Dynamics Of Double Pendulum Research Articles

Correction to: Vibrational and stability analysis of planar double pendulum dynamics near resonance

Correction to: Vibrational and stability analysis of planar double pendulum dynamics near resonance

Vibrational and stability analysis of planar double pendulum dynamics near resonance

AbstractThe focus of this paper is to examine the motion of a novel double pendulum (DP) system with two degrees of freedom (DOF). This system operates under specific constraints to follow a Lissajous curve, with its pivot point moving along this path in a plane. The nonlinear differential equations governing this system are derived using Lagrange's equations. Their analytical solutions (AS) are subsequently calculated using the multiple-scales method (MSM), which provides higher-order approximations. These solutions are considered new, as the traditional MSM has been applied to this novel system for the first time. Additionally, the accuracy of these solutions is validated through numerical results obtained using the fourth-order Runge–Kutta method. The solvability conditions and characteristic exponents are determined based on resonance cases. The Routh–Hurwitz criteria (RHC) are employed to assess the stability of the fixed points corresponding to the steady-state solutions. They are also used to demonstrate the frequency response curves. The nonlinear stability analysis is performed by examining the stability and instability ranges. Resonance curves and time history plots are presented to analyze the behavior of the system for specific parameter values. The investigation delves into a comprehensive analysis of bifurcation diagrams (BDs) and Lyapunov exponent spectra (LEs), aiming to uncover the various types of motion present within the system. Systematic examination of these charts reveals critical insights into transitions between stable, quasi-stable, and chaotic dynamical behaviors. This work has practical applications in various fields, such as robotics, pump compressors, rotor dynamics, and transportation devices. It can be used to study the vibrational motion of these systems.

Lyapunov approach for the control of overhead crane systems with double-pendulum dynamics and uncertain disturbances

In this paper, a new composite control strategy, consisting of a disturbance observer and a nonlinear anti-swing control approach, is presented for underactuated overhead crane systems. Compared with existing control strategies for overhead crane systems, both the double-pendulum dynamics and unknown disturbances of which are taken into consideration. In addition, the unknown disturbances are handled by a feedforward compensation control method. Specifically, first, based on the system dynamic equations, an auxiliary signal is constructed and a nonlinear disturbance observer is presented, which can exactly estimate the unknown disturbances in finite time. Next, an elaborate Lyapunov function is introduced and a disturbance-observer-based control approach is presented for the overhead crane system with double-pendulum dynamics. Then, rigorous theoretical analysis is given to prove the convergence of the system states. At last, simulation tests are included to verify the effectiveness and robustness of the devised approach.

Dynamics of a double pendulum with viscous friction in the joints. I. Mathematical model of motion and construction of the regime diagram

The paper discusses the dynamic behavior of a double mathematical pendulum with identical parameters of links and end loads, which is under the influence of viscous friction in both of its joints with generally different dissipative coefficients. A linear mathematical model of the system motion for small deviations is given, and a characteristic equation containing two dimensionless dissipative parameters is derived. For the case of low damping, approximate analytical expressions are found that make it possible to evaluate and compare with each other the damping factors during the motion of the system on each of the oscillation modes. A diagram of dissipative motion regimes is constructed, which arises when the plane of dimensionless parameters is divided by discriminant curves into regions with a qualitatively different character of the system motion. It is noted that a dissipative internal resonance can take place in the system under consideration, and the conditions for its existence in an analytical form are established, as well as their graphic illustration is also given. This article is the first part of the study of the dynamics of a dissipative double pendulum, the continuation of which will be presented as a separate work “Dynamics of a double pendulum with viscous friction in the joints. II. Dissipative oscillation modes and optimization of damping parameters”.

Finite time disturbance observer design and Lyapunov-based control design for overhead cranes with double-pendulum dynamics

In this paper, a novel disturbance observer and a Lyapunov-based control approach are developed for the crane control. For existing works, the assumption on the disturbances is strong and the motivation of these methods is to reject disturbances by feedback control. To tackle these problems, this paper proposes a novel composite control strategy consisting of a nonlinear control method and a finite time disturbance observer. Specifically, first, based on the system dynamic equations, a disturbance estimator is put forward, which can exactly identify the unknown disturbances in finite time. Next, an elaborate Lyapunov function is constructed and a disturbance-compensation-based control approach is presented for the overhead crane system with double-pendulum dynamics. Then, rigorous theoretical analysis is given to prove that all of the states of the closed-loop system asymptotically converge to the origin. At last, simulation tests are included to verify the effectiveness and robustness of the proposed method.

