System Identification and Adaptive Tracking Control for a 3-DOF LCD Glass-handling Robot

System Identification and Adaptive Tracking Control for a 3-DOF LCD Glass-handling Robot

Rong-Fong Fung* and Sin-Syun You

Department of Mechanical and Automation Engineering, National Kaohsiung First University of Science and Technology, Taiwan

(Received 30 November 2010; Accepted 14 June 2011; Published on line 1 September 2011)
*Corresponding author: rffung@ccms.nkfust.edu.tw
DOI: 10.5875/ausmt.v1i1.56

Abstract: In this paper, a physical model of a 3 Degree-Of-Freedom (DOF) LCD glass-handling robot driven by three Permanent Magnet Synchronous Motors (PMSMs) is established. The completely mathematical model includes two parts: the electrical equations of the three PMSMs and mechanical equations of the system in three directions. In numerical simulations, MATLAB software was employed to solve the mathematical models, while Real-coded Genetic Algorithm (RGA) was used to identify the system parameters which include those not easily measured from experiments. Numerical simulations and experimental results were compared to show the identified parameters of the mechatronic system. In the adaptive control design, Lyapunov function was employed and the external loading and unknown disturbances were considered. Two different commands of angular-displacement curve of the trapezoidal and double S speed profiles were used as the tracking trajectories for comparison. It was concluded that the double S speed profile has smaller tracking errors than those of the trapezoidal speed profile.

Keywords: 3-DOF LCD Glass-handling robot; double S speed profile; Lyapunov function; permanent magnet synchronous motors (PMSMs); Real-coded Genetic Algorithm (RGA).

1. Introduction

In recent years, flat panel displays such as liquid crystal display (LCD) and wafer have replaced the cathode ray tube (CRT) display due to their small volume, lighter weight and low power consumption. As such, the electronics industry has produced a great deal of products using this technology, like LCD-TVs, monitors, mobile displays, etc. For the LCD-TV, many LCD manufacturers have made the large size and thin film mother glass in order to meet consumers' demands. For transport of mother glasses among these equipments, the LCD glass-handling robot, stocker crane, conveyor, and automatically guided vehicle are used, with the LCD glass-handling robot as being the most essential transport device. Many dynamic models [1, 2] of mechanical systems were formulated in industrial applications. For example, Hamilton's principle was employed to derive Lagrange's equations for a LCD glass-handling robot [1] driven by a Permanent Magnet Synchronous Motor (PMSM). Fung et al. [2] developed the dynamic models for motor currents, angular speeds and input voltages of the PMSM driving the XY precision positioning table.

System parameters could be identified by the Real-coded Genetic Algorithm (RGA). Genetic algorithm was first defined by John Holland in 1975 and is a search process based on natural selection now being used as a tool for searching in large, poorly understood spaces that arise in many application areas of science and engineering. RGA is an optimization searching algorithm which simulates an evolution mechanism on a computer-based platform in conjunction with natural selection and genetic mechanism. The chromosomes are expressed by vectors and each element is called a gene. The initial real-valued genes in the chromosomes are obtained through generating a sequence of real-valued variable by a randomly limited range. All chromosomes are composed of a population and are evaluated according to the pre-given evaluation index and given fitness values.

However, it is more natural to represent the genes directly as real numbers, because the method is calculated by real code and has high calculation efficiency. In uses where RGA is employed to find the identified parameters of the system, Fung and Lin [3] identified a novel 6 Degree-Of-Freedom (DOF) precision positioning table, which is assembled by two different 3-DOF precision positioning tables. Tian and Collins [4] proposed a novel trajectory planning method for a robot manipulator whose workspace includes several obstacles.

For the motion trajectory control, Hsu et al. [5] designed the trapezoidal and double S trajectory profiles for the electric injection molding machine during the closing operation, and the effectiveness of the Adaptive Tracking Control (ATC) was demonstrated by numerical simulations and experiments. Biagiotti and Melchiorri [6] developed the trajectory planning for automatic machines and robots. In a plant model, the system performance can be easily deteriorated by external disturbances, nonlinearities, and some parametric uncertainties due to unavoidable modeling errors. The adaptive control is able to eliminate these external disturbances and nonlinearities [7-13]. In these studies, Hu et al. [7] designed a modified version of the controller based on adaptive variable structure output feedback control, which satisfies the global reaching condition of sliding mode and adapts the unknown upper bounds of the lumped uncertainties and perturbations. Shanhnazi and Akbarzadeh-T [8] developed the PI-adaptive fuzzy control architecture for a class of uncertain nonlinear systems to provide robustness in the presence of large and fast but bounded uncertainties and disturbances. Chuang et al. [9] used the inertia-related Lyapunov function to design an adaptive controller for the motor-toggle servo mechanism. Lin et al. [10] formulated a robust controller with the adaptive uncertainty observer to increase the robustness of the motor drive system. Lee et al. [11] studied an adaptive control scheme with a pre-specified HN property for the tracking control of the permanent magnet synchronous motor (PMSM) drives. Fung and Lin [12] based on the state-space model; the adaptive backstepping control was proposed to track contour trajectories. Fung et al. [13] developed the adaptive backstepping fuzzy control method to compensate for the piezoelectric hysteresis and achieve motion tracking control.

