Flexible Inverted Pendulum Research Articles

Vibration control of a flexible inverted pendulum using the planned flywheel motion

Flexible inverted pendulums have found wide applications in mechanical systems, but their stability is susceptible to external disturbances. In this study, a new strategy that employs a flywheel to suppress vibration of flexible inverted pendulums is proposed. We develop the theoretical dynamic model of flexible pendulums with a flywheel by the Euler-Bernoulli beam theory, where the axial displacement of the pendulum and geometric nonlinearities are considered. An active control method based on the Bézier trajectory is proposed, where the acceleration of the flywheel is set as the input. Additionally, the equivalent damping ratio of the system including flywheel's motion is derived. The relationship between the parameters in the control equation and the suppression effect is obtained through the simulation of the equivalent damping ratio. Then, an experimental platform is built. The experimental results show that the present control strategy has good suppression effect on the vibration of the flexible inverted pendulums when selecting appropriate parameters. Meanwhile, the results of anti-disturbance test show that the present method has a certain robustness. Finally, the simulation results show that this strategy is also effective for nonlinear forced vibration under different excitation frequencies.

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

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.

Development of a fuzzy-state feedback regulator for stabilizing a flexible inverted pendulum system

Accurate modeling and efficient control of inverted pendulums have always been a challenge for researchers. So, the current research aims to achieve the following objectives: (I) proposing a comprehensive dynamic model for the inverted pendulums which accounts for the flexibility of the pendulum bar and (II) suggesting an appropriate supervisory fuzzy-pole placement control strategy for stabilizing the pendulum system. Using a Lagrangian formulation, the equations of motion are derived and linearized. Then, a state feedback controller with a reduced-order observer is designed to stabilize the system. Closed-loop simulations reveal that at least six modes shall be considered in the dynamic equations. To improve the quality of the transient response, a novel fuzzy system is developed for real-time assignment of the controller poles. Simulation results demonstrate that the control quality is significantly improved by adding a supervisory fuzzy system to the control loop. The developed approach for dynamic modeling of the system, and the idea of multi-level fuzzy-pole placement control architecture developed in this paper, may be successfully applied to improve the response specifications in other dynamic systems.

Control of a flexible inverse pendulum based on the singular perturbation method

In this article, a partial differential equation model for a flexible inverted pendulum system is derived by using the Hamilton principle. Specifically, problems of stabilization and the optimization of such a system are considered. In addition, the singular perturbation method has been used to divide the partial differential equation model for a fast and a slow subsystem. For a fast subsystem stabilization, the control algorithm proposed a boundary force applied at the free end of the beam which proved that the closed-loop subsystem is appropriate and exponentially stable. To stabilize the slow subsystem, a sliding mode control method was used to design the controller, while the method of linear matrix inequality was used in designing the sliding surface. In conclusion, it was shown that the slow subsystem is exponentially stable.

Unstable Oscillating Systems with Hysteresis: Problems of Stabilization and Control

The work is devoted to studying the dynamics of unstable oscillating systems (in the form of an inverted pendulum) controlled by the action of a hysteretic type. The results for different types of motion of a suspension point are presented, in particular, for vertical and horizontal motion. A mathematical model of the inverted pendulum with an oscillating suspension is considered. For this pendulum the explicit criteria of stability are obtained using the linearized equations of motion. The dependences between the initial conditions and the value of the control parameters providing periodic oscillations of the pendulum are described. A mathematical model of the inverted pendulum with feedback control is given under the conditions of the horizontal motion of the suspension point. The conditions that guarantee the stabilization of the considered system are obtained; the conditions are formulated in terms of constraints on the initial conditions. The solution to the problem of the optimal control of an oscillating system is presented in the sense of minimization of a quadratic goal functional. The stabilization problem for an unstable system with distributed parameters, the flexible inverted pendulum, is also considered, and the stabilization conditions are formulated. Fulfilment of these conditions ensures the boundedness of the phase coordinates in the infinite interval of time. The optimal parameters (in the sense of minimization of a quadratic goal functional) corresponding to stabilization of the distributed system are identified.

