Rotary Pendulum Research Articles

Multiple scales method for analyzing a forced damped rotational pendulum oscillator with gallows

This study reports the analytical solution for a generalized rotational pendulum system with gallows and periodic excited forces. The multiple scales method (MSM) is applied to solve the proposed problem. Several types of rotational pendulum oscillators are studied and talked about in detail. These include the forced damped rotating pendulum oscillator with gallows, the damped standard simple pendulum oscillator, and the damped rotating pendulum oscillator without gallows. The MSM first-order approximations for all the cases mentioned are derived in detail. The obtained results are illustrated with concrete numerical examples. The first-order MSM approximations are compared to the fourth-order Runge–Kutta (RK4) numerical approximations. Additionally, the maximum error is estimated for the first-order approximations obtained through the MSM, compared to the numerical approximations obtained by the RK4 method. Furthermore, we conducted a comparative analysis of the outcomes obtained by the used method (MSM) and He-MSM to ascertain their respective levels of precision. The proposed method can be applied to analyze many strong nonlinear oscillatory equations.

Energy Harvester Based on a Rotational Pendulum Supported with FEM

The proposed energy harvesting system is based on a rotational pendulum-like electromagnetic device. Pendulum energy harvesting systems can be used to generate power for wearable devices such as smart watches and fitness trackers, by harnessing the energy from the human body motion. These systems can also be used to power low-energy-consuming sensors and monitoring devices in industrial settings where consistent ambient vibrations are present, enabling continuous operation without any need for frequent battery replacements. The pendulum-based energy harvester presented in this work was equipped with additional adjustable permanent magnets placed inside the induction coils, governing the movement of the pendulum. This research pioneers a novel electromagnetic energy harvester design that offers customizable potential configurations. Such a design was realized using the 3D printing method for enhanced precision, and analyzed using the finite element method (FEM). The reduced dynamic model was derived for a real-size device and FEM-based simulations were carried out to estimate the distribution and interaction of the magnetic field. Dynamic simulations were performed for the selected magnet configurations of the system. Power output analyses are presented for systems with and without the additional magnets inside the coils. The primary outcome of this research demonstrates the importance of optimization of geometric configuration. Such an optimization was exercised here by strategically choosing the size and positioning of the magnets, which significantly enhanced energy harvesting performance by facilitating easier passage of the pendulum through magnetic barriers.

Stabilization control of rotary inverted pendulum using a novel EKF-based fuzzy adaptive sliding-mode controller: design and experimental validation

This article methodically develops an improved self-regulating fuzzy-adaptive Sliding Mode Controller (SMC) that strengthens the disturbance compensation capacity of the nonlinear rotary pendulum systems while effectively attenuating the chattering content and curbing the control energy consumption. The article contributes to augmenting the SMC with online adaptation tools to achieve the said objectives. It employs the conventional Gao’s power-rate reaching law as the baseline. The scaling gain and the power rate of the said reaching law are adaptively modulated via a pre-calibrated two-input state-error-driven fuzzy nonlinear function. Additionally, the sign function in the law is also replaced with an odd-symmetric nonlinear fuzzy function to address the hard limits imposed by the former. Finally, the membership functions of the fuzzy function are self-regulated using the Extended-Kalman-Filter to improve the compensator’s adaptability in handling the system’s rapidly changing control requirements under exogenous disturbances. The aforementioned propositions are verified by performing customized and reliable hardware-in-loop experiments on the Quanser single-link rotary pendulum platform. As compared to baseline SMC law, the proposed control procedure contributes a ∼45.2%, ∼48.5%, and ∼34.8% reduction in position-regulation errors, control energy consumption, and peak overshoots, respectively. The experimental assessment validates the proposed control system’s enhanced robustness and chattering-suppression capability.

