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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

The measurement and control strategy of a piezo-based platform by using strain gauge sensors (SGS) and a robust composite controller is investigated in this paper. First, the experimental setup is constructed by using a piezo-based platform, SGS sensors, an AD5435 platform and two voltage amplifiers. Then, the measurement strategy to measure the tip/tilt angles accurately in the order of sub-μrad is presented. A comprehensive composite control strategy design to enhance the tracking accuracy with a novel driving principle is also proposed. Finally, an experiment is presented to validate the measurement and control strategy. The experimental results demonstrate that the proposed measurement and control strategy provides accurate angle motion with a root mean square (RMS) error of 0.21 μrad, which is approximately equal to the noise level.

Ultra-precision angle motion and positioning have significant importance in free-space laser communication, space telescopes, staring cameras and some other space optical instruments [

Additionally, high bandwidth angle measurement on the order of sub-μrad is necessary in free-space optics applications. For example, angle acquisition and tracking with a bandwidth of hundreds of Hz is required in the acquisition, tracking and pointing (ATP) systems of space laser communications. Generally, the tip and tilt (also named pitch and yaw) angles are measured and determined by attitude sensors, e.g., sun sensors, star sensors, integrating gyros and fiber optic gyros, but the accuracy is generally worse than 10 μrad and the measurement bandwidth is generally less than 5 Hz. To measure tip and tilt angles on the order of sub-μrad, SGS sensors are employed in this paper to determine the angles with high accuracy and high bandwidth by measuring the length changes of piezo actuators. Then, the tip and tilt angles are computed according to these length changes. Meanwhile, electric bridges can be chosen for better stability and resolution [

To achieve fast angle motion in the order of sub-μrad, closed loop control is also necessary because of the linear dynamics and hysteresis effects in the piezoelectric actuators [_{∞} feedback controller and derivative feedforward controller is designed to efficiently compensate the linear dynamics and hysteresis effects. Instead of complex hysteretic dynamics, linear dynamics is used to represent the piezo-based platform. The composite controller is designed from an application perspective. Compared with most other _{∞} works, the trade-offs between the control bandwidth, measurement noise and control limitation have been quantitatively considered in our work. Moreover, a realtime physical simulation platform (

This paper is organized as follows: firstly, the experimental setup and working principle are presented in Section 2. Next, the measurement scheme of the tip/tilt platform is proposed in Section 3. A piezo-based platform, an AD5435 realtime platform, two amplifiers and SGS sensors are employed. Then, the composite control system of the piezo-based platform is presented in Section 4. Finally, the experimental results and discussions are presented.

The experimental setup, the measurement strategy and the angle calibration are presented in this section. The constructed piezo platform system consists of a piezo-based platform, SGS sensors, voltage amplifiers, an AD5435 platform and a signal conditioning unit, as shown in

The experimental setup is shown in

The working principle of the proposed piezo-based platform is different from that of other piezo applications. First, a preloading spring is used to maintain the structure stability. Then, a constant voltage and two varying inputs are used to drive four independent piezo stacks. The working principle of the piezo-based platform in X-axis is shown in _{1} and _{3} are the tilt angle, the translational displacements of PZT I and PZT III, respectively. A constant voltage +100 V is added to PZT I. Thus, the input voltage of CH 1 can be used to drive both PZT I and PZT III to produce the tilt angle

To further introduce the working principle,

In this section, the measurement strategy to provide angle measurements with high accuracy on the order of sub-μrad is presented. Instead of direct measurements of the angle motion, the positions of PZT stacks are measured first. Then the angle motion is computed. To evaluate the accuracy of the measurement strategy, the angle calibration is also presented.

SGS sensors are widely applied to measure piezo displacements. In this paper, one piezo actuator (PZT stack) is used to show the measuring principle of SGS sensors. Two resistive films are bonded to the PZT stack, as shown in

The strain gauges are used to measure the mechanical length change, because the length change results in the resistance change and the generation of voltage signal. More details of the electric bridges are presented by Huang

The tip/tilt angles of the piezo-based platform are indirectly measured by comparing the length changes of the PZT stacks. First, the SGS sensor gives the length change of the PZT stack. The relationship among _{1} and _{3} in

The travel span of the piezo stacks in this paper is 15 μm. The max tip/tilt angles are 0.002 rad. Thus, tan

Finally, there is a linear relationship between the tilt angle and the piezo displacements and it will be validated in the next calibration section.

To validate the measurement strategy the angle calibration is presented. For simplification, only the tilt angle calibration is presented. A ZYGO ZMI-2000 interferometer is used as the measurement setup to calibrate angles. The calibration temperature is 20.8 °C and the calibration humidity is equal to 39%.