This present article follows the mathematical modeling of the dynamic equations [1, 2], which include electrical and mechanical equations, and extends the 1-DOF system [1] to the 3-DOF by adding the yaw and vertical directions for real industry applications. Kinematic equations are used to describe the end point of the arm of the 3-DOF LCD glass-handling robot. The parameters are identified by the RGA method. In controller design, two tracking curves of trapezoidal and double S speed profiles are employed for the commands of the PMSMs. In the adaptive control, we consider the dynamic performance, robustness of parameter variations, trajectory-tracking errors, external loading and unknown disturbances compensation.

In this paper, the complete mechatronic formulation including the electrical and mechanical equations is performed. In control performance, it is concluded that using the double S speed profile could have a better working cycle than that of the trapezoidal speed profile. The proposed adaptive controller can compensate for the external loading and unknown disturbances, which are considered in the nonlinear time-variant mechatronic system.

2. Systems Formulation

Figure 1 shows the 3-DOF LCD glass-handling robot driven by three PMSMs. Figure 1(a) shows the side view, Figure 1(b) shows the three PMSMs, and Figure 1(c) shows the three arms from the top view. In order to complete the dynamic equations, we used both the electrical and mechanical equations to describe them.

Figure 1. Three-DOF LCD glass-handling robot. (a) Side view;(b) The three PMSMs; (c) The three arms from top view.

2.1 Electrical equations

The desirable features of the PMSM are its compact structure, high air-gap flux density, high power density, high torque-to-inertia ratio, and high torque capability. The PMSM drive plays a vitally important role in motion-control applications and in the low-to-medium power ranges. To achieve fast four-quadrant operation and smooth starting and acceleration, the field-oriented control is used in the design of the PMSM drive. However, its control performance is still influenced by the uncertainties of the plant, including unpredictable plant parameter variations, external loading disturbances, uncertainties and nonlinear dynamics.

The PMSM is used to drive the timing belts, screws and gears, and then to drive the mechanical system. The electrical models of the three PMSMs can be described in the rotating coordinates as follows:

(1)
(2)

where

(3)
(4)

The subscripts d and q represent the parameters on the d - and q -axes, and the subscript i=1, 2 and 3 represent the parameter in PMSM1, 2 and 3, respectively. Vdi and Vqi are the stator voltages, idi and iqi are the stator currents, Ldi and Lqi are inductances, λdi and λqi are the stator flux linkages, Rsi and ωsi are the stator resistances and inverter frequencies, respectively. In Equations (3) and (4), Ifdi and Lmdi are the equivalent d-axis magnetizing currents and inductances, respectively.

The electromagnetic torques are

(5)

where p is the number of pole pairs. Equations for the motor dynamics are

(6)

where τmi, Bmi, ωri, ωri and Jmi are the load torques, damping coefficients, angular speeds, angular accelerations, moments of inertia of the rotors, respectively. In addition, the relations between the inverter frequencies (ωsi) and rotor angular speeds (ωri) can be expressed as

(7)

The basic principle in controlling the PMSM drive is based on the field orientation. The flux position in the d-q coordinate can be determined by the shaft-position sensor because the magnetic flux generated from the rotor permanent magnet is fixed in relation to the rotor shaft position. In Equation (5), if idi=0 the d-axis flux linkages λdi are fixed since Lmdi and Ifdi are constants for the surface-mounted PMSMs, and the electromagnetic torques τei of Equation (5) are then proportional to iqi, which are determined by the closed-loop control.

The rotor flux is produced in the d-axis only, and the current vector is generated in the q-axis for the field-oriented control. The generated motor torque is linearly proportional to the q-axis currents iqi, when the d-axis rotor flux linkages λdi are constants, and the maximum torques per ampere will be achieved. With the implementation of field-oriented control, the electric torques of Equation (5) can be simplified as

(8)
(9)

where kti are the motor torque constants.

