Numerical Simulation Results

Here, a result of regulation simulation done by MATLAB is shown in Fig. 5-2, where the designed controller parameters are listed as follows: Λ_x =200, Q_x =130,

0.05,N_x = 70000000, 20000,Γ =_1,x Γ =_2,x Γ =_3,x 5, 20Γ =_4,x

Fig. 5-2 Simulation of adaptive sliding mode controller’s regulation.

In Fig. 5-2, it is not hard to find the transient time is within 0.04 seconds. Additionally, this simulation result also reveals that all the estimated signals are converged and bounded as shown in Fig. 5-3. Instead, for convenience, here we only show the first elements of K^ˆ₀, , ,B^ˆ₀ M^ˆ ₀ and wˆ_c, respectively.

(a) (b)

(c) (d) Fig. 5-3 The estimation of (a) kx, (b) bx, (c) m, (d) wc

Next, we will perform a number of experiments in the next chapter and meanwhile, discuss the experimental results based on the proposed controller.

Chapter 6

Experimental Results

Equation Chapter (Next) Section 1

6.1 Hardware and Experimental Environment

To make the positioning stage compact and to utilize the advantages of monolithic construction, the flexure mechanism stage is fabricated through precision wire electrical discharge machining (EDM). The width of the flexure is only 0.3mm, which is shown in Fig. 6-1. It also shows the inlaid structure which is mentioned in Section 3.3. The outer frame is made of copper, which is non-ferromagnetic, and the inner movable stage and the flexure are made of steel, which is ferromagnetic. Then the inner flexure is inlaid into the outer copper frame.The overall dimension of the positioning stage is 110×110×35 mm³.

Fig. 6-1 The flexure mechanism and inlaid structure

In order to avoid the electromagnetic interference effect and meet the requirements of long travel range and high resolution, the optical sensors are therefore chosen in our system. They are manufactured by OMRON, with 10 kHz sampling frequency, 8mm active range, and 10μm accuracy. So far, we have achieved satisfactory performance using this kind of sensors. Figure 6-2 shows the noise of the displacement sensor to measure a static object. The root mean square value of the 10000 sampled data is 2.3μm.

Fig. 6-2 The noise of the displacement sensor

The amplifiers, manufactured by Copley Controls Corp., are linear servo amplifiers for DC brush motors. We use the torque mode to transform the voltage drive into current drive. Table 6-1 lists the specifications of the microcomputer and the used A/D, D/A cards, manufactured by National Instruments.

Table 6-1 Specifications of the PC-based controller.

CPU PentiumIV 2.4 GHz

RAM 512MB Operating System Windows XP

Language Matlab Real-Time Windows Target Ver.2.5.1 D/A card NI 6733 (16 bits)

A/D card NI 6034E (16 bits)

The software environment is the Matlab Real-Time Windows Target of Version 2.5.1, which is a PC-based solution for real-time application. We can generate executable code with Real-Time Workshop, State-flow Coder, and the C/C++ compiler.

Then, we can run the real-time application with “Simulink” in external mode.

The compact positioning stage is shown in Fig. 6-3. The figure also indicates that three displacement sensors are mounted at the side of the positioning stage to measure the position and posture of the stage.

Fig. 6-3 The positioning stage and sensors

6.2 The Vibration Suppression

The vibration of flexure mechanism is a serious challenge to the control of the positioning stage utilizing flexure suspension. In the proposed system, the electromagnetic eddy current damper is used to provide additional damping in this system to attenuate the vibration. Through some simple experiments, the effect of the proposed damper can be easily verified. Figure 6-4 shows open-loop impulse responses of the positioning stage without and with the eddy current damper. The blue curve in Fig.

6-4 indicates the response without the eddy current damper. The vibration takes much longer time to decay than the one of the red line, which indicates the response without eddy current damper. The decay time is shortened from about 17 seconds to 0.8 seconds.

The same phenomenon can be found for the case of step response as shown in Fig. 6-5.

Fig. 6-4 The vibration suppression in impulse response

Fig. 6-5 The vibration suppression in step response

Fig. 6-6 The vibration suppression in frequency response

Responses in the frequency domain with and without the eddy current damper are shown in Fig. 6-6. The frequency responses from 1Hz to 100Hz are measured and compared. At the natural frequency, 16.5Hz, of the flexure mechanism, the magnitude of response is reduced about 26dB. According to the above three experiments, the vibration can be significantly suppressed by our purposed eddy current damper.

6.3 The Verification of Linear Actuator

The experiment to verify the effectiveness of the design of the linear electromagnetic is designed and shown in Fig. 6-7. The actuator is connected to a load cell and then is fixed on an X-Y-Z positioning stage. The load cell can measure the force exerted by the actuator. The X-Y-Z positioning stage is used to change the position of the movable steel structure relative to the fixed coil mount to check the output force of the actuator at different position.

Fig. 6-7 The measurement equipment of the force variation

Figure 6-8 shows the measurement results of the output force on the desired motion plane. The traveling range is 3mm x 3mm. These forces are normalized by

dividing the maximum value. From this figure, the results show that on our desired traveling plane, the variance of the forces does not exceed 2%. In the view of practical, the variance can be ignored and the electromagnetic actuator can be regarded as an approximated linear actuator.

