Closed-loop control of microgripper system based on compliant mechanism

The efficient, accurate, and reliable assembly of numerous small parts is feasible with the assistance of microassembly and microoperation technology. In the assembly process, the microgripper system, as the end-effector of microassembly and micro-operating system, is in direct contact with the object being manipulated, and its performance directly determines the result of the assembly task.^1–3 How to precisely grip, handle, and release the manipulated objects is the key to the microassembly task.^4–6 In most microassembly tasks, the size of the object being manipulated is generally several hundred micrometers, and sometimes even reaches the millimeter level, so the designed microgripper system needs to have a large tension to accommodate objects of different sizes.^7,8 Moreover, due to the variable and irregular shapes of the clamped objects, in the actual clamping process, the clamp will be affected by the longitudinal force and the relative sliding trend of the clamp arm, increasing the instability of clamping and thereby affecting the efficiency and accuracy of the assembly task. Flat clamping can solve this issue, so the designed microgripper system should possess the characteristics of flat clamping.^9,10

The exploration of compliant units and their mechanisms began in the mid-twentieth century. In the late 1980s, Purdue University conducted systematic research on mechanisms with compliant units and named them compliant mechanisms. After ∼30 years of development, contemporary compliant mechanisms have become a crucial element of modern institutional science and are widely employed in precision engineering, robotics and intelligent materials, and numerous other domains.^11–14 In contrast to rigid mechanisms that utilize motion pairs to achieve motion transfer, compliant mechanisms are a novel type of mechanism that employs motion, force, or energy to transfer or convert the elastic deformation of its own flexible parts.^15,16 The design of microgripper systems based on compliant mechanisms not only reduces the overall size of the system but also improves the accuracy of motion and the dependability of gripping.^17,18 Nevertheless, since the deformations of compliant mechanisms are typically nonlinear, conventional linear equations are no longer applicable, which makes the analysis and design highly challenging. Moreover, compliant mechanisms are realized via the bending deformations of flexible units to transfer displacements, and these bending deformations will generate stresses, and the constant repetition of the same motion will inevitably lead to fatigue loads. Hence, fatigue failure is also a significant factor that cannot be ignored in the design of compliant mechanisms.¹⁹ 

In Ouyang’s paper, a topology based on a symmetric five-bar structure for displacement amplification was proposed, and a compliant mechanism was implemented for the amplifier. In short, a mechanical amplifier is called a compliant mechanical amplifier. The proposed CMA can achieve a large amplification ratio and a high natural frequency.^20,21 In Liu’s paper, the concept of a compliant mechanism is applied to a prosthetic hand. A fully compliant monolithic prosthetic finger was designed and constructed with 3D printing technology. A new design process was proposed, in which the flexure hinge in the context of the prosthesis finger is regarded as a 3D object, contrary to the literature that treats it as a 2D object. Then, a new prosthetic finger was constructed with 3D printing technology.²² In another paper by Liu, a fully compliant finger with a monolithic structure and flexure hinge was built. Subsequently, finite element analysis for the compliant finger was implemented, and the results were compared with the experimental result to verify the design. Finally, the complaint finger was applied in a prosthetic hand design and worked excellently with the hand.²³ In Tawk’s article, a three-dimensional (3-D) printed omni-purpose soft gripper (OPSOG) was presented, which can grasp a wide variety of objects with different weights, sizes, shapes, textures, and stiffnesses. The soft gripper has a unique design that incorporates soft fingers and a suction cup that operate either separately or simultaneously to grasp specific objects. A bundle of 3-D-printable linear soft vacuum actuators (LSOVA) that generate a linear stroke upon activation is employed to drive the tendon-driven soft fingers.²⁴ 

Taking into account the variety in size and shape of the objects being operated on, the designed microgripper system should have a large tension. Moreover, considering the small size and susceptibility to damage of the gripped objects, it is necessary to detect and control the finger displacements and gripping forces of the microgripper system during the actual gripping process. The effective detection and control of finger displacements and gripping forces has always been a key research focus. The design of microgripper systems with favorable structural characteristics, high tension, and accurate detection and control of finger displacements and gripping forces is of great significance for promoting the development of microassembly and microoperation technologies.^25,26

