Rotary Pendulum Research Articles

Indirect Adaptive State-Feedback Control of Rotary Inverted Pendulum Using Self-Mutating Hyperbolic-Functions for Online Cost Variation

This paper presents the development of an indirect adaptive state-feedback controller to improve the disturbance-rejection capability of under-actuated multivariable systems. The ubiquitous Linear-Quadratic-Regulator (LQR) is employed as the baseline state-feedback controller. Despite its optimality, the LQR lacks robustness against parametric uncertainties. Hence, the main contribution of this paper is to devise and retrofit the LQR with a stable online gain-adjustment mechanism that dynamically adjusts the state weighting-coefficients of LQR's quadratic cost-function via state-error dependent nonlinear-scaling functions. An original self-mutating phase-based adaptive modulation scheme is systematically formulated in this paper to self-adjust the state weighting-coefficients. The scheme employs pre-calibrated secant-hyperbolic-functions whose waveforms are dynamically reconfigured online based on the variations in magnitude and polarity of state-error variables. This augmentation dynamically alters the solution of the Riccati-Equation which modifies the state-feedback gains online. The proposed adaptation flexibly manipulates the system's control effort as the response converges to or diverges from the reference. The efficacy of proposed adaptive controller is validated by conducting hardware-in-the-loop experiments to vertically stabilize the QNET 2.0 Rotary Pendulum system. As compared to the standard LQR, the proposed adaptive controller renders rapid transits in system's response with improved damping against oscillations, while maintaining its asymptotic-stability, under bounded exogenous disturbances.

Piecewise sampled-data control of rotational pendulum with reference input along vibration manipulation function

Piecewise sampled-data control of rotational pendulum with reference input along vibration manipulation function

Reference-input modification for finite-time settling control of rotational pendulum along vibration manipulation function

Reference-input modification for finite-time settling control of rotational pendulum along vibration manipulation function

Stabilization of Furuta Pendulum Using Nonlinear MPC

Abstract This paper is devoted to design of a non-linear model predictive controller (NMPC), which will swing-up and stabilize an inverse rotary pendulum known as the Furuta Pendulum. This paper presents a simulation validation of the NMPC strategy using a full-fidelity non-linear mathematical model of the Furuta pendulum obtained from the Euler-Lagrange motion equations. The NMPC strategy was implemented in MATLAB using the MATMPC toolbox.

A rotational pendulum based electromagnetic/triboelectric hybrid-generator for ultra-low-frequency vibrations aiming at human motion and blue energy applications

There is abundance of low-frequency biomechanical and blue energy that can be harvested from our ambient environment. Utilization of these energy plays a significant role for the sustainability of the wearable devices and environmental monitoring devices. Here, we presented a rotational pendulum triboelectric-electromagnetic hybrid generator (RPHG) which is able to scavenge low-frequency (<5 Hz) vibration energy from both arbitrary human motion and ocean waves. Four stacked disk-shaped rotor magnets are introduced into the generator for the rotational motion. The pendulum rotor can easily rotate around the bearing by gravity or inertia force with a broadband frequency excitation, and coupled with surrounding coils to operate electromagnetic generator (EMG). In the meantime, triboelectric nanogenerator (TENG) is realized by utilizing the rotor to trigger the contact-separation mode of the copper ring and surrounding fluorinated ethylene propylene (FEP) film. The hybridized generator is very sensitive to irregular human motion and ultra-low-frequency ocean waves. After the optimization of the configuration of the hybrid generator, the maximum power density of 3.25 W/m2 and 79.9 W/m2 are obtained at a driving frequency of 2 Hz and amplitude of 14 cm by the TENG and EMG, respectively, which can charge a capacitor of 22 μF to 7 V rapidly in 30 s. The RPHG can light up about more than hundreds of commercial lights emitting diodes (LEDs). Then, the RPHG was placed on human body and tested by running on treadmill and rope skipping. The open-circuit voltage of the EMG can be about 9 V and that of TENG can reach 150 V. Finally, the RPHG was tested on a homemade buoy in Tai Lake. When the wave frequency is about 1 Hz and the wave height is 20 cm, the open-circuit voltage of EMG can reach over 3 V and that of TENG can reach 80 V regardless of whether the RPHG is placed vertically or horizontally. Generally, the proposed device demonstrates the effectiveness towards energy harvesting from ambient environment of multidirectional and wide frequency range, i.e., human motion and blue energy, showing the great potential of being a sustainable power source for those wearable devices and monitoring devices in blue ocean.

