Double Pendulum Research Articles

Flight Dynamics and Control of an Unmanned Helicopter with Underslung Double Pendulum

This paper presents systematic modeling and simulation of the dynamics and control of a carrier helicopter unmanned aerial vehicle (UAV) with a suspended net used for capturing target UAVs. The goal is to study the effect of key design parameters such as the weight of the net end rod, its length, and point of impact of the UAV on the swing angle of the underslung net. First, a net with an entangled target UAV is modeled as a planar double pendulum. Next, the coupled flight dynamic model of carrier UAV with underslung net with entangled target UAV is simulated and validated using the test data gathered for this purpose. From the simulations, it is identified that the length of the net and point of capture of the UAV on the net are critical parameters that govern the maximum swing angles of the double-pendulum system. Moving the point of impact of the target UAV by 10% of net length resulted in up to 23 and 43% reduction in swing angles of pendulums 1 and 2. The active swing suppression using proportional derivative (PD) controller is effective in attenuating the swing angles in the simulation for successful capture.

Swing-up design of double inverted pendulum by using passive control method based on operator theory

Swing-up design of double inverted pendulum by using passive control method based on operator theory

Optimization of oscillation damping modes of spatial double pendulum. I. Formulation of the problem

The paper discusses the issues of optimal damping of oscillations of a spatial double pendulum, whose joint axes are not collinear to each other. As options for damping, both simply passive damping associated with the influence of viscous friction, and combined passive and active damping are considered, and active influences are formed according to the principle of collinear control. The analytical solution of the system motion equations is given for both cases in the framework of the linear model, and it clearly demonstrates the damping of motions on the natural oscillation modes of the original conservative model. The optimization criteria characterizing the efficiency of the damping processes of system movements are considered. It is noted that in order to obtain the most strongly marked damping modes, the degree of stability should be maximized or the integral energy-time indicator should be minimized. In addition, the main advantages and disadvantages of these optimization criteria are discussed. This article is the basis for further research, which will be presented as a separate article “Optimization of oscillation damping modes of spatial double pendulum. II. Solving the problem and analyzing the results”.

Frequency-Modulation Input-Shaping Strategy for Double-Pendulum Overhead Cranes Undergoing Simultaneous Hoist and Travel Maneuvers

Large payloads hoisted by overhead cranes exhibit the same behavior as that of a double pendulum. Changes in the hoisting cable length during payload transfer maneuvers gives rise to time-dependent system frequencies rendering input-shaped commands ineffective. In this paper, a frequency-modulation input-shaping strategy that enables simultaneous hoist and travel maneuvers using single-mode input-shaping is presented. This strategy utilizes model-based feedback and partial feedback linearization techniques. Simulations are carried out using arbitrary travel and hoist commands combined with the common single-mode zero-vibration (ZV) and zero-vibration-derivative (ZVD) input-shapers. Sensitivity analysis reveals robust performance in the presence of high system uncertainties.

Features Related to Initial Energy Dependence in the Transition to Chaos of the Double Pendulum

Features Related to Initial Energy Dependence in the Transition to Chaos of the Double Pendulum

Studying the influence of external torques on the dynamical motion and the stability of a 3DOF dynamic system

This article explores the planar motion of a novel dynamical system of three degrees of freedom (DOF) triple pendulum, consisting of a double rigid pendulum attached to an un-stretched one, in which its suspension point is considered to be fixed. Applying Lagrange's equations, the controlling system of the equations of motion (EOM) is derived. The multiple scales method (MSM) is used to achieve the analytic solutions of these equations up to higher order of approximation as novel approximate solutions. The accuracy of these solutions is demonstrated by comparing them with the numerical results of the EOM. The system's resonance cases are characterized, and its modulation equations are established. In light of solvability conditions, the steady-state solutions are analyzed. A graphic depiction of the dynamical behavior regarding the motion’s time histories and resonance curves is presented. The stability zones are discussed by analyzing their graphs to evaluate the positive effect of different parameters on the dynamic behavior. The importance of the studied dynamical system is evident from its many applications in various fields such as engineering and physics. However, the vibrational motion can be taken into account in engineering applications like shipbuilding and ships motion, structure vibration, automotive devices, designing robots, and human walking analysis.