In our paper, a three-stage displacement amplification microgripper system was designed based on the triangle amplification mechanism, the lever amplification mechanism, and the parallelogram mechanism. The microgripper system mainly includes a grip body, piezoelectric ceramics, displacement sensors (resistive strain gauges), force sensors (resistive strain gauges), preload bolts, a gasket, and a base. With the symmetrical design, the two piezoelectric ceramics are symmetrically mounted inside the microgripper system to make full use of the space. In our manuscript, on the one hand, the microgripper system was designed, and more importantly, its performances were subjected for simulation and experimental verification, including good leveling performance, finger displacement, and gripping force.

Compliant mechanisms make use of the elastic deformations of their own flexible parts to transfer motions and forces, and they have the advantages of low wear, less need for lubrication, and reduced return. Hence, compliant mechanisms were employed in the structural design of the microgripper system. Moreover, due to the fewer elements in compliant mechanisms, the overall size of the microgripper system can be kept within a small range. A three-stage displacement amplification microgripper system was designed based on the triangle amplification mechanism, the lever amplification mechanism, and the parallelogram mechanism, as shown in Fig. 1. The microgripper system primarily consists of a grip body, piezoelectric ceramics, displacement sensors (resistive strain gauges), force sensors (resistive strain gauges), preload bolts, a gasket, and a base. Among them, A₁, A₂, A₃, A₄, C, D₁, and D₂ are single-circle flexible hinges, E and F are double-circle flexible hinges, and B is a rectangular flexible hinge. The displacement sensors are attached to the outer sides of the flexible hinge A₄ to detect the output displacements of the gripping arms of the microgripper system. The force sensors are attached to the roots of the gripping arms to detect the gripping forces. Thanks to the symmetrical design, the two piezoelectric ceramics are symmetrically installed inside the microgripper system to optimally utilize the space.

To accurately analyze the displacement amplification of the microgripper system, an input displacement of 10 µm was applied, and the output displacement cloud of the microgripper system is shown in Fig. 2. It is shown that under the effect of the 10 µm input displacement, the maximum output displacement of the left gripping arm of the microgripper system is 189.73 µm, and that of the right gripping arm is 189.66 µm. The output displacements of the two gripping arms are basically the same. Therefore, the displacement amplification of the microgripper system is 19 times.

To verify the leveling gripping effect of the designed microgripper system, an input displacement of 10 µm was applied to the microgripper system. Figure 3 shows the output displacement cloud of the right gripping arm of the microgripper system. It is illustrated that under the influence of 10 µm input displacement, the output displacements of the right gripping arm of the microgripper system increase gradually from the top to the bottom. The maximum output displacement of 189.64 µm occurs at the root of the gripping arm, and the minimum output displacement of 189.36 µm occurs at the tip. The entire gripping arm only generates a leveling error of 0.28 µm. Therefore, it is indicated that the designed microgripper system has excellent leveling performance and can achieve leveling gripping of the operated objects.

To ensure that the designed microgripper system will not experience yield failure during the actual application, it is necessary to carry out a simulation analysis on the stresses and strains of the microgripper system. The aluminum alloy is chosen as the material for the microgripper system. Figure 4(a) shows the stress cloud diagram of the microgripper system under the effect of a 10 µm input displacement. It is shown that the stresses in the microgripper system are concentrated at the flexible hinges. The stress at the rectangular flexible hinge B is the largest, reaching 81.5 MPa, which is much smaller than the yield limit of the aluminum alloy, which is 503 MPa. Therefore, the microgripper system will not experience yield failure during actual use. Figure 4(b) shows the equivalent variation of strains in the microgripper system under the action of a 10 µm input displacement. From the figure, it can be observed that the equivalent variation of strains is concentrated at the flexible hinges, and the equivalent variation at hinge B is still the largest.

