1. Introduction
In the present era of advanced engineering and scientific advancements, manipulating and observing objects at the nano and microscale levels have emerged as the fundamental building blocks for further discoveries and advancements. To realize this, compliant mechanisms have been extensively explored recently to generate sub-nano/micrometer motion quality. Compliant mechanisms provide displacement, force, or energy through the elastic deformation of its flexural hinges [
1]. This reduces friction, backlash, wear, and the need for lubrication [
2,
3]. In addition, as such mechanisms are usually manufactured monolithically, the manufacturing cost, assembly error, and complexity are reduced considerably [
4]. Due to their indispensable properties of precision and accuracy, compliant mechanisms have been researched for use in various fields, including the medical sector [
5,
6,
7], space and aerospace industries [
8,
9], micro/nano manipulation [
10,
11,
12,
13,
14], robotics [
15,
16,
17,
18], and energy harvesting [
19].
Recently, to meet the demand for a large range of precise translational motion, various kinds of parallel XY mechanisms have been synthesized. In designing a large range of micropositioning stages, cross-coupling, larger footprint size, modeling complexity arising from stiffness non-linearity, complex control algorithms, minimized resolution, and low dynamic performance are the main challenges. A four-PP (P-prismatic) parallel mechanism with spatial supporting linkages was proposed to increase workspace and payload capacity in [
20,
21]. In [
22,
23], a similar approach was used by using passive rigid links to constrain the unwanted motion in the orthogonal direction. However, such approaches limit the stroke and increase manufacturing complexity. In [
24], the same configuration was proposed with a modular design using four double parallelogram mechanisms. Due to the longer and thinner leaf flexure used, a low resonant frequency of 15 Hz was achieved. To further increase the range with the same approach, a triple-stage compound double parallelogram flexure approach was introduced in [
25]. Although this mechanism helps to provide a large stroke with pure translational motion, it degrades the dynamic performance of the mechanism due to the unconstrained intermediate stage [
26,
27]. To eliminate this problem, a nested linkage mechanism was used in [
28]. However, the first resonant frequencies appeared lower than 5 Hz, and the stage had a large footprint size. Unlike serial mechanisms, parallel mechanisms exhibit higher cross-coupling, leading to degraded motion accuracy and complex control problems. Towards this, redundant constraints are often used to minimize cross-coupling and parasitic motion, but they limit motion range and add complexity to modeling and manufacturing [
29,
30]. Particularly, the typical manufacturing methods of monolithic flexure-based mechanisms are applicable to planar mechanisms with simple structures. Thus, having a simple design configuration is preferred to minimize the cost and manufacturing process. Concerning this, a modular architecture design is implemented in [
31] to improve the functionality of flexure-based positioning stages by manufacturing reconfigurable design modules separately. It is typically challenging to attain a large workspace, higher bandwidth, and enhanced compactness simultaneously, so a better mechanism design is often required to strike a balance among these conflicting performance parameters [
32].
To ensure precise and accurate motion, a mechanism with an appropriate kinematic configuration and carefully chosen constraints is not enough. The presence of inevitable factors such as cross-coupling, stress stiffening, parameter variation, and external disturbances often necessitates the implementation of an effective control strategy to compensate for these limitations and enhance the motion accuracy and precision of the mechanism [
33,
34].
In order to address these challenges, various control strategies have been implemented to drive flexure-based mechanisms. To improve the motion accuracy of a one-DOF mechanism, a sliding mode control (SMC) based on a PID sliding surface with an adaptive fuzzy disturbance observer (AFDO) was employed in [
35]. The chattering effect was attenuated as the implemented observer provided an approximate estimation of the disturbance. The same approach was used with a feedforward control in [
20] to drive an XY mechanism. The inverse of a Bouc–Wen-based hysteresis model was used in the feedforward control to compensate for the non-linearity of the actuator. Similarly, a nonlinear disturbance observer (NDO) was integrated with SMC to control a large-range three-DOF micropositioning stage [
36]. Moreover, H-infinity [
37], enhanced model predictive control [
38], disturbance observer-based repetitive control [
39], adaptive sliding mode control [
40], hysteresis-compensated model reference adaptive control [
41], and others were implemented on various flexure-based mechanisms. In prior studies, sliding mode control has commonly been employed to control large-range mechanisms due to its simplicity and robustness. However, this method suffers from chattering, which hinders smooth motion and shortens the lifespan of the actuator. It is essential to note that the effectiveness of the control methods implemented in previous studies largely depends on the accuracy of the mechanism’s model. However, accurately characterizing compliant mechanisms is difficult due to their coupled kinematics and elastomechanical behaviors [
42]. Moreover, stress stiffening and parameter uncertainties pose additional challenges in accurately capturing the system dynamics. Some studies have investigated the modeling of stress stiffening [
43] and actuator hysteresis [
44,
45,
46] to enhance controller performance. However, this is a demanding approach that is typically effective for dealing with deterministic variations. A more efficient and promising alternative is to integrate disturbance estimation techniques to address unmodeled dynamics and parameter uncertainties [
47]. Towards this issue, active disturbance rejection control (ADRC) is being proposed in different studies to enhance motion accuracy in the presence of parameter variations and disturbances. This control method employs simple strategies and requires minimal model information. Recently, ADRC has been implemented in many studies to address the hysteresis problem of the piezoelectric actuator in flexure-based mechanisms [
48,
49]. However, limited research has been conducted to explore the potential application of ADRC in voice coil motor (VCM)-driven large-range flexure-based mechanisms.
