Development of a Parallel Dual-Stage Compliant Nanopositioning System

: This paper presents a novel parallel dual-stage compliant nanopositioning system (PDCNS), aimed at nanoscale positioning for microscale manipulation. In the developed PDCNS, the coarse stage actuated by the voice coil motor and the ﬁne stage driven by the piezoelectric actuator are integrated in a parallel manner by a specially devised A-shaped compliant mechanism, which leads to many excellent performances, such as good resolution and large stroke and broadband. To enhance the closed-loop-positioning capability of the proposed PDCNS, a double-servo cooperative control (DSCC) strategy is specially constructed. The performance of the proposed PDCNS is evaluated by analytical model, ﬁnite element analysis, and experimental research. Results show that the ﬁrst-order resonance frequency of the designed A-shaped compliant mechanism can reach 99.7 Hz. Combined with the designed DSCC, the developed PDCNS prototype is demonstrated to provide a stroke of 1.49 mm and a positioning resolution of ≤ 50 nm.


Introduction
Precision-positioning platforms with nanometer resolution play more and more important roles in many scientific and industrial applications [1][2][3][4][5][6]. In previous research, piezoelectric actuators were often adopted to configure these nanopositioning stages because of their fast speed and nanometer-level resolution [7][8][9][10]. Nevertheless, the maximum stroke of the piezoelectric actuator is only about 0.1% of its length [11]. Piezoelectric motors which can realize continuous nanoscale movement without stroke limit have been applied in nanopositioning systems as well [12][13][14][15]. However, the piezoelectric motors encounter severe nonlinear friction. To overcome these drawbacks, the dual-stage nanopositioning systems are employed to deliver high-resolution motion over a long range.
In the literature, some types of dual-stage positioning systems were proposed. To improve the areal density of the magnetic hard disk, a dual-stage positioning mechanism was designed to regulate the position of read-write head, in which an additional piezoelectric actuator was mounted on the primary stage for fast and fine positioning [16]. Fast tool servo units based on the piezoelectric actuator or electromagnetic actuator were used to cooperate with the feed shaft of machine tools for high-speed and high-accuracy nanocutting [17][18][19]. A piezo-driven compliant injector was configured at the terminal of a traditional XYZ-positioning stage to accomplish the automatic injection and batch micromanipulation of cells [20,21]. However, the previous design concepts of the dual-stage system were limited to stacking a fine stage on top of a coarse stage in a serial manner, which encountered many technical bottlenecks, such as large moving mass, assembly error,

Design of PDCNS
The PDCNS consists of an A-shaped compliant parallel mechanism (ACPM), a voice coil motor and a piezoelectric actuator, as shown in Figure 1. The ACPM can transmit the motion of the voice coil motor (VCM) to the end effector through two parallel limbs. At the same time, the piezoelectric actuator (PEA) contributes to the motion of the end effector via a half bridge-type compliant chain. For compactness, the upper part of the ACPM and the half bridge-type chain share the same flexural structure. To eliminate parasitic motions, two guiding flexible beams are designed for the end effector which help to improve the resonant frequency as well. In addition, two input decoupling compliant mechanisms are devised to protect the piezoelectric actuator from undesired lateral forces. A support bearing is adopted to guide the actuation motion of the voice coil motor. error, and uncertain disturbance from the cable. Therefore, new configurations of dual stage positioning systems are still required.
Apart from the structure design, the control strategy also significantly influences the performance of the dual-stage positioning system. Extensive research on the closed-loop control of the dual-stage positioning system was carried out in the past few years. Fo example, Xu et al. [22] and Michellod et al. [23] designed single-feedback control scheme for the dual-stage nanopositioning system that determines the relative position between two stages using observers. By employing the position feedbacks from coarse and fine stages, Dong et al. [24] and Zhu et al. [25] proposed multiple-feedback control systems fo the dual-stage nanopositioning system. Despite this, it is still a challenging job to contro a dual-stage positioning system.
In this paper, a novel parallel dual-stage compliant nanopositioning system (PDCNS with a corresponding double-servo cooperative control (DSCC) strategy is developed. The PDCNS employs a new A-shaped compliant parallel mechanism to combine the coarse motion from the voice coil motor and the fine motion from the piezoelectric actuator. Con sidering the uncertain disturbances and the double-input-single-output configuration, a DSCC is designed in this paper to enhance the motion capability of the proposed PDCNS The static and dynamic performances of the PDCNS are investigated by the analytical model-based calculation and the finite element analysis. Moreover, a series of experi mental studies are carried out to demonstrate the kinematic properties and closed-loop positioning performance of the PDCNS.
The rest of this paper is organized as follows: Section 2 presents the mechanical de sign of the PDCNS. An analytical model of the PDCNS is established in Section 3. With the analytical model and finite element analysis, the static and dynamic performances o the proposed PDCNS are analyzed in Section 4. Moreover, experiments are conducted to evaluate the static and dynamic performances of the proposed PDCNS in Section 5. Fi nally, conclusions are given in Section 6.