Furthermore,

The comprehensive design of the composite control of the piezo-based platform to achieve precision angle motion is proposed in this section. In this section, only the tilt motion (_{∞} feedback controller is designed. Sequentially, the derivative feedforward is incorporated to enhance the tracking performance. Finally, the proposed composite controller is applied to the piezo-based platform to demonstrate the tracking performance.

Modeling and identification are necessary for the design of a modern robust control system. Various methods for modeling and identifiying piezo systems have been investigated [_{∞} controller, time-invariant linear dynamics are used to represent the main dynamics of the PZT stacks. Also, the un-modeled dynamics is represented by an input uncertainty.

The time-invariant linear dynamics of the piezo-based platform is identified by using the square wave input (only X-axis is investigated). By employing armax method (

Multi-objective robust _{∞} optimal control is investigated in this section in order to achieve precision angle motion in the face of modeling uncertainty and measurement noise. Weighting functions and loop shaping technique are used to design the robust _{∞} controller.

_{∞} control system, in which _{∞} controller and the linear dynamics of the PZT stacks, respectively, and _{1} is the performance weighting function. To enhance the disturbance suppressing at low frequencies and limit the feedback gain at high frequencies simultaneously, an integral action is added to _{1}, and _{2} is the control weighting function which limits the control gain at high frequencies. To guarantee the robust stability at high frequencies, the control signal is limited to slower than 2,000 Hz. The terms _{r}_{n}_{u}_{u}_{r}_{u}_{1}, _{2}_{n}

The _{∞} optimal controller [_{∞}_{∞}_{∞}

To further enhance the tracking performance of the robust _{∞} feedback control in Section 4.2, a derivative feedforward of the reference signal is added to the tacking error. The composite control is thus designed, as shown in _{∞} feedback controller, respectively, and

Finally,

In this paper, the derivative feedforward _{∞} composite controller. To represent the tracking performance in frequency domain, the Bode diagram of the sensitivity functions of the _{∞} controller and composite controller is shown in _{∞} feedback controller at frequencies higher than 5,000 rad/s.

In this section, the simulation study of a triangle wave with the amplitude of 100 μrad and the frequency of 10 Hz is presented. _{∞} controller. The tracking error of the proposed composite control is less than 36% of the _{∞} feedback control. The tracking error RMS of the proposed composite control is 0.205 μrad, which is almost equal to the electrical noise. The simulation result indicates that the proposed composite controller presents satisfactory tracking performance.

In this section, the experimental studies to demonstrate the measurement and control strategy are described. Both triangle and sinusoidal waves are selected as the reference signal. Firstly, the triangle wave at 10 Hz is chosen as the reference signal

The tracking error of the proposed composite control is shown in _{∞} and composite control is negligible. The tracking error of the proposed composite control is less than 0.8 μrad with the RMS error is 0.21 μrad. The relative tracking error of the proposed composite control is 0.36% of the reference signal RMS. The following

The following

where _{r}

The experimental results reveal that the precision angle motion of the piezo-based platform can be achieved by employing the proposed composite controller. In addition, for angle motion and positioning with high accuracy, linearity and repeatability in the order of sub-μrad, closed loop operation is necessary for the piezo-based platform, though the piezo-based platform is commonly regarded as a precision system. Thus a robust composite controller is designed in closed loop to compensate the linear dynamics and hysteresis effects. It can be seen from the results that the tracking accuracy of the triangle wave at 10 Hz approaches the level of measurement noise.

As for comparison, the experimental results in this paper are compared with that in Reference [

Precision angle motion and positioning in space optics are increasingly required. A piezo-based platform system is constructed in addition to a measurement and control strategy to resolve this problem. A measurement strategy providing effective measurement with high bandwidth and accuracy is investigated. A comprehensive design of the composite control strategy is developed. The effectiveness of precision angle motion under the proposed composite controller is validated through the experimental results. It can be demonstrated from the results that the measurement and control strategy in this paper will have significant benefit for future applications in space optics.

This work is supported by the Fundamental Research Funds for the Central Universities of China (DUT13RC(3)09), and the China Postdoctoral Science Foundation (2013M530123).

The authors declare no conflict of interest.

Experimental components for precision angle motion.

Experimental setup.

Working principle of the piezo-based platform in X-axis.

Working principle of the piezo-based platform in both X-axis and Y-axis.

Measuring principle of the PZT stack position using SGS sensors.

Measuring principle of the PZT stack position using SGS sensors.

Relationship between the sensor output voltage and its angle displacement.

Nonlinearity of the measurement of SGS sensors.

Design sketch of the robust _{∞} controller.

Composite control sketch of the piezo-based platform.

Bode diagrams of the sensitivity function of the _{∞} controller and composite controller.

Tracking errors in simulation study.

Tracking performance of the triangle wave.

Tracking error of the triangle wave in the piezo experiments.

Tracking performance of sinusoidal wave at 100 Hz under the composite control.