Pendulum Problem Research Articles

Design of reduced-order observer based steady state optimal linear-quadratic feedback controllers

In this paper we formulate and solve a problem encountered in engineering applications when a linear-quadratic (LQ) optimal feedback controller uses state estimates obtained via a reduced-order observer. Due to the use of state estimates instead of the actual state variables, the optimal quadratic performance is degraded in a pretty complex manner. In the paper, we show how to find the exact formula for the optimal performance degradation (optimal performance loss) in linear time invariant systems for the steady state case (infinite horizon optimization problem). The optimal performance loss is obtained in terms of solution of a reduced-order algebraic Lyapunov equation whose dimension is equal to the dimension of the reduced-order observer. The quantities that impact the performance criterion loss are identified. Practical examples (an inverted pendulum on a cart and an aircraft) show that the optimal performance loss can be very significant in some applications, and even very high in the presented linear-quadratic optimal control of an inverted pendulum problem. We have shown, using the derived formula, how the optimal performance loss can be considerably reduced by properly choosing the reduced-order observer initial conditions via the least square method.

Trajectories of two-dimensional harmonic oscillators in a rotating frame: application to Foucault pendulum problem

Trajectories of two-dimensional harmonic oscillators in a rotating frame: application to Foucault pendulum problem

A robust Gated-PINN to resolve local minima issues in solving differential algebraic equations

Physics-Informed Neural Network (PINN) has emerged as a promising tool for solving various physical problems with differential equations. However in practice, PINN often suffers from the local minima issue while solving problems with minimal initial conditions. In this paper, we propose a robust Gated-PINN to address this issue, which blends numerical methods and PINN. The Gated-PINN overcomes the local minima issue by controlling the flow of information of the neural network similar to a numerical method. We first investigate the use of conventional PINN to the Cartesian-coordinate simple pendulum problem with minimal initial conditions, as one of basic differential algebraic equation (DAE) problem. Our results show the advantage of PINN over existing numerical methods in that it does not require sophisticated mathematical techniques such as index reduction. But we also observe that conventional PINN can lead to inaccurate solutions; such solutions partially satisfy the differential equation requirement, but does not meet the given initial condition and fail to further improve. We demonstrate the effectiveness of the proposed Gated-PINN by showing that it yields accurate solutions, such that for a pendulum of length 1, the mean Euclidean error between the Gated-PINN model and the traditional numerical method model is less than 0.01 and pointwise maximum Euclidean error is less than 0.04. Moreover, Gated-PINN can operate without any complicated index reduction, and unlike conventional PINN, accurate solution can be obtained consistently without falling into a local minima. Overall, our study presents the potential of the Gated-PINN for solving DAE problems and provides a valuable insight into the challenges and limitations of using PINN for solving physical problems.

A discrete adjoint gradient approach for equality and inequality constraints in dynamics

The optimization of multibody systems requires accurate and efficient methods for sensitivity analysis. The adjoint method is probably the most efficient way to analyze sensitivities, especially for optimization problems with numerous optimization variables. This paper discusses sensitivity analysis for dynamic systems in gradient-based optimization problems. A discrete adjoint gradient approach is presented to compute sensitivities of equality and inequality constraints in dynamic simulations. The constraints are combined with the dynamic system equations, and the sensitivities are computed straightforwardly by solving discrete adjoint algebraic equations. The computation of these discrete adjoint gradients can be easily adapted to deal with different time integrators. This paper demonstrates discrete adjoint gradients for two different time-integration schemes and highlights efficiency and easy applicability. The proposed approach is particularly suitable for problems involving large-scale models or high-dimensional optimization spaces, where the computational effort of computing gradients by finite differences can be enormous. Three examples are investigated to validate the proposed discrete adjoint gradient approach. The sensitivity analysis of an academic example discusses the role of discrete adjoint variables. The energy optimal control problem of a nonlinear spring pendulum is analyzed to discuss the efficiency of the proposed approach. In addition, a flexible multibody system is investigated in a combined optimal control and design optimization problem. The combined optimization provides the best possible mechanical structure regarding an optimal control problem within one optimization.