Almost periodic motion planning and control for double rotary pendulum with experimental validation

AbstractThe aim is to develop a systematic procedure for planning feasible motions for a double rotary pendulum. This pendulum has one directly actuated horizontal link and two passive links, moving in a rotating vertical plane. We plan a nontrivial oscillatory motion for the passive links that is consistent with the horizontal link rotating at a given average speed and also design a stabilizing controller to approximately induce such a motion. For the motion planning, a numerical optimization procedure is proposed in the form of a sequence of three simpler problems to systematically derive initial guesses for the final optimization search. For the controller design, firstly the system is linearized along a nominal trajectory, and then a parametrized family of candidate stabilizing controllers is designed. For each set of parameters, a necessary and sufficient stability condition can be checked for the derived linear time varying periodic system. Therefore, a numerical optimization procedure is used to find the controller gain for the linear system based on the stability condition. The performance of the closed‐loop system is illustrated via numerical simulations and verified via experiments with an educational platform produced by PendCon, demonstrating achieving oscillations with required characteristics. However, the formal proofs for convergence and even for existence of almost periodic solutions are left for future studies.

Stabilization of a double inverted rotary pendulum through fractional order integral control scheme

Rotary double inverted pendulum is a highly nonlinear complex system and requires a high performance controller for its control. Using gain matrix which is obtained through state feedback technique may create complexity while removing the steady-state error for all states. Incorporating an integral action can be an alternative for these errors. Therefore, a state space–based fractional order controller with fractional integral action is designed and tested on a rotary double inverted pendulum in this article. The fractional integral controller is designed based on Bode’s ideal transfer function. Two degree of freedom is considered for tuning purpose as well. The integer order controller based on the state space approach is also shown for comparison. Simulation and experimental results are presented for both controllers of this rotary double inverted pendulum system.

Adaptive sliding mode control for asymptotic stabilization of underactuated mechanical systems via higher-order nonlinear disturbance observer

The development of a nonlinear controller of stabilization of underactuated mechanical systems (UMSs) is a challenging endeavor due to a larger number of output variables to be controlled than the control input space. This paper proposes an adaptive sliding mode control based on a higher-order nonlinear disturbance observer (HONDO) for stabilizing the rotational pendulum (RP) system falling under the class of UMSs. Firstly, the HONDO is designed in such a way that it can improve accuracy in estimations with its incremental order. As a result, the proposed controller obtained from the sliding surface which is developed with system’s states and estimations, forces the states attaining the sliding mode and hence keeps them to their origin forever against disturbances. To achieve this, the sliding coefficients are obtained using inertia matrix of the system. The zero dynamics is stabilized by the proposed controller. This alleviates the chattering problem in the control input. Finally, numerical performance on the underactuated RP model is analyzed to show the efficiency of the proposed controller and it is compared with the established control technique found in the literature.

Periodic motion planning and control for double rotary pendulum via virtual holonomic constraints

Periodic motion planning for an under-actuated system is rather difficult due to differential dynamic constraints imposed by passive dynamics, and it becomes more difficult for a system with higher underactuation degree, that is with a higher difference between the number of degrees of freedom and the number of independent control inputs. However, from another point of view, these constraints also mean some relation between state variables and could be used in the motion planning. We consider a double rotary pendulum, which has an underactuation degree 2. A novel periodic motion planning is presented based on an optimization search. A necessary condition for existence of the whole periodic trajectory is given because of the higher underactuation degree of the system. Moreover this condition is given to make virtual holonomic constraint &#x0028 VHC &#x0029 based control design feasible. Therefore, an initial guess for the optimization of planning a feasible periodic motion is based on this necessary condition. Then, VHCs are used for the system transformation and transverse linearization is used to design a static state feedback controller with periodic matrix function gain. The controller gain is found through another optimization procedure. The effectiveness of initial guess and performance of the closed-loop system are illustrated through numerical simulations.