Based on the above simulation, the overall size of the microgripper system is 52 × 88 mm², the single arm output displacement is 189.73 µm, and the maximum tension can reach 379.5 µm. The clamping arm has excellent leveling performance, and the strain of the force and displacement sensor is more noticeable. It is demonstrated that the indicators of the microgripper system satisfy the design requirements and can be processed, as shown in Fig. 5.

To achieve precise control of the finger displacement and gripping force, in this section, feedback control of the microgripper system was carried out through the PID controller on the Simulink platform, and the actual control effect of the PID controller was tested with typical signals, such as sinusoidal signal, variable frequency signal, and variable amplitude signal.

The block diagram of the PID control principle is shown in Fig. 6, and the PID controller is divided into three loops: the proportion loop [k_p × e(t)], the integration loop [$ki×∫0te(t)dt$], and the differentiation loop [$kd×e(t)dt$].

To precisely control the finger displacements of the microgripper system, eliminate the influence of the hysteresis nonlinearity of the piezoelectric ceramic on the finger displacements, and make the output displacements linearly related to the input voltages, a PID controller was employed to control the microgripper system with closed-loop feedback. To examine the actual control effect of the PID controller on the finger displacements of the microgripper system, a sinusoidal signal, a variable amplitude signal, and a variable frequency signal were utilized as reference input signals to trace the finger displacements of the microgripper system under closed-loop conditions.

A sinusoidal signal with a frequency of 0.1 Hz and an amplitude of 50 µm was applied as the reference signal to the microgripper system, and Fig. 7 shows the displacement tracking results of the microgripper system in the closed-loop case.

Figure 7(a) represents the displacement tracking curve of the microgripper system in the closed-loop case. Figure 7(b) represents the closed-loop displacement tracking error. Figure 7(c) represents the histogram of the displacement tracking error. Figure 7(d) represents the closed-loop displacement hysteresis curve. From the figures, it can be seen that in the closed-loop case, the actual output displacement curve of the microgripper system and the reference displacement curve match extremely well. The displacement tracking error is small and follows a normal distribution, and the 95% confidence interval of the displacement tracking error is −0.0291 ± 0.3616 µm. Since the tracking error is a random error and follows a normal distribution, the standard deviation can be used to represent the magnitude of the error. The standard deviation of the sinusoidal signal tracking error in the closed-loop case is calculated to be 0.184 µm, which is only 0.37% of the maximum value of the reference signal. In addition, in the closed-loop case, the hysteresis of the microgripper system displacements is effectively improved, and the output displacements are linearly related to the input voltages.

The variable amplitude signal was employed as the reference signal to track the finger displacements of the microgripper system, and Fig. 8 displays the displacement tracking results of the microgripper system in the closed-loop case.

Figure 8(a) shows the displacement tracking curve of the microgripper system in a closed-loop situation. Figure 8(b) shows the closed-loop displacement tracking error. Figure 8(c) shows the histogram of the displacement tracking error. Figure 8(d) shows the closed-loop displacement hysteresis curve. It is demonstrated that in the closed-loop case, the tracking effect of the finger displacements of the microgripper system is excellent, and the actual output displacement curve and the reference displacement curve match fairly well. The closed-loop displacement tracking error is small and conforms to the normal distribution, and the 95% confidence interval of the displacement tracking error is −0.0229 ± 0.2168 µm. The standard deviation of the closed-loop displacement tracking error is calculated to be 0.111 µm, which is only 0.222% of the maximum value of the reference signal. Moreover, in the closed-loop case, the hysteresis of the displacements of the microgripper system is effectively improved, and the output displacements are linearly related to the input voltages.

To test the actual control effect of the PID controller on the variable frequency signal, a signal y = sin(2π · 0.1t + 4.71) + sin(2π · 0.05t + 4.71) + sin(2π · 0.2t + 4.71) was employed as the reference input signal for the displacement tracking experiment. Figure 9 shows the displacement tracking results of the microgripper system in the closed-loop case.