Active Exploration Deep Reinforcement Learning for Continuous Action Space with Forward Prediction

The application of reinforcement learning (RL) to the field of autonomous robotics has high requirements about sample efficiency, since the agent expends for interaction with the environment. One method for sample efficiency is to extract knowledge from existing samples and used to exploration. Typical RL algorithms achieve exploration using task-specific knowledge or adding exploration noise. These methods are limited to current policy improvement level and lack of long-term planning. We propose a novel active exploration deep RL algorithm for the continuous action space problem named active exploration deep reinforcement learning (AEDRL). Our method uses the Gaussian process to model dynamic model, enabling the probability description of prediction sample. Action selection is formulated as the solution of the optimization problem. Thus, the optimization objective is specifically designed for selecting samples that can minimize the uncertainty of the dynamic model. Active exploration is achieved through long-term optimized action selection. This long-term considered action exploration method is more guidance for learning. Enable intelligent agents to explore more interesting action spaces. The proposed AEDRL algorithm is evaluated on several robotic control task including classic pendulum problem and five complex articulated robots. The AEDRL can learn a controller using fewer episodes and demonstrates performance and sample efficiency.

Study of damped oscillations using Phyphox and Arduino controlled Hall-sensor

The paper presents physics education activities organized around the topic of damped oscillations. We used the Phyphox smartphone application for secondary school physics classes. These activities served as a basis for a physics education workshop, where an Arduino-controlled Hall-sensor and the Phyphox Magnetometer were presented. The problem of a damped pendulum, a vertical oscillation in water, and an LCr oscillating circuit was examined as part of a Phyphox project. Mechanical and electromagnetic damped oscillations can be demonstrated with our devices. Using our data, we could compare Hall-sensors of different devices, estimate some characteristics of the waves and help plan an LCr oscillating circuit. Activities for secondary school physics classes are suggested, based on the pedagogical goals.

On stability analysis of nonlinear ADRC-based control system with application to inverted pendulum problems

On stability analysis of nonlinear ADRC-based control system with application to inverted pendulum problems

Nonlinear optimal control for the rotary double inverted pendulum

AbstractThe control problem of the rotary double inverted pendulum (double Furuta pendulum) is nontrivial because of underactuation and strong nonlinearities in the associated state‐space model. The system has three degrees of freedom (one actuated and two unactuated joints) while receiving only one control input. In this article, a novel nonlinear optimal (H‐infinity) control approach is developed for the dynamic model of the rotary double inverted pendulum. First, the dynamic model of the double pendulum undergoes approximate linearization with the use of first‐order Taylor series expansion and through the computation of the associated Jacobian matrices. The linearization process takes place at each sampling instance around a temporary operating point which is defined by the present value of the system's state vector and by the last sampled value of the control inputs vector. At a next stage a stabilizing H‐infinity feedback controller is designed. To compute the controller's feedback gains an algebraic Riccati equation has to be solved at each time‐step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis. To implement state estimation‐based control without the need to measure the entire state vector of the rotary double‐pendulum the H‐infinity Kalman filter is used as a robust state observer. The nonlinear optimal control method achieves fast and accurate tracking of setpoints by all state variables of the rotary double inverted pendulum under moderate variations of the control input.

On the sample complexity of actor-critic method for reinforcement learning with function approximation

Reinforcement learning, mathematically described by Markov Decision Problems, may be approached either through dynamic programming or policy search. Actor-critic algorithms combine the merits of both approaches by alternating between steps to estimate the value function and policy gradient updates. Due to the fact that the updates exhibit correlated noise and biased gradient updates, only the asymptotic behavior of actor-critic is known by connecting its behavior to dynamical systems. This work puts forth a new variant of actor-critic that employs Monte Carlo rollouts during the policy search updates, which results in controllable bias that depends on the number of critic evaluations. As a result, we are able to provide for the first time the convergence rate of actor-critic algorithms when the policy search step employs policy gradient, agnostic to the choice of policy evaluation technique. In particular, we establish conditions under which the sample complexity is comparable to stochastic gradient method for non-convex problems or slower as a result of the critic estimation error, which is the main complexity bottleneck. These results hold in continuous state and action spaces with linear function approximation for the value function. We then specialize these conceptual results to the case where the critic is estimated by Temporal Difference, Gradient Temporal Difference, and Accelerated Gradient Temporal Difference. These learning rates are then corroborated on a navigation problem involving an obstacle and the pendulum problem which provide insight into the interplay between optimization and generalization in reinforcement learning.

Synthesis of a Calibration-Free Visual Feedback Controller for an Inverted Pendulum Using a Fisheye Lens

This article considers the synthesis of feedback controllers for a class in which strong robust stability is required against sensing errors. The motivation dealing with this class arises from the stabilization problem of a cart-driven inverted pendulum by visual feedback control using a fisheye lens. While fisheye lenses guarantee a wide-angle of view, a large image distortion at off-center area is inevitable. Therefore this study first derives a geometric measurement model of a pendulum angle using a general fisheye lens model. A new design method for robust controllers for our problem is then proposed. Finally, a robust feedback controller that allows lens distortion is designed. The effectiveness of the controller based on the proposed model and the proposed design method is verified through numerical simulations and experiments.

