Design and control of a novel asymmetrical piezoelectric actuated microgripper for micromanipulation

Abstract Microgripper is an important tool in high precision micromanipulation task, which directly affects the quality and efficiency of micromanipulation. This paper presents the design and control of a novel asymmetrical microgripper driven by a piezoelectric (PZT) actuator. The developed microgripper is designed as an asymmetrical structure with one movable jaw, so it has the advantages of no dense mode and fixed locating datum compared with the symmetrical microgripper with two movable jaws. The main body of microgripper is a compact flexure-based mechanical structure with a three-stage amplification mechanism. Based on the three-stage amplification structure, large-range, robust and parallel grasping operation can be realized. The characteristics analyses of the developed microgripper are carried out by finite element analysis (FEA). A position-force switching control strategy is utilized to regulate the position and grasping force of movable jaw. Discrete-time sliding model (DSM) controller is designed to control the position and grasping force. Experimental studies are conducted and the experiment results show that the microgripper exhibits good performance and high precision grasping operations can be realized through the developed control strategy.


INTRODUCTION
Recently, the demands of micromanipulation in terms of precision and efficiency have been growing with the development of biological engineering, microelectronics industry et.al [1][2][3]. As the operated objects in micromanipulation are developing toward ultra-micronization, the difficulty of micromanipulation is increasing continuously.
Microgripper, as a key component of precision micromanipulation systems, plays an important role during the operation process [4,5]. The microgripper contacts the operated objects directly in the automated grasp-hold-release operations, so the performance of microgripper will directly affect the quality, efficiency and accuracy of micromanipulation [6][7].
Nowadays various types of microgrippers have been developed according to different actuations, such as electrothermal, coil voice actuated, shape memory alloy (SMA) actuated and piezoelectric (PZT) microgrippers [8][9][10][11]. Particularly PZT actuator has the advantages of small volume, large output force, fast response and zero backlash, so it has been widely used in microgripper [12][13][14]. Since the output displacement of PZT actuator is quite small which is usually several tens micrometers, displacement amplification mechanism (DAM) has an important effect on the PZT-actuated microgripper which is utilized to amplify the output displacement of actuator and transmit the displacement from the actuator to the jaws. Based on different types of DAMs several PZT-actuated microgrippers have been developed. Sun et al. designed a PZT driven compliant-based microgripper for micromanipulation which consists of Scott-Russell mechanism and leverage mechanism [15]. Xu designed a PZT-actuated microgripper with a two-stage flexure-based DAM [16]. Wang et al. designed a monolithic compliant PZT driven microgripper with a larger displacement amplification ratio [17]. Zubir et al. designed an asymmetrical PZT driven microgripper with two grasping jaws movable to obtain large range of grasping operation [18]. Most present PZT actuated microgrippers are designed with two movable grasping jaws. The advantage of this kind microgripper is that large displacement amplification ratio can be realized which can meet the requirement of grasping large scale objects. However, the inevitable error during fabrication and assembly may lead to asymmetric motion of movable jaws, which will make the grasping objects not in the center line of the jaws. Therefore the operation precision will be reduced inevitably. In addition the first two mode shapes of the microgripper with two jaws movable are usually too close to each other. The dense modes will lead bad dynamic characteristics and bring troubles to the control of microgripper. The microgripper with one movable jaw has a specific and fixed locating datum which can improve the operation accuracy. Moreover, since just one grasping jaw is movable, only one side of the microgrippers is flexible. Therefore there is no dense modes problem in the asymmetrical microgrippers with one movable jaw.
For the reason that the micro-objects are usually small and easily broken, both precise position and stable grasping force are required to be guaranteed during micromanipulation process. Therefore high performance position-force controller is needed during working process. Unfortunately, most of the present microgrippers focus on position control and the researches dedicate to both position and grasping force control are relatively rare to find. There are several effective and practical control methodologies in terms of position and force control, including hybrid control [19,20], impedance control [21,22] and switching control [23,24]. Hybrid control and impedance control are applicable to the system that the force and the position are coupled, while the switching control is suitable to the force and position decoupled control system. Compared with the other control methodologies the switching control has a concise structure and is easy to implement, especially in force and position decoupled control system. As the most common controller, proportional-integral-derivative (PID) controllers are the most dominating form of feedback in use today [25]. But due to the high-frequency dynamic vibrations during the fast grasping and releasing operations, traditional PID control can not satisfy the requirement of precision position-force regulation. To overcome this problem, several advanced controllers