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We discuss the Lewis and Riesenfeld invariant method for cases where the invariant has continuous eigenvalues and use it to find the Schrödinger wave functions of an inverted pendulum under time-dependent gravitation. As a particular case, we consider an inverted pendulum with exponentially increasing mass and constant gravitation. We also obtain the exact solutions for a generalized time-dependent inverted pendulum.

In this paper, the inverted pendulum model is proposed to describe a pedestrian’s walking motion by considering that the pivot point vibrates periodically up and down. The stability, periodic solutions and oscillations of the inverted pendulum are theoretically investigated, the correctness of which is illustrated by numerical simulations. According to frequency spectrum analysis, the inverted pendulum can exhibit periodically or quasi-periodically stable oscillations. However, we demonstrate that the inverted pendulum will maintain the ratio between the lateral and vertical vibration frequencies near $1∕2$ as an optimizing selection of stability. The theoretical result agrees with the measurement result for a normal pedestrian such that the lateral step frequency is always half the vertical step frequency, which means that it is feasible and reasonable to describe a pedestrian’s walking motion using the inverted pendulum with the pivot vibrating harmonically in the vertical direction. The inverted pendulum model suggested in this paper could contribute to the study of pedestrian–footbridge interaction, which overcomes the difficulty of directly determining the expression of the lateral force induced by pedestrians.

Wind turbines are slender and flexible structures which can undergo large amplitude vibrations when subjected to external loads and, an important way to reduce these excessive vibrations is to apply structural control. In this work, the vibration control of the coupled wind tower–blade system subjected to external lateral loads and rotating blade is studied. To model the tower and blade, the nonlinear Euler–Bernoulli beam theory was applied, and the structural control device considered is a nonlinear inverted tuned mass damper pendulum (ITMDP), located at the top of the tower. The Rayleigh–Ritz method, together with Hamilton’s principle, is applied to obtain a set of coupled nonlinear ordinary differential equations of motion which are in turn solved by the Runge–Kutta method. First, the dynamic instability of the system is studied by considering the variation of the natural frequencies of the tower, as a function of the speed of rotation of the blade and the veering phenomenon is observed. The optimum parameters for the ITMDP are obtained and, a nonlinear dynamic analysis is performed to evaluate the influence of the tuned mass damper (TMD) in the nonlinear regime, by obtaining the time responses, resonance curves, Poincaré maps and basins of attraction. The obtained results show the importance of considering the coupled system and the good performance of the structural control in the dynamic behavior of the system.

Floating offshore wind turbines (FOWTs) are slender structures that are sensitive to hazardous conditions, mainly to winds. Wind action can cause undesirable vibrations that may affect safety and operability. These vibrations can be controlled in numerous ways through passive, active, semiactive or hybrid control. An example of passive control is a tuned mass damper (TMD) that dissipates mechanical energy from the main system when well-tuned to the structure. An inverted pendulum is a device that can be used as a TMD that vibrates out of phase in relation to the main system, reducing the vibrations in comparison to a classical TMD. This work describes a procedure for controlling a FOWT, which is subjected to wind actions, using a proposed tuned mass damper by means of an inverted pendulum (TMD-IP). TMD-IP is evaluated by two specific metrics: tower peak rotation and tower variance rotation. Optimization of the parameters of the inverted pendulum achieved reductions of approximately 95% for the rotation of the tower and the translation of the barge when a harmonic excitation is applied. Using the Monte Carlo method, random wind loading is applied to the FOWT; the reduction in the tower rotation is approximately 70%, and the reduction in the barge translation is approximately 80%. In addition, it is verified that changes in blade stiffness and damping alter the response of the structure very little when the FOWT is under extreme wind conditions. Finally, the results obtained through time and frequency domain analysis are closer to those.

Event-driven intermittent feedback control is a form of feedback control in which the variables of interest are observed continuously, but corrective control action is only initiated intermittently based on certain threshold criteria. According to the literature, the human central nervous system (CNS) adopts event-driven intermittent control strategy while performing various tasks such as stabilize upright posture, control of saccadic eye movement or while driving. In this paper, we examine whether event-driven intermittent control when applied to a system wherein both open-loop dynamics and continuous closed-loop dynamics are unstable can yield bounded overall response (as has been observed in some situations of human postural control). With the help of both illustrative example simulations and experimental results from a table-top experiment inspired from human postural control, we show that this is indeed possible. We further provide insights using passivity theory and phase space analysis. Our analysis shows that bounded response is possible even if the phase trajectories of unstable open-loop and unstable closed-loop systems do not diverge in opposite directions or when Hurwitz convex combinations are not feasible. This study does not necessitate any assumption regarding the stabilizability or passivity of any of the constituent subsystems.

This paper proposes an analysis of the effect of vertical motion of the center of mass (COM) during humanoid walking. The linear inverted pendulum (LIP) model is classically used to deal with humanoid balance during walking. The effects on energy consumption of the COM height remaining constant for humanoid robots, or varying like human beings are studied here. Two approaches are introduced for the comparison: the LIP which offers the great advantage of analytical solving (i.e., fast and easy calculations), and a numerical solving of the IP dynamics, which allows varying the height of the center of mass during walking. The results are compared using a sthenic criterion in a 3D dynamics simulation of the humanoid robot Romeo (Aldebaran Robotics Company) and show a consequent reduction of the robot torque solicitation when the COM oscillates vertically.

This paper presents a novel compositional method for finding fuzzy rules in a three-layered hierarchical fuzzy structure. The proposed method incorporates a multi-objective evolutionary algorithm and a large set of initial conditions, including dynamical conditions of the system under investigation. The proposed method is focused on handling the large set of initial conditions by a multi-objective evolutionary algorithm and it can be applied to a wide range of dynamical control systems in robotics. The method has been evaluated on a dynamical system such as the inverted pendulum. The experimental results and analysis showed that the proposed method is much better than the existing methods such as amalgamation and single objective evolutionary algorithm based methods.

In nature, humans can always find a way to maintain the balance when there is an external push in a large magnitude. Most of present bipedal robots walk properly on the even terrain without any external disturbances. However, they cannot deal with the unexpected disturbance very well, such as, uneven terrain or abrupt strong push. The Capture Point^1,2 has been presented to compute the region on the ground where the bipedal robots must land so as to complete a safe stop. Except for the model-based control approaches, model free control methods have been recently researched, such as, a biologically motivated control for a biped has been researched by.^3,4 The integrated behavior-based control framework⁵ has been as well included in this simulated biped which simplifies the complexity of control system in a large scale and enhances its adaptability to different environments. This paper will describe, firstly, how to establish the model of a linear double link inverted pendulum with a flywheel. Afterwards, the calculated torques will be integrated into the motor patterns of biologically motivated control to enhance the capability of the biped in more complex environments. Finally the simulation results will be demonstrated and analyzed.

This paper proposes an analysis of the effect of vertical motion of the CoM during humanoid walking. The linear inverted pendulum (LIP) model is classically used to deal with humanoid balance during walking. The effects on energy consumption of the CoM height remaining constant for humanoid robots, or varying like human beings, are badly known and are studied here. Two approaches are introduced for the comparison: the LIP which offers the great advantage of analytical solving (i.e. fast and easy calculations), and a numerical solving of the IP dynamics, which allows varying the height of the center of mass during walking. The results are compared in a dynamics simulation of a 2D walking biped.