Design of PDCNS
The PDCNS consists of an A-shaped compliant parallel mechanism (ACPM), a voice coil motor and a piezoelectric actuator, as shown in Figure 1. The ACPM can transmit the motion of the voice coil motor (VCM) to the end effector through two parallel limbs. A the same time, the piezoelectric actuator (PEA) contributes to the motion of the end effec tor via a half bridge-type compliant chain. For compactness, the upper part of the ACPM and the half bridge-type chain share the same flexural structure. To eliminate parasitic motions, two guiding flexible beams are designed for the end effector which help to im prove the resonant frequency as well. In addition, two input decoupling compliant mech anisms are devised to protect the piezoelectric actuator from undesired lateral forces. A support bearing is adopted to guide the actuation motion of the voice coil motor. With the above designs, both the voice coil motor and the piezoelectric actuator can contribute to the output motion. As the voice coil motor has a large-range motion capability and the piezoelectric actuator features a high-resolution motion characteristic, the proposed PDCNS is expected to position in the millimeter range with nanometer resolution. Moreover, as the bases of the voice coil motor and the piezoelectric actuator remain static to the ground, low-moving mass and high dynamics can be realized.

Analytical Modelling
In this section, the kinematic and stiffness characteristics of PDCNS are investigated. Based on the finite element theory [26,27], an analytical model is built for the ACPM, as shown in Figure 2. We can see from the figure, rigid nodes with three degrees of freedom are defined at the two ends of the beams to describe the deformation of the flexible beam. A global coordinate system O-xy is defined to analyze the overall deformation of ACPM. With the above designs, both the voice coil motor and the piezoelectric actuator can contribute to the output motion. As the voice coil motor has a large-range motion capa bility and the piezoelectric actuator features a high-resolution motion characteristic, the proposed PDCNS is expected to position in the millimeter range with nanometer resolu tion. Moreover, as the bases of the voice coil motor and the piezoelectric actuator remain static to the ground, low-moving mass and high dynamics can be realized.

Analytical Modelling
In this section, the kinematic and stiffness characteristics of PDCNS are investigated Based on the finite element theory [26,27], an analytical model is built for the ACPM, as shown in Figure 2. We can see from the figure, rigid nodes with three degrees of freedom are defined at the two ends of the beams to describe the deformation of the flexible beam A global coordinate system O-xy is defined to analyze the overall deformation of ACPM According to the structural mechanics theory [26,28], in the local coordinate system of the flexible beam, the relationship between the applied force and the induced displace ment of the flexible beam can be expressed as are the x-, y-directional forces and z-directional torque applied on node i in the local coordinate system, , , are the x-, y-directional forces and z-direc tional torque applied on node j in the local coordinate system, , , are the x-, y directional translations and z-directional angular displacement of node i in the local coor dinate system, , , are the x-, y-directional translations and z-directional angula displacement of node j in the local coordinate system, E is Young's modulus, b, h are the section sizes of the flexible beam, l is the length of the flexible beam, and I is the momen of inertia of the flexible beam which can be calculated by = ℎ 3 12 ⁄ , is the loca stiffness matrix of the flexible beam.
With Equation (1), the loads applied on the two end ports of the flexible beam can be calculated according to the displacements of the corresponding two rigid nodes. Never theless, the stiffness matrix in Equation (1) is expressed in the local coordinate system which relates the deformations and loads of the flexible beam in the local coordinate According to the structural mechanics theory [26,28], in the local coordinate system of the flexible beam, the relationship between the applied force and the induced displacement of the flexible beam can be expressed as where f lx i , f ly i , τ lθ i are the x-, y-directional forces and z-directional torque applied on node i in the local coordinate system, f lx j , f ly j , τ lθ j are the x-, y-directional forces and z-directional torque applied on node j in the local coordinate system, δ lx i , δ ly i , φ lθ i are the x-, y-directional translations and z-directional angular displacement of node i in the local coordinate system, δ lx j , δ ly j , φ lθ j are the x-, y-directional translations and z-directional angular displacement of node j in the local coordinate system, E is Young's modulus, b, h are the section sizes of the flexible beam, l is the length of the flexible beam, and I is the moment of inertia of the flexible beam which can be calculated by I = hb 3 /12, k loc is the local stiffness matrix of the flexible beam.
With Equation (1), the loads applied on the two end ports of the flexible beam can be calculated according to the displacements of the corresponding two rigid nodes. Nevertheless, the stiffness matrix in Equation (1) is expressed in the local coordinate system, which relates the deformations and loads of the flexible beam in the local coordinate system. To combine the deformations and loads of all the flexible beams, a rotation transformation is Actuators 2022, 11, 136 4 of 14 employed to transfer the local stiffness matrix, k loc , to be a global stiffness matrix, k glo , as shown in Equation (3) [26,29].
where R(θ) is the rotation matrix, θ is the angle between the local and global coordinate systems (defined as the angle from the x axis of the local coordinate system to the x axis of the global coordinate system around the o point of the local coordinate system).
With the global stiffness matrix in (3) and the analytical model in Figure 2a, an overall stiffness matrix, K, can be established for the ACPM with the finite element theory [26,30], which is able to depict the relationship between the external forces and induced deformations as follows.
where f x i and f y i are the xand y-directional external forces applied on node i, τ θ i is the external torque around the z-axis applied on node i, δ x i , and δ y i are the xand y-directional translations of node i, and φ θ i is the rotational angle (around z-axis) of node i. While two input forces, f PA , are applied on nodes 7 and 20, the resulting input displacements, δ y 1 , from static equilibrium Equation (4) can be set to zero (unchanged with respect to the ground) by searching for a suitable actuation force, f VCM , for node 1. Under the above conditions, the input displacements, δ x 7 and δ x 20 , as well as the output displacement, δ y 26 , can be calculated.
(1) The input stiffness for the PEA can be derived as: (2) The displacement amplification ratio for the PEA can be derived as: (3) The maximum stress coefficient of the beam 1 contributed by the input displacement of PEA can be calculated by [30,31].
where k c is the stress concentration factor which is set to 2 in this paper. By applying one input force, f VCM , on node 1 and in the meantime imposing a suitable actuation force, f PA , on nodes 7 and 20, the resulting input displacement δ x 7 + δ x 20 from static equilibrium Equation (4) can be set to zero. Under the above conditions, the induced input displacement, δ y 1 , and output displacement, δ y 26 , can be calculated. (4) The input stiffness for the VCM can be derived as: (6) The maximum stress coefficient of beam 2 contributed by the input displacement of the VCM can be calculated by [30,31].
(7) For the PDCNS, the maximum stress usually occurs in beam 1 or beam 2. Hence, a maximum stress matrix is established for the PDCNS as follows: where σ 1 max and σ 2 max are the maximum stresses of beam 1 and beam 2, respectively. According to (11), the compressive stress of beam 2 induced by the PEA is ignored since it is small compared to the bending stress induced by the maximum displacement of the VCM. Due to the good input decoupling, the bending stress of beam 1 induced by the VCM is ignored as well.