Output feedback control for overhead cranes subject to double-pendulum swing effects and uncertain disturbances

In this paper, a disturbance-observer–based control approach is developed for overhead crane systems. Different from existing control strategies, the issues consisting of the output feedback, input saturation, double-pendulum dynamics, and uncertain disturbances are taken into consideration here. In particular, a disturbance observer is designed first, which can exactly estimate uncertain disturbances. Next, to enhance the performance of the controller, a virtual position signal is constructed and a corresponding Lyapunov function is introduced. Then, based on the provided Lyapunov function and the designed disturbance observer, a composite control approach is developed for overhead crane systems with double-pendulum dynamics and the convergence of the system states is proved via rigorous theoretical analysis. Finally, the effectiveness and robustness of the proposed control approach are verified by simulation tests.

Time-Polynomial-Based Optimal Trajectory Planning for Double-Pendulum Tower Crane With Full-State Constraints and Obstacle Avoidance

Double-pendulum dynamics increase the difficulty of the control for the tower cranes. Especially, it would be a greater challenge with obstacle avoidance while satisfying full-states constraints. A time-polynomial-based optimal trajectory planning strategy is proposed for double-pendulum tower crane with full-state constraints and obstacle avoidance. First, with the dynamic model of double-pendulum tower crane system, the trajectory planning problem is formulated by taking positioning, anti-swing, state constraints, and obstacle avoidance into account. Then, a group of auxiliary signals are introduced to express the state variables. The trajectory planning problem for jib slewing is transformed into that for the auxiliary signals. The time-based polynomial functions are introduced for trajectory planning of the auxiliary signals. By planning proper trajectories for the auxiliary signals and trolley translation displacement, we obtain time suboptimal trajectory for the jib slewing and the trolley transferring. The jib and the trolley can reach their destination accurately with suppressing swing of the payload and the hook while avoiding obstacles. The time suboptimal solution not only makes transportation time as short as possible, but also ensures that full-states constraints and obstacles avoidance are satisfied. Finally, hardware experiments demonstrate that the proposed trajectory planning strategy has better control performance than existing state-of-the-art methods.

Positioning and Swing Elimination Control of the Overhead Crane System with Double-Pendulum Dynamics

Positioning and Swing Elimination Control of the Overhead Crane System with Double-Pendulum Dynamics

A Time-Varying PD Sliding Mode Control Method for the Container Crane Based on a Radial-Spring Damper

For the multi-rope structure of the container crane system and its large mass payload anti-swing positioning problem, an equivalent double-swing model based on radial spring-dampers is established and a time-varying PD sliding mode controller (TVPD-SMC) with improved transient performance is proposed. In particular, the dynamics of the container crane system are first analyzed using the Lagrange method, and an equivalent double pendulum dynamics model of the crane is established. Compared with the traditional double pendulum model, this model ensures the accuracy of modeling without measuring the second pendulum angle. On this basis, an enhanced coupled time-varying sliding mode control method is designed to eliminate the sliding mode control method’s reaching phase and improves the robustness of the controller. Finally, the convergence and stability of the closed-loop system are proved using the Lyapunov technique and the Lasalle invariance theorem, and the simulation results demonstrate the effectiveness of the method.