including the adaptive robust control [26], iterative learning control [27] and neural networks control [28] et.al, are proposed to control precision positioning systems. However expensive algorithms and calculations, together with complex parameters tuning process make the applications of these controllers limited. Since the discrete-time sliding mode controller (DSMC) has the advantages of fast response and strong robustness, DSMC becomes one of the most promising candidates for the control of microgrippers. This paper is motivated to design a novel high performance microgripper, which is designed asymmetrically with one movable jaw. Through the flexure-based mechanical structure with a three-stage amplification mechanism, the input displacement can be Due to the output displacement of the PZT actuator is very small, the microgripper should have enough large displacement range to grasp different sizes of micro-objects; thus a flexure-based DAM with large amplification ratio is designed. The motion transmission from the PZT actuator to the moveable jaw is realized through the DAM, while the output displacement of the PZT actuator is magnified. As the first-stage amplification the leverage mechanism can amplify the output displacement of the PZT actuator based on the deformation of flexure hinge as shown in Fig. 3(a). The bridge-type mechanism is designed as the second-stage amplification which consists of a connecting rod mechanism connected by four elliptical flexure hinges. The working principle of the bridge-type mechanism is that once driven by an input displacement, the device produces a vertical output displacement as shown in Fig. 3(b). The parallelogram mechanism shown in Fig. 3(c) can work like a leverage mechanism which is the third-stage amplification. Moreover pure jaw translations of moveable jaw can be realized based on the parallelogram mechanism, which can guarantee stable and robust grasping manipulations.   During the working process, the PZT actuator will expand and push the leverage mechanism; then the leverage mechanism will swing upward and pull the bridge-type mechanism; finally the parallelogram mechanism pulled by the bridge-type mechanism will swing and cause the moveable gripping jaw to close to grasp the manipulated micro-object. After power is switched off, the PZT actuator will retract to its original length and the flexible motion transmission mechanism will return to its initial position, which causes the moveable grasping jaws to release the micro-object.  One advantage of the designed asymmetrical microgripper with just one movable jaw is that it has a specific and fixed locating datum as shown in Fig. 4(a). The locating datum of the symmetrical microgripper with two jaws movable may not be fixed and has a drift due to the inevitable error during microgripper fabrication and assembly as shown in Fig. 4(b). Therefore the asymmetrical microgripper with one movable jaw is more suitable for application of high precision micromanipulation, such as high precision micro assembly. Another advantage of the designed asymmetrical microgripper with one movable jaw is that it has no dense modes and better dynamic performance. Figure  out that the displacement of output terminal of the parallelogram mechanism is larger than that of the movable jaw. The reason is that the force applied on the input terminal of parallelogram mechanism is not just in x-axis direction which is also has a component in y-axis direction. The parasitic displacements in y-axis of the movable jaw, flexible beam and output terminal of parallelogram mechanism are also obtained which is shown in Fig.   6(a). The maximum parasitic error is located in the movable jaw which is 1.47μm and the maximum relative parasitic error is 0.98%. Moreover the results also show that the maximum stress occurs at the flexure hinge which connects the bridge-type mechanism and the parallelogram mechanism. The maximum stress is 198.9MPa, which is far less than the yield strength (503 MPa) of the material of the microgripper, AL7075-T651. It is known that the transfer functions of position response can be represented as: where ai and bi are the coefficients and x(k) and u(k) present output position and input voltage at time step k, respectively. In addition, p(k) is perturbation which includes piezoelectric hysteresis, drift, external force, parameter uncertainties, and other model disturbances.
The perturbation can be estimated by Then the dynamic model of Eq. (1) can be expressed as represents the error between the estimated perturbation and real perturbation.
It can be deduced that pe(k) is also bounded where  is a positive constant and Ts is the sampling time.
In this paper, the output tracking error is defined as follows: where xr(k) is the desired output.
Based on the error in Eq (5), the PID-type sliding function is defined as follows: where c1 and c2 are positive parameters.
In addition, the integral error item (k) is defined as Designing the sliding control law as , λ1 and λ2 are positive constant parameters, c=c1+c2+1 and sgn is sign function.
Substituting Eq. (8) into Eq. (3), the following equation can be obtained: Substituting Eq. (9) into the equation of PID-type sliding function (6) yields  (10) To evaluate the stability of the control system, a Lyapunov function is defined as The following conditions should be satisfied: Since the sampling time is short, Eq. (12) can be written as (13) For the DSMC with the exponential reaching law, the following relationship can be deduced: If the gain λ2 is designed to meet the condition      2 , where  is an arbitrary positive constant, the following relationship can be obtained: When the sampling time Ts is short, the following relationship can be obtained: Based on Eq. (15) and (16), the system can be verified to be stable.