Simulation Analysis
Based on the working principles of the PDCNS, a 3D virtual prototype is built by using the SOLIDWORKS software. The kinematic and stiffness performances of the PDCNS virtual prototype are evaluated by the established analytical model. The key parameters of the analytical model are listed in Table 1. The dimensions of the PDCNS virtual prototype are indicated in Figure 2. The material utilized for analysis is TC4 titanium alloy with Young's modulus E, Poisson's ratio µ, and mass density ρ [32,33]. To validate the static and dynamic performances of the PDCNS virtual prototype, ANSYS Workbenchbased FEA simulation is carried out as well in this section. The computational models for FEA analysis are configured as follows: mesh physics preference = mechanical, mesh size function = proximity and curvature, mesh relevance = −75, inner surfaces of six bolt holes is fixed, solver type = program controlled.  First, by applying a driving force (38.36 N) to the input end of the VCM and fixing the input end of the PEA in the ANSYS Workbench, the resulting overall deformation of the proposed ACPM can be obtained as shown in Figure 3a. The slight displacement difference between the two input ends of PEA can be attributed to asymmetric probe location, asymmetric mesh generation, computational convergence error, etc. The displacement amplification ratio for the VCM can be derived by relating the displacements of the VCM and the end effector. The input stiffness for VCM can be revealed by identifying the slope of the applied force over the induced displacement. As seen in Table 2, the amplification ratio for the VCM is approximately 1, which means that the motion of the VCM is transmitted to the end effector almost directly. The input stiffness for the VCM is revealed to be only 0.026 N/µm, which helps to achieve a large stroke. Good agreement between the results of the analytical model and the ANSYS Workbench are obtained, confirming the performance of the proposed ACPM. PEA are revealed to be 0.52 and 2.02 N/μm, respectively, which are highly consistent with the analytical predictions, as shown in Table 2.  Corresponding to the deformations shown in Figure 3, the stress distributions of ACPM are also resolved in the ANSYS Workbench, as illustrated in Figure 4. It can be observed that the beam between nodes 20 and 23 (which has the same stress state as beam 2) is revealed to have the maximum stress, while the VCM is working at its maximum stroke of 1.5 mm. The beam between nodes 16 and 17 (which has the same stress state as beam 1) tends to show the maximum stress, while the PEA is working at the displacement of 60 μm. The obtained maximum stress coefficients are tabulated in Table 2, which agree well with the analytical results. To reveal the maximum stress while the proposed ACPM is actuated by the VCM and PEA simultaneously, two actuation displacements of 1.5 mm (| 1 |) and 60 μm (| 7 | + | 20 |) are applied on the ACPM, as shown in Figure 5. It can be observed that beam 2 shows the maximum stress with a value close to the maximum stress shown in Figure 4a. Furthermore, the maximum stresses of the node16-node17 beams (beam between nodes 16 and 17) in Figure 4b and Figure 5 are close to each other. Therefore, for the proposed ACPM, the maximum stress of beam 2 is mainly contributed by the VCM. Because of the  Similarly, by applying a driving force of 121 N to the input ends of the PEA and fixing the input end of the VCM in the ANSYS Workbench, the resulting deformation of the proposed ACPM is shown in Figure 3b. The amplification ratio and input stiffness for the PEA are revealed to be 0.52 and 2.02 N/µm, respectively, which are highly consistent with the analytical predictions, as shown in Table 2.
Corresponding to the deformations shown in Figure 3, the stress distributions of ACPM are also resolved in the ANSYS Workbench, as illustrated in Figure 4. It can be observed that the beam between nodes 20 and 23 (which has the same stress state as beam 2) is revealed to have the maximum stress, while the VCM is working at its maximum stroke of 1.5 mm. The beam between nodes 16 and 17 (which has the same stress state as beam 1) tends to show the maximum stress, while the PEA is working at the displacement of 60 µm. The obtained maximum stress coefficients are tabulated in Table 2, which agree well with the analytical results. PEA are revealed to be 0.52 and 2.02 N/μm, respectively, which are highly consistent with the analytical predictions, as shown in Table 2.  Corresponding to the deformations shown in Figure 3, the stress distributions of ACPM are also resolved in the ANSYS Workbench, as illustrated in Figure 4. It can be observed that the beam between nodes 20 and 23 (which has the same stress state as beam 2) is revealed to have the maximum stress, while the VCM is working at its maximum stroke of 1.5 mm. The beam between nodes 16 and 17 (which has the same stress state as beam 1) tends to show the maximum stress, while the PEA is working at the displacement of 60 μm. The obtained maximum stress coefficients are tabulated in Table 2, which agree well with the analytical results. To reveal the maximum stress while the proposed ACPM is actuated by the VCM and PEA simultaneously, two actuation displacements of 1.5 mm (| 1 |) and 60 μm (| 7 | + | 20 |) are applied on the ACPM, as shown in Figure 5. It can be observed that beam 2 To reveal the maximum stress while the proposed ACPM is actuated by the VCM and PEA simultaneously, two actuation displacements of 1.5 mm ( δ y 1 ) and 60 µm ( δ x 7 + δ x 20 ) are applied on the ACPM, as shown in Figure 5. It can be observed that beam 2 shows the maximum stress with a value close to the maximum stress shown in Figure 4a. Furthermore, the maximum stresses of the node16-node17 beams (beam between nodes 16 and 17) in Figures 4b and 5 are close to each other. Therefore, for the proposed ACPM, the maximum stress of beam 2 is mainly contributed by the VCM. Because of the input-decoupling compliant mechanisms, the VCM has limited influence on the stress of beam 1.  Finally, modal analysis of the proposed ACPM is carried out in the ANSYS Work bench software. Considering the guiding function of the bearing of the VCM, a cylindrica support with the Y-translational freedom is applied on the input end of the VCM. Unde sired out-of-plane resonances can be attenuated by the VCM bearing. Figure 6 shows the revealed first-order modal shape of the proposed ACPM. It can be seen that the first res onant resonance occurs along the actuation direction of the VCM with a frequency of 99.7 Hz. Therefore, while in open-loop control mode, the VCM actuation should be operated below 99.7 Hz to avoid mechanical resonance.