White Noise Tests on the LSTM Model Trained with Double Pendulum

This paper describes the intrinsic qualities of a simple double pendulum (DP), with a visual representation, a rigorous deduction of the Lagrangian equation, and a concrete factor analysis. LSTM model was utilized to simulate the double pendulum’s periodic and chaotic behaviors and evaluates the effectiveness of the model. The auto-correlation coefficients was calculated. Meanwhile, Box-Pierce test and Ljung-Box tests for various state-dependent time series were conducted to give various initial conditions to explore the DP system’s random characteristics. The research results are as follows: 1) Chaos did not lead to direct randomness; 2) seasonality could coexist with chaos; 3) the highly auto-regressive nature of DP’s time series data are found. Therefore, it can be concluded that the chaos in a double pendulum has particular patterns (such as the positive relationship with the likelihood of being a random white noise series) that could be further explored.

A Finite Element Model for Compressive Ice Loads Based on a Mohr-Coulomb Material and the Node Splitting Technique

Abstract This paper presents a finite element model for the simulation of ice–structure interaction problems, which are dominated by crushing. The failure mode of ice depends significantly on the strain rate. At low strain rates, the ice behaves ductile, whereas at high strain rates it reacts in brittle mode. This paper focuses on the brittle mode, which is the dominating mode for ship–ice interactions. A multitude of numerical approaches for the simulation of ice can be found in the literature. Nevertheless, the literature approaches do not seem suitable for the simulation of continuous ice–structure interaction processes at low and high confinement ratios in brittle mode. Therefore, this paper seeks to simulate the ice–structure interaction with the finite element method (FEM). The objective of the here introduced Mohr-Coulomb Nodal Split (MCNS) model is to represent the essential material behavior of ice in an efficient formulation. To preserve mass and energy as much as possible, the node splitting technique is applied, instead of the frequently used element erosion technique. The intention of the presented model is not to reproduce individual cracks with high accuracy, because this is not possible with a reasonable element size, due to the large number of crack fronts forming during the ice–structure interaction process. To validate the findings of the model, the simulated maximum ice forces and contact pressures are compared with ice extrusion and double pendulum tests. During validation, the MCNS model shows a very good agreement with these experimental values.

Machine-learning nonconservative dynamics for new-physics detection.

Energy conservation is a basic physics principle, the breakdown of which often implies new physics. This paper presents a method for data-driven "new physics" discovery. Specifically, given a trajectory governed by unknown forces, our neural new-physics detector (NNPhD) aims to detect new physics by decomposing the force field into conservative and nonconservative components, which are represented by a Lagrangian neural network (LNN) and an unconstrained neural network, respectively, trained to minimize the force recovery error plus a constant λ times the magnitude of the predicted nonconservative force. We show that a phase transition occurs at λ=1, universally for arbitrary forces. We demonstrate that NNPhD successfully discovers new physics in toy numerical experiments, rediscovering friction (1493) from a damped double pendulum, Neptune from Uranus' orbit (1846), and gravitational waves (2017) from an inspiraling orbit. We also show how NNPhD coupled with an integrator outperforms both an LNN and an unconstrained neural network for predicting the future of a damped double pendulum.

Comparative response assessment of base-isolated braced-frame buildings considering effects of ductility design

This paper evaluates seismic responses of steel concentrically bracing frames isolated by double friction pendulum bearings with two levels of ductility. The seismic responses of base-isolated buildings are compared to their comparable fixed-base buildings. For this purpose, two isolated and two conventional concentrically braced-frame low-rise buildings are seismically designed, and their three-dimensional finite element models are developed in OpenSees. To investigate the impact of ductility, seismic responses of both ordinary and special levels of ductility are computed and analyzed. The results indicate that the ductility levels significantly affect seismic responses of base-isolated buildings. The superstructure with special level of ductility decreases the peak floor acceleration compared to the ordinary ones. The maximum reduction is calculated by about 20%. Moreover, special ductility level increases peak drift demand compared to the ordinary ones up to 75% in base-isolated buildings. Furthermore, the results show that the ordinary superstructure reduces the maximum and residual displacement of double friction pendulum bearings compared to the special ones. • Effects of level of ductility on seismic responses of isolated concentrically braced-frame buildings. • Seismic assessment of buildings with double friction pendulum bearings. • Comparative response evaluation of base-isolated and conventional steel frames. • Considering different hazard levels in assessment of base-isolated buildings.

Methodology of robust inverted pendulum controllers on a vehicle

In the theory of controllers, the simple and inverted pendulum play an important role due to the equations that result from them, which imply non-linearities and perturbations, thus, in this article, a brief classification of inverted pendulums is presented: inverted pendulum, inverted double pendulum, inverted rotary pendulum (Furuta pendulum). Subsequently, a mathematical model of the inverted pendulum is described through the deduction of the equations of motion that represent the dynamics of the system. Robust control is presented that allows expanding the richness of the mathematical equations, for this case, a control with output feedback is presented and applied to the inverted pendulum to control the unstable dynamics of this model. The results are compared with a post placement control and a robust control using a norm that analyses the characteristics of the system.