Self-tuning anti-sway control for shipboard cranes providing combined world and deck-frame compensation

This paper presents a novel self-tuning anti-sway control system for shipboard cranes that provides full, six degree-of-freedom (DOF) motion compensation with respect to both the world (ocean) coordinate frame and the ship deck coordinate frame. The system can either manually or automatically switch between compensation modes during operation, and is capable of providing anti-sway action while tracking a time-varying operator input. Rather than requiring the operator to provide individual joint control, the entire system can track a simple three-axes Cartesian input from the operator, while simultaneously providing full anti-sway compensation in either coordinate frame.High-fidelity simulation case studies were performed with a nine-DOF shipboard knuckle boom crane, which included double pendulum dynamics between the hook and payload, in the presence of full six-DOF ship motion at sea-state six. The self-tuning anti-sway system showed at least a 92.70% reduction in payload tracking error when the corresponding compensation mode (either deck or world-frame) was activated, a significant reduction in undesired payload motion. Wind disturbances were added to the simulation using the Dryden wind model, where the system showed a maximum decrease in performance of only 0.73%; additional tests were run with self-tuning disabled, which resulted in a decrease in performance of up to 101%, indicating self-tuning is highly effective at minimizing the effect of wind disturbances.

A single and double closed-loop compound anti-sway control method for double-pendulum bridge cranes locating at random position

In the assembly workshops of some heavy special equipment, the bridge cranes for payload lifting often needs to be located frequently. However, the locating position is often determined by the operator, which is random and results in significant payload oscillation and difficulties in trolley positioning. Furthermore, in practice, the bridge crane always exhibits more complicated double-pendulum dynamics compared with single-pendulum crane. To solve these problems, this paper establishes the double-pendulum model of bridge crane. Derived from the proportional-derivative (PD) control, the single closed-loop is designed based on the hook oscillation during acceleration and transporting; when locating, the double closed-loop is presented by utilizing the position and the hook oscillation. Combining the two control methods, a single and double closed-loop compound anti-sway control (SDCAC) method for the bridge crane is proposed. On this basis, to improve the performance of the SDCAC system, the sequential quadratic optimization (SQP) method is adopted to optimize PD parameters. Besides, a novel bumpless transfer control method is proposed to realize the smooth transition between the two control modes. Finally, the simulations and experiments are conducted. The results demonstrate the effectiveness of the proposed method.

Aerodynamic and hydrodynamic force simulation for the dynamics of double-pendulum articulated offshore tower

Articulated towers are one of the class of compliant offshore structures that freely oscillates with wind and waves, as they are designed to have low natural frequency than ocean waves. The present study deals with the dynamic response of a double-pendulum articulated tower under hydrodynamic and aerodynamic loads. The wind field is simulated by two approaches, namely, single-point and multiple-point. Nonlinearities such as instantaneous tower orientation, variable added mass, fluctuating buoyancy, and geometrical nonlinearities are duly considered in the analysis. Hamilton's principle is used to derive the nonlinear equations of motion (EOM). The EOM is solved in the time domain by using the Wilson-Θ method. The maximum, minimum, mean, and standard deviation and salient power spectral density functions (PSDF) of deck displacement, bending moment, and central hinge shear are drawn for high and moderate sea states. The outcome of the analyses shows that tower response under multiple-point wind-field simulation results in lower responses when compared to that of single-point simulation.

SINDy-PI: a robust algorithm for parallel implicit sparse identification of nonlinear dynamics.

Accurately modelling the nonlinear dynamics of a system from measurement data is a challenging yet vital topic. The sparse identification of nonlinear dynamics (SINDy) algorithm is one approach to discover dynamical systems models from data. Although extensions have been developed to identify implicit dynamics, or dynamics described by rational functions, these extensions are extremely sensitive to noise. In this work, we develop SINDy-PI (parallel, implicit), a robust variant of the SINDy algorithm to identify implicit dynamics and rational nonlinearities. The SINDy-PI framework includes multiple optimization algorithms and a principled approach to model selection. We demonstrate the ability of this algorithm to learn implicit ordinary and partial differential equations and conservation laws from limited and noisy data. In particular, we show that the proposed approach is several orders of magnitude more noise robust than previous approaches, and may be used to identify a class of ODE and PDE dynamics that were previously unattainable with SINDy, including for the double pendulum dynamics and simplified model for the Belousov–Zhabotinsky (BZ) reaction.