Finally, the load torques [1, 2] can be obtained by substituting Equation (8) into Equation (6) as follows:

(10)

2.2 Mechanical equations

In the dynamic formulation, three PMSMs are independent and the timing belts are assumed rigid. The PMSM1 is used to rotate the robot in the yaw (z) direction, and the dynamic equation is

(11)

where , , are the angular displacement, speed and acceleration, respectively, J1 is the moment of inertia of the rotor, B1 is the damping coefficient, K1 is the equivalent stiffness, and kt1 is the motor torque constant of the PMSM1.

For the PMSM2, the motor is used to drive the robot, and the dynamic equation could be derived from previously published research [1] as follows:

(12)

where

, , are the angular displacement, speed and acceleration of the PMSM2, respectively. n2 is the radius ratio of the two gears that belted together in Arm 1 and it is assumed that the radius ratio in Arm 2 is also n2. Jm2 is the motor moment inertia of the roto. Bm2 is the motor damping coefficient. μ is the nonlinear damping coefficient. I1I3 are the moments of inertia of Arms 1–3 respectively. I4I6 are the moments of inertia of Gear 2-4 respectively. m2 and m3 are the masses of Arms 2 and 3 respectively. m4m6 are the masses of Gears 2–4 respectively. b is the moment Arm of Arm 2 with respect to the joint center of Arm 1 and Arm 2.

The PMSM3 is used to translate the robot in the vertical direction via a screw with a lead . The dynamic equation is

(13)

where , , are the angular displacement, speed and acceleration, respectively, and τ30 is the torque due to gravity. The vertical displacement Z can be obtained as follows:

(14)

2.3 The combined equation

The electrical equation (1) for the three PMSMs and the dynamic equations (11-13) for the mechanical system for the 3-DOF LCD glass-handling robot can be combined into a matrix form as follows:

(15)
(16)

where

E, A, B and C are the constant matrices of the system parameters. x(t) and y(t) are the state and output variable vectors, respectively. u(t) is the control voltage vector, ƒ(t, θ2) is the nonlinear part of the dynamic equation from PMSM2, and DL is the torque due to gravity. The parameters in Equation (15) are to be identified in the following section. Equations (15) and (16) are used to design the adaptive control in Section 5.

3. Identification based on RGA — Fitness function

This study adopts the RGA to identify the system parameters. Through the RGA identification, all unknown parameters of the electrical and mechanical equations were identified. The detailed parameters are listed in Table 1. There are seven parameters for PMSM1, nine parameters for PMSM2 and eight parameters for PMSM3. Therefore, a total of 24 parameters in Equation (15) are to be identified simultaneously. The procedure of identification based on the RGA is the same as [1-3].

Since Equation (15) is a set of ordinary differential equations, the fitness function [1-3] must include total summation of the errors during the operation time and is designed as follows:

(17)
(18)

where N is the total number of sampling times in numerical simulations, the superscripts (j) and *(j) represent the experimental and simulation values at the jth sampling time, respectively. Ej represents the errors of the angular displacements for PMSMs.

4. Motion Profile Design

4.1 Kinematic equation

In order to describe the kinematic equation of the 3-DOF LCD glass-handling robot, the position vector r3 of Arm 3 in Figure 5(a) should be defined. This position vector indicates the end point of the robot, and is the resultant position due to three angular departments (θi, i = 1, 2, 3), which rotate by three PMSMs.

Table 1. System parameters of the 3-DOF LCD glass-handling robot.
Electrical
equations
Feasible
domains
Identified
values
Mechanical
equations
Feasible
domains
Identified
values
Rs1(Ω) 0.017-0.0185 0.0180 j1(kg-m2) 0.0001-0.001 0.0008
Rs2(Ω) 0.017-0.0185 0.0182 j02(kg-m2) 0.001-0.1 0.0030
Rs3(Ω) 0.017-0.0185 0.0177 j1(kg-m12) 0.001-0.1 0.0400
Lq1(H) 0.016-0.02 0.1785 jm2(kg-m2) 0.0001-0.01 0.00012
Lq2(H) 0.016-0.02 0.1860 j3(kg-m2) 0.001-0.05 0.0100
Lq3(H) 0.016-0.02 0.1920 B1(kgm-rad) 0.0001-0.05 0.0170
λd1 0.01-0.03 0.0184 Bm2(kgm-rad) 0.0001-0.05 0.0002
λd2 0.01-0.03 0.0210 B3(kgm-rad) 0.0005-0.01 0.0081
λd3 0.01-0.03 0.0205 k1(kgm2-rad) 0.001-0.05 0.0012
kt1(kgm/A) 0.88-0.93 0.8950 k3(kgm2-rad) 0.001-0.05 0.0122
kt2(kgm/A) 0.88-0.93 0.9014 μ 0.1-5 2
kt3(kgm/A) 0.88-0.93 0.9220 m3 0.1-20 3.7