Experimental Analysis
A PDCNS prototype is fabricated to verify its kinematic performance as well as study its closed-loop motion capability, as shown in Figure 7. The ACPM with the dimension shown in Table 1 is monolithically fabricated from TC4 titanium alloy. A commercia VCM (TM motion, model: TMEH0250, with a support bearing) and a commercial PEA (model: NAC2015-H38, maximum displacement: 59.4 μm, maximum thrust: 4200 N) are adopted to actuate the ACPM. To evaluate the motion of the ACPM, two laser displace ment sensors (KEYENCE, model: LK-G10, max. measurement range: ±1mm) are used to measure the displacements of the VCM and the end effector, and two capacitance dis placement sensors (MTI Instruments, model: ASP-125M, max. measurement range: 125 μm) are employed to detect the input displacement of the PEA. The following experiment are carried out on a PC with a PCI-6259 data acquisition card. Finally, modal analysis of the proposed ACPM is carried out in the ANSYS Workbench software. Considering the guiding function of the bearing of the VCM, a cylindrical support with the Y-translational freedom is applied on the input end of the VCM. Undesired outof-plane resonances can be attenuated by the VCM bearing. Figure 6 shows the revealed first-order modal shape of the proposed ACPM. It can be seen that the first resonant resonance occurs along the actuation direction of the VCM with a frequency of 99.7 Hz. Therefore, while in open-loop control mode, the VCM actuation should be operated below 99.7 Hz to avoid mechanical resonance.  Finally, modal analysis of the proposed ACPM is carried out in the ANSYS Workbench software. Considering the guiding function of the bearing of the VCM, a cylindrica support with the Y-translational freedom is applied on the input end of the VCM. Unde sired out-of-plane resonances can be attenuated by the VCM bearing. Figure 6 shows the revealed first-order modal shape of the proposed ACPM. It can be seen that the first resonant resonance occurs along the actuation direction of the VCM with a frequency of 99.7 Hz. Therefore, while in open-loop control mode, the VCM actuation should be operated below 99.7 Hz to avoid mechanical resonance.

Experimental Analysis
A PDCNS prototype is fabricated to verify its kinematic performance as well as study its closed-loop motion capability, as shown in Figure 7. The ACPM with the dimensions shown in Table 1 is monolithically fabricated from TC4 titanium alloy. A commercia VCM (TM motion, model: TMEH0250, with a support bearing) and a commercial PEA (model: NAC2015-H38, maximum displacement: 59.4 μm, maximum thrust: 4200 N) are adopted to actuate the ACPM. To evaluate the motion of the ACPM, two laser displace ment sensors (KEYENCE, model: LK-G10, max. measurement range: ±1mm) are used to measure the displacements of the VCM and the end effector, and two capacitance displacement sensors (MTI Instruments, model: ASP-125M, max. measurement range: 125 μm) are employed to detect the input displacement of the PEA. The following experiments are carried out on a PC with a PCI-6259 data acquisition card.

Experimental Analysis
A PDCNS prototype is fabricated to verify its kinematic performance as well as study its closed-loop motion capability, as shown in Figure 7. The ACPM with the dimensions shown in Table 1 is monolithically fabricated from TC4 titanium alloy. A commercial VCM (TM motion, model: TMEH0250, with a support bearing) and a commercial PEA

Kinematic Investigation
The displacement amplification ratio for the VCM is identified by maintaining th position of the PEA with a closed-loop control and increasing the control signal of th VCM at the same time. Figure 8a shows the obtained displacement relationship betwee the end effector and the VCM while the displacement of the PEA is maintained at 0μm The displacement amplification ratio for the VCM is calculated to be 0.992. The small di ference between the measured and calculated displacement amplification ratios of VCM confirms the effectiveness of the established analytical model.