Figure 9(a) shows the displacement tracking curve of the microgripper system in a closed-loop situation. Figure 9(b) presents the closed-loop displacement tracking error. Figure 9(c) displays the histogram of the displacement tracking error. Figure 9(d) shows the closed-loop displacement hysteresis curve. It is illustrated that the tracking effect of the finger displacements of the microgripper system in the closed-loop situation is excellent, and the actual output displacement curve and the reference displacement curve coincide. The closed-loop displacement tracking error is small and conforms to the normal distribution. The 95% confidence interval of the displacement tracking error is −0.0194 ± 0.263 µm. The standard deviation of the closed-loop displacement tracking error is calculated to be 0.134 µm, which only accounts for 0.256% of the maximum value of the reference signal. In addition, in the closed-loop situation, the hysteresis of the displacements of the microgripper system is effectively improved, and the output displacements are linearly related to the input voltages.

Based on the above experimental results, it can be observed that the displacement tracking error is relatively small under closed-loop control. Moreover, the PID controller effectively eliminates the adverse effects of the hysteresis nonlinearity of the piezoelectric ceramic on the finger displacements of the microgripper system. The output displacements are linearly related to the input voltages, and the PID controller can accurately control the finger displacements of the microgripper system.

To accurately control the gripping forces of the microgripper system, eliminate the influence of the piezoelectric ceramic hysteresis nonlinearity on the gripping forces of the microgripper system, and make the output gripping forces linear with the input voltages, a PID controller was adopted to control the microgripper system with closed-loop feedback. To test the actual control effect of the PID controller on the gripping forces of the microgripper system, sinusoidal signals, variable amplitude signals, and variable frequency signals were used as reference input signals to track the gripping forces of the microgripper system in the closed-loop case.

A sinusoidal signal having a frequency of 0.1 Hz and an amplitude of 100 mN was applied as the reference input signal for the microgripper system, and Fig. 10 shows the actual control results of the gripping forces of the microgripper system in the closed-loop situation.

Figure 10(a) represents the output gripping forces curve of the microgripper system in the closed-loop case. Figure 10(b) represents the closed-loop tracking error of the gripping forces. Figure 10(c) represents the histogram of the gripping force tracking error. Figure 10(d) represents the hysteresis curve of the gripping forces. It is shown that the gripping forces curve of the microgripper system in the closed-loop case and the reference signal basically coincide. The tracking error of the gripping forces is small and obeys normal distribution, and the 95% confidence interval of the tracking error of the gripping forces is −0.0897 ± 0.463 mN. The standard deviation of the tracking error of the sinusoidal signal in the closed-loop case is calculated to be 0.236 mN, which is only 0.236% of the maximum value of the reference signal. In addition, in the closed-loop case, the hysteresis of the gripping forces is effectively improved and the output gripping forces are linearly related to the input voltages.

The tracking experiment of the gripping forces of the microgripper system was carried out by taking the variable amplitude signal as the reference signal, and Fig. 11 shows the results of the gripping forces tracking of the microgripper system in the closed-loop case.

Figure 11(a) shows the gripping forces curve of the microgripper system in the closed-loop case. Figure 11(b) presents the closed-loop tracking error of the gripping forces. Figure 11(c) shows the histogram of the gripping forces tracking error. Figure 11(d) exhibits the hysteresis curve of the gripping forces. From these figures, it can be observed that the tracking effect of the gripping forces of the microgripper system in the closed-loop case is excellent, and the actual output gripping forces curve matches well with the reference curve. The closed-loop gripping forces tracking error is small and conforms to the normal distribution, and the 95% confidence interval of the gripping forces tracking error is −0.0391 ± 0.632 mN. The standard deviation of the closed-loop gripping forces tracking error is calculated to be 0.322 mN, which is only 0.322% of the maximum value of the reference signal. Furthermore, in the closed-loop case, the hysteresis of the gripping forces of the microgripper system is effectively improved, and the output gripping forces are linearly related to the input voltages.