Differential design based on double pendulum anti-galloping device for important line

Based on the differential anti-galloping concept and Double Pendulum Anti-galloping Device, optimization analysis was carried out for the ordinary span and long span of 1000 kV Ultra-High-Voltage AC transmission project. Based on the full investigation of the design conditions of ordinary span and long span of the UHV AC line, and the stability mechanism, the influence of the theoretical pendulum length values of Double Pendulum Anti-galloping Device on the instability characteristics of iced conductor with different icing thickness and ambient wind speed were calculated and analyzed. The results showed that 900 mm should be taken as the theoretical pendulum length of the ordinary span, and 800 mm for the long span, which can obtain the best anti-galloping effect, which provides a reliable reference for the anti-galloping design of subsequent UHV transmission lines.

Tensegrity system dynamics based on finite element method

This study presents a finite element analysis approach to nonlinear and linearized tensegrity dynamics based on the Lagrangian method with nodal coordinate vectors as the generalized coordinates. In this paper, nonlinear tensegrity dynamics with and without constraints are first derived. The equilibrium equations in three standard forms (in terms of nodal coordinate, member force density, and member force vectors) and the compatibility equation are also given. Then, we present the linearized dynamics and modal analysis equations with and without constraints. The developed approach is capable of conducting the following comprehensive dynamics studies for any tensegrity structures accurately: performing rigid body dynamics with acceptable errors, which is achieved by setting relatively high stiffness for bars in the simulation; simulating FEM dynamics accurately, where bars and strings can have elastic or plastic deformations; dealing with various kinds of boundary conditions, for example, fixing or applying static/dynamic loads at any nodes in any direction (i.e., gravitational force, some specified forces, or arbitrary seismic vibrations); conducting accurate modal analysis, including natural frequency and corresponding modes. Three examples, a double pendulum, a cantilever truss with external force, and a double prism tensegrity tower, are carefully selected and studied. The results are compared with rigid body dynamics and FEM software ANSYS. This study provides a deep insight into structures, materials, performances, as well as an interface towards integrating control theories.

Analyzing the Stability for the Motion of an Unstretched Double Pendulum near Resonance

This work looks at the nonlinear dynamical motion of an unstretched two degrees of freedom double pendulum in which its pivot point follows an elliptic route with steady angular velocity. These pendulums have different lengths and are attached with different masses. Lagrange’s equations are employed to derive the governing kinematic system of motion. The multiple scales technique is utilized to find the desired approximate solutions up to the third order of approximation. Resonance cases have been classified, and modulation equations are formulated. Solvability requirements for the steady-state solutions are specified. The obtained solutions and resonance curves are represented graphically. The nonlinear stability approach is used to check the impact of the various parameters on the dynamical motion. The comparison between the attained analytic solutions and the numerical ones reveals a high degree of consistency between them and reflects an excellent accuracy of the used approach. The importance of the mentioned model points to its applications in a wide range of fields such as ships motion, swaying buildings, transportation devices and rotor dynamics.

Characterization of a double torsion pendulum used to detect spin-induced torque based on Beth's experiment.

We characterized a double torsion pendulum system, including measurements of the photon-spin-induced torque. Our experimental strategy was similar to that used in Beth's experiment, which was performed in 1936 to measure photon-spin-induced torque using forced oscillation caused by polarization modulation of light incident on a suspended object. Through simple passive isolation of the suspended object from external vibration noise, the achieved torque sensitivity was 2 × 10-17 N m in a measurement time of 104 s, which is close to the thermal noise limit and one order smaller than the minimum torque measured in Beth's experiment. The observed spin-induced torque exerted on the light-absorbing optics is consistent with the angular momentum transfer of ℏ per photon.

A study of the double pendulum using polynomial optimization.

In dynamical systems governed by differential equations, a guarantee that trajectories emanating from a given set of initial conditions do not enter another given set can be obtained by constructing a barrier function that satisfies certain inequalities on the phase space. Often, these inequalities amount to nonnegativity of polynomials and can be enforced using sum-of-squares conditions, in which case barrier functions can be constructed computationally using convex optimization over polynomials. To study how well such computations can characterize sets of initial conditions in a chaotic system, we use the undamped double pendulum as an example and ask which stationary initial positions do not lead to flipping of the pendulum within a chosen time window. Computations give semialgebraic sets that are close inner approximations to the fractal set of all such initial positions.