High-Precision Inertial Sensor Charge Ground Measurement Method Based on Phase-Sensitive Demodulation

Inertial sensors are the key payloads in space gravitational wave detection missions, and they need to ensure that the test mass (TM), which serves as the inertial reference, freely floats in the spacecraft without contact, so that the TM is not disturbed by the satellite platform and the cosmic environment. Space gravitational wave detection missions require that the residual acceleration of the TM should be less than 3×10−15ms−2Hz−1/2. However, the TM with charges will interact with surrounding conductors and magnetic fields, introducing acceleration noise such as electrostatic force and Lorentz force. Therefore, it is necessary to carry out charge management on the TM, in which the high-precision measurement of charge is crucial. Space gravitational wave detection missions require a residual charge measurement accuracy of 3×10−13C for the TM. In this paper, we design a high-precision inertial sensor charge measurement method based on phase-sensitive demodulation (PSD). By establishing a torsion pendulum rotation model based on the force modulation method, the characteristics of the TM torsion angle signal are analyzed. The PSD is used to extract the amplitude of the specific frequency signal component containing the charge information, and then to calculate the value of the accumulated charges. The method is compared with the Butterworth band-pass filtering method, and the simulation results show that the method has a higher measurement accuracy, shorter settling time, and stronger anti-interference ability, meeting the TM residual charge measurement accuracy index requirement.

Damping of Oscillations of a Rotary Pendulum System

This paper describes an innovative design based on the spectral approach of a novel shaper that eliminates frequency components that induce unwanted residual oscillations in various flexible mechanical systems, such as tower cranes or chain carousels, which are vital to many manufacturing and material-handling processes. However, their physical structure leads to flexible effects that limit their usefulness. Apart from the circular motion problem, control is provided by a single actuator, which makes it a so-called underactuated system. The input signal needs to be modified so that the spectral components from several interconnected degrees of freedom are considered together during shaper design, which increases the complexity of this task since one of its components induces nonlinear behavior. This means that traditional shaping techniques, based on linear theory, fail to provide good performance over the whole input range. The underdamped dynamics of the model and the effect of nonlinearities on the spectrum of the final signal are examined; the proposed method for application as a command shaping control technique is applied; and its effectiveness is analyzed by simulation and real-time implementation. The theoretical results verified on an experimental crane system confirm the expected oscillation phenomenon and show that the designed nonlinear shaper can reduce the payload swing significantly.

Real-Time Control of Underactuated Systems in a Mechatronics Kit Via a Novel LMI-Based Sliding Mode Approach

Real-time implementation of a sliding mode control scheme based on the unit vector approach, for a variety of underactuated configurations of a mechatronics kit (rotational, inertia, and double pendulum) is presented in this work. Details are given to perform diffeomorphisms leading to the required normal form, based on which design conditions are cast as linear matrix inequalities, thus improving numerical systematicness of the traditional methodology. Once the control law is designed, its implementation requires reliable estimates of the velocities since the different plants under consideration do not measure them; to this end, Levant's robust differentiator is employed. Results are provided that show the effectiveness of the proposal.

On the Orbital Stability of Pendulum Periodic Motions of a Heavy Rigid Body with a Fixed Point, the Main Moments of Inertia of which are in the Ratio 1 : 4 : 1

The motion of a heavy rigid body with a fixed point in a uniform gravitational field is considered. It is assumed that the main moments of inertia of the body for the fixed point satisfy the condition of D.N. Goryachev–S.A. Chaplygin, i.e., they are in the ratio 1 : 4 : 1. In contrast to the integrable case of D.N. Goryachev–S.A. Chaplygin, no additional restrictions are imposed on the position of the center of mass of the body. The problem of orbital stability of pendulum periodic motions of the body is investigated. In the neighborhood of periodic motions, local variables are introduced and equations of perturbed motion are obtained. On the basis of a linear analysis of stability, the orbital instability of pendulum rotations for all values of the parameters has been concluded. It has been established that, depending on the values of the parameters, pendulum oscillations can be both orbitally unstable and orbitally stable in a linear approximation. For pendulum oscillations that are stable in the linear approximation, based on the methods of KAM theory, a nonlinear analysis is performed and rigorous conclusions about the orbital stability are obtained.

A comparative study of anti-swing radial basis neural-fuzzy LQR controller for multi-degree-of-freedom rotary pendulum systems

A comparative study of anti-swing radial basis neural-fuzzy LQR controller for multi-degree-of-freedom rotary pendulum systems

A robust variable-structure LQI controller for under-actuated systems via flexible online adaptation of performance-index weights.

