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The problem of the elastic vibration control for a translational flexible manipulator system (TFMS) under variable load conditions is studied. The input shaper can effectively filter out the vibration excitation components for the flexible manipulator in the driving signals, but the adaptability and rapidity of the conventional input shaper are poor because it is essentially an open-loop control mode and there are time-lag links inevitably. Thus, by combining the state feedback with the input shaping, a master-slave integrated controller of the TFMS is proposed. Moreover, in order to solve the time-lag effect of the conventional input shaper, based on the optimal algorithm, a two-mode vibration cascade shaper for the TFMS is designed. Then, under variable load conditions, the control effects of the conventional input shapers, the two-mode vibration cascade shaper, and the combination of the state feedback integral controller (SFIC) with the above shapers are investigated. The results show that the designed master-slave integrated controller has high robustness under variable load conditions and takes good account of the requirements of system response time and overshoot for achieving the goal of nonovershoot under fast response speed. Simulation experiment results verify the effectiveness of the designed controller.

The efficient encapsulation of the integrated circuit (IC) board depends on the precision of the SMT assembly devices. On the SMT assembly line, the packaging equipment is the device whose actuators are mainly assembled by the mechanical manipulator [

The existing high-precision and high-dynamic response manipulators are mostly driven by the permanent magnet servo system. The translational flexible manipulator system (TFMS) driven by the permanent magnet motor is a complex electromechanical system with a multimodule structure and multiphysical process [

Fortunately, the full-state feedback controller can significantly improve the system dynamic characteristics by arbitrarily configuring the system closed-loop poles. On the basis of a zero set concept, a robust state feedback controller was presented for the STATCOM state feedback design [

The input shaping method is a typical feedforward control method. Through the convolution of the system input with a series of pulse signals, the new system input control signal, among which the frequency component that can arouse the system oscillation is filtered, can be generated to drive the original system. With the exact boundary conditions considered, the input shaping method was combined with the PD controller to suppress the residual vibration of a flexible-link parallel manipulator [

Above all, for the purpose of improving the robustness, rapidity, and accuracy of the vibration controller under variable load conditions, the combination mode of the state feedback controller and the input shaper is investigated to constitute the master-slave integrated controller for the elastic vibration of the TFMS. In the specific design process, with the inherent static difference considered, the state feedback controller is integrated with the integral controller to form the state feedback integral controller (SFIC). Moreover, based on the optimal algorithm and the cascade method, the two-mode vibration cascade shaper of the TFMS is designed, which can effectively relieve the time-delay effect of the conventional input shaper. The structure of this paper is organized as follows: Section

The motion diagram of the TFMS is shown in Figure _{y}(_{t} indicates the quality of terminal load, and

Motion diagram of the TFMS.

Taking the center of the base as the coordinate origin, the transverse absolute coordinates of the

The kinetic energy of the TFMS includes the kinetic energy of the base, the kinetic energy of the flexible manipulator, and the kinetic energy of the terminal load, which can be shown as_{b} is the mass of the base and

The potential energy of the TFMS mainly considers the elastic potential energy generated by the deformation of the flexible manipulator, which can be represented as

The external force of the TFMS mainly includes the driving force transmitted by the AC servomotor through the ball screw pair and the friction resistance between the base and the moving guide rail. In addition, the structural damping force of the flexible manipulator itself also needs to be considered. The direction of the driving force is along the motion direction of the base, while the directions of the friction and the structural damping force are opposite. Then, according to the virtual work principle [_{f} is the friction coefficient between the base and the moving guide rail, and _{s} indicates the structural damping of the flexible manipulator itself.

Based on Hamilton’s principle [

Substitution equations (

Equations (

Because the root of the flexible manipulator is rigidly connected with the base, the boundary condition at the root is a fixed constraint: the deflection and the rotation angle are both 0. On the contrary, the end of the flexible manipulator is regarded as a free boundary condition: the bending moment and the shear force are both 0. Thus, the mode shape function of the flexible manipulator can be further shown as [

With equation (

Substituting equations (_{si} is the structural damping of the

Through multiplying both sides of equation (

According to the existing research, on the flexible manipulator, whose length is much larger than its cross-sectional area, the first several low-order modes play a leading role in the system vibration responses. And taking the first two order modes can completely meet the control requirements for the TFMS [

Based on the system controllability and observability theory, one can obtain that the state variables of the TFMS are completely controllable and observable [_{r}(_{z} is the gain matrix of the state feedback controller, _{i} is the coefficient of the integral controller, and

State feedback integral controller of the TFMS.

The error vector of the desired displacement at the end of the TFMS is defined as

Then, the output control quantity of the SFIC can be expressed as

In order to prove that the designed SFIC can drive the TFMS to a specified position without static error, the end positioning error of the flexible manipulator is added as a new state variable. And the augmented system can be shown as

Equation (

According to equation (

Through the above analysis, by designing a suitable state feedback control law for the augmented TFMS, the purpose of positioning the flexible manipulator to a specified position without deviation is achieved.

Based on the pole assignment method, the control parameters of the SFIC are designed. Because the first two modes of the transverse vibration of the flexible manipulator are considered, there are 7 poles in the augmented TFMS. Moreover, owing to the fact that the vibration frequency and damping of the transverse vibration mode coordinates are much smaller than the vibration frequency and damping of the base, the response of the augmented TFMS is mainly determined by the two pairs of poles corresponding to the vibration mode coordinates, and the other three poles are fast poles. According to equation (

The expected poles for the augmented TFMS are assumed as

In the augmented TFMS, the feedback gain matrix can be expressed as

Finally, according to Ackermann’s formula, the gains of the SFIC can be expressed as

Because of the time-delay link, the conventional input shaper will significantly affect the response speed of the TFMS. Thus, based on the input shaping method and the optimal theory, the two-mode vibration cascade shaper of the TFMS is designed for reducing the time-lag effect introduced by the input shaper [

The two-mode vibration cascade shaper is cascaded by the first-mode vibration shaper and the second-mode vibration shaper of the TFMS. Moreover, the first two modes of vibration shapers for the TFMS are designed separately, and the first mode is illustrated as an example. The system state variables in equation (

According to Duhamel’s integral, the time-domain expression of equation (

Combined with the requirement of the terminal positioning error, the quadratic objective function of the TFMS is set as_{m} is the weighted matrix.

