Pendulum Model Research Articles

Morphing control of a new bionic morphing UAV with deep reinforcement learning

With rapid development of aviation technology, materials science and artificial intelligence, aircraft design is pursuing higher requirements both in civil and military fields. The new generation of aircraft should have the autonomous capable of performing a variety of tasks (such as take-off and landing, cruising, maneuvering, hover, attack, etc.) under a highly variable flight environment (height, Mach number, etc.) and meanwhile maintaining good performance. Morphing aircraft can use smart materials and actuators to autonomously deform the shape according to the changes in flight environment and mission, and always maintain an optimal aerodynamic shape, therefore get flourished developments. Based on the ability of birds to stretch wings when flying at low speed and to constrict wings at high speed, a new bionic morphing UAV has been designed and developed as the study model by our team. In order to make this new aircraft be able to complete rapid autonomous morphing and aerodynamic performance optimization under different missions and flight conditions, we developed deep neural networks and reinforcement learning techniques as a control strategy. Considering the continuity of the state and action spaces for model, the Deep Deterministic Policy Gradient (DDPG) algorithm based on the actor-critic, model-free algorithm was adopted and verified on the classic nonlinear Pendulum model and Cart Pole game. After the feasibility was verified, morphing aircraft model was controlled to complete prescribed deformation using DDPG algorithm. Furthermore, on the condition that the DDPG algorithm can control morphing well, through training and testing on model using simulation data from wind tunnel tests and actual flight, the autonomous morphing control for the shape optimization of the bionic morphing UAV model could be realized.

Three-dimensional phase structures of mean motion resonances

For coplanar or nearly coplanar orbits, the widths in semimajor axes of mean motion resonances (MMRs) with planets have been analytically characterized as functions of eccentricity, but no satisfactory analytical models exist to consider the resonance widths for the inclined case. The present work intends to develop an analytical model with the aim of producing 3D phase structures of MMRs. For a p:q resonance at an arbitrary inclination, we define the characteristic resonance argument as σ = pλp − qλ + (q − p)Ω, whose amplitude in the disturbing function is a good indicator of representing the total resonance strength. We adopt the characteristic argument as the resonant angle and formulate a multi-harmonics pendulum model, governing the evolution of σ, under the assumption that the mean eccentricity, inclination, and argument of pericentre remain constant during the resonant time-scale. The pendulum model is then applied to the 3:1 and 2:1 resonances with Jupiter, and the phase space structures, dynamical separatrices, and resonance widths are identified as functions of the mean eccentricity, inclination, and argument of pericentre. For the 3:1 resonance, the analytical separatrices are compared to the corresponding numerical structures, and an excellent agreement between them is observed, indicating that the multi-harmonics pendulum model with σ as the resonant angle is applicable in producing 3D phase structures of MMRs.

Human-Cooperative Control of a Wearable Walking Exoskeleton for Enhancing Climbing Stair Activities

In this paper, a human-cooperative control strategy is proposed for locomotion assistance along a physiologically meaningful path through a wearable walking exoskeleton. First, a kinematics model for climbing stairs with the step length, walking speed, and direction is developed. Then, the trajectory of the center of gravity (CoG) in both sagittal plane (SP) and coronal plane (CP) of the human–robot system is designed using virtual slope method (VSM) and dual length linear inverted pendulum model (DLLIPM), which are generated in parallel to the gradient vector of the virtual slope and the horizontal plane, respectively. Then, the task workspace of climbing stairs can be divided into a human domination region and a robot assistance region through a novel designed barrier energy function, such that the motion of human's legs can be constrained within a compliant region around the desired trajectories. Moreover, based on a smooth transition between the human and robot regions, an adaptive controller is designed to simultaneously incorporate robot motion and human's capabilities. Actual experiments involving several human subjects have been conducted and its results demonstrate the performance of the control strategy.