Dynamic analysis and time optimal anti-swing control of double pendulum bridge crane with distributed mass beams

The existing anti-swing control of crane is mostly used for the single pendulum system, but in practical application, specific payload shape and hoisting mechanism will induce to complex double pendulum dynamics. In this paper, the double pendulum crane with distributed mass beams(DMB) is considered. Based on the dynamic analysis, the DMB oscillations composed of the quasi-periodic single pendulum around the drum and the second-mode pendulum around the hook are obtained. And their period calculation equations can be acquired. On this basis, the time optimal control trajectory is planed to eliminate the oscillation of the quasi-periodic single pendulum by using the minimum principle. And the oscillation suppression strategy of the second-mode pendulum is designed with the step response law of the oscillation system. Finally, combining the time optimal control trajectory and the oscillation suppression strategy, the time optimal anti-swing control method is established. Simulation and experimental results demonstrate that the proposed method can suppress the DMB oscillation in short time. The adaptability of the control method to the parameters of suspension length and DMB quality is verified. Besides, the accelerated/decelerated time and distance traveled by the proposed controller is less than that of input shaping and command smoothing. Finally, the presented method is applied to the bridge crane in the rolling mills.

Is it really chaos? The complexity of transient dynamics of double pendula

In this paper, we re-examine the dynamics of double pendulum in numerical simulations and experimental observations. Typical types of behaviors of the parametrically excited double pendula are presented, including chaos, rotations and periodic oscillations, and the bifurcation analysis is performed, exhibiting complex transitions from one type of motion into another. The character of the observed dynamics is analyzed using Lyapunov exponents, which confirms the hyperchaotic nature of the system. Particular attention is paid to the transient behaviors, showing that the length of the irregular motion can be extremely sensitive to both parameters and initial conditions. Apart from the single double pendulum, we consider also the case of two coupled double pendula, connected by a typical linear scheme. Our results show that depending on the network’s parameters, one can observe the phenomenon of a transient chaotic synchronization, during which the units spontaneously synchronize and desynchronize. The loss of coherence is strictly related to the motion of the pendula around the unstable equilibrium of the system, which has been confirmed in the scenario of pure chaotic oscillations. We determine the regions of the occurrence of transient synchronization in the coupling parameters’ plane, as well as study the statistical properties of the observed patterns. We show that the problem of determining the final dynamical attractor of the system is not straightforward.

Semi-Supervised Manifold Alignment Using Parallel Deep Autoencoders

The aim of manifold learning is to extract low-dimensional manifolds from high-dimensional data. Manifold alignment is a variant of manifold learning that uses two or more datasets that are assumed to represent different high-dimensional representations of the same underlying manifold. Manifold alignment can be successful in detecting latent manifolds in cases where one version of the data alone is not sufficient to extract and establish a stable low-dimensional representation. The present study proposes a parallel deep autoencoder neural network architecture for manifold alignment and conducts a series of experiments using a protein-folding benchmark dataset and a suite of new datasets generated by simulating double-pendulum dynamics with underlying manifolds of dimensions 2, 3 and 4. The dimensionality and topological complexity of these latent manifolds are above those occurring in most previous studies. Our experimental results demonstrate that the parallel deep autoencoder performs in most cases better than the tested traditional methods of semi-supervised manifold alignment. We also show that the parallel deep autoencoder can process datasets of different input domains by aligning the manifolds extracted from kinematics parameters with those obtained from corresponding image data.

Dynamics and Embedded Internet of Things Input Shaping Control for Overhead Cranes Transporting Multibody Payloads.

