Slocum, “Constraint-based design of parallel kinematic XY flexure mechanisms,” J. Mech. Des., vol. 129, no. 8, pp. 816–830, 2007. [29] S. Fatikow, T. Wich, H. Hulsen, T. Sievers, and M. Jahnisch, “Microrobot system for automatic nanohandling inside a scanning electron microscope,” IEEE/ASME Trans. Mechatronics, vol. 12, no. 3, pp. 244–252, Jun. 2007. [30] J. B. Hag and D. S. Bernstein, “Nonminimum-phase zeros - much to do about nothing - classical control - revisited—Part II,” IEEE Control Syst. Mag., vol. 27, no. 3, pp. 45–57, Jun. 2007. [31] J. A. Butterworth, L. Y. Pao, and D. Y. Abramovitch, “The effect of nonminimum-phase zero locations on the performance of feedforward model-inverse control techniques in discrete-time systems,” in Proc. Amer. Control Conf., Seattle, WA, 2008, pp. 2696–2702. [32] M. Ruzbehani, “A new tracking controller for discrete-time SISO nonminimum phase systems,” Asian J. Control, vol. 12, no. 1, pp. 89–95, 2010. [33] X.-G. Yan, C. Edwards, and S. K. Spurgeon, “Output feedback sliding mode control for non-minimum phase systems with non-linear disturbances,” Int. J. Control, vol. 77, no. 15, pp. 1353–1361, 2004. [34] M. Deng, A. Inoue, A. Yanou, and Y. Hirashima, “Continuous-time antiwindup generalized predictive control of non-minimum phase processes with input constraints,” in Proc. 42th IEEE Conf. Decision Control, Maui, HI, 2003, pp. 4457–4462. [35] A. G. Wills, D. Bates, A. J. Fleming, B. Ninness, and S. O. R. Moheimani, “Model predictive control applied to constraint handling in active noise and vibration control,” IEEE Trans. Control Syst. Technol., vol. 16, no. 1, pp. 3–12, Jan. 2008. [36] V. A. Neelakantan, G. N. Washington, and N. K. Bucknor, “Model predictive control of a two stage actuation system using piezoelectric actuators for controllable industrial and automotive brakes and clutches,” J. Intell. Mater. Syst. Struct., vol. 19, no. 7, pp. 845–857, Jul. 2008.

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[37] Y. Li and Q. Xu, “Design and analysis of a totally decoupled flexurebased XY parallel micromanipulator,” IEEE Trans. Robot., vol. 25, no. 3, pp. 645–657, Jun. 2009. [38] J.-X. Xu and K. Abidi, “Discrete-time output integral sliding-mode control for a piezomotor-driven linear motion stage,” IEEE Trans. Ind. Electron., vol. 55, no. 11, pp. 3917–3926, Nov. 2008. [39] Q. Xu and Y. Li, “Micro-/nanopositioning using model predictive output integral discrete sliding mode control,” IEEE Trans. Ind. Electron., vol. 59, no. 2, pp. 1161–1170, Feb. 2012. [40] A. J. Koshkouei and A. S. I. Zinober, “Sliding mode state observers for discrete-time linear systems,” Int. J. Syst. Sci., vol. 33, no. 9, pp. 751–758, 2002.

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Qingsong Xu (M’09) received the B.S. degree (Hons.) in mechatronics engineering from Beijing Institute of Technology, Beijing, China, in 2002 and the M.S. and Ph.D. degrees in electromechanical engineering from the University of Macau, Macao, China, in 2004 and 2008, respectively. He was a visiting scholar with the Swiss Federal Institute of Technology, Zurich, Switzerland. He is currently an Assistant Professor with the Department of Electromechanical Engineering, University of Macau. His current research interests include parallel manipulators, microelectromechanical devices, micro-/nanorobotics, micro-/nanomanipulation, smart materials and structures, and computational intelligence. Dr. Xu is a member of the American Society of Mechanical Engineers.