To the first-mode vibration of the flexible manipulator, the number of pulses of the first-mode vibration shaper is set as 2 and the amplitude and the time lag of the first pulse are set as

Based on the optimal theory, in order to minimize the objective function, two conditions,

According to the amplitude and time lag of the first pulse in the first-mode vibration shaper, by combining with equation (

The amplitude of the corresponding pulse in equation (

On the contrary, it is obvious that the zero points of equation (

Similarly, the control parameters of the second-mode vibration shaper for the flexible manipulator can be determined. According to equation (

Based on the cascade method, the two-mode vibration cascade shaper of the TFMS can be deduced as

If the two-mode vibration cascade shaper is set in the closed loop of the SFIC, the time-lag link will affect the stability of the closed-loop system. Thus, in the designed master-slave integrated controller, the two-mode vibration cascade shaper is placed outside the closed loop of the SFIC, which is illustrated in Figure

Elastic vibration integrated controller of the TFMS.

The design process of the master-slave integrated controller for the TFMS is as follows:

Step 1: with the angular frequencies and damping ratios of the first two modes of the flexible manipulator combined, the system expected poles are set up. And according to equation (

Step 2: among the configured system poles, two pairs of poles that play a leading role in the transverse vibration of the flexible manipulator are extracted.

Step 3: the angular frequencies and damping ratios of the closed-loop system are calculated.

Step 4: the first-mode vibration shaper and the second-mode vibration shaper of the TFMS are designed by equation (

Step 5: the two-mode vibration cascade shaper of the TFMS is determined by equation (

Step 6: the master-slave integrated controller of the TFMS is constructed by the combination mode which is shown in Figure

In this section, the master-slave integrated controller is verified by simulation experiment analysis. The physical parameters of the flexible manipulator are set as ^{3}, ^{2}. The simulation platform is constructed by MATLAB 2015b/Simulink.

Firstly, the movement for the base of the TFMS is set as trapezoidal velocity movement and the specific motion parameters are set as follows: a constant acceleration motion of 1250 mm/s^{2} during ^{2} during

Positioning base velocity and displacement curves before and after the effect of the two-mode vibration cascade shaper: (a) velocity curve; (b) displacement curve.

It is seen from Figure

Figure

Control effect of the two-mode vibration cascade shaper on the elastic vibration of the flexible manipulator: (a) comparison of control effect before and after shaping; (b) control effect under different terminal loads.

According to the natural frequencies and damping ratios for the first two modes of the TFMS, the closed-loop poles of the augmented TFMS are configured. With the system response speed considered, the natural frequencies and damping ratios, which correspond to the two pairs of the system dominant poles, are set as 10.00 rad/s, 0.30 and 25.00 rad/s, 0.20. Then, the closed-loop poles of the augmented TFMS can be calculated as

Furthermore, the parameters of the two-mode vibration cascade shaper of the TFMS can be calculated as

Similarly, for the TFMS, in accordance with [

The target displacement of the base for the TFMS is set as 200 mm, and the control target is to locate the base to the designated position by suppressing the elastic vibration of the flexible manipulator quickly. The control effects of the above shapers are shown in Figure

Control effect of the SFIC with different input shapers combined on the TFMS: (a) control effect of the base displacement; (b) control effect of the elastic vibration.

Similarly, as is shown in Figure

Figure

Control effect of the master-slave integrated controller under different terminal loads: (a) control effect of the base displacement; (b) control effect of the tip elastic vibration.

Above all, through combining the two-mode vibration cascade shaper and the SFIC of the TFMS, the robustness of the two-mode vibration cascade shaper is improved and the contradictions of system rapidity and system overshoot for the SFIC can be effectively alleviated. The designed master-slave integrated controller can achieve the synchronous optimization of the system rapidity and accuracy.

With the time-lag filtering effect of the input shaper and the configuration capability of the state feedback for the closed-loop system poles considered, the SFIC and the two-mode vibration cascade shaper are combined to constitute the master-slave integrated controller for the elastic vibration of the TFMS for improving the robustness, the rapidity, and the accuracy of the system under variable load conditions. The following results are drawn:

Based on the input shaping method and the cascade method, the designed two-mode vibration cascade shaper can effectively suppress the excitation of the motor output to the mode vibration of the flexible manipulator, but it is less robust when used alone.

Through comparing the control results of the master-slave integrated controller and the conventional input shapers with the SFIC, it is seen that the response speed of the two-mode vibration cascade shaper is better than that of the conventional input shapers and the time-lag effect of the conventional input shapers is effectively solved. Furthermore, the system response time and overshoot of the SFIC can be well taken into account to achieve a faster response speed without overshoot.

Under variable load conditions, the master-slave integrated controller can effectively control the elastic vibration of the TFMS and realize the complementary advantages and disadvantages of the SFIC and the two-mode vibration cascade shaper in the elastic vibration suppression of the TFMS.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This research work was supported by the National Natural Science Foundation of China (No. 51805001), in part by the Anhui Provincial Natural Science Foundation (Nos. 1908085QE193 and 1808085QE137), and in part by the Research Starting Fund Project for Introduced Talents of Anhui Polytechnic University (No. 2018YQQ005).