Fig. 6-8 The variance of the force in the traveling range

6.4 Results of Adaptive Sliding Mode Control

This section shows the experimental results of the adaptive sliding mode control.

The designed parameters are:

80 0 0

8

6.4.1 Step and regulation response

Step and regulation responses are shown here in order to test the dynamic property and the positioning precision of the positioning stage. Figure 6-9 shows the measured data of the step response, where at t = 5, a step command is given to the stage. Fig. 6-10 shows the transient response, where the rise time is 65ms (from 0 to 98% of the step size), there is no overshoot, and the settling time is 70ms.

Fig. 6-9 1mm step response on X-axis

Fig. 6-10 The transient of the step response

The steady state error is shown in Fig. 6-11. The root mean square value of the error is 3.25μm, which is close to the limit of the sensors used in the system. The other two states, y and θ, are shown in Fig. 6-12. The unexpected couplings in the other axes are not serious.

Fig. 6-11 The steady state of step response

(a) (b) Fig. 6-12 The remainder states of step response, (a) displacement in y-axis, and (b)

rotation in θ-axis

6.4.2 Sinusoidal tracking response

Figure 6-13 shows experimental result of the sinusoidal tracking with 1mm sinusoidal amplitude at the frequency 0.5Hz along the x-axis. The experimental result shows the fine tracking ability, with both tiny positioning error and little time delay.

Figure 6-14 shows the tracking error. Notice that due to the modeling error, the tracking error is large at the beginning. With the adaptive law to on-line estimate system parameters and tune the gain of the sliding mode controller to more adequate values, the tracking error decrease with time. The root mean square value of the tracking error from 15 seconds to 20 seconds is only 10.6μm. Figure 6-15 shows the other two states of the positioning system.

Fig. 6-13 0.5Hz sinusoidal tracking in x-axis

Fig. 6-14 Sinusoidal tracking error

(a) (b) Fig. 6-15 The remainder states of sinusoidal tracking, (a) displacement in y-axis, and (b)

rotation in θ-axis

6.4.3 Circular contouring

The other way to show the tracking capability is to profile a desired contour. In the experiment of this section, a circle which radius equal to 1mm is our desired reference contour, and the frequency of circling is 0.5Hz. Figure 6-16 shows the tracking trajectory of the positioning stage between 20 seconds and 30 seconds. The root mean square value of the tracking error shown in Fig. 6-17 is 23.1μm.

Fig. 6-16 Trajectory of circular contouring

Fig. 6-17 Contouring error of circular motion

6.4.4 Spiral motion

In one of the future applications of the proposed positioning stage, atomic force microscope (AFM), the spiral scanning is useful to replace the conventional raster

scanning in order to avoid the high acceleration at the sharp turn [41]. In the section, the experimental result of the spiral motion is shown to verify the capability of the proposed positioning stage serving as the scanner of AFM. Figure 6-18 shows the tracking trajectory. The reference spiral curve starts at the origin and then increase the radius with time. The velocity is designed as a constant to keep the motion smooth. The tracking error is shown in Fig. 6-19, and the root mean square value of the tracking error is 8μm.

Fig. 6-18 Trajectory of spiral contouring

Fig. 6-19 Contouring error of spiral motion

Chapter 7 Conclusions

Equation Chapter (Next) Section 1

In this thesis, we have first surveyed several kinds of research results relevant to the current precision positioning techniques, especially for the positioning utilizing flexure mechanism. Then, we have designed and implemented a precision positioning stage utilizing parallel flexure mechanism, electromagnetic actuator, and damper. This proposed system consists of frictionless flexure mechanism as the guidance, linear electromagnetic actuators to generate the desired 3-DOF motion, noncontact eddy current damper to suppress the vibration of the flexure structure. Three optical displacement sensors are used to obtain the present position and posture of the stage.

The positioning stage possesses smooth linear motion capability by involving only pure and linear magnetic forces.

Next, the dynamics of our 3-DOF positioning system has been thoroughly analyzed and its mathematical model with complete three DOFs has also been detailed derived. After that, an advanced adaptive sliding-mode controller is designed and implemented in a personal computer to regulate the three DOFs up to a precision level and to track a particular desired trajectory.

Finally, satisfactory performance of the precision positioning motion can be obtained in the actual experiments. The travel range of this proposed positioning stage is 3mm×3mm with the accuracy close to the limit of the sensor. Furthermore, based on the adaptive sliding-mode controller, the system’s response is also fast enough. Several experimental results show that the adaptive sliding-mode controller has satisfactory performance of the robustness when dealing with uncertainties and disturbances. The successful performances of the linear actuator and noncontact electromagnetic damper are also revealed by the experimental results. As a consequence, various goals which are set in the beginning of the thesis have been successfully fulfilled. To sum up, a novel 3-DOF high-precision compact positioning stage has been successfully brought to practice in this thesis.

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