Robust linear quadratic regulator applied to an inverted pendulum

AbstractThe natural instability of an inverted pendulum and its dynamics richness, in terms of nonlinearity, provide a nice apparatus to reproduce behaviors of analogous systems. In this way, it is useful to perform benchmark tests for new control approaches developed. In this paper, we address the main inverted pendulum problems: pendulum stabilization, tracking, and catching swing‐up control. We show how robust recursive, control and filtering, techniques improve the system performance. They are developed to solve stochastic problems based on deterministic approaches, in order to decrease the worst influence of uncertainties. Experimental results of the proposed robust approach provide robust stability and performance despite parametric uncertainties, disturbances, and noise effects.

On The Power Series Solution to The Nonlinear Pendulum

Summary The exact solution to the simple pendulum problem has long been known in terms of Jacobi elliptic functions, of which an efficient numerical evaluation is standard in most scientific computing software packages. Alternatively, and as done in V. Fairén, López and L. Conde (Power series approximation to solutions of nonlinear systems of differential equation, Am. J. Phys. 56 (1988) 57–61], the pendulum equation can be analytically solved exactly by the power series solution method. Although recursive formulae for the series coefficients were provided in V. Fairén et al., the series itself—as well as the optimal location about which an expansion should be chosen to maximize the domain of convergence—has not yet been examined, and this is provided here. By virtue of its representation as an elliptic function, the pendulum function has singularities that lie off of the real axis in the complex time plane. This, in turn, imposes a radius of convergence on the power series solution in real time. By choosing the expansion point at the top of the trajectory, the power series converges all the way to the bottom of the trajectory without being affected by these singularities. We provide an exact resummation of the pendulum series that accelerates the series’ convergence uniformly from the top to the bottom of the trajectory. We also provide the formulae needed to extend the pendulum series for all time via symmetry. The pendulum problem, in its relative simplicity, provides an explicit demonstration of the effect of singularity structure and initial condition location on convergence properties of power series solutions.

Dynamical Stabilization of Multiplet Supercurrents in Multiterminal Josephson Junctions.

The dynamical properties of multiterminal Josephson junctions (MT-JJs) have attracted interest, driven by the promise of new insights into synthetic topological phases of matter and Floquet states. This effort has culminated in the discovery of Cooper multiplets in which the splitting of a Cooper pair is enabled via a series of Andreev reflections that entangle four (or more) electrons. Here, we show that multiplet resonances can also emerge as a consequence of the three-terminal circuit model. The supercurrent appears due to correlated phase dynamics at values that correspond to the multiplet condition nV1 = -mV2 of applied bias. Multiplet resonances are seen in nanofabricated three-terminal graphene JJs, analog three-terminal JJ circuits, and circuit simulations. The stabilization of the supercurrent is purely dynamical, and a close analog to Kapitza's inverted pendulum problem. We describe parameter considerations that optimize the detection of the multiplet lines both for design of future devices.

Integrability technique for fluid flow induced deformation of a boundary hair

A solution method for a nonlinear integro-differential equation is proposed and conducted on the problem of a boundary hair exposed to shear flow. While bearing resemblance to the pendulum problem from mechanics, this equation has not been analytically solved until now. We obtain this solution by treating the integral term as a parameter which we later constrain after obtaining the prospective trajectories. Our analytic solution circumvents the difficulties one can have with finite difference along with other numerical approaches and we argue that it could be used as a basis for understanding weakly time-dependent systems.

Angular Momentum Control for Preventing Rollover of a Heavy Vehicle

In this article, gyroscopes and flywheels are used to prevent the rollover of a vehicle due to external forces. The rollover preventing performance of the flywheels for a heavy trailer (an inverted pendulum problem) is investigated at a high road bank angle risk. Two control moment gyroscopes (CMG) and a reaction wheel are controlled by proportional torques to keep the vehicle in a stable motion in the vertical position. The reaction wheel was used only to eliminate the dissipation energy of damper. By using the energy stored in the flywheels of gyroscopes, the sprung mass of a vehicle can make a stable oscillating motion of small amplitude, standing upright without tipping over. The optimum flywheel speed was derived from the frequency equations with the appropriate controller gain. Most importantly, the required angular momentum to keep the vehicle upright without tipping over depends on the amplitude of the gimbal oscillation and the frequency. It has also been observed that Matlab simulations of Lagrangian equations and simulations modeled in RecurDyn software are in perfect harmony.

