Precision Control of a Piezo-Actuated Micro Telemanipulation System

Piezoelectric actuators are widely used in micro manipulation applications. However hysteresis nonlinearity limits accuracy of these actuators. This paper presents a novel approach for utilizing a piezoelectric nano-stage as slave manipulator of a teleoperation system. The Prandtl-Ishlinskii (PI) model is used to model actuator hysteresis in feedforward scheme to cancel out this nonlinearity. To deal with the influence of parametric uncertainties, unmodeled dynamics, and PI identification error a perturbation term is added to the slave model and apply a sliding mode based impedance control with perturbation estimation. The stability of the entire system is guaranteed by Llewellyn's absolute stability criterion. Performance of the proposed controllers is verified through experiments. Manuscript received: / Accepted: 2 / INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol., No. The dynamic hysteresis relation between the applied voltage and the actuator displacement originates from a cascaded combination of a static hysteresis. This is between the applied voltage and the induced charge into the actuator. There is a linear electromechanical coupling between the induced charge and the excitation force. A linear dynamic relation between the excitation force and the actuator displacement is also present . Although the effect of hysteresis could be bypassed with control of the induced charge, costly instrumentations are required for the measurement and amplification of the induced charge. 11 Voltage drive strategies are thus preferred despite their limiting hysteresis nonlinearity. Many methods have been proposed to compensate hysteresis on the actuation of piezoelectric actuators. Preisach 8, 12 and Maxwell 13,14 are well known hysteresis models, however their approximation accuracy is limited and suffer from a complicated identification procedure. Kuhenen proposed a modified Prandtl-Ishlinskii (PI) model for the hysteresis nonlinearity .This model has been extended for ratedependent and load-dependent hysteresis . PI model is less complex and its inverse can be computed analytically. In this study a modified PI model is applied and its inverse is utilized to cancel out the hysteresis effect. Accurate tracking of piezoelectric actuators with voltage steering strategy are extensively carried out in both feedforward and feedback control operations. Most feedforward controllers cascade an inverse hysteresis model in series with a piezoelectric actuator plant to cancel out the effect of nonlinearity and achieve a relatively linear response . Cahyadi et al. developed a 3-DOF macro-micro teleoperation system using piezoelectric actuator. The proposed controller for slave side did not consider uncertainty of model. Moreover, hysteresis effect after compensation (Figure 12) depicts that remained nonlinearity is considerable. Bilateral telemanipulation systems attempt to provide force tracking in master side. (Figure 14) depict proper position tracking and contact force of the slave. The paper did not provide any plot to explain force tracking. In this paper, the nonlinear piezoelectric actuator is linearized using feedforward inverse hysteresis. The linearized uncertain model is then used to design the controller. The sliding-mode based impedance control with perturbation estimation scheme is used. With an impedance control for the master, a desired dynamic behavior between human operator and master device can be realized. Stability of the teleoperation system against time delay is performed using Llewellyn criterion and proper controller gains are adjusted to achieve stability and performance simultaneously. 2. Teleoperator Modeling Fig. 1 shows the master slave system for a micro telemanipulation setup. To design an efficient controller for this system the dynamics equations of motion of the teleoperation system are first derived. 2.1. Dynamic Modeling for the Master Robot In this research the master is a 1 DOF manipulator which utilizes a DC servo motor. A load cell is installed on the shaft of the motor to measure the force exerted on the master. Dynamic model of the motor can be considered as follows: j θ̈ (t) + b θ̇ (t) + k θ (t) = u (t) + L F (t) (1) Fig.1 Macro-micro Telemanipulation setup Where θ denote rotation angle,j , b and k are moment of inertia of the rotating system, damping and stiffness respectively. F is the force exerted by human operator and L is the effective length between the force and motor shaft. u is control signal that is applied to the master robot. 2.2. Dynamic Modeling for the Slave Robot The slave manipulator consists of a 1-DOF stage actuated by a piezo stack actuator. Hysteresis effect of piezoelectric actuators which is revealed in their response to an applied electric field is the main drawback in precise positioning. Therefore, the development of a dynamic model which describes the hysteresis behavior is very important. This is for the improvement of the control performance of the piezo-positioning mechanism. In many investigations, a secondorder linear dynamics has been utilized for describing the system dynamics. As shown in figure 2, this model combines mass-springdamper ratio with a nonlinear hysteresis function appearing in the input excitation to the system. The following equation defines the model: m ẍ (t) + b ẋ (t) + k x (t) = H v(t) (2) Where x (t) is the salve position, m , b and k are mass, viscous coefficient and stiffness respectively. H v(t) denotes the hysteretic relation between input voltage and excitation force. Fig.2 Piezoelectric actuator equivalent dynamic model Piezoelectric actuators have very high stiffness, and consequently, possess very high natural frequency. In low-frequency operations, the effects of actuator damping and inertia could be safely neglected. Hence, the governing equation of motion is reduced to the following static hysteresis relation between the input voltage and actuator displacement: INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol., No. / 3 x(t) = 1 k H v(t) = H v(t) {m ẍ (t) ≪ b ẋ ( ) ≪ ( )} (3) Equation (3) facilitates the identification of the hysteresis function ( ) between the input voltage and the excitation force. This is performed by first identifying the hysteresis map between the input voltage and the actuator displacement, H v(t) . It is then, scaled up to k to obtain H v(t) . m ẍ (t) + b ẋ (t) + k x (t) = k H v(t) (4) To consider the influence of parametric uncertainties, unmodeled dynamics, and identification error, a perturbation term P(t) is added to the slave model. Thus the slave model (2) can be rewritten as the following: m ẍ (t) + b ẋ (t) + k x (t) = H v(t) + P(t) = k H v(t) + P(t) (5) To consider interaction with environment, the force F exerted by the environment is inserted into the model. Therefore dynamic model of the slave manipulator can be written as follows: m ẍ (t) + b ẋ (t) + k x (t) = k H v(t) + P(t) − F (6) 3. Hysteresis Modeling In this section hysteresis modeling using Prandtl-Ishlinskii (PI) is described. This model can approximate hysteresis loop accurately and its inverse could be obtained analytically. Therefore it facilitates the inverse feedforward control design. 3.1 Prandtl-Ishlinskii (PI) There is a backlash operator in the PI hysteresis model (Fig.3) that is defined by: y(t) = H [x, y ](t) = max {x(t) − r, min{x(t) + r, y(t − T)}}