Input shaping is an Optimal Control feedforward strategy whose ability to define how and when a flexible dynamical system defined by Ordinary Differential Equations (ODEs) and computer controlled would move into its operative space, without command induced unwanted dynamics, has been exhaustively demonstrated. This work examines the issue of Embedded Internet of Things (IoT) Input Shaping with regard to real time control of multibody oscillatory systems whose dynamics are better described by differential algebraic equations (DAEs). An overhead crane hanging a double link multibody payload has been appointed as a benchmark case; it is a multibody, multimode system. This might be worst scenario to implement Input Shaping. The reasons can be found in the wide array of constraints that arise. Firstly, the reliability of the multibody model was tested on a Functional Mock-Up Interface (FMI) with the two link payload suspended from the trolley by comparing the experimental video tapping signals in time domain faced with the signals extracted from the multibody model. The FFTs of the simulated and the experimental signal contain the same frequency harmonics only with somewhat different power due to the real world light damping in the joints. The application of this approach may be extended to other cases i.e., the usefulness of mobile hydraulic cranes is limited because the payload is supported by an overhead cable under tension that allows oscillation to occur during crane motion. If the payload size is not negligible small when compared with the cable length may introduce an additional oscillatory mode that creates a multibody double pendulum. To give the insight into the double pendulum dynamics by Lagrangian methods two slender rods as payloads are analyzed dealing with the overhead crane and a composite revolute-revolute joint is proposed to model the cable of the hydraulic crane, both assumptions facilitates an affordable analysis. This allows developing a general study of this type of multibody payloads dynamics including its normal modes, modes ratios plus ranges of frequencies expected. Input Shapers were calculated for those multimodes of vibration by convolving Specified Insensitivity (SI) shapers for each mode plus a novel Direct SI-SI shaper well suited to reduce the computational requirements, i.e., the number of the shaper taps, to carry out the convolution sum in real time by the IoT device based on a single microcontroller working as the command generator. Several comparisons are presented for the shaped and unshaped responses using both the multibody model, the experimental FMI set-up and finally a real world hydraulic crane under slewing motion commanded by an analog Joystick connected by two RF modules 802.15.4 to the IoT device that carry out the convolution sum in real time. Input Shaping improves the performances for all the cases.

Optimized Input-Shaped Model Reference Control on Double-Pendulum System

This paper presents an optimized input-shaped model reference control (OIS-MRC) for limiting oscillation of multimode flexible systems. The controller is analyzed by using it to control an uncertain, time-varying double pendulum using a linear single-pendulum reference model. Single- and double-pendulum dynamics are presented, and the significant natural frequency ranges of the double pendulum are calculated. A Lyapunov control law using only the first mode states of the plant is obtained. An optimization technique is used to obtain the OIS-MRC controller parameters that realizes the shortest time duration, while meeting a set of design constraints. The oscillation suppression, control effort reduction, and disturbance rejection performances of the proposed OIS-MRC controller are tested via numerical simulations and experiments. The OIS-MRC achieves a robust oscillation suppression performance, while reducing the rise time.

An energy-optimal solution for transportation control of cranes with double pendulum dynamics: Design and experiments

Abstract Underactuated cranes play an important role in modern industry. Specifically, in most situations of practical applications, crane systems exhibit significant double pendulum characteristics, which makes the control problem quite challenging. Moreover, most existing planners/controllers obtained with standard methods/techniques for double pendulum cranes cannot minimize the energy consumption when fulfilling the transportation tasks. Therefore, from a practical perspective, this paper proposes an energy-optimal solution for transportation control of double pendulum cranes. By applying the presented approach, the transportation objective, including fast trolley positioning and swing elimination, is achieved with minimized energy consumption, and the residual oscillations are suppressed effectively with all the state constrains being satisfied during the entire transportation process. As far as we know, this is the first energy-optimal solution for transportation control of underactuated double pendulum cranes with various state and control constraints. Hardware experimental results are included to verify the effectiveness of the proposed approach, whose superior performance is reflected by being experimentally compared with some comparative controllers.

Adaptive tracking control for double-pendulum overhead cranes subject to tracking error limitation, parametric uncertainties and external disturbances

In a practical application, overhead cranes are usually subjected to system parameter uncertainties, such as uncertain payload masses, cable lengths, frictions, and external disturbances, such as air resistance. Most existing crane control methods treat the payload swing as that of a single-pendulum. However, certain types of payloads and hoisting mechanisms result in double-pendulum dynamics. The double-pendulum effects will make most existing crane control methods fail to work normally. Therefore, an adaptive tracking controller for double-pendulum overhead cranes subject to parametric uncertainties and external disturbances is developed in this paper. The proposed adaptive tracking control method guarantees that the trolley tracking error is always within a prior set of boundary conditions and converges to zero rapidly. The asymptotic stability of the closed-loop system׳s equilibrium point is assured by Lyapunov techniques and Barbalat׳s Lemma. Simulation results show that the proposed adaptive tracking control method is robust with respect to system parametric uncertainties and external disturbances.