Adaptive-Horizon Iterative UFIR Filtering Algorithm With Applications

The unbiased finite impulse response (UFIR) filter has strong engineering features for industrial applications, because it does not require the noise statistics and initial values. This filter minimizes the mean square error (MSE) on the optimal horizon of $N_\mathrm{opt}$ points, and the determination of $N_\mathrm{opt}$ is an important issue. In this paper, a new strategy is proposed to adaptively estimate $N_\mathrm{opt}$ in real time. A concept of the maximum allowed horizon is introduced, referring to the fact that the current iteration with large horizon contains data from the previous iterations with small horizons. That allows selection of the target horizon in a single cycle of iterations and a design of the adaptive-horizon UFIR (AUFIR) filter. The proposed AUFIR filter is tested by a rotary pendulum system and a 3-degree-of-freedom (DOF) helicopter system. Higher accuracy and robustness of the AUFIR filter are demonstrated in a comparison with the Kalman filter (KF), adaptive KF, and UFIR filter.

Periodic Regime of Motion of a Vibratory Robot under a Control Constraint

A class of motions of a vibratory robot consisting of a body with a pendulum inside it, on the angular acceleration of which a constraint is imposed, is considered. The limits of the angular acceleration correspond to the technical characteristics of the motor providing the pendulum rotation. Some phase constraints are imposed on the system and the control is proposed which satisfies them and provides periodic motions of the robot. The control during each period consists of seven intervals; choice of control at each of them is conditioned by velocity maximization. For considered control law numerical experiments are carried out and comparison with other controls is presented.

Coupled State-Dependent Riccati Equation Control for Continuous Time Nonlinear Mechatronics Systems

This paper considers a novel coupled state-dependent Riccati equation (SDRE) approach for systematically designing nonlinear quadratic regulator (NLQR) and H∞ control of mechatronics systems. The state-dependent feedback control solutions can be obtained by solving a pair of coupled SDREs, guaranteeing nonlinear quadratic optimality with inherent stability property in combination with robust L2 type of disturbance reduction. The derivation of this control strategy is based on Nash's game theory. Both finite and infinite horizon control problems are discussed. An under-actuated robotic system, Furuta rotary pendulum, is used to examine the effectiveness and robustness of this novel nonlinear control approach.

Sliding mode control for underactuated mechanical systems via nonlinear disturbance observer: stabilization of the rotational pendulum

This paper presents design of the sliding mode controller (SMC) via a nonlinear disturbance observer (NDO) for stabilization of the rotational pendulum. The mathematical model is depicted by four first-order nonlinear differential equations with two outputs and single input. The linear model is obtained by linearizing nonlinear equations. This linearized version is found to be highly unstable. The applied NDO is capable of estimating extremely nonlinear terms (disturbance) present in the model. Then, the NDO based SMC is designed using linear model and disturbance estimation, and hence is applied to the original nonlinear system. The developed controller is enough to drive a sliding surface (which holds both actuated and unactuated dynamics) attaining the sliding mode against severe nonlinearities existing in the model, in addition large parameter variations and/or external disturbance. The sliding coefficients are determined using the linear quadratic regulator (LQR) technique. The obtained controller parameters can establish the zero dynamics as of sliding mode dynamics asymptotically. Also, the proposed control algorithm has potential to alleviate the chattering phenomenon occurred in the control signal. Overall, the control scheme expands the stability region of an arm (in horizontal half-plane) and a pendulum (over vertical half-plane) both for large initial conditions. Lastly, numerical simulations and quantitative measurements are validated with an established controller in the literature.