An optimization method for the inverted pendulum problem based on deep reinforcement learning

The inverted pendulum problem is a classical problem. The inverted pendulum starting at a random position keeps moving upwards and aims to reach an upright position. The problem has been solved through some methods based on deep reinforcement learning (DRL) such as Deep Deterministic Policy Gradient (DDPG). However, DDPG also has disadvantages. Deterministic policy is not conducive to action exploration. Moreover, the Q value needs to be estimated reasonably accurately for the policy to be accurate. Nevertheless, at the beginning of the learning, there is a certain difference in the Q value estimation, and the parameters learned at this time are easy to deviate. Therefore, this paper combining AdaBound with DDPG algorithm proposes an optimization method for the inverted pendulum problem, and compares the performance with that of four published baselines. The experimental results show that for the inverted pendulum problem, the proposed method outperforms the above four baselines to a certain extent.

An anti-inertial motion bias explains people discounting inertial motion of carried objects

In this paper we propose an anti-inertial motion (AIM) bias that can explain several intuitive physics beliefs including the straight-down belief and beliefs held concerning the pendulum problem. We show how the AIM bias also explains two new beliefs that we explore - a straight-up-and-down belief as well as a straight-out/backward bias that occurs for objects traveling in one plane that are then thrown in another plane, ostensibly affording a greater opportunity for perception of canonical motion. We then show how the AIM bias in general is invariant across perceived/imagined speed of the object carrier, only altering percentages of straight-out from backward responses, and why occluding the carrier once the object is released into a second plane does not result in more veridical perception. The AIM bias serves as a simple explanation for a family of beliefs including those in the current paper as well as those shown in previous work.

Pendulum: the partial and global approach

The pendulum was an important scientific instrument in the 17th century. It became a typical textbook problem in the 18th century. After the introduction of vectors in physics in the 1890s, the pendulum problem started to be progressively solved in the manner we know nowadays from introductory mechanics courses. Starting from F = ma and using the tangential component of weight, we arrive at the equation of motion. The other part of F = ma, which concerns the weight radial component, tension and centripetal force, has in no way deserved the same attention from textbook writers, if any. Physics education research has, however, pointed out relevant aspects of the phenomenon related with the latter part of the equation. The present paper presents an experiment, which reflects those relevant aspects, and provides a mathematical insight into the phenomenon suitable for the high school student.

Direct adaptive control for nonlinear systems using a TSK fuzzy echo state network based on fractional-order learning algorithm

This paper presents a new Takagi-Sugeno-Kang fuzzy Echo State Neural Network (TSKFESN) structure to design a direct adaptive control for uncertain SISO nonlinear systems. The proposed TSKFESN structure is based on the echo state neural network framework containing multiple sub-reservoirs. Each sub-reservoir is weighted with a TSK fuzzy rule. The adaptive law of the TSKFESN-based direct adaptive controller is derived by using a fractional-order sliding mode learning algorithm. Moreover, the Lyapunov stability criterion is employed to verify the convergence of the fractional-order adaptive law of the controller parameters. The evaluation of the proposed direct adaptive control scheme is verified using two case studies, the regulation problem of a torsional pendulum and the speed control of a direct current (DC) machine as a real-time application. The simulation and the experimental results show the effectiveness of the proposed control scheme.

The kinematics behaviour of coupled pendulum using differential transformation method

The topic of coupled oscillations is rich in physical content which is both interesting and complex. In this research work we aimed to study the kinematics behaviour of an important physical system known as coupled pendulum, in which two identical pendulums were coupled together by a spring. The Lagrangian of the system was first constructed, then the equations of motion have been derived. Analytical solutions were obtained for these equations, and in addition we apply a reliable algorithm based on an adaptation of the standard differential transformation method is presented, which is the multi-step differential transformation method to obtain numerical solutions for the considered system for some selected initial conditions. The solutions of the equations of motion were obtained by the multi-step differential transform method. Of course, this method is needed if it predicts the motion of the considered system. For this reason figurative comparisons between the multi-step differential transformation method, the standard differential transformation method and the classical fourth-order Runge–Kutta method are given. The results reveal that the proposed technique is a promising tool to predict the behaviours of the considered system in long time interval. Moreover, the motion of curves represents the initial conditions that sensitive depend and how much oscillatory motion. In conclusion, the proposed technique is a reliable method to solve the considered double pendulum problem in long time interval. We believe that the system considered in this work along with the method used is interesting for physicists, mathematicians and engineers.