1. Introduction
Compliant mechanisms (CMs) represent a significant area in modern mechanism design, characterized by high precision, high reliability, and absence of clearance, friction, or wear [
1,
2,
3,
4]. By enhancing the reduction ratio of CMs, it is possible to avoid friction and backlash errors typically associated with gear reducers, thereby effectively improving the resolution while maintaining accuracy [
5]. Recently, mechanisms based on flexible designs have garnered considerable attention for applications in microscopic imaging [
6,
7] and micromanipulation [
8,
9,
10,
11,
12,
13].
The synchrotron radiation facility (SRF) marks a significant milestone in the development of human light sources, providing a new, high-performance, highly collimated, pure, and bright light source [
14,
15,
16]. A monochromator, a core component of the synchrotron radiation device, separates the polychromatic light emitted by the source into the required monochromatic light [
17], as depicted in
Figure 1.
To achieve angular positioning of the monochromator splitter crystal, extensive research has been conducted. The Argonne National Laboratory’s advanced photon source (APS) has developed a series of nano-positioning instruments [
18,
19,
20,
21,
22,
23,
24]. Among these, CMs utilized in the artificially slotted crystal monochromator and the hard X-ray polarizer can attain angular resolutions of 20–40 nrad and 80 nrad, respectively, in angular adjustments. The European synchrotron radiation facility (ESRF) employs traditional slow-walking silk-cutting technology to fabricate the four-link straight circular flexhinge bending mechanism in a single piece, achieving a focused light spot of less than 50 nm [
25]. Utilizing CMs, the Shanghai synchrotron radiation facility (SSRF) has demonstrated the feasibility of creating micro displacement adjustment mechanisms for synchrotron radiation devices using 3D printing technology [
26].
The multi-objective optimization problem (MOOP) aims to addres scenarios with multiple conflicting objectives by finding a solution that meets all optimization criteria. This is crucial for both theoretical research and practical engineering applications. Traditional methods for multi-objective optimization include the weighted sum method, constraint method, objective programming method, distance function method, and minimax method [
27]. Traditional optimization techniques often struggle to guarantee excellent outcomes when solving MOOPs due to their inherent limitations. Consequently, the study of intelligent optimization algorithms, which are essentially stochastic search algorithms that mimic the behavior of natural biological groups, has garnered significant interest. These algorithms are particularly effective for tackling complex problems characterized by discontinuity, nonlinearity, and multiple variables.
In the application research of multi-objective optimization, Ehsan Naderi et al. developed a hybrid algorithm combining the random frog leaping algorithm (RFLA) and particle swarm optimization (PSO), achieving high solution quality [
28]. Nguyen et al. utilized the nondominated sorting genetic algorithm-II (NSGA-II) to optimize the design of a compliant linear guide mechanism with a high-precision feed drive, achieving a closed-loop control error in output displacement of less than 0.02
m [
29]. Their team also applied NSGA-II to the design of micromachining feed drive mechanisms, producing and evaluating the optimization results [
9]. Additionally, WANG et al. implemented the multi-objective particle swarm optimization (MOPSO) algorithm to design a planar parallel 3-DOF nano-positioner for scanning probe microscopy (SPM). The effectiveness of the proposed modeling and optimization methods was corroborated by both simulation and experimental results [
30].
From the investigations discussed, it is evident that as the requirements for the monochromatic beam of synchrotron radiation devices continue to increase, so too do the technical demands on nano-scale micro-displacement tables. To meet the ultra-high resolution micro-displacement needs of the SSRF, the compliant mechanism offers a solution capable of achieving ultra-precision motion, which is challenging to obtain with conventional motor servo systems. This paper employs a multi-objective optimization algorithm with the aim to minimize the driving force of the mechanism while maintaining a sufficiently high inherent frequency. The primary contribution of this study lies in the modeling and optimal design method for the nano-locator. The proposed method utilizes the NSGA-II to derive the Pareto front, and the optimal design parameters are subsequently selected based on this front. Following the determination of design parameters, a simulated and manufactured experimental prototype is used to validate the optimal design results.
The remainder of this paper is structured as follows:
Section 2 establishes the mathematical model, capturing both the static and dynamic attributes of the mechanism.
Section 3 presents the design metrics, evaluates the design parameters, and conducts a preliminary analysis.
Section 4 outlines the objective equations and associated constraints.
Section 5 details the solution computation using the MOOP algorithm and analyzes the simulation results.
Section 6 displays the prototype validation compared to the design’s optimization results. Finally,
Section 7 offers a concise summary.
2. Optical Angle Nano-Positioning Mechanism Configuration
The machine developed in this study is designed to have a simple and reliable structure, extensive range of movement adjustment, and high inherent frequency. To fulfill these criteria, a trapezoidal four-link is implemented for high-precision adjustment and positioning within a restricted range. The design features include a fixed platform, four flexible link rods, a moving platform, and a driver. The fixed platform is securely mounted to the base platform. Upon application of a driving force
by the driver, the moving platform undergoes a micrometric displacement, resulting in an angular displacement around the pivot point, as depicted in
Figure 2.