On the Nonlinear Energy Interactions in Harmonically Excited Post-Buckled Flexible Inverted Pendulum in the Presence of Friction

Abstract Nonlinear energy interaction is a fascinating feature of nonlinear oscillators and has been drawing the attention of researchers since the last few decades. Omnipresent friction in mechanical systems can play a crucial role in modifying these interactions. Using post-buckled flexible inverted pendulum as a candidate system we characterize here, theoretically and experimentally, significant changes in the nonlinear energy transfer in the presence of friction at the input side. Particularly, even with relatively low friction, the energy gets transferred in the higher harmonics of excitation close to a resonant mode as against the transfer to higher modes reported previously. We term this new phenomenon as “excitation harmonic resonance locking.” Theoretical modeling and simulations, considering large deformations, based on assumed modes method, and using a simple friction model reasonably capture the experimental observation. In summary, the paper explicates the role of friction in shifting energy transfer frequencies and can be useful in understanding and designing of oscillators and nonlinear vibrating systems.

Control of a flexible inverted pendulum with a gain scheduling control

Control of a flexible inverted pendulum with a gain scheduling control

Boundary Control for a Flexible Inverted Pendulum System Based on a PDE Model with Input Saturation

AbstractThis research considers the control problem of a flexible inverted pendulum system (FIPS) in the presence of input saturation. The model for a flexible inverted pendulum system (FIPS) is derived via the Hamilton principle. The FIPS model is divided into a fast subsystem and a slow subsystem via the singular perturbation method. We introduce an auxiliary system to deal with the input saturation of a fast subsystem. To stabilize the fast subsystem, a boundary anti‐windup control force is applied at the free end of the beam. It is proven that the closed‐loop subsystem is stable. For the slow subsystem, a sliding mode control method is employed to design a controller and a new decoupling method to design the sliding surface. Then it is shown that the slow subsystem is stable. Finally, simulation results are provided to confirm the efficacy of the proposed controller.

Boundary Control for A Flexible Inverted Pendulum System Based on A Pde Model

AbstractA partial differential equation (PDE) model for a flexible inverted pendulum system (FIPS) is derived by the use of the Hamilton principle. To solve the coupling system model, a singular perturbation method was adopted. The PDE model was divided into a fast subsystem and a slow subsystem using the singular perturbation method. To stabilize the fast subsystem, a boundary control force was applied at the free end of the beam. It then was proven that the closed‐loop subsystem is appropriate and exponentially stable. For the slow subsystem, a sliding mode control method was employed to design a controller and the Linear Matrix Inequality (LMI) method was used to design the sliding surface. It then was shown that the slow subsystem is exponentially stable.

Multi-Level Linear Flexible Inverted Pendulum Modeling

Linear flexible inverted pendulum is a very good platform for testing control algorithm. A new algorithm will usually be used to control linear inverted pendulum in order to detect its control performance. Before that, you need to know the exact mathematical model of the inverted pendulum. Known modeling of multi-level linear flexible inverted pendulum used Newton's second law, in the case of ignoring the mass obtained, therefore, there are imprecise situation. Here the use of Lagrange equations with dissipation function, consider the quality of the mass of the multi-level linear flexible inverted pendulum modeling to obtain a more accurate mathematical model.

Rigid–flexible interactive dynamics modelling approach

Dynamics modelling of multi-body systems composed of rigid and flexible elements is elaborated in this article. The control of such systems is highly complicated due to severe underactuated conditions caused by flexible elements and an inherent uneven non-linear dynamics. Therefore, developing a compact dynamics model with the requirement of limited computations is extremely useful for controller design, simulation studies for design improvement and also practical implementations. In this article, the rigid–flexible interactive dynamics modelling (RFIM) approach is proposed as a combination of Lagrange and Newton–Euler methods, in which the motion equations of rigid and flexible members are separately developed in an explicit closed form. These equations are then assembled and solved simultaneously at each time step by considering the mutual interaction and constraint forces. The proposed approach yields a compact model rather than a common accumulation approach that leads to a massive set of equations in which the dynamics of flexible elements is united with the dynamics equations of rigid members. The proposed RFIM approach is first detailed for multi-body systems with flexible joints, and then with flexible members. Then, to reveal the merits of this new approach, few case studies are presented. A flexible inverted pendulum is studied first as a simple template for lucid comparisons, and next a space free-flying robotic system that contains a rigid main body equipped with two manipulating arms and two flexible solar panels is considered. Modelling verification of this complicated system is vigorously performed using ANSYS and ADAMS programs. The obtained results reveal the outcome accuracy of the new proposed approach for explicit dynamics modelling of rigid–flexible multi-body systems such as mobile robotic systems, while its limited computations provide an efficient tool for controller design, simulation studies and also practical implementations of model-based algorithms.