Chebyshev cardinal wavelets for nonlinear stochastic differential equations driven with variable-order fractional Brownian motion

Abstract This paper is concerned with a computational approach based on the Chebyshev cardinal wavelets for a novel class of nonlinear stochastic differential equations characterized by the presence of variable-order fractional Brownian motion. More precisely, in the proposed approach, the solution of a nonlinear stochastic differential equation is approximated by the Chebyshev cardinal wavelets and subsequently the intended problem is transformed to a system of nonlinear algebraic equations. In this way, the nonlinear terms are significantly reduced, due to the cardinal property of the basis functions used. The convergence analysis of the expressed method is theoretically investigated. Moreover, the reliability and applicability of the approach are experimentally examined through the numerical examples. In addition, the presented method is implemented for some famous stochastic models, such as stochastic logistic problem, stochastic population growth model, stochastic Lotka–Volterra problem, stochastic Brusselator problem, stochastic Duffing-Van der Pol oscillator problem and stochastic pendulum model. As another new finding, a procedure is established for constructing the variable-order fractional Brownian motion. Indeed, the standard Brownian motion together with the block pulse functions and the hat functions are utilized for generating the variable-order fractional Brownian motion.

Development, Analysis, and Control of Series Elastic Actuator-Driven Robot Leg.

The mass-spring system-like behavior is a powerful analysis tool to simplify human running/locomotion and is also known as the Spring Loaded Inverted Pendulum (SLIP) model. Beyond being just an analysis tool, the SLIP model is utilized as a template for implementing human-like locomotion by using the articulated robot. Since the dynamics of the articulated robot exhibits complicated behavior when projected into the operational space of the SLIP template, various considerations are required, from the robot's mechanical design to its control and analysis. Hence, the required technologies are the realization of pure mass-spring behavior during the interaction with the ground and the robust position control capability in the operational space of the robot. This paper develops a robot leg driven by the Series Elastic Actuator (SEA), which is a suitable actuator system for interacting with the environment, such as the ground. A robust hybrid control method is developed for the SEA-driven robot leg to achieve the required technologies. Furthermore, the developed robot leg has biarticular coordination, which is a human-inspired design that can effectively transmit the actuator torque to the operational space. This paper also employs Rotating Workspace (RW), which specializes in the control of the biarticulated robots. Various experiments are conducted to verify the performance of the developed robot leg with the control methodology.

Nonlinear analysis and control of a reaction wheel pendulum: Lyapunov-based approach

This paper presents a nonlinear analysis, control, and comparison of controllers based on the dynamical model of the reaction wheel pendulum (RWP) in a tutorial style. Classical methodologies such as proportional integral derivative (PID) control and state variables feedback control are explored. Lyapunov’s method is proposed to analyze the stability of the proposed nonlinear controllers, and it is also used to design control laws guaranteeing globally asymptotically stability conditions in closed-loop. A swing up strategy is also included to bring the pendulum bar to the desired operating zone at the vertical upper position from an arbitrary initial location. Simulation results show that it is possible to obtain the same dynamical behavior of the RWP system adjusting the control gains adequately. All simulations were conducted via MATLAB Ordinary Differential Equation packages.

Usage of Computer Mechanics in the Theory of Inertial Navigation Systems

An idea of using computer mechanics for inertial navigation systems is given and examples are presented. The equations, algorithms, and properties of the pendulum-type strapdown inertial navigation system are analyzed. As a result, it has been stated that such a system is analogous to analytical systems. A similar comparison for both semi-analytical and strapdown inertial navigation systems, in which the Schuler’s pendulum models are described in a horizontal coordinate system, is carried out. An analogy of their properties is established. By comparing the analytical system, platform axes of which are directed along the axes of the inertial coordinate system (orientation of the Schuler’s pendulum is also described in the inertial coordinate system), and the strapdown inertial navigation system with the same orientation of the platform’s computer model (the comparison is also made for the operation algorithms of such systems), an analogy of such systems has been established. The degree of use of computer mechanics in all types of strapdown inertial navigation systems is much greater than in platform ones. According to the degree of utilization for principles of computer mechanics, types of inertial navigation systems can be arranged in the following order: strapdown pendulum, other strapdown systems, semi-analytical, analytical, and geometric platform inertial navigation systems. Their accuracy depends on the degree of sophistication of the element base, that is, on sensitive elements and on-board computers. We claim that the smaller the volume and mass of the mechanical part of the system the better its weight-and-dimensional characteristics and cost.