Kinematic Investigation
The displacement amplification ratio for the VCM is identified by maintaining the position of the PEA with a closed-loop control and increasing the control signal of the VCM at the same time. Figure 8a shows the obtained displacement relationship between the end effector and the VCM while the displacement of the PEA is maintained at 0 µm. The displacement amplification ratio for the VCM is calculated to be 0.992. The small difference between the measured and calculated displacement amplification ratios of VCM confirms the effectiveness of the established analytical model.

Kinematic Investigation
The displacement amplification ratio for the VCM is identified by maintaining the position of the PEA with a closed-loop control and increasing the control signal of the VCM at the same time. Figure 8a shows the obtained displacement relationship between the end effector and the VCM while the displacement of the PEA is maintained at 0μm The displacement amplification ratio for the VCM is calculated to be 0.992. The small dif ference between the measured and calculated displacement amplification ratios of VCM confirms the effectiveness of the established analytical model. By applying a triangular-waveform signal with a frequency of 0.1 Hz and amplitude of 5 V to the PEA (maintaining the control voltage of the VCM at 0V), the induced dis placement of the end effector and the coupling displacement of the VCM are measured and recorded. By calculating the output displacement induced by the VCM (product o the identified amplification ratio and measured displacement of the VCM) and subtracting it from the measured displacement of the end effector, the PEA-induced output dis placement can be obtained. Figure 8b shows the measured relationship between the displacement of PEA and the output displacement induced by PEA. Because of measuremen noise, the measured raw data are distributed around the linear fitting line. Despite this the kinematic of the PEA actuation can be identified according to the slope ratio of fitted linear line. The displacement amplification ratio for PEA is obtained to be 0.433. According to the established analytical model, the amplification ratio for the PEA is 0.514. By means of finite element analysis (based on ANSYS Workbench), the amplification ratio for the PEA is evaluated to be 0.524. The deviation between the proposed method and the finite element analysis is only 2.0%. Nevertheless, the deviation between the proposed By applying a triangular-waveform signal with a frequency of 0.1 Hz and amplitude of 5 V to the PEA (maintaining the control voltage of the VCM at 0 V), the induced displacement of the end effector and the coupling displacement of the VCM are measured and recorded. By calculating the output displacement induced by the VCM (product of the identified amplification ratio and measured displacement of the VCM) and subtracting it from the measured displacement of the end effector, the PEA-induced output displacement can be obtained. Figure 8b shows the measured relationship between the displacement of PEA and the output displacement induced by PEA. Because of measurement noise, the measured raw data are distributed around the linear fitting line. Despite this, the kinematic of the PEA actuation can be identified according to the slope ratio of fitted linear line. The displacement amplification ratio for PEA is obtained to be 0.433. According to the established analytical model, the amplification ratio for the PEA is 0.514. By means of finite element analysis (based on ANSYS Workbench), the amplification ratio for the PEA is evaluated to be 0.524. The deviation between the proposed method and the finite element analysis is only 2.0%. Nevertheless, the deviation between the proposed method and the experiments is 15.7%. This obvious deviation of the PEA displacement amplification ratio can be attributed to manufacturing error, assembly error, mismatched material parameters, etc.

Stroke and Hysteresis Investigation
The stroke and hysteresis characteristics of the proposed PDCNS are further investigated. First, two 0.1 Hz triangular control signals with amplitudes of 10 V and 5 V are separately applied to the VCM. The displacement responses of the PDCNS are obtained, as in Figure 9a. It can be seen that the maximum stroke of the PDCNS can reach 1.49 mm. While the output displacement of PDCNS is calibrated to be 1.49 mm, the maximum stress of the ACPM is simulated to be 431 Mpa, which is below the yield stress of TC4 titanium alloy (>790 Mpa [32]), as shown in Figures 3a and 4a. Hence, the stroke of the PDCNS prototype is limited by the maximum driving force of the voice coil motor. Besides, due to the friction of the VCM's bearing, a significant dead zone exists in the response curve of PDCNS. Then, another two triangular control signals are applied to the PEA, as illustrated in Figure 9b. Compared to the VCM-actuated PDCNS, the PEA-actuated PDCNS is free of the friction-induced dead zone. In the meantime, the PEA-actuated PDCNS displays a reduced hysteresis behavior.
Actuators 2022, 11, x FOR PEER REVIEW method and the experiments is 15.7%. This obvious deviation of the PEA displa amplification ratio can be attributed to manufacturing error, assembly error, mism material parameters, etc.

Stroke and Hysteresis Investigation
The stroke and hysteresis characteristics of the proposed PDCNS are further gated. First, two 0.1 Hz triangular control signals with amplitudes of 10 V and separately applied to the VCM. The displacement responses of the PDCNS are o as in Figure 9a. It can be seen that the maximum stroke of the PDCNS can reach 1 While the output displacement of PDCNS is calibrated to be 1.49 mm, the maximu of the ACPM is simulated to be 431 Mpa, which is below the yield stress of TC4 t alloy (>790 Mpa [32]), as shown in Figures 3a and 4a. Hence, the stroke of the prototype is limited by the maximum driving force of the voice coil motor. Besid to the friction of the VCM's bearing, a significant dead zone exists in the respons of PDCNS. Then, another two triangular control signals are applied to the PEA, trated in Figure 9b. Compared to the VCM-actuated PDCNS, the PEA-actuated PD free of the friction-induced dead zone. In the meantime, the PEA-actuated PDC plays a reduced hysteresis behavior.