Position-force switching control strategy
The grasping task of microgripper can be classified into three phases, namely the closing phase, contact phase and opening phase. In closing and opening phases position control is essential to ensure high position precision of moveable jaw, while in contact phase force control is required to make the grasping force controllable. Therefore a position-force control strategy is utilized in which the final voltage in previous phase should be the base value for the next phase to guarantee a stable and smooth transition.
Once the moveable jaw contacts the object and the grasping force exceeds the threshold value, the control system switches to the force control from the position control. When it is time to release the object, the control system will switch to the position control again.
The flow chart of the position-force switching control strategy is shown in Fig. 8. In the flow chart Fr is the desired grasping force of force control in contact phase, while dr1 and dr2 are the desired displacements of position control in closing phase and opening phase, respectively. Fs is the threshold force, tg is the time to release the object and Tg is overall time of grasping operation. usp is the switching voltage from position control to force control and usf is the switching voltage from force control to position control.

Prototype development and experimental setup
In order to guarantee the geometrical accuracy wire electro-discharge machining (WEDM) technique is utilized to manufacture the flexure-based mechanism. Figure 9 shows the prototype of microgripper made by AL7075-T651. AL7075-T651 has the property of high elasticity, yield strength, and light mass for the excellent physical and thermal properties. The initial gap between the grasping jaws can be adjustable by the preload bolt to grasping different sizes of objects. A number of tests have been conducted on the designed microgripper which is mounted on a vibration-isolated Newport RS-4000 optical table to reduce the external vibration disturbance. Two laser displacement sensors, which provide a 50 nm resolution within a 20 mm measuring range, are employed to measure the input displacement and output displacement of the microgripper, respectively. As shown in Fig. 10 (a) the input displacement is measured at the point where the PZT actuator pushes the leverage mechanism and the output displacement is measured at the point on the output terminal of the parallelogram leverage mechanism. For the purpose to measure the grasping force a strain gage is glued on the on the base end of the flexible beam, which is adopted form a quarter Wheatstone bridge for the grasping force measurement. A dynamic strain gauge system is adopted to measure the signal of Wheatstone bridge. The strain gage is calibrated by hanging several objects on the movable jaw, whose weights are measured by electronic balance. At the same time the output voltage of the sensor are acquired by NI data acquisition card. The gain is then calculated as 0.559 mN/mV, which is used to convert the voltage into force value. A dSPACE DS1103 controller is adopted to implement the control algorithm, which picks up the displacement and force signals to determine the current state of the microgripper. In addition, a voltage amplifier (E505.00 from PI, Inc) is utilized to amplify the output voltage of controller for the PZT actuator drive and the gain factor is 10. All the devices are also placed on the vibration-isolated table and experimental setup is displayed in Fig. 10 (b).

Characteristics test
In order to measure the displacement amplification ratio, a sinusoidal signal with the amplitude of 60 V and frequency of 1 Hz is applied on the PZT actuator. The input and output displacements can be measured by two laser displacement sensors at the same time; therefore the relationship between the input and output displacements can be obtained. Identification Toolbox is used to process the data. The results are summarized in Fig. 12.
In Fig. 12(a)  Step response is investigated and the desired displacement trajectory is defined as a step signal with a final value as 20 µm. The result of step response by position control is shown in Fig. 13. From the result it can be seen that the settling time is 12 ms, the overshoot is 2.5%, and the steady-state error is ±0.2 µm. The sinusoidal response is investigated using a sinusoidal reference input with the amplitude of 10 μm, frequency of 20 Hz and bias of 10 μm, which is shown in Fig. 14(a). The corresponding position tracking errors are illustrated in Fig. 14(b). It is found that the tracking errors are within ±1.3 μm. In order to verify the performance of the force controller, step response with a final value as 50 mN is investigated while the threshold force is set as 5 mN. The results are shown in Fig.15. From the results it can be seen that the settling time is 30 ms, the overshoot is 1.3%, and the steady-state error is ±0.4 mN. For the purpose to achieve a complete and efficient grasp-hold-release operation, the position and force trajectories are planned as shown in Fig.16. In order to ensure smooth switching, fast close and slow contact is essential in closing phase, a homothetic trapezoidal velocity planning is employed to control the jaw displacement which leads to the contact of the jaws and grasped wire with a constant velocity. In the contact phase a step force signal is utilized as the reference signal of the grasping force. In order to make the jaw return to its initial position, the trapezoidal velocity planning is used based on the final position in the contact phase. 2. We can find that the developed microgripper outperforms the others in terms of no dense mode, and faster overall operation time.

CONCLUSION
A novel piezoelectric actuated microgripper has been reported in this paper, which is designed asymmetrically with just one jaw movable. Through a three-stage flexure-based mechanical structure a large displacement amplification ratio has been achieved. The