Evaluation of various benchmark processes with appropriate controller design in LabVIEW platform

Engineering education needs simulation studies and it is the best way of understanding engineering concepts with minimal cost and energy. The main focus of this present work is to provide a workspace especially in LabVIEW for analyzing the performance of the system and to operate them in stable and controlled regions. LabVIEW is a simulation platform which facilitates the simulation of various systems and their response with various controllers programmed using functional blocks that are easy to understand. In this work, the benchmark test systems like DC series motor as first order system, general second order system and a Real time rotary Inverted Pendulum (RIP) model which has been reduced with high order system reduction technique are considered. The test systems are controlled with PID controller tuned by various tuning methods. The PID controller is tuned with conventional methods like Ziegler-Nichols (ZN), Internal Model Control (IMC), and Direct Synthesis (DS) and the responses of the systems are analysed in LabVIEW platform. The common platform has been developed and tested for supporting the performance and analysis of the system with adjustable PID controller parameters like proportional gain Kp, integral time constant Ti, and derivative time constant Td. The performance specifications are phase margin, gain margin, bandwidth, settling time, rise time and integral errors. This LabVIEW platform facilitates the learners to analyze the system effectively with various controllers and their performance parameters are compared easily.

Planning Trajectory of Anthropomorphic Walking Robot (Biped Robot)

Humanoid robots have been a topic of great interest for a long time. Among the various motions of a humanoid robot, the most basic and important motion is bipedal walking. Bipedal walking is probably the most appropriate way for robots to move around in a real environment. However Building trajectories for biped robot walking is a complex task considering all degrees of freedom (DOFs) commonly bound within the mechanical structure. To study the stable motion of any bipedal robot we should study the Kinematics, Dynamics, design a control system and of most important the trajectories. This work deals with planning trajectory of humanoid robots (AK Biped robot), especially for planning trajectory of center of mass (CoM), zero moment point (ZMP)a and planning trajectory of swing foot. diffident method used in planning trajectory like 3D linear inverted pendulum (LIMP), polynomial interpolation. The object of study is the process of automatic control walking of an anthropomorphic walking robot (AK biped robot), for that this paper proposes an approach of joint trajectory generation for the biped robot using the three Three-dimensional linear inverted pendulum for planning trajectory of CoM which keep the ZMP to be inside the support polygon which and guarantee stable dynamic walking of AK biped robot, then trajectory of swing foot derived using the polynomial interpolation function for smooth motion. The aim of this paper is to present a full 3D walking strategy using dynamic model of 3D inverted pendulum for generation references joint trajectories with simulation also design a control system that granites a stable walking of Biped Robot. This is done by first reviewing the literature about different walking strategies. During this literature review the 3D-Linear Inverted Pendulum Model appeared to be the most interesting strategy for further research. The idea of the strategy is simple. It models the human as a linear inverted pendulum with massless rods, which represent the legs, and a point mass at the end of the rods representing the total mass of the body. During walking, there is always at least one foot on the ground which can be seen as stance leg. This stance leg is then modeled as an inverted pendulum. The general closed form solution of the dynamics of the linear inverted pendulum are used to design a trajectory for the center of mass (CoM ) for stance leg. In the trajectory generator, the general solution of the 3D-LIPM is used to prescribe a trajectory. for the CoM of the biped robot AK. The trajectory of the swing leg is designed by a cosine velocity profile interpolation function. The joint trajectories are used as the input for a dynamic model of biped robot in SimMechanics. The proposed control system in simulation are carried out to tune the trajectory such that the dynamic model is able to walk balanced and the robustness of the gait was verified by adding of disturbances. For example, the position of the CoM was increased or decreased and the steps were simulated with different ground levels. These simulations showed that the trajectory is relatively robust. Ref. 10, fig. 4.