Motion Resolution Investigation
Consecutive-step-positioning experiments are carried out to investigate the resolution of the proposed PDCNS, as shown in Figure 10. A clear motion resoluti a step size of 0.74 μm can be achieved while the PDCNS is actuated by PEA, w mainly limited by the resolution of the displacement sensor. Compared to the PE ated PDCNS, the VCM-actuated PDCNS exhibits a much worse resolution with age step size of 9.04 μm. The unstable response of the VCM-actuated PDCNS ca tributed to the nonlinearity and friction effects. Hence, the PEA-actuated PDC sesses an obvious advantage in high-resolution motion.

Motion Resolution Investigation
Consecutive-step-positioning experiments are carried out to investigate the motion resolution of the proposed PDCNS, as shown in Figure 10. A clear motion resolution with a step size of 0.74 µm can be achieved while the PDCNS is actuated by PEA, which is mainly limited by the resolution of the displacement sensor. Compared to the PEA-actuated PDCNS, the VCM-actuated PDCNS exhibits a much worse resolution with an average step size of 9.04 µm. The unstable response of the VCM-actuated PDCNS can be attributed to the nonlinearity and friction effects. Hence, the PEA-actuated PDCNS possesses an obvious advantage in high-resolution motion.

Closed-Loop Motion Investigation
(1) DSCC Design. Considering that the proposed PDCNS is a two-inputs-onesystem with severe hysteresis nonlinearity, a double-servo cooperative control ( strategy is designed in this section. Figure 11 depicts the block diagram of the pr DSCC. The DSCC consists of three main parts: position observer, VCM servo contr PEA servo control. The position observer is designed to estimate the virtual cente end effector, , (the position of the end effector if the control voltage of PEA i according to the displacement feedback of end effector as follows: the output di ment induced by PEA is calculated by multiplying the control voltage, , and th age-displacement coefficient, ; by removing the PEA-induced displacement fr measured position (with the laser displacement sensor) of the end effector, the center of the end effector can be obtained. Using the virtual center of the end effe feedback, a PID-based servo control strategy with adaptive parameters is designed trol the VCM according to the reference position, . The VCM servo control c only converge the end effector to the reference position with coarse resolution b ensure that the PEA is working within its rated range. To cooperate with the VCM tion, a PID-based PEA servo control method is employed to achieve the fine moti assure the position accuracy. Additionally, an initial 5 V is added to the control vol PEA, , to ensure that the output signal of PEA servo control, , can be reg within −5 V~+5 V (the control voltage range of PEA driver is 0~10 V). With the sp designed DSCC, the PDCNS is expected to realize nm-level positioning over mm range.

Closed-Loop Motion Investigation
(1) DSCC Design. Considering that the proposed PDCNS is a two-inputs-one-output system with severe hysteresis nonlinearity, a double-servo cooperative control (DSCC) strategy is designed in this section. Figure 11 depicts the block diagram of the proposed DSCC. The DSCC consists of three main parts: position observer, VCM servo control, and PEA servo control. The position observer is designed to estimate the virtual center of the end effector, x vir , (the position of the end effector if the control voltage of PEA is zero) according to the displacement feedback of end effector as follows: the output displacement induced by PEA is calculated by multiplying the control voltage, V PEA , and the voltage-displacement coefficient, γ PEA ; by removing the PEA-induced displacement from the measured position (with the laser displacement sensor) of the end effector, the virtual center of the end effector can be obtained. Using the virtual center of the end effector as feedback, a PID-based servo control strategy with adaptive parameters is designed to control the VCM according to the reference position, x re f . The VCM servo control can not only converge the end effector to the reference position with coarse resolution but also ensure that the PEA is working within its rated range. To cooperate with the VCM actuation, a PID-based PEA servo control method is employed to achieve the fine motion and assure the position accuracy. Additionally, an initial 5 V is added to the control voltage of PEA, V PEA , to ensure that the output signal of PEA servo control, V PEA , can be regulated within −5 V~+5 V (the control voltage range of PEA driver is 0~10 V). With the specially designed DSCC, the PDCNS is expected to realize nm-level positioning over mm-level range.