COLLINEAR CONTROL OF OSCILLATION MODES OF SPATIAL DOUBLE PENDULUM WITH VARIABLE GAIN

This article is devoted to the study of controlled movements of spatial double pendulum with non-parallel cylindrical joints axes. The collinear control is used to swinging of the system by feedback. The most important property of collinear control is the ability of increasing system oscillations only on one oscillation mode. A modification of the collinear control law with variable gain depending on the energy level is investigated. It allows to control the system motions more flexible than in the case of constant gain. As a result, it is possible to observe a smooth transition from a linear oscillation mode to a nonlinear one with a gradual output to a steady oscillation motion with a given energy level. The obtained results are clearly illustrated by graph dependencies that demonstrate the swinging of the system on one oscillation mode from small to finite amplitudes.

INVESTIGATION OF MATHEMATICAL MODEL OF A VIBRATING MIXER WITH A FREEZERY PENDULUM MECHANISM

An extensive review of vibrating mixers, identified their advantages and disadvantages. The application of the action of low-frequency oscillations on the process of mixing bulk inhomogeneous products is substantiated. All considered schemes of vibrating mixers, as well as designs in which mixing devices are driven by vibration-pulse transducers (in which the energy of the main or master system is used to create the technological motion of the driven system) can be attributed to energy-saving technological equipment. The article describes the design, the principle of operation of the developed vibrating mixer with a double pendulum freewheel mechanism for driving the agitator blades in rotational motion. The calculation scheme and laws of motion of this system in the form of Lagrange equations of the second kind (mathematical model) are presented. Using the obtained 2nd order multiple regression equations, recommendations were determined regarding the main parameters of the operating mode of the studied vibration mixing process. Analyzing the results of vibration mixing of bulk inhomogeneous raw materials, we can say that the most influential factors in determining the natural oscillation frequency of the system pendulum, which is one of the main objectives of our study is spring stiffness, spring attachment distance from axis of rotation, rotation angles, pendulum weight, length and the moment of inertia. The paper proves that among various forms of mechanical action on dispersed systems in technological processes vibration action occupies an important place as one of the most effective means to create the necessary dynamic state of dispersed systems.

Frequency-dependent force direction elucidates neural control of balance

BackgroundMaintaining upright posture is an unstable task that requires sophisticated neuro-muscular control. Humans use foot–ground interaction forces, characterized by point of application, magnitude, and direction to manage body accelerations. When analyzing the directions of the ground reaction forces of standing humans in the frequency domain, previous work found a consistent pattern in different frequency bands. To test whether this frequency-dependent behavior provided a distinctive signature of neural control or was a necessary consequence of biomechanics, this study simulated quiet standing and compared the results with human subject data.MethodsAiming to develop the simplest competent and neuromechanically justifiable dynamic model that could account for the pattern observed across multiple subjects, we first explored the minimum number of degrees of freedom required for the model. Then, we applied a well-established optimal control method that was parameterized to maximize physiologically-relevant insight to stabilize the balancing model.ResultsIf a standing human was modeled as a single inverted pendulum, no controller could reproduce the experimentally observed pattern. The simplest competent model that approximated a standing human was a double inverted pendulum with torque-actuated ankle and hip joints. A range of controller parameters could stabilize this model and reproduce the general trend observed in experimental data; this result seems to indicate a biomechanical constraint and not a consequence of control. However, details of the frequency-dependent pattern varied substantially across tested control parameter values. The set of parameters that best reproduced the human experimental results suggests that the control strategy employed by human subjects to maintain quiet standing was best described by minimal control effort with an emphasis on ankle torque.ConclusionsThe findings suggest that the frequency-dependent pattern of ground reaction forces observed in quiet standing conveys quantitative information about human control strategies. This study’s method might be extended to investigate human neural control strategies in different contexts of balance, such as with an assistive device or in neurologically impaired subjects.

Design and Optimization of an Omnidirectional Humanoid Walk: A Winning Approach at the RoboCup 2011 3D Simulation Competition

This paper presents the design and learning architecture for an omnidirectional walk used by a humanoid robot soccer agent acting in the RoboCup 3D simulation environment. The walk, which was originally designed for and tested on an actual Nao robot before being employed in the 2011 RoboCup 3D simulation competition, was the crucial component in the UT Austin Villa team winning the competition in 2011. To the best of our knowledge, this is the first time that robot behavior has been conceived and constructed on a real robot for the end purpose of being used in simulation. The walk is based on a double linear inverted pendulum model, and multiple sets of its parameters are optimized via a novel framework. The framework optimizes parameters for different tasks in conjunction with one another, a little-understood problem with substantial practical significance. Detailed experiments show that the UT Austin Villa agent significantly outperforms all the other agents in the competition with the optimized walk being the key to its success.