A rapid stabilization method of the flexible inverted pendulum based on constrained boundary circumferential motion

https://doi.org/10.1016/j.ymssp.2022.109895Get rights and content

Highlights

  • A new method to control the circumferential motion of the boundary is proposed.

  • The theory of suppression is derived from dynamic analysis.

  • An experiment platform corresponding to the theoretical model is built.

  • The control parameters are optimized by simulation and experimental results.

Abstract

This paper proposed a method to control the circumferential motion of the constraint boundary of the flexible inverted pendulum to achieve its fast stability. Firstly, the dynamic equation of the system is obtained by the beam theory. The displacement of the flexible inverted pendulum is used as the input to explore the relationship between the output of the constrained boundary and the displacement. The phase delay control method is used to control the circumferential displacement. Then, the influence of the control parameters delay time and motion range on the suppression efficiency is obtained through the damping analysis. The same results as the theory are obtained through simulation. Finally, the corresponding experimental platform is built. The experimental results show that the method can effectively suppress the vibration of the flexible inverted pendulum.

Introduction

Compared with rigid mechanical systems, flexible mechanical systems have attracted the attention of academia because of their lightweight, low energy, and lower manufacturing costs. It also has some other properties, such as frictionless, repeatable, and higher velocity. Therefore, flexible mechanical systems are widely used in human–computer interaction and some unstructured environments. Generally speaking, the flexible machinery acts as a connecting structure and an actuator in the system, such as in flexible robots and flexible micro-positioning systems [1], [2]. As shown in Fig. 1(a), in aerospace engineering, due to the low energy consumption requirements and the limitation of the carrying capacity of space rockets. The satellite antenna must use lightweight materials. In Fig. 1(b), the flexible robotic arm is used in engineering applications because they have more degrees of freedom. However, the elasticity of this structure can cause unexpected vibration problems, which have to account for the instability [3]. For the research of the above problems, the flexible inverted pendulum structure is an important simplified research object. But it is usually under-actuated because it has more degrees of freedom than the driver. So how to solve the vibration problem of that is worth studying.

For the inverted flexible pendulum model, some scholars have done some research on its modeling and stability in recent years. Patil et al. [4] considered the typical case of a flexible inverted pendulum with cutting-edge mass in a trolley system. The dynamic model of the system is established by using the Lagrange equation. The results show that the proposed kinetic equation has an odd number of multiple equilibria. Gandhi et al. [5] considered a super-flexible inverted pendulum on the car and proposed a new nonlinear energy shaping controller to keep the super-flexible inverted pendulum in an upward position while the car is parked in an ideal position. The constrained delay range equation obtains the model for this design, and the controller design consists of two PID controllers. Franco et al. [6] studied the equilibrium control problem of a flexible inverted pendulum system and studied the relationship between system parameters and disturbance robustness. An energy-shaping controller with adaptive disturbance compensation is proposed for a class of under-actuated mechanical systems. Then, a method to identify key system parameters affecting the robustness of closed-loop systems is proposed. The method is applied to the flexible vibration car system, and a simulation study is carried out to verify the effectiveness of the method. Peng et al. [7] established the partial differential equation model of the flexible inverted pendulum system by using the Hamilton principle. The coupled system model is solved by the singular perturbation method. Using the singular perturbation method, the PDE model is divided into fast subsystems and slow subsystems. To stabilize the fast subsystem, they applied a constrained control force to the free end of the pendulum. The rationality and exponential stability of the closed-loop system are proved.