The optical angle nano-positioning mechanism incorporates two sets of symmetrically arranged flexible hinges, which effectively constrain translational movements along the z-axis and rotational movements around the x-axis. When the piezoelectric (PZT) actuator exerts a driving force along the x-axis, the four sets of rigid rods pivot around the flexible hinges anchored to the fixed platform. This rotational motion is subsequently transmitted to the moving platform via the flexible hinges connected to it.
2.1. Physical Design
The mechanism features a uniaxial straight circular hinge, constituting a four-link flexure hinge mechanism as shown in
Figure 3, and is simplified using the pseudo-rigid body model. The four links, denoted
(
i = 1–4), are interconnected via four flexible hinges. Here,
is equal to
, and
acts as the fixed installation surface. The lines of
and
converge at the waist, forming an isosceles triangle with
O as its vertex, where
represents the initial angle of the lower bar
with respect to
. The swing arm is rigidly connected to
, maintaining a constant perpendicular angle to
. An external force is applied to the swing arm at a distance
L from the point
O. When the PZT actuator drives the moving platform to displace by
, the rotation angles of
and
are denoted as
, and the rotation angle of
is
, which is the mechanism output angle.
Following a rotation of
by an angle
, it can be maneuvered based on its geometric relationship with
, which is as follows:
2.2. Statics Analysis
The straight circular flexure hinge, characterized by high precision, stable adjustment, and wear resistance, is illustrated in
Figure 4. Within this figure,
r represents the circle radius of the flexure hinge,
h stands for the height of the flexure hinge,
t denotes the thickness between the hinges, and
b indicates the width of the flexible hinge.
The adjustment mechanism consists of eight straight circular flexible hinges, each identical in design and function. A force analysis is performed on one of these flexible hinges. Assuming the left end of the flexible hinge is fixed, the relationship between the angle
around the
z-axis and the external torque
is established as follows:
where
E is the elastic modulus of the material, and:
The PZT actuator’s thrust output,
F, is entirely transformed into the flexible hinge’s elastic potential energy,
V. Consequently, the total energy input of the mechanism is as follows:
where
is the displacement corresponding to the angular displacement
of the flexible hinges pushed by the PZT actuator. The total potential energy
U of the mechanism can be obtained using the energy method:
Furthermore, based on energy conservation, we have the following:
2.3. Kinematics Analysis
Dynamic characteristics are critical benchmarks for the optical angle nano-positioning mechanism. The inclusion of a flexible structure means that design parameters significantly affect the mechanism’s inherent frequency. To enhance the mechanism’s overall anti-vibration capabilities, it is crucial to optimize the design of parameters that influence dynamic stiffness. In this configuration, the flexure hinge functions as a rotating spring, while other structural elements are considered rigid bodies. The dynamic model of the optical angle nano-positioning mechanism is illustrated in
Figure 5.
When the output angle is sufficiently small, the movement of the moving platform can be approximated as translational motion. Under these conditions, the kinetic energy, denoted by
T, in the direction of motion is defined as follows:
where
represents the mass of the moving platform of the optical angle nano-positioning mechanism and
J signifies the moment of inertia of the rigid rod with respect to the endpoints in the direction of motion. Here,
is the mass of the rigid rod and
denotes the length of the rigid rod:
Since the output angle is small enough, there is
. By plugging in Equation
, we can get:
Considering that the optical angle nano-positioning mechanism consists of eight identical hinges, the potential energy, denoted by
U, in the direction of motion is defined as follows:
For a system with
n degrees of freedom, the motion under the influence of an external force
F adheres to the Lagrangian equation, which is defined as follows:
where
represents the generalized coordinates adopted by the system. By considering
as the generalized coordinates, we can derive the following by substituting Equations (10) and (11):
Moreover, the first-order inherent frequency, denoted as
f, of the optical angle nano-positioning mechanism can be calculated as follows:
7. Conclusions
In this paper, we design an optical angle nano-positioning mechanism based on CMs to achieve ultra-high resolution nano-displacement, as necessitated by the specific engineering requirements of the SSRF. This study focuses on the design parameters of the flexible hinge, using the driving force and the inhernet frequency of the mechanism as the optimization objectives to conduct multi-objective optimization. The following conclusions can be drawn from this study:
By establishing the kinematic model based on the geometric dimensions of the mechanism, the relationship between the mechanism size and the output motion is determined.
Analyzing the driving force required by the mechanism at the limit output angle begins with the mechanical characteristics of a single hinge. A mapping relationship between the stiffness of a single hinge and that of the entire mechanism is established.
Utilizing the Lagrangian kinematics theory, the dynamic model of the mechanism is developed based on geometric size and mass data. The first-order inherent frequency is determined by solving the partial differential equation.
The NSGA-II multi-objective optimization algorithm optimizes the design parameters and addresses the conflict inherent in multi-objective optimization for optical angle nano-positioning mechanisms.
The theoretical model is validated through finite element calculations and prototype testing. The results indicate that the first-order inherent frequency of the theoretical model differs by 3.33% from the finite element analysis result and by 2.63% from the real object, substantiating the effectiveness of the proposed method.
Future studies will target more complex CMs mechanisms, aiming to achieve higher motion response, higher output resolution, and enhanced mechanism stability as objectives for further optimization.