This article presents flexible online adaptation strategies for the performance-index weights to constitute a variable structure Linear-Quadratic-Integral (LQI) controller for an under-actuated rotary pendulum system. The proposed control procedure undertakes to improve the controller's adaptability, allowing it to flexibly manipulate the control stiffness which aids in efficiently rejecting the bounded exogenous disturbances while preserving the system's closed-loop stability and economizing the overall control energy expenditure. The proposed scheme is realized by augmenting the ubiquitous LQI controller with an innovative online weight adaptation law that adaptively modulates the state-weighting factors of the internal performance index. The weight adaptation law is formulated as a pre-calibrated function of dissipative terms, anti-dissipative terms, and model-reference tracking terms to achieve the desired flexibility in the controller design. The adjusted state weighting factors are used by the Riccati equation to yield the time-varying state-compensator gains.

Complex Fractional-Order LQIR for Inverted-Pendulum-Type Robotic Mechanisms: Design and Experimental Validation

This article presents a systematic approach to formulate and experimentally validate a novel Complex Fractional Order (CFO) Linear Quadratic Integral Regulator (LQIR) design to enhance the robustness of inverted-pendulum-type robotic mechanisms against bounded exogenous disturbances. The CFO controllers, an enhanced variant of the conventional fractional-order controllers, are realised by assigning pre-calibrated complex numbers to the order of the integral and differential operators in the control law. This arrangement significantly improves the structural flexibility of the control law, and hence, subsequently strengthens its robustness against the parametric uncertainties and nonlinear disturbances encountered by the aforementioned under-actuated system. The proposed control procedure uses the ubiquitous LQIR as the baseline controller that is augmented with CFO differential and integral operators. The fractional complex orders in LQIR are calibrated offline by minimising an objective function that aims at attenuating the position-regulation error while economising the control activity. The effectiveness of the CFO-LQIR is benchmarked against its integer and fractional-order counterparts. The ability of each controller to mitigate the disturbances in inverted-pendulum-type robotic systems is rigorously tested by conducting real-time experiments on Quanser single-link rotary pendulum system. The experimental outcomes validate the superior disturbance rejection capability of the CFO-LQIR by yielding rapid transits and strong damping against disturbances while preserving the control input economy and closed-loop stability of the system.

Event-driven adaptive intermittent control applied to a rotational pendulum

Intermittent control combines open-loop trajectories with feedback at discrete time instances determined by events. Among other applications, it has recently been used to model quiet standing in humans where the system was assumed to be time-invariant. This article expands this work to the time-variant case by introducing an adaptive intermittent controller that exploits the well-known self-tuning architecture of adaptive control with a Kalman filter to perform online state and parameter estimation. Simulation and experimental results using a rotational inverted pendulum show advantages of the intermittent controllers compared to continuous feedback control since the former can provide persistent excitation due to their internal triggering mechanism, even when no external reference changes or disturbances are applied. Moreover, the results show that the event thresholds of intermittent control can be used to adjust the degree of responsiveness of the adaptation in the system, becoming a tool to balance the trade-off between steady-state performance and flexibility against parametric changes, addressing the stability–plasticity dilemma of adaptation and learning in control.

Closed-Form Solutions to a Forced Damped Rotational Pendulum Oscillator

In this investigation, some analytical solutions to both conserved and non-conserved rotational pendulum systems are reported. The exact solution to the conserved oscillator (unforced, undamped rotational pendulum oscillator), is derived in the form of a Jacobi elliptical function. Moreover, an approximate solution for the conserved case is obtained in the form of a trigonometric function. A comparison between both exact and approximate solutions to the conserved oscillator is examined. Moreover, the analytical approximations to the non-conserved oscillators including the unforced, damped rotational pendulum oscillator and forced, damped rotational pendulum oscillator are obtained. Furthermore, all mentioned oscillators (conserved and non-conserved oscillators) are linearized, and their exact solutions are derived. In addition, all obtained approximations are compared with the four-order Runge–Kutta (RK4) numerical approximations and with the exact solutions to the linearized oscillators. The obtained results can help several authors for discussing and interpreting their results.