Modeling and Simulation of a Flexible Inverted Pendulum System

This paper presents a dynamic model of a planar flexible inverted pendulum system under the frame of multi-body dynamics by floating frame of reference formulation (FFRF). By proper simplification and linearization, the state space equation of the system was established for linear analysis and control design. The simulation method for such a coupled system by multi-body dynamics program was also provided. The designed controller with a simple low-pass filter for the flexible inverted pendulum was validated by the simulation of a simple flexible pendulum sample. The result demonstrates a new method of designing and verifying a feedback controller of a flexible multi-body system.

이족보행 로봇의 무게중심 실시간 추정에 관한 연구

In this paper, a closed-loop observer to extract the center of mass (CoM) of a bipedal robot is suggested. Comparing with the simple conversion method of just using joint angle measurements, it enables to get more reliable estimates by fusing both joint angle measurements and F/T sensor outputs at ankle joints. First, a nonlinear-type observer is constructed to estimate the flexible rotational motion of the biped in the extended Kalman filter framework. It adopts the flexible inverted pendulum model which is appropriate to address the flexible motion of bipeds, specifically in the single support phase. The predicted estimates of CoM in terms of the flexible motion observer are combined with measurements (that is, output of the CoM conversion equation with joint angles). Then, we have final CoM estimates depending on the weighting values which penalize the flexible motion model and the CoM conversion equation. Simulation results show the effectiveness of the proposed algorithm.

Positioning Control of Flexible Inverted Pendulum Using Frequency-Dependent Optimal Servo System.

This paper presents a control method for the positioning of the flexible inverted pendulum and cart system. The distributed parameter model is derived in considering the elasticity of the pendulum. The frequency dependent optimal servo system is designed using the reduced order model, which is derived by eliminating the dynamics of flexible modes. The proposed design method enables not only to eliminate the spillover phenomenon caused by the residual elastic modes but to compensate the friction of the cart. Some experimental results show the effectiveness of the proposed control system.

Nonstationary Robust Control of Flexible Structures using Frequency-Shaped Time-Varying Criterion Function

In this method, for a robust control to reduce influences of the unstructured uncertainties, low-pass property is given to the controller by using a frequency-shaped type of criterion function. Moreover, an integrated design of both the H2 control and the H∞ control is established by applying simultaneously the time-varying property to the criterion function. As an example, a positioning control problem of flexible inverted pendulum is considered. The usefulness of the method is verified by theoretical calculations and experiments.

周波数成形時変評価関数を用いた柔軟構造物の非定常ロバスト制御

Abstract In this method, for a robust control to reduce influences of the unstructured uncertainties, low-pass property is given to the controller by using a frequency-shaped type of criterion function. Moreover, an integrated design of both the H 2 control and the H ∞ control is established by applying simultaneously the time-varying property to the criterion function. As an example, a positioning control problem of flexible inverted pendulum is considered. The usefulness of the method is verified by theoretical calculations and experiments.

The initiation of normal walking.

Lower extremity electromyograms (EMGs), ground reaction forces, and body motion were measured during the brisk initiation of forward walking performed by 12 healthy adults, aged 20 to 82 years. Gait was initiated 20 times in response to a visual cue. During gait initiation, the body rotated about the ankles like a flexible inverted pendulum. The muscles of the lower extremities were activated stereotypically so as to create moments of force about the ankles that propelled the body toward the stance foot and into forward motion. All volunteers exhibited similar patterns of gait initiation, which were so reproducible that computer averaging of multiple steps by each person was possible. Gait initiation is a stereotyped sequence of postural shifts that culminates in a forward step. Disturbances of gait initiation could result from abnormalities in postural control, movement, or their integration.

Stabilizing Control of Flexible Inverted Pendulum by Means of H.INF. Control.

This study shows that the stabilizing and positioning control of a flexible inverted pendulum of which the pivot is mounted on a cart is experimentally possible by means of H∞ control theory. Since the control system has an eigenvalue of zero, the design method of H∞ control utilizing a settling function is applied, and one controller is designed with consideration of the rigid and the flexible first modes and another is designed neglecting all flexible modes. By numerical calculations and experiments, we have investigated the positioning control characteristics and the robust stability against disturbance to the pendulum of each controller, and it has been made clear that the robust stability of the controller designed considering the flexible first mode is superior to the one designed neglecting all flexible modes.