4.2 Trapezoidal speed profile

A very common method [4, 5] to obtain trajectories with a continuous speed profile is to use linear motions with parabolic blends, where the trajectories can be characterized by the typical trapezoidal speed profiles. In this case, a motion is defined by assigning the initial and final time instants as t0 and t1 and their conditions on displacement and speed at t0 and t1. From a mathematical point of view, the problem is then to find a function θ = θ(t), t∈[t0, t1] where θ(t) is the profile of angular displacements such that the given conditions are satisfied.

The trapezoidal trajectories are divided into three parts: acceleration phase, constant speed phase and deceleration phase. The detailed mathematical descriptions of displacements and speeds can be seen in [5, 6].

4.3 Double S speed profile

A trapezoidal speed profile presents a discontinuous acceleration. For this reason, this kind trajectory may generate efforts and stresses on mechanical systems and may result in detrimental or generate undesired vibration effects. Therefore, a smoother motion profile can be defined, for example by adopting a continuous, linear piece-wise, speed profile. In this manner, the resulting speed profile is composed by linear segments connected by parabolic blends. The shape of the speed profile is the reason of the name double S for this trajectory [6], known also as bell trajectory or seven-segment trajectory. It is composed by several tracts with constant jerk, and the detailed mathematical description of displacements and speeds can be seen in [6].

5. Adaptive Controller Design

In order to achieve good tracking control of the robot actuated by three PMSMs, the adaptive control [7] is established here, with the control variable defined as follows:

(19)

where z(t) = [ω1 ω2 ω3]T is a vector of control variables and GR3×9. Equation (15) can be rewritten as follows:

(20)

where a = E-1A, b = E-1B are parameter matrices, N(t, θ2)=E-1ƒ(t, θ2) , and d(t)=E-1(DL + Ddis(t)) is the non-constant disturbance, where the unknown disturbance Ddis(t) is added. We set Δa and Δb are the parameter variations of a and b, respectively, and illustrate d(t) as follows:

(21)

Following [8], we define the PI structure via the following two assumptions:

Assumption 1: We define d(t) as bounded by an unknown constant D,

(22)

Assumption 2: We define PI structure of d(t) as follows:

(23)
(24)

where dp, d1, dDR3×1 are constant vectors.

The method proposed in this paper is motivated by the adaptive control, which is defined by the adaptive law via Lyapunov function. Let us first follow [9-13] and consider the first-order liner time-invariant plant as shown in Equation (24).

We design the adaptive controller as follows:

(25)

where u1(t) is used to control the system state, u2(t) is used to eliminate the nonlinear part of PMSM2 and u3(t) is used to eliminate the external loading and unknown disturbance.

The controllers are designed as follows:

(26)
(27)
(28)

where λR3×3, IαR9×1, IdR3×1 are constant vectors, , and are constant matrices, and are the estimations of the unknown parameters α, and dp, d1, dD, respectively. To develop the adaptive laws for α and dp, d1, dD, their parameter errors: , , and are needed.

Let and , i=1, 2, 3, be the errors of angular displacements and angular speeds, respectively, then the error equations (si, i=1, 2, 3) are defined as follows:

(29)

where λ = diag[λ1 λ2 λ3] is a positive matrix, and .

The error dynamics [9] of the system equation (20) is

(30)

The global asymptotic stability of Equation (20) can be guaranteed by using the adaptive laws [10-13] and Lyapunov function defined as

(31)

By taking the time derivative of Lyapunov function (31), we have

(32)

where , γ1,γ2,γ3,γ4R3×3 are constant matrices, γn1 and γn2R are constant values. hR3×3 is a constant matrix such that h = hT > 0 and satisfies

(33)

for any chosen constant matrix QR3×3 such that Q = QT > 0. From Equation (32), we have by using the fact that , , and . The adaptive laws are as follows:

(34)
(35)
(36)
(37)

6. Numerical Simulations and Experimental Results

6.1 Numerical simulations

In order to obtain numerical simulations, Runge-Kutta method [1] with tolerance error 10-6 was employed to simulate the variations of angular displacements θ1, θ2 and θ3. By using the RGA method to identify system parameters as shown in Table 1, and substituting the assigned and identified parameters into dynamic equations (15, 16), we can obtain the compared numerical simulations (not shown here) by the commercial MATLAB software. It was found that the response errors are below ±0.5 %, ±1.5 %, and 1.0 % for PMSM1, PMSM2 and PMSM3, respectively. The RGA method was thusly validated as feasible for parameter identification of the 3-DOF LCD glass-handling robot.

