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The piezoelectric actuators are used to investigate the active vibration control of flexible manipulators in this paper. Based on the assumed mode method, piezoelectric coupling model, and Hamilton’s principle, the dynamic equation of the single flexible manipulator (SFM) with surface bonded actuators is established. Then, a singular perturbation model consisted of a slow subsystem and a fast subsystem is formulated and used for designing the composite controller. The slow subsystem controller is designed by fuzzy sliding mode control method, and the linear quadratic regulator (LQR) optimal control method is used to design fast subsystem controller. Furthermore, the changing trends of natural frequencies along with the changes in the position of piezoelectric actuators are obtained through the ANSYS Workbench software, by which the optimal placement of actuators is determined. Finally, numerical simulations and experiments are presented. The results demonstrate that the method of optimal placement is feasible based on the maximal natural frequency, and the composite controller presented in this paper can not only realize the trajectory tracking of the SFM and has a good result on the vibration suppression.

In recent years, the robotic technology has been widely used in many areas, such as aerospace, industry, and medical treatment. Due to the characteristics of high speed, high precision, and high loading weight ratio, flexible structures have received great attention in the robot areas [

The dynamical model of flexible manipulators has the features of being time-varying, strong coupling, and highly nonlinear which brings insuperable difficulties of the single controller design and it is hard to get ideal control effect [

Moreover, experiments and simulations results have proved that the positions of the actuators have important influence on vibration suppression effect [

This paper aims to investigate the modelling and composite control of the SFM with piezoelectric actuators and realizes the trajectory tracking and vibration suppression of flexible manipulators based on the composite controller which is designed by singular perturbation theory. Furthermore, the optimal placement of piezoelectric actuators is done through the index of the maximal natural frequency. The paper is organized as follows: “Dynamic Model of the SFM” is illuminated in the following section. “Singular Perturbation Decomposition of Dynamic Equation” presents the reduction of dynamic equation. Subsequently, a compound controller is designed for trajectory tracking and vibration suppression of the SFM in “Design of the Composite Controller” of this paper. Then, the location optimization of piezoelectric actuators, simulation, and experiments results of the composite control method are given in “Optimal Placement and Composite Control of the SFM.” The paper is concluded with a brief summary in last section.

The structural diagram of the SFM in the horizontal plane studied in this paper is shown in Figure

Structural diagram of the SFM.

As shown in Figure

The velocity vectors of point

The kinetic energy of the SFM is written as

Based on Euler–Bernoulli theory, the potential energy is given by

As shown in Figure

Model of the SFM with surface bonded actuators.

Because the model is symmetrical, taking the upper part of the SFM with surface bonded actuators as an example to empirical analysis and the moment equilibrium equation about the neutral plane is given as [

The bending moment of the

The virtual work carried out by the torque

In accordance with the assumed mode method,

The governing equation of motion can be obtained through the application of the Hamilton’s principle

Since the inertia matrix

Assuming that

To derive the boundary layer correction,

Combining (

To derive the fast subsystem, introduce a fast time scale

Because the slow time scale and fast time scale are independent of each other, near boundary layer region

The control objective is to track the desired trajectory and suppress the vibration of the SFM. As shown in Figure

Structure diagram of the composite controller.

Assuming the desired trajectory as

Obviously, if the slow subsystem states stay on the sliding mode surface, it ensures that

A sliding mode controller has two phases; the first is called the reaching phase. During this phase the controller is guiding the system state onto the sliding surface. The second phase is called the sliding phase, during which the system state “slides” along the sliding surface [

Generic sliding surface.

The adaptive fuzzy sliding mode controller is illustrated in Figure

Adaptive fuzzy sliding mode controller.

Seven linguistic levels (NB, NM, NS, ZO, PS, PM, and PB) are used to represent the input domain and output domain with their membership values lying between 0 and 1 [

Membership functions of fuzzy controller ((a) linguistic variable

Table

Fuzzy inference rules of fuzzy controller.

| | ||||||
---|---|---|---|---|---|---|---|

NB | NM | NS | ZO | PS | PM | PB | |

| PB | PM | PS | ZO | PS | PM | PB |

| PM | PS | ZO | NS | ZO | PS | PM |

| PS | ZO | NS | NM | NS | ZO | PS |

| ZO | NS | NM | NB | NM | NS | ZO |

| PS | ZO | NS | NM | NS | ZO | PS |

| PM | PS | ZO | NS | ZO | PS | PM |

| PB | PM | PS | ZO | PS | PM | PB |

Surface of the fuzzy rules.