모델기반 swing-up 제어를 위한 2단 도립진자의 구조 설계 및 계수 추정에 대한 연구

Mathematical models and parameters with high accuracy are required for the swing-up control of a double inverted pendulum. This study proposes a mechanical structure of a double inverted pendulum having excellent matching characteristics with a mathematical model. The implementation of a parameter estimation method based on nonlinear optimization, which is essential for swing-up control, is also proposed. The proposed structure of a double inverted pendulum eliminates backlash using the direct drive of a BLDC motor, minimizes the unmodeled frictional force through bearing cleaning and oiling, minimizes unnecessary load torque using the double support of a timing pulley shaft and power transmission through a coupling, and constrains the pendulum rotation to one degree-of-freedom by adopting a dual-bearing joint. In addition, both 3D printing and CNC milling are appropriately used to satisfy the requirements for each mechanical part. The swing motion of the manufactured double inverted pendulum is measured using the lab-built light-weight rapid control prototyping 02 (LW-RCP02). The proposed parameter estimation method based on nonlinear optimization is implemented to match the response trajectory of the actual double inverted pendulum with that of a Simulink model implemented as an S-function. Finally, the effectiveness of the proposed double inverted pendulum design and parameter estimation method is confirmed by showing that the mathematical model that, based on the parameters estimated using the proposed method, accurately describes the response characteristics of the double inverted pendulum.

Advanced Control Strategies for Rotary Double Inverted Pendulum

This paper presents the stabilization control of the rotational double inverted pendulum system. It is generally a non-linear, non-minimum phase, unstable, and underactuated system of high order. It has some significant real-life applications such as position control, aerospace vehicle control, and robotics. The objective is to determine the control law to the motor’s output torque such that the inverted pendulum motion will be stabilized about a vertical axis and position the rotary arm to a commanded angular position. In this study, multi-PID controllers, Linear-Quadratic Regulator (LQR), and Internal Model Control (IMC) controllers are designed and MATLAB-based simulations are performed to understand and compare the performance of the three control schemes.

Wavefront Sensing with a Coupled Cavity for Torsion-Bar Antenna

Torsion-Bar Antenna (TOBA) is a ground-based gravitational wave detector using torsion pendulums. TOBA can detect intermediate-mass black hole binary mergers, gravitational wave stochastic background, and Newtonian noise, and is useful for earthquake early warning. A prototype detector Phase-III TOBA with 35 cm-scale pendulums is under development to demonstrate noise reduction. The target strain sensitivity is set to $1\times10^{-15}\,{/\sqrt{\rm Hz}}$ between 0.1 Hz--10 Hz. A new scheme of wavefront sensing with a coupled cavity was proposed to measure the pendulum rotation as low as $5\times10^{-16}\,{{\rm rad}/\sqrt{\rm Hz}}$ for Phase-III TOBA. In our method, an auxiliary cavity is used to enhance the first-order Hermite--Gaussian mode in a main cavity. Experimental demonstration is ongoing to confirm the feasibility of angular signal amplification and establish a method for locking a coupled cavity. We evaluated the performance of the coupled cavity and concluded that angular signal amplification would be feasible with this sensor. The coupled cavity was successfully locked to the resonance by the Pound--Drever--Hall technique with two modulation frequencies.

Energy harvesting from a rotational nonlinear energy sink in vortex-induced vibrations

The concept of nonlinear energy sinks (NES) has been studied recently for passive fluid-induced vibration mitigation. A NES presents a fast and irreversible capacity of efficiently dissipating energy for various vibration problems. Besides, energy harvesting (EH) applications are suitable for generating electricity in vibrating mechanical structures. This work investigates the combination of a rotational NES with EH for the vortex-induced vibrations (VIV) of a cylinder exposed to airflow. The study considers an energy harvesting device comprising an electric generator ideally attached to the rotational NES mass pendulum shaft, resulting in a cylinder-NES-EH assembly. The model consists of an elastically mounted cylinder subjected to transverse airflow. The NES-EH is attached to the cylinder as a rigid pendulum arm with a mass at its tip that can freely oscillate, simultaneously absorbing the cylinder vibration and generating electricity. The fluid modeling is based on a wake model represented by the Van der Pol oscillator. The study is carried out by computational simulations of the differential equations of motion, and the rotational NES parameters optimization is done with the particle swarm method. Results comprise the analyses of the cylinder-NES-EH nonlinear dynamics characterization, identification of response regimes of the NES-EH system through a parametric study, and comparison with results from the literature. An optimal configuration was presented to maximize the generation of electric energy. Combining an EH with a rotational NES has shown an adequate potential for the design of energy extraction devices.

Robustification of the state-space MRAC law for under-actuated systems via fuzzy-immunological computations.