State Space Based Linear Controller Design for the Inverted Pendulum

In a previous survey paper the detailed PID controller design to stabilize the inclination angle as well as the horizontal movement of an inverted pendulum system has been presented. In this paper the linear controller design based on the state space representation is shown step by step. Pendulum model is based on EulerLagrange modeling, and the nonlinear state space model is linearized in the unstable upward position, finally pole placement by Ackermann formula and Bass–Gura equation, moreover linear quadratic optimal control are presented. The pendulum has been inserted into a virtual reality laboratory, which is suitable to use in model based control teaching.

Nonlinear Optimal Control for the Wheeled Inverted Pendulum System

SummaryThe article proposes a nonlinear optimal control method for the model of the wheeled inverted pendulum (WIP). This is a difficult control and robotics problem due to the system’s strong nonlinearities and due to its underactuation. First, the dynamic model of the WIP undergoes approximate linearization around a temporary operating point which is recomputed at each time step of the control method. The linearization procedure makes use of Taylor series expansion and of the computation of the associated Jacobian matrices. For the linearized model of the wheeled pendulum, an optimal (H-infinity) feedback controller is developed. The controller’s gain is computed through the repetitive solution of an algebraic Riccati equation at each iteration of the control algorithm. The global asymptotic stability properties of the control method are proven through Lyapunov analysis. Finally, by using the H-infinity Kalman Filter as a robust state estimator, the implementation of a state estimation-based control scheme becomes also possible.

Capturability-Based Pattern Generation for Walking With Variable Height

Capturability analysis of the linear inverted pendulum (LIP) model enabled walking with constrained height based on the capture point . In this paper, we generalize this analysis to the variable-height inverted pendulum (VHIP) and show how it enables 3-D walking over uneven terrains based on capture inputs . Thanks to a tailored optimization scheme, we can compute these inputs fast enough for real-time model predictive control. We implement this approach as open-source software and demonstrate it in dynamic simulations.

Stability Analysis of a Pedestrian’s Walking Motion Based on the Inverted Pendulum Model

In this paper we use an inverted pendulum with harmonic vibrations of the pivot point to model the walking motion of a pedestrian. The stability analysis of the inverted pendulum is carried out to understand dynamics of a pedestrian. Our theoretical research shows that not only the ratio between the lateral and vertical amplitudes of the pivot point, but also the ratio between the lateral and vertical frequencies of the pivot point has significant effects on the stability of the inverted pendulum. The theoretical results are consistent with everyday experiences and observations as well as numerical simulations. Our research indicates that it is proper to applying the inverted pendulum model to analyze to dynamics of a pedestrian’s walking motion.

Stability Analysis of Pedestrian-Footbridge Interaction Based on a Plane Pendulum Model

Stability Analysis of Pedestrian-Footbridge Interaction Based on a Plane Pendulum Model

Adaptation to Coriolis force perturbations of postural sway requires an asymmetric two-leg model.

In the companion paper (Bakshi A, DiZio P, Lackner JR. J Neurophysiol. In press, 2019), we reported how voluntary forward-backward sway in a rotating room generated medial-lateral Coriolis forces that initially deviated intended body sway paths. Pure fore-aft sway was gradually restored over per-rotation trials, and a negative aftereffect occurred during postrotation sway. Force plate recordings showed that subjects learned to compensate for the Coriolis forces by executing a bimodal torque, the distribution of which was asymmetric across the two legs and of opposite sign for forward vs. backward sway. To explain these results, we have developed an asymmetric, nonparallel-leg, inverted pendulum model to characterize upright balance control in two dimensions. Fore-aft and medial-lateral sway amplitudes can be biomechanically coupled or independent. Biomechanical coupling occurs when Coriolis forces orthogonal to the direction of movement perturb sway about the ankles. The model includes a mechanism for alternating engagement/disengagement of each leg and for asymmetric drive to the ankles to achieve adaptation to Coriolis force-induced two-dimensional sway. The model predicts the adaptive control underlying the adaptation of voluntary postural sway to Coriolis forces. A stability analysis of the model generates parameter values that match those measured experimentally, and the parameterized model simulations reproduce the experimentally observed sway trajectories. NEW & NOTEWORTHY This paper presents a novel nonparallel leg model of postural control that correctly predicts the perturbations of voluntary sway that occur in a rotating environment and the adaptive changes that occur to restore faithful movement trajectories. This engaged leg model (ELM) predicts the asymmetries in force distribution and their patterns between the two legs to restore accurate movement trajectories. ELM has clinical relevance for pathologies that generate postural asymmetries and for altered gravitoinertial force conditions.