Optimistic planning with an adaptive number of action switches for near-optimal nonlinear control

We consider infinite-horizon optimal control of nonlinear systems where the control actions are discrete, and focus on optimistic planning algorithms from artificial intelligence, which can handle general nonlinear systems with nonquadratic costs. With the main goal of reducing computations, we introduce two such algorithms that only search for constrained action sequences. The constraint prevents the sequences from switching between different actions more than a limited number of times. We call the first method optimistic switch-limited planning (OSP), and develop analysis showing that its fixed number of switches S leads to polynomial complexity in the search horizon, in contrast to the exponential complexity of the existing OP algorithm for deterministic systems; and to a correspondingly faster convergence towards optimality. Since tuning S is difficult, we introduce an adaptive variant called OASP that automatically adjusts S so as to limit computations while ensuring that near-optimal solutions keep being explored. OSP and OASP are analytically evaluated in representative special cases, and numerically illustrated in simulations of a rotational pendulum. To show that the algorithms also work in challenging applications, OSP is used to control the pendulum in real time, while OASP is applied for trajectory control of a simulated quadrotor.

Rotations of Pendulum When Its Pivot Oscillates Chaotically

This paper investigates the possibility of energy generation via pendulum rotations when the source of vertical excitation is chaotic in nature. The investigations are conducted using an additional height-adjustable mechanism housing a secondary spring to optimize a configuration of experimental pendulum setup. Chaotic oscillations of the pendulum pivot are made possible at certain excitation conditions due to a piecewise-linear stiffness characteristic introduced by the modification. A velocity control method is applied to maintain the rotational motion of the pendulum as it interacts with the vertical oscillator. The control input is affected by a motor, and a generator is used to quantify the energy extraction. The experimental results imply the feasibility of employing a pendulum device in a chaotic vibratory environment for energy harvesting purpose.

Enhancing power generation of floating wave power generators by utilization of nonlinear roll-pitch coupling

We propose utilization of the nonlinear coupling between the roll and pitch motions of wave energy harvesting vessels to increase their power generation by orders of magnitude. Unlike linear vessels that exhibit unidirectional motion, our vessel undergoes both pitch and roll motions in response to frontal waves. This significantly magnifies the motion of the vessel and thus improves the power production by several orders of magnitude. The ocean waves result in roll and pitch motions of the vessel, which in turn causes rotation of an onboard pendulum. The pendulum is connected to an electric generator to produce power. The coupled electro-mechanical system is modeled using energy methods. This paper investigates the power generation of the vessel when the ratio between pitch and roll natural frequencies is about 2 to 1. In that case, a nonlinear energy transfer occurs between the roll and pitch motions, causing the vessel to perform coupled pitch and roll motion even though it is only excited in the pitch direction. It is shown that co-existence of pitch and roll motions significantly enhances the pendulum rotation and power generation. A method for tuning the natural frequencies of the vessel is proposed to make the energy generator robust to variations of the frequency of the incident waves. It is shown that the proposed method enhances the power output of the floating wave power generators by multiple orders of magnitude. A small-scale prototype is developed for the proof of concept. The nonlinear energy transfer and the full rotation of the pendulum in the prototype are observed in the experimental tests.

Periodic motion planning and control for underactuated mechanical systems

ABSTRACTWe consider the problem of periodic motion planning and of designing stabilising feedback control laws for such motions in underactuated mechanical systems. A novel periodic motion planning method is proposed. Each state is parametrised by a truncated Fourier series. Then we use numerical optimisation to search for the parameters of the trigonometric polynomial exploiting the measure of discrepancy in satisfying the passive dynamics equations as a performance index. Thus an almost feasible periodic motion is found. Then a linear controller is designed and stability analysis is given to verify that solutions of the closed-loop system stay inside a tube around the planned approximately feasible periodic trajectory. Experimental results for a double rotary pendulum are shown, while numerical simulations are given for models of a spacecraft with liquid sloshing and of a chain of mass spring system.

Design and simulation of a vertical broadband seismometer

In this paper, a new vertical broadband seismometer is designed, simulated, and evaluated. The proposed seismometer has a bandwidth from 40 s to 50 Hz and sensitivity to ground velocity of 1500 V’/m/s. The mechanical suspension used in the proposed design is a mass-leaf spring rotational pendulum. The displacement of the seismic mass is sensed by a laser interferometer and then forwarded to an electronic force feedback system which feeds a linear actuator to keep the mass almost fixed in its position. The current needed to fix the mass in its position is an indication of the ground acceleration. The performance of the simulated seismometer is evaluated by applying a data record of two actual earthquakes to the simulated model and comparing its response to the earthquakes with that of a real seismometer (Trillium-40) having the same bandwidth and sensitivity. The proposed seismometer has performed within 0.4 % residual variance for the local event and 0.03 % residual variance for the regional one compared to Trillium-40.