This paper formulates an enhanced Model-Reference-Adaptive-Controller (MRAC) that is augmented with a fuzzy-immune adaptive regulator to strengthen the disturbance-attenuation capability of closed-loop under-actuated systems. The proposed scheme employs the conventional state-space MRAC and augments it with a pre-configured fuzzy-immune mechanism that acts as a superior regulator to dynamically modulate the adaptation gains of the Lyapunov gain-adjustment law. The immunological computations increase the controller's adaptability to flexibly manipulate the damping control effort under exogenous disturbances. The efficacy of the proposed Immune-MRAC law is comparatively analyzed under practical disturbance conditions by conducting real-time hardware experiments on the QNET rotary pendulum. The experimental outcomes validate the faster transient-recovery behavior and stronger damping effort of the proposed control law against the exogenous disturbances while preserving the system's asymptotic stability and control energy efficiency.

Solution for a rotational pendulum system by the Rach–Adomian–Meyers decomposition method

Abstract In this article, we report for the first time the application of a novel and extremely valuable methodology called the Rach–Adomian–Meyers decomposition method (MDM) to obtain numerical solutions to the rotational pendulum equation. MDM is a tool for solving nonlinear differential equations that combines both series solution and the Adomian decomposition method efficiently. We present a simple and highly accurate MDM-based algorithm and its numerical implementation via a one-step recurrence approach for obtaining periodic solutions to the rotational pendulum equation. Finally, numerical simulations are performed to demonstrate the efficiency and accuracy of the proposed technique for both large and small amplitudes of oscillation.

Design, Modeling, and Evaluation of a Class-A Triaxial Force-Balance Accelerometer of Linear Based Geometry

Abstract A new class-A force-balance accelerometer (FBA) is designed, simulated, and evaluated. The focus of this work was to design a low-cost but high-performance instrument. The FBA has output voltage proportional to ground acceleration, flat response from direct current to 200 Hz, output range ± 10 V differential (40Vpp, peak-to-peak) and sensitivity 2.5 V/g, which provides a ± 4g range. Unlike other well-established designs with rotational pendulum systems and single-coil actuators that present partially nonlinear performance, the proposed design is based on a linear motion spring-mass mechanism with two parallel leaf springs and a double symmetrical magnet-coil force actuator. This architecture ensures that the displacement transducer’s response is linear and that an acting force on the seismic mass is not disturbed by any cross-axial motion. This force depends on displacement only, as imposed by the electronic control circuit, which is implemented on a small high-density printed circuit board (PCB) mounted on top of the mechanical construction. The plates of the variable capacitance displacement transducer consist of ordinary PCBs for cost efficiency. The coils of the force actuator are placed on each side of the moving plate of the capacitive transducer and the magnets are placed on the aluminum rigid frame of the device. The central moving plate of the variable capacitor and the attached force actuator coils, along with some extra aluminum mass, consist the accelerometer’s seismic mass. The performance of the accelerometer is evaluated in terms of earthquake data records and in comparison of its response with that of a commercial FBA with corresponding specifications. The instrument’s self-noise was also measured over a long period of operation and proved to comply with typical FBA application requirements and commercial product standards.

An enhanced adaptive-LQR procedure for under-actuated systems using relative-rate feedback to dynamically reconfigure the state-weighting-factors

This paper synthesizes an innovative nonlinear-type self-adaptive Linear Quadratic Regulator (LQR) to enhance the robustness of under-actuated systems against bounded exogenous disturbances by using the relative-rate and state-error feedback. The control procedure is realized by augmenting the generic LQR with an online reconfiguration block that acts as a superior regulator to dynamically adjust the state-weighting-factors associated with the quadratic-performance index. The Riccati Equation uses the modified state-weighting-factors to re-compute the LQR problem and delivers the time-varying state-feedback gains. The said weighting-factors are adaptively modulated via a conventional state-error-magnitude-driven adaptation law that is retrofitted with auxiliary relative-rate-driven reconfiguration blocks. These blocks utilize error acceleration to deduce the response speed and thus re-adjust the variation rate of the weights in real time. This arrangement reinforces the controller’s adaptability to flexibly manipulate the tightness of the applied control effort. The proposed scheme is analyzed by conducting real-time hardware experiments on the QNET Rotary Pendulum. The experimental outcomes validate the superior position-regulation and disturbance-rejection behavior of the proposed scheme.