Closed-Loop Motion Investigation
(1) DSCC Design. Considering that the proposed PDCNS is a two-inputs-one-outpu system with severe hysteresis nonlinearity, a double-servo cooperative control (DSCC strategy is designed in this section. Figure 11 depicts the block diagram of the proposed DSCC. The DSCC consists of three main parts: position observer, VCM servo control, and PEA servo control. The position observer is designed to estimate the virtual center of the end effector, , (the position of the end effector if the control voltage of PEA is zero according to the displacement feedback of end effector as follows: the output displace ment induced by PEA is calculated by multiplying the control voltage, , and the volt age-displacement coefficient, ; by removing the PEA-induced displacement from th measured position (with the laser displacement sensor) of the end effector, the virtua center of the end effector can be obtained. Using the virtual center of the end effector a feedback, a PID-based servo control strategy with adaptive parameters is designed to con trol the VCM according to the reference position, . The VCM servo control can no only converge the end effector to the reference position with coarse resolution but also ensure that the PEA is working within its rated range. To cooperate with the VCM actua tion, a PID-based PEA servo control method is employed to achieve the fine motion and assure the position accuracy. Additionally, an initial 5 V is added to the control voltage o PEA, , to ensure that the output signal of PEA servo control, , can be regulated within −5 V~+5 V (the control voltage range of PEA driver is 0~10 V). With the specially designed DSCC, the PDCNS is expected to realize nm-level positioning over mm-leve range. Figure 11. Block diagram of the double-servo cooperative control (DSCC) strategy. (2) Positioning in mm-level range. A consecutive-step reference trajectory is adopted to guide the PDCNS (with DSCC) from −0.7 mm to 0.65 mm. For comparison, the VCM (with traditional PID) positioning experiments are also conducted based on the PDCNS prototype (while the control voltage of PEA is maintained at zero). The parameters of the PDCNS positioning (cooperation of voice coil motor and piezoelectric actuator) are tuned to be: VCM proportional coefficient = 0.03, VCM integral coefficient is adaptive, VCM derivative coefficient = 1 × 10 −4 , voltage-displacement coefficient = 2.5, PEA proportional coefficient = 0.2, PEA integral coefficient = 150, and PEA derivative coefficient = 1 × 10 −5 . In contrast, the parameters of the VCM positioning (only using the voice coil motor) are tuned to be: VCM proportional coefficient = 0.03, VCM integral coefficient = 0.3, and VCM derivative coefficient = 1 × 10 −4 . The trajectory control results of the PDCNS (with DSCC) and the VCM (with PID) are compared in Figure 12. It can be observed that both PDCNS and VCM are effective for the mm-level positioning. Compared to the VCM, the PDCNS can realize a faster response and eliminate the steady-state error. Due to the fast response, PDCNS also induces an obvious overshoot. Figure 13 depicts the motion-tracking experiments of VCM and PDCNS while facing a consecutive-step trajectory from 0.7 mm to −0.65 mm, which confirms the global-positioning capability of the PDCNS. (2) Positioning in mm-level range. A consecutive-step reference trajectory is adopted to guide the PDCNS (with DSCC) from −0.7 mm to 0.65 mm. For comparison, the VCM (with traditional PID) positioning experiments are also conducted based on the PDCNS prototype (while the control voltage of PEA is maintained at zero). The parameters of the PDCNS positioning (cooperation of voice coil motor and piezoelectric actuator) are tuned to be: VCM proportional coefficient = 0.03, VCM integral coefficient is adaptive, VCM derivative coefficient = 1 × 10 −4 , voltage-displacement coefficient = 2.5, PEA proportional coefficient = 0.2, PEA integral coefficient = 150, and PEA derivative coefficient = 1 × 10 −5 . In contrast, the parameters of the VCM positioning (only using the voice coil motor) are tuned to be: VCM proportional coefficient = 0.03, VCM integral coefficient = 0.3, and VCM derivative coefficient = 1 × 10 −4 . The trajectory control results of the PDCNS (with DSCC) and the VCM (with PID) are compared in Figure 12. It can be observed that both PDCNS and VCM are effective for the mm-level positioning. Compared to the VCM, the PDCNS can realize a faster response and eliminate the steady-state error. Due to the fast response, PDCNS also induces an obvious overshoot. Figure 13 depicts the motion-tracking experiments of VCM and PDCNS while facing a consecutive-step trajectory from 0.7 mm to −0.65 mm, which confirms the global-positioning capability of the PDCNS.  (3) Positioning with μm-level steps. To evaluate the precision positioning capability of the proposed PDCNS, consecutive-step-positioning tests with step sizes of ±1 μm are performed, as shown in Figures 14 and 15. Compared to the VCM (with PID), faster setting and smaller error can be achieved by the PDCNS (with DSCC). Because severe hysteresis exists in the VCM, it is time consuming for the PID to provide a control signal that can start the motion of the VCM. Due to the existence of friction, it is also a challenging job for the VCM to converge the end effector towards the reference position steadily. Different from the VCM, the PDCNS guarantees the convergence of the tracking error by (2) Positioning in mm-level range. A consecutive-step reference trajectory is adopted to guide the PDCNS (with DSCC) from −0.7 mm to 0.65 mm. For comparison, the VCM (with traditional PID) positioning experiments are also conducted based on the PDCNS prototype (while the control voltage of PEA is maintained at zero). The parameters of the PDCNS positioning (cooperation of voice coil motor and piezoelectric actuator) are tuned to be: VCM proportional coefficient = 0.03, VCM integral coefficient is adaptive, VCM derivative coefficient = 1 × 10 −4 , voltage-displacement coefficient = 2.5, PEA proportional coefficient = 0.2, PEA integral coefficient = 150, and PEA derivative coefficient = 1 × 10 −5 . In contrast, the parameters of the VCM positioning (only using the voice coil motor) are tuned to be: VCM proportional coefficient = 0.03, VCM integral coefficient = 0.3, and VCM derivative coefficient = 1 × 10 −4 . The trajectory control results of the PDCNS (with DSCC) and the VCM (with PID) are compared in Figure 12. It can be observed that both PDCNS and VCM are effective for the mm-level positioning. Compared to the VCM, the PDCNS can realize a faster response and eliminate the steady-state error. Due to the fast response, PDCNS also induces an obvious overshoot. Figure 13 depicts the motion-tracking experiments of VCM and PDCNS while facing a consecutive-step trajectory from 0.7 mm to −0.65 mm, which confirms the global-positioning capability of the PDCNS.  (3) Positioning with μm-level steps. To evaluate the precision positioning capability of the proposed PDCNS, consecutive-step-positioning tests with step sizes of ±1 μm are performed, as shown in Figures 14 and 15. Compared to the VCM (with PID), faster setting and smaller error can be achieved by the PDCNS (with DSCC). Because severe hysteresis exists in the VCM, it is time consuming for the PID to provide a control signal that can start the motion of the VCM. Due to the existence of friction, it is also a challenging job for the VCM to converge the end effector towards the reference position steadily. Different from the VCM, the PDCNS guarantees the convergence of the tracking error by (3) Positioning with µm-level steps. To evaluate the precision positioning capability of the proposed PDCNS, consecutive-step-positioning tests with step sizes of ±1 µm are performed, as shown in Figures 14 and 15. Compared to the VCM (with PID), faster setting and smaller error can be achieved by the PDCNS (with DSCC). Because severe hysteresis exists in the VCM, it is time consuming for the PID to provide a control signal that can start the motion of the VCM. Due to the existence of friction, it is also a challenging job for the VCM to converge the end effector towards the reference position steadily. Different from the VCM, the PDCNS guarantees the convergence of the tracking error by coordinating the PEA and VCM. The VCM is responsible for driving the end effector close to the reference position as well as decreasing the control effort of the PEA in a slow manner. Meanwhile, the PEA is in charge of eliminating the tracking error in a fast manner. Therefore, the proposed PDCNS can realize large-range, high-speed, and high-resolution positioning.
Actuators 2022, 11, x FOR PEER REVIEW coordinating the PEA and VCM. The VCM is responsible for driving the end effect to the reference position as well as decreasing the control effort of the PEA in a slo ner. Meanwhile, the PEA is in charge of eliminating the tracking error in a fast m Therefore, the proposed PDCNS can realize large-range, high-speed, and high-res positioning.  (4) Positioning with nm-level steps. Finally, the nanometer-level positioning c ity of the proposed PDCNS (with DSCC) is investigated by two consecutive-step re signals with step sizes of ±50 nm. In this subsection, to improve the signal-to-nois the laser displacement sensor is configured to have a measurement range of −10 μ μm (corresponding to an output voltage of −10 V to 10 V). A 50 nm height staircas tory response is clearly obtained in Figure 16a. Moreover, a continuous stepping ment with amplitude of −50 nm can be observed in Figure 16b. Therefore, the close positioning resolution of the proposed PDCNS is better than 50 nm. Actuators 2022, 11, x FOR PEER REVIEW coordinating the PEA and VCM. The VCM is responsible for driving the end effect to the reference position as well as decreasing the control effort of the PEA in a slo ner. Meanwhile, the PEA is in charge of eliminating the tracking error in a fast m Therefore, the proposed PDCNS can realize large-range, high-speed, and high-res positioning.  (4) Positioning with nm-level steps. Finally, the nanometer-level positioning ity of the proposed PDCNS (with DSCC) is investigated by two consecutive-step re signals with step sizes of ±50 nm. In this subsection, to improve the signal-to-noi the laser displacement sensor is configured to have a measurement range of −10 μ μm (corresponding to an output voltage of −10 V to 10 V). A 50 nm height staircas tory response is clearly obtained in Figure 16a. Moreover, a continuous stepping ment with amplitude of −50 nm can be observed in Figure 16b. Therefore, the close positioning resolution of the proposed PDCNS is better than 50 nm. (4) Positioning with nm-level steps. Finally, the nanometer-level positioning capability of the proposed PDCNS (with DSCC) is investigated by two consecutive-step reference signals with step sizes of ±50 nm. In this subsection, to improve the signal-to-noise ratio, the laser displacement sensor is configured to have a measurement range of −10 µm to 10 µm (corresponding to an output voltage of −10 V to 10 V). A 50 nm height staircase trajectory response is clearly obtained in Figure 16a. Moreover, a continuous stepping movement with amplitude of −50 nm can be observed in Figure 16b. Therefore, the closed-loop-positioning resolution of the proposed PDCNS is better than 50 nm.