By using these identified parameters as shown in Table 1, the dynamic responses by numerical simulation can be obtained. The comparisons between the experimental and identified results of the angular displacements are shown in Figure 2. The input voltage as shown in Figure 2(a) is 0.25sin0.5t + 0.05sin5t with two different frequencies to drive the PMSMs. For the PMSM drive, we set idi=0, i=1, 2, 3 thus the d-axis flux linkages λdi, i=1, 2, 3 are fixed. The rotor flux is produced in the d-axis only and the current vector is generated in the q-axis for the field-oriented control.

Figure 2. Comparisons between the numerical simulations and experimental results for the three PMSMs. (a) Input voltages. (b-d) Angular displacements. (e-g) Angular-displacement errors.

In numerical simulations of the proposed adaptive controllers, Runge–Kutta method [1] with tolerance error 10-6 was also employed with MATLAB software. From Equation (25), it is known that the adaptive control needs 17 gain values of , , , and in Vqi. The gain values are adjusted by adaptive laws, and the block diagram of the adaptive control is shown in Figure 3, where the constant vectors and values are

γ1 = [100×10-4  5×10-4  3×10-4],
γ2 = [10×10-4  50×10-4  70×10-4],
γ3 = [25×10-4  20×10-4  75×10-4],
γ4 = [2×10-4  2×10-4  2×10-4],
η = diag(h1,h2,h3) = diag(10,15,5),
λ = diag(0.85,0.7,0.75), and D = 100.

By setting these values into , , , , λ and h of adaptive laws of Equations (34-37), we obtain the adaptive control of Equation (25).

Figure 3. Block diagram of the proposed adaptive control.

6.2 Experimental setup

An experimental setup of the 3-DOF LCD glass-handling robot is shown in Figure 4, where three PMSMs were used as actuators. The Japanese-made PMSM with MITSUBISHI HC-KFS series, and the specifications include a rated output of 400 W and rated rotation speed of 3000 rpm. The waveform of the input voltage is generated by LabVIEW software, which is a Windows supported graphical programming language. The D/A converter (NI PCI-6052) with a resolution of 12 bits is used to transform the voltage waveform to the PMSM drive with MITSUBISHI HC-KFS series, and reads the voltage of the potential meter. The encoder (MR-J2S) signal is transformed from the A/D converter to the motion card (NI PCI-7344), which will read the digital single of PMSM's encoders. By using LabVIEW software, the human machine interface (HMI) is also developed in Figure 4. The related calculation of the proposed controller is handled within the HMI.

Figure 4. Experimental setup.

In order to measure the angular displacements of the PMSMs, the interface device is implemented by a motion control card (NI PCI-7344) and an A/D card (NI PCI-6052), which can measure angles with a sampling time Δt= 0.005 sec. In order for system identification, we give the input voltages 0.25sin0.5t + 0.05sin5t to drive PMSMs, and use the PC to store angular displacements of the PMSMs.

The trapezoidal and double S speed profiles of angular displacements [6] are used to validate the proposed control method. They are designed for all three PMSMs, but with different time durations. Figure 5(a) shows the three arms in the X-Y coordinates, while Figure 5(b) shows three commands for the corresponding three PMSMs, and is the complete working cycle for duration time of nine seconds. There are eight steps included in Figure 5(a). The first graphic of Figure 5(b) is for PMSM 2 rotating the three arms and the position vector r3 moves on the x axis. The second and third graphics of Figure 5(b) are for the PMSMs rotating in the yaw direction and translating in the vertical direction, respectively.

Figure 5. Complete working cycle of the 3-DOF glass-handing robot. (a) Physical model of a LCD glass-handing robot. (b) Angular-displacement commands for the three PMSMs.
Figure 6. Torque disturbances in the numerical simulations. (a) Assigning D1 (100 Nm) to PMSM1 and PMSM2. (b) Assigning D2 (3.3 Nm) to PMSM3.