To test the actual control effect of the PID controller on variable frequency signals, a certain signal y = sin(2π · 0.1t + 4.71) + sin(2π · 0.05t + 4.71) + sin(2π · 0.2t + 4.71) was used as the reference input signal for the gripping forces tracking experiment. Figure 12 shows the tracking results of the gripping forces of the microgripper system in the closed-loop case.

Figure 12(a) shows the output gripping forces curve of the microgripper system in the closed-loop case. Figure 12(b) shows the closed-loop tracking error of the gripping forces. Figure 12(c) shows the histogram of the closed-loop tracking error. Figure 12(d) shows the hysteresis curve of the gripping forces. From these figures, it can be seen that in the closed-loop case, the gripping forces tracking effect of the microgripper system is satisfactory, and the actual output gripping forces curve basically coincides with the reference curve. The closed-loop tracking error of the gripping forces is small and satisfies the normal distribution. The 95% confidence interval of the gripping forces tracking error is −0.0218 ± 0.391 mN. The standard deviation of the closed-loop tracking error of the gripping forces is calculated to be 0.199 mN, which is merely 0.199% of the maximum value of the reference signal. In addition, in the closed-loop case, the hysteresis of the output gripping forces of the microgripper system is effectively improved, and the output gripping forces and the input voltages are approximately linear.

Based on the above experimental results, it can be concluded that under closed-loop control, the tracking error of the gripping forces of the microgripper system is rather small, and the PID controller effectively eliminates the negative effect of the hysteresis nonlinearity of the piezoelectric ceramics on the gripping forces of the microgripper system. The output gripping forces are linearly associated with the input voltages, and the PID controller is capable of accurately controlling the gripping forces of the microgripper system.

To test the actual gripping effect of the microgripper system, the experimental platform was built, mainly comprising the microgripper system, the driving power supply, the data acquisition card, the capacitive displacement sensor, the piezoelectric ceramic, the signal conditioning circuit, and the strain sensor. A sinusoidal driving voltage with a frequency of 0.1 Hz and an amplitude of 100 V was applied to the microgripper system, and a Φ0.9 × 8 mm² micro-axis was gripped. Figure 13 shows the change curves of the output displacements and gripping forces of the microgripper system during the gripping process. From Fig. 9(a), it is revealed that the maximum output displacement of the microgripper system is 102.3 µm, and the corresponding input displacement at this time is 6.1 µm, so the displacement amplification factor of the microgripper system can be obtained as 16.8 times. From Fig. 9(b), it is demonstrated that the maximum gripping force is 227.7 mN when the collet of the microgripper system is fixed under the action of 100 V sinusoidal voltage.

A piezoelectric actuated microgripper system was developed based on flexible hinges. The static mechanical model of the microgripper system was established, and thereafter, the output displacements, stresses, and strains of the microgripper system were simulated using ANSYS software. At last, the closed-loop characteristics of the microgripper system were tested. It is illustrated that under the driving of 100 V sinusoidal voltage, the maximum output displacement of the gripping finger is 102.3 µm, and the maximum gripping force is 227.7 mN. The closed-loop control effectively eliminates the hysteresis nonlinearity of the piezoelectric actuator. The standard deviation of the closed-loop error of the displacements is no more than 0.2 µm, and that of the gripping forces is no more than 0.35 mN.

This work was supported by the Key Research Project in Colleges and Universities of Henan Province (Grant No. 23A460010).

The authors have no conflicts to disclose.

D.W. provided technical guidance, analyzed the data, and ultimately completed the manuscript; Y.Z. carried out the overall design; H.Y. conceived, designed, and performed the experiments as well as drafted the original manuscript; K.H. analyzed the finite element models.

Dongsheng Wang: Formal analysis (equal). Yanru Zhao: Funding acquisition (equal). Huimeng Yang: Data curation (equal). Kunpeng Hong: Software (equal).

The data that support the findings of this study are available within the article.