Define boundary layer variables as

Optimal control block diagram of the fast subsystem.

To damp out the flexible vibration and decrease the control effort, the quadratic objective function is chosen as

To verify the composite control strategy on the trajectory tracking and vibration suppression of the SFM, numerical simulations are conducted in this section. Owing to the lower order modes play leading role in the vibration of the SFM, only the first two order modes are considered, and the active control simulations are performed using the independent mode space control method. The structural parameters of the SFM and actuators are shown in Tables

Structural parameters of SFM.

Parameter | Value | Unit |
---|---|---|

Length | 0.400 | m |

Width | 0.035 | m |

Thickness | 0.0027 | m |

Density | 2.03 × 10^{3} | kg^{3} |

Elastic modulus | 25.24 | GPa |

Rotational inertia | 2.5 × 10^{−4} | kg⋅m^{2} |

Hub radius | 0.0175 | m |

Tip mass | 0.03 | kg |

Structural parameters of actuators.

Parameter | Value | Unit |
---|---|---|

Length | 0.04 | m |

Width | 0.01 | m |

Thickness | 0.0008 | m |

Density | 7.65 × 10^{3} | kg^{3} |

Elastic modulus | 57.3 | GPa |

The three-dimensional model of the SFM with surface bonded actuators is established by ANSYS Workbench software, and it is shown as Figure

Three-dimensional model of the SFM with surface bonded actuators.

Changing trends of natural frequencies along with the position of piezoelectric actuator ((a) first-order natural frequency; (b) second-order natural frequency).

The angular displacement trajectory of the SFM is shown as

The simulation curves of composite control for trajectory tracking and vibration suppression of the SFM are shown in Figure

Simulation curves of composite control for trajectory tracking and vibration suppression of the SFM ((a) time-varying control variable; (b) control torque; (c) control voltage; (d) tip angle; (e) first-order modal coordinate; (f) second-order modal coordinate; (g) tip deflection; (h) strain time history).

In order to better demonstrate the effectiveness of the composite controller, the experiments on composite control of the SFM are constructed. The overall experimental setup and the flowchart of composite control experiments are depicted in Figures

Photograph of the experimental apparatus.

Flowchart of the composite control experiments.

The comparison between the simulated and experimental results of the SFM is shown in Figure

Comparison between the simulated and experimental results of the SFM ((a) time-varying control variable; (b) control torque; (c) control voltage; (d) tip angle; (e) tip deflection; (f) strain trend).

This paper mainly discusses the modelling and composite control of the SFM with piezoelectric actuators. The dynamic model of the SFM with surface bonded actuators is described by means of the assumed mode method, piezoelectric coupling model, and Hamilton’s principle. Based on the singular perturbation method, the coupled dynamic equation is decomposed into slow (rigid) and fast (flexible) subsystems. Then, the composite controller for the slow subsystem and fast subsystem is achieved using the fuzzy sliding mode control and LQR technique. Furthermore, the optimal placement of actuators is determined by the index of natural frequencies of SFM.

By the analysis of numerical simulation, the optimal placement of actuators is feasible based on the maximal natural frequency. Meanwhile, the composite controller presented in this paper can not only realize the trajectory tracking of the SFM and can successfully suppress the simultaneous elastic vibration in movement of the SFM, and the closed-loop stability of the SFM is achieved. In addition, the structural strain history along the length of SFM demonstrates that the highest strain experienced occurs in the root of the SFM in the early stage and the strain is reduced gradually in the control of composite controller. Finally, the experiments demonstrate that the control effect of the composite controller reaches the expectation and the experiment results are consistent with simulation results, which further demonstrated the feasibility of the proposed composite control method.

The authors declare that there are no competing interests regarding the publication of this paper.

This research work was partially supported by the National Natural Science Foundation of China (no. 51305444 and no. 51307172), the Scientific and Technological Projects of Jiangsu Province (BY2014028-06), the Six Talent Peaks Project in Jiangsu Province (ZBZZ-041), the Postgraduate Cultivation and Innovation Project of Jiangsu Province (KYLX16_0523), and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).