6.3 Comparisons between numerical simulations and experimental results

In order for comparisons, the results are performed for two working cycles, and each cycle takes nine seconds. In numerical simulations, Figure 6 shows the external loading and unknown disturbances added to the system. Figure 6(a) shows the external loading resulting in the jumping for the PMSM1 and PMSM2. Figure 6(b) shows the effect of external loading in the step of the PMSM3. Two examples of jumping torques are shown in the PMSM1 and PMSM2, but their time durations are much shorter than with the PMSM3.

Figure 7 shows the comparison results of numerical simulations between the trapezoidal and double S speed profiles. Figures 7(a-c) show the control voltages of the three PMSMs, and it is seen that the trapezoidal speed profile has more jumping phenomenon than that of the double S speed profile. Figures 7(d-f) show the angular displacements of the three PMSMs, and Figures 7(g-i) show their angular speeds. In these figures it is seen that the designed controller can be used to control the robot system, and the response results of θ0, θ1 and θ2 are well in agreement with the commands in the first working cycle, and they are convergent in the second working cycle.

Figure 7. Comparisons of numerical simulations between the trapezoidal and double S speed profiles for the three PMSMs. (a-c) Control voltages. (d-f) Angular displacements. (g-i) Angular speeds. (T: trapezoidal speed profile; S: double S speed profile)

Moreover, Figure 8 shows the comparison of response errors between the trapezoidal and double S speed profiles. Figures 8(a-c) show the angular-displacement errors of the three PMSMs, and it is seen that the double S speed profile is more suitable than the trapezoidal speed profile. Figures 8(d-f) show the angular-speed errors of the PMSMs, while Figures 8(g-i) show the position error s in the x-, y- and z- axis for the Arm 3.

Figure 8. Comparisons of numerical simulations between the trapezoidal and double S speed profiles. (a-c) Angular-displacement errors of PMSMs. (d-f) Angular-speed errors of PMSMs. (g-i) The position errors in the x-axis, y-axis and z-axis of Arm3, respectively. (T: trapezoidal speed profile; S: double S speed profile).

Figures 9(a-o) show comparisons between numerical simulations and experimental results of the proposed adaptive controller in controlling the 3-DOF LCD glass-handling robot system. In this case, external loading and unknown disturbance are not considered in the numerical simulations, but are considered in the experimental setup. In experiments, the constant vectors and constant values are taken as follows:

γ1 = [100×10-4  5×10-4  3×10-4],
γ2 = [20×10-4  80×10-4  1×10-4],
γ3 = [50×10-4  60×10-4  5×10-4],
γ4 = [10×10-4  40×10-4  1×10-4],
h = diag(h1,h2,h3) = diag(10,15,5),
λ = diag(0.85,0.7,0.75), and D = 100.

By setting these values into , , , , λ and h of adaptive laws of Equations (34-37), we obtain the adaptive control of Equation (25).

Figure 3. Block diagram of the proposed adaptive control.

Figures 9(a-c) show control voltages and it is seen that the experimental values are larger than those of numerical simulations. Figures 9(d-f) and Figures 9(j-l) show the angular displacements and their corresponding errors, respectively, in numerical simulations and experimental results. Figures 9(g-i) and Figures 9(m-o) show the angular speed and their corresponding errors, respectively, and it is seen that the experimental errors are larger than those of numerical simulations. It is noted that only the angular displacements are obtained by encoders, and the angular speeds in experiments are obtained by time differentiating the angular displacements through numerical manipulations.

Figure 9. Comparisons between the numerical simulations and experimental results for the three PMSMs. (a-c) Control voltages. (d-f) Angular displacements. (g-i) Angular-displacement errors. (j-l) Angular speeds. (m-o) Angular-speed errors.

Conclusion

One main objective for this study is to formulate the dynamic mechatronic equations including the electrical and mechanical equations. The RGA method was employed to identify the parameters of a 3-DOF LCD glass-handling robot, and the errors are all below 1.5 %. From these results it is validated that the RGA method is feasible for the mechatronic system. The proposed adaptive control scheme was presented with unknown parametric uncertainties and disturbances, and then used to control the robot system to track the designed trajectories for comparisons in numerical simulation and experimental results. From these comparisons, since the trapezoidal speed command generates large jumping in the turning points, it was determined that the double S speed trajectory is therefore more suitable for the robot system.

Acknowledgment

The financial support from National Science Council of the Republic of China with contract number NSC-97-2221-E-327-016-MY3 is gratefully acknowledged.

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