An vibration energy harvester with broadband and frequency-doubling characteristics based on rotary pendulums

This paper describes a vibration energy harvester with broadband and frequency-doubling characteristics based on rotary pendulums. The harvester is composed of a Terfenol-D/PMN-PT/Terfenol-D magnetoelectric transducer and a rotary pendulum embedded with six magnets. An analytical model is developed to analyze the nonlinear vibration, the frequency response and electrical-output characteristics of the harvester. A prototype is fabricated and tested. The experimental results are in agreement with the analytical results. The results show that the harvester has broadband and frequency-doubling characteristics and can produce high output power at low frequency. The prototype has a 3 db bandwidth of 3.2Hz and an output power of 970.2μW at 0.5g (1g=9.8m/s2) RMS acceleration with a resonant frequency of 14.8Hz.

A broadband vibration energy harvester using double transducers and pendulum-type structures

As cantilever-based vibration energy harvesters are easily fractured under large amplitude vibration excitation, in this paper we present a vibration energy harvester based on a pendulum-type structure with broadband and frequency-doubling characteristics. The harvester consists of two Terfenol-D/PMN-PT/Terfenol-D magnetoelectric transducers and a rotary pendulum embedded with six magnets. These six magnets are arranged into an optimum configuration and can produce a concentrated flux gradient which makes the magnetoelectric transducers generate a high power. While the two transducers are used to further improve the output power and power density of the harvester without increasing the volume of the harvester. The rotary pendulum of the harvester changes linear vibration into a back-and-forth swing of the rotary pendulum. When the rotary pendulum swings, the stress is hardly generated in the interior of the rotary pendulum. Therefore the rotary pendulum is not easily fractured under the large amplitude vibration. Therefore the proposed pendulum-based vibration energy harvester is suitable for scavenging the large amplitude ambient vibration energy. The swing equation of the rotary pendulum is established. The nonlinear dynamic equation of the rotary pendulum is solved by the Lindstedt-Poincar method. The frequency response characteristic and the mechano-magneto-electric transduction characteristic of the harvester at resonance are analyzed by combining the swing equation of the harvester with the magnetoelectric characteristics of the magnetoelectric transducers. The spectrum of the output voltage waveform of the harvester is discussed. The analytical and experimental results indicate that the harvester has broadband and frequency-doubling characteristics. The broadband characteristic of the harvester is derived from the nonlinear magnetic force between the magnets and magnetoelectric transducers. The voltage frequency-doubling characteristic is derived from the nonlinearity of the magnetic field produced by the magnets. It does not need frequency conversion mechanism for the proposed harvester, so the proposed harvester has some advantages, such as simple structure and easy manufacture. Under 1 g (1 g = 9.8 m/s2) RMS vibration acceleration excitation, the measured maximum RSM voltage and the resonant frequency of the prototype are 90.9 V and 16.9 Hz, respectively. The 3 dB bandwidth for the sweep-down condition is 4.8 Hz from 16.9 Hz to 21.7 Hz and that for the sweep-up condition is 2.1 Hz from 22.8 Hz to 24.9 Hz. Compared with other harvesters, the proposed harvester has a wide relative bandwidth. The load output power of the prototype reaches 3.569 mW across a 1.9 M optimal resistor at resonant frequency of 16.9 Hz with 1 g RMS vibration acceleration. The output RMS powers of the prototype across 1.9 M resistor are 0.156 mW, 0.6863 mW, 1.777 mW at 0.3 g, 0.5 g and 0.7 g with resonance, respectively. The proposed harvester can effectively improve the output powers at lower frequency vibrations for its two transducers, broadband and frequency-doubling characteristics.