To this end, this paper introduces a novel large-range two-DOF mechanism by utilizing a hybrid configuration incorporating both corrugated and leaf flexures. A corrugated flexure possesses higher compliance and reduced stress concentration compared to a straight segment of equal longitudinal length [
50,
51,
52]. This allows the development of the stage with an improved workspace while maintaining a compact size. Besides, redundant leaf flexures are employed to minimize the cross-coupling and enhance the rotational stiffness. In addition, a LADRC controller is developed to enhance motion precision and robustness with limited model information and a simple control structure. Due to its capability of providing accurate measurements over an extended range, the laser interferometry-based sensing and measurement (LISM) approach is adopted in this paper [
53]. The experimental investigation reveals that the proposed mechanism, coupled with LADRC, achieves significant tracking accuracy and disturbance rejection capabilities.
This paper is organized as follows: The mechanism design is presented in
Section 2, followed by analytical modeling in
Section 3.
Section 4 describes the computational analysis conducted using FEA.
Section 5 presents the experimental setup, system identification, and controller design. In
Section 6, experimental results are presented. Finally, the conclusion is provided in
Section 7.
4. Computational Analysis
In order to validate the developed analytical models, FEA is performed using ANSYS 2020 R1 software. From the computational analysis, the output compliance, achievable workspace in the x and y directions, natural frequencies, and maximum stress are evaluated. The mechanism is manufactured using acrylonitrile butadiene styrene (ABS) with its properties given in
Table 3.
In order to evaluate the output compliance and maximum range of a mechanism, a series of input displacements were applied while imposing fixed constraints at the outer holes. The output compliance of the mechanism is determined by taking the ratio of displacement to the recorded reaction force. A force of 1 N is applied to the input compliant beam, which results in a displacement of 0.36 mm. The comparison between the analytical and FEA results is provided in
Table 4, showing good agreement. The maximum displacement is determined to be 3mm with an equivalent von Mises stress of 16.8 Mpa, as shown in
Figure 4 and
Figure 5, respectively. The recorded maximum stress occurring in the leaf spring is less than the yield strength of ABS. The safety factor is determined to be more than 1.8, ensuring the mechanism operates in the elastic range with good repeatability. A linear buckling analysis was conducted to obtain an estimation of the critical load, prior to manufacturing and physical experiments. The analysis indicated that buckling typically occurs at approximately 113 N, a value significantly higher than the maximum actuation force of the VCM (87 N).
Modal analysis is carried out to determine the natural frequencies of the mechanism. Determining the dynamic performance is important, as the natural frequency determines the system’s overall response speed and bandwidth. In the analysis, the weight of the reflector mirror holder, the weight of the VCM connector, and the weight of the moving coil mass are taken into consideration.
Figure 6 illustrates the first six natural frequencies and their respective mode shapes. The first two modes occur at 33.3 Hz and correspond to translational motion in the working direction along the
x and
y axes. The third mode (106 Hz) is the out-of-plane translational motion along the
z-axis, indicating a higher out-of-plane stiffness and better payload capability. The sixth mode is due to the in-plane rotation that occurs at 170 Hz. From this, it can be inferred that the over-constrained leaf flexures maximize the rotational stiffness, effectively suppressing the parasitic in-plane rotation. By considering the FEA result as a benchmark, the error with the analytical result has a slight deviation, as shown in
Table 4.