Conclusions
In this paper, the mechanical design, analytical modeling, and closed-loop co a novel PDCNS are conducted. The designed PDCNS combines the coarse motion voice coil motor and the fine motion from a piezoelectric actuator through a new which breaks through many barriers encountered in traditional serial dual-stage s To resolve the kinematic characteristics of the proposed ACPM, an analytical m specially established. Furthermore, a DSCC strategy with good robustness to non ties is devised to enhance the positioning capability of the PDCNS.
Simulations and experiments are conducted to demonstrate the static and d performance of the PDCNS. The results show that the maximum stroke of the PDC reach 1.49 mm, and the positioning resolution can be better than 50 nm. Moreo designed DSCC can eliminate the steady-state position error of PDCNS, despit nonlinear factors.

Conclusions
In this paper, the mechanical design, analytical modeling, and closed-loop control of a novel PDCNS are conducted. The designed PDCNS combines the coarse motion from a voice coil motor and the fine motion from a piezoelectric actuator through a new ACPM, which breaks through many barriers encountered in traditional serial dual-stage systems. To resolve the kinematic characteristics of the proposed ACPM, an analytical model is specially established. Furthermore, a DSCC strategy with good robustness to nonlinearities is devised to enhance the positioning capability of the PDCNS.
Simulations and experiments are conducted to demonstrate the static and dynamic performance of the PDCNS. The results show that the maximum stroke of the PDCNS can reach 1.49 mm, and the positioning resolution can be better than 50 nm. Moreover, the designed DSCC can eliminate the steady-state position error of PDCNS, despite many nonlinear factors.