In addition, the flexible inverted pendulum model can be regarded as a kind of flexible robotic arm. Cao et al. [8] proposed an adaptive constrained boundary iterative learning control (ILC) scheme for a double-link rigid-flexible manipulator with parameter uncertainty. Using the Hamilton principle, the coupled dynamic model of the system is established. Based on traditional joint torque control, a constrained boundary control strategy is added. Zhao et al. [9] proposed a pneumatic-piezoelectric hybrid flexible manipulator system. Combining a proportional valve-based pneumatic actuator with a piezoelectric actuator bonded to a flexible beam enables a hybrid actuation scheme. Based on the system identification method, the system dynamics model is established by using the established experimental system, and pneumatic drive vibration control, piezoelectric vibration control, and hybrid vibration control experiments were carried out. Gao et al. [10] studied a more complex multi-degree-of-freedom coupled flexible beam. There are two kinds of vibration of flexible beams, namely bending vibration and torsional vibration. Based on the coupled partial differential equation (PDE) model of flexible beams, the author designs an event-triggered control algorithm to solve the vibration control problem of mechanical systems. Based on the distributed parameter model, Endo et al. [11] discussed the force control of an unconstrained single-link flexible manipulator affected by gravity. A simple controller is proposed that uses the bending moment at the root of the flexible manipulator as input. The performance of the proposed controller is determined by numerical simulation. Abdelhafez et al. [12] used a positive position feedback controller to control the vibration of a forced self-excited nonlinear beam and considered the loop delay. The external excitation is the harmonic excitation caused by the support motion of the cantilever beam. The first-order approximate solution is obtained by applying the multi-time-scale perturbation technique. The influence of time delay on the system is deeply studied, and the optimal conditions for system operation are deduced. Raoufi et al. [13] conducted an experimental study on the control scheme of the flexible manipulator. A novel fractional-order controller is proposed for the nonlinear dynamics and vibration problems of flexible manipulators. The tracking control of the flexible manipulator can be effectively performed even in the presence of noise and total disturbance. Zhang et al. [14] aimed at a flexible double-link manipulator with a variable free end load. An adaptive constrained boundary control scheme is designed to adjust the joint position while compensating for parameter uncertainty and suppressing vibration. The overflow instability problem that may be caused by ignoring the flexible mode is avoided. Pereira et al. [15] proposed a control design method for a single-link flexible manipulator. The technique is based on an Integral Resonant Control (IRC) scheme. The controller consists of two nested feedback loops. The internal loop controls the joint angles, making the system robust to joint friction. The external loop based on IRC technology damps vibrations. Shitole et al. [16] proposed an estimation method based on sliding discrete Fourier transform (SDFT) for vibration mode estimation of a single-link flexible manipulator (SLFM). Deif et al. [17] pointed out that flexible manipulators have received extensive attention because they are more realistic than rigid manipulators in many practical conditions. Thus, a modified genetic algorithm (MGA) is proposed to optimize the proportional derivative (PD) controller for controlling the position and vibration of the flexible manipulator. Compared with the simple genetic algorithm, the new algorithm has a faster convergence speed and higher accuracy. They et al. [18] adopted a new obstacle Lyapunov function in the control design and stability analysis for the vibration control problem of flexible Euler-Bernoulli beams. Then an adaptive control is designed to deal with the uncertainty of the system parameters. Numerical simulations demonstrate that the method can achieve vibration suppression. On the other hand, some scholars' control theory of simple pendulum has also inspired this paper. Hurel et al. [19] aimed at a pendulum system that can cause vibration in two directions subjected to a generalized external force. A smooth absorber whose main body consists of a mass-slider that can move between two springs is proposed and coupled to a pendulum system, and the dynamic equations are analyzed by a multi-scale method. For the pendulum structure, Su et al. [20] applied the event-triggered fuzzy control design to the inverted pendulum system. Gritli et al. [21] proposed a robust feedback controller using Linear Matrix Inequalities (LMIs) formulation for the stabilization of the Inertia Wheel Inverted Pendulum (IWIP). Then, he [22] studied self-generated limit cycle tracking of IWIP under an interconnection and damping assignment passivity-based control. Li et al. [23] proposed a control scheme of boundary lateral displacement for its system stability and verified the effectiveness of the scheme from experiments. Chu et al. [24] proposed to use a mass that can move along the pendulum to suppress the vibration of the inverted pendulum. The influence of the mass movement on system damping is analyzed. Kovacic et al. [25] studied a double pendulum system and its associated regular behavior at low energy levels. The effect of the relevant initial conditions on the frequency is also discussed. An accurate equation of motion is established to describe the nonlinear vibration, and the entire moving process is analyzed through energy diagrams and bifurcation diagrams. Lu et al [26] and Sun et al [27] respectively propose self-learning interval type-2 fuzzy neural network controllers and predefined-time nonsingular terminal sliding-mode approach for trajectory planning of pendulum-type robots.

It can be seen from the above references that scholars mainly adopt the automatic control theory, whether in the research on the stability control of an inverted flexible pendulum or a flexible manipulator, and there are few methods for active control based on the dynamic mechanism. Therefore, it is necessary to propose a robust active control method based on the suppression mechanism.

This paper proposes a circumferential control method of constrained boundary to achieve the fast suppression of the flexible inverted pendulum. The Euler-Bernoulli beam theory is used first, the beam function is used as the mode shape function to be solved, and the dynamic equation, including the circumferential motion is deduced. Then, the influence of the circumferential motion on the system's energy is analyzed. A phase delay control method for the circumferential movement of the constrained boundary is proposed. The relationship between the suppression efficiency and each control parameter is theoretically derived. Then the specific response of the pendulum under the control method is obtained through simulation and compared with the theory. Finally, the corresponding experimental platform is built to complete the experiment corresponding to the simulation.

Section snippets

Energies expression

As shown in Fig. 2, the constrained boundary circumferential motion means the pendulum with clamped boundary conditions can move circumferentially around the bottom end, that is pivot O, and the motion state quantity of the bottom end is used as input. The total length of the pendulum is L. The angle deviating from the vertical direction is θ. The position of any point on the pendulum is x. The displacement due to vibration at the position x and time t is v(x, t). The flexible inverted pendulum

Parameter identification and correction

To verify the correctness of the dynamic equation and the reliability of the simulation results, in this section, we mainly focus on identifying the initial damping of the system and comparing the simulation results with the experimental results. The experimental hardware part is introduced in Section 5.1. All the following displacements are the top displacements of the flexible inverted pendulum. The measured voltage response of the flexible inverted pendulum under free vibration in the

Simulation results and analysis

According to the initial control method proposed in Section 2.3, this section will study the influence of the constrained boundary circumferential phase delay time and motion range on the vibration from the aspect of simulation. The inherent parameters taken in the simulation are the same as those in the experiment, as shown in Table 1. The initial condition for the simulation is to bend the pendulum to the initial displacement. The displacement of the top end of the pendulum is vint=0.08m and

Experiment preparation

In this experiment, the experimental system includes the experimental device part and the control part. The circumferential motion of the constraint boundary is achieved by driving the rotation of the motor, and its velocity is also realized by controlling the motor. The experimental device part includes a flexible inverted pendulum, a motor fixed seat, and a flexible inverted pendulum fixture. Where, the material selected for the pendulum is a 65 manganese steel blade, because it has good

Conclusion

Aiming at the vibration control problem of a flexible inverted pendulum, we proposed an effective control method of the circumferential phase delay of the constrained boundary in this paper. The method's effectiveness is theoretically deduced, and the effect is manifested in numerical simulations. The relevant experimental platform is built, and the experimental results further understand the vibration control based on the constrained boundary circumferential phase delay control method. The

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This project is supported by the National Natural Science Foundation of China (No. 52075086) and the Fundamental Research Funds for the Central Universities (No. N2203021).

References (27)

Cited by (2)

View full text