The third operating mode exhibits a frequency that exceeds three times that of the first two modes, which suggests that the proposed positioning stage possesses two degrees of freedom. This design ensures that higher frequency modes are less likely to be excited, thereby enhancing the in-plane translational capabilities. Considering the lower Young’s modulus of ABS, the mechanism has a competitive dynamic performance.
6. Results
A range of experimental tests were conducted to assess the effectiveness of the LADRC control in trajectory tracking and disturbance rejection capability. The experimental results validate the absence of buckling or yielding phenomena, ensuring that the mechanism operates within a safe operational range. The positioning stage’s tracking ability, resolution, workspace, and robustness were evaluated through various experimental tests.
The designed mechanism is evaluated using a 2.5 mm pseudo-step command as the reference trajectory to assess its tracking performance and achievable workspace.
Figure 11a shows that the LADRC exhibited a prompt response with no overshoot for the target positioning. The rise time and the 2% settling time are presented in
Table 5. Additionally, the steady-state tracking error is kept within the range of ±0.5
m, which is less than 0.02% of the total motion range, as demonstrated in
Figure 11b.
Several periodic trajectories are commanded to the VCM to assess the trajectory accuracy of the mechanism. A sinusoidal signal with an amplitude of 1 mm and a frequency of 0.5 Hz is applied to drive the VCM along the
x-axis, as illustrated in
Figure 12a. The root mean square error (RMSE) is approximately 9.15
, which corresponds to 0.915% of the total motion range. The maximum tracking error (MAXE) is restricted to 14.6
, which is 1.46% of the total stroke, as depicted in
Figure 12b.
A triangular trajectory is commonly used in image scanning applications. Hence, a 1 mm peak-to-peak triangular signal with a frequency of half hertz is utilized, as shown in
Figure 12c. The RMSE and MAXE are limited to 9.26 and 11.3
, respectively, accounting for 0.926% and 1.1% of the total range, respectively. It is worth noting that the highest error is found at the sharp turning corners. To overcome this issue, a smoothed triangular signal is implemented to ensure smooth trajectory tracking. Moreover, periodic square trajectories are used to evaluate the tracking performance of the positioning stage, as it is a commonly adopted trajectory scenario in object manipulation tasks. As shown in
Figure 12f, LADRC tracks the command with a slight overshoot and minimum tracking error.
To further assess the tracking performance of LADRC, a superimposed trajectory by combining different sinusoidal signals with varying frequency and amplitude is employed, as illustrated in
Figure 13. The RMSE and MAXE errors are 24.3 and 45.5
, accounting for 2% and 3.79% of the total range, respectively. The system’s response to the dynamically varying reference command, especially with higher-frequency sinusoidal components, contributes to the observed higher tracking errors. This limitation is attributed to LESO, which effectively estimates only constant and slowly varying disturbances. In forthcoming studies, we aim to address this limitation by integrating adaptive disturbance estimators and improved control law techniques within the ADRC framework.
To test the resolution of the designed mechanism, a consecutive multi-step signal with a time duration of 2 s and a step size of 0.4
is applied, as shown in
Figure 14. Despite the mechanism being 3D printed and having a large range, a competitive resolution is achieved when compared with recently developed mechanisms.
In order to evaluate the robustness of the designed control strategy in the presence of model uncertainties, metal bars are added to the positioning stage, as shown in
Figure 15a. A 0.5 mm peak-to-peak sinusoidal trajectory with 0.5 Hz is tracked with and without the mass. The results demonstrate that LADRC can effectively track the signal with minimal error difference, as shown in
Figure 15b. The improved estimation performance of the observer and real-time rejection of the lumped total disturbance ensures consistent performance in the presence of modeling uncertainty, as demonstrated in the experiment.
A comparison of the designed mechanism’s performance with recently manufactured positioning stages is presented in
Table 6. As the manufacturing method and material properties play a significant role in the mechanism’s performance, the comparison is limited to additively manufactured stages. With regards to workspace, the proposed mechanism exhibits a greater range while maintaining enhanced compactness. In contrast, the design presented in reference [
66] achieved improved natural frequency but at the cost of decreased range. It is important to note that the first natural frequency of the mechanisms discussed in references [
67,
68] is derived from finite element analysis and does not take into account certain factors such as the mass of the moving coil of the motor, motor connectors, and sensor frames. As a result, these designs are expected to exhibit a lower natural frequency. Overall, the designed mechanism is expected to provide an improved dynamic performance and enhanced motion repeatability if the stage is manufactured with a wire electrical discharge machining (WEDM).