The Validity of an Oculus Rift to Assess Postural Changes During Balance Tasks.

To investigate whether shifts in head position, measured via an Oculus Rift head-mounted display (HMD), is a valid measure of whole-body postural stability. The inverted single-link pendulum model of balance suggests shifts in whole-body center of mass can be estimated from individual body segments. However, whether head position describes postural stability such as center-of-pressure (COP) remains unclear. Participants (N = 10) performed six conditions while wearing an HMD and performing a previously validated virtual reality (VR)-based balance assessment. COP was recorded with a Wii Balance Board force plate (WBB), while an HMD recorded linear and angular head displacement. Visual input was presented in the HMD (stable scene, dark scene, or dynamic scene) and somatosensory information (with or without foam) was varied across each condition. The HMD time series data were compared with the criterion-measure WBB. Significant correlations were found between COP measures (standard deviation, range, sway area, velocity) and head-centered angular and linear displacements (roll, pitch, mediolateral and anteroposterior directions). The Oculus Rift HMD shows promise as a measure of postural stability without additional posturography equipment. These findings support the application of VR HMD technology for assessment of postural stability across a variety of challenging conditions. The human factors and ergonomic benefit of such an approach is in its portability, low cost, and widespread availability for clinic and home-based investigation of postural disturbances. Fall injury affects millions of people annually, so assessment of fall risk and treatment of the underlying causes has enormous public health benefit.

USING GENETIC ALGORITHM IN OPTIMIZING THE MATRIX OF LQR CONTROLLER FOR NONLINEAR INVERTED PENDULUM SYSTEM

In this article, the author used the matlab software to simulate and then compared the results between the classical LQR (Linear Quadratic Regulator) controller and another method to adjust the matrix parameters toward optimization of the LQR controller. It is the GA (Genetic Algorithm) method to optimize the matrix of the LQR controller, and the results have been verified on the nonlinear pendulum model. The Genetic Algorithm is a modern control algorithm, which is widely applied in research and practice. The main objective of this article is to use the GA algorithm in order to optimize the matrix parameters of LQR controller, whichcontrolled the position and angle of the nonlinear inverted pendulum at the stable balance point. The matlab-based simulating results showed that the system has operated properly to the requirements and the output response has reached an equilibrium position of about 2.5 seconds.

Quiet standing: The Single Inverted Pendulum model is not so bad after all.

In the study of balance and postural control the (Single) Inverted Pendulum model (SIP) has been taken for a long time as an acceptable paradigm, with the implicit assumption that only ankle rotations are relevant for describing and explaining sway movements. However, more recent kinematic analysis of quiet standing revealed that hip motion cannot be neglected at all and that ankle-hip oscillatory patterns are characterized by complex in-phase and anti-phase interactions, suggesting that the SIP model should be substituted by a DIP (Double Inverted Pendulum) model. It was also suggested that DIP control could be characterized as a kind of optimal bi-axial active controller whose goal is minimizing the acceleration of the global CoM (Center of Mass). We propose here an alternative where active feedback control is applied in an intermittent manner only to the ankle joint, whereas the hip joint is stabilized by a passive stiffness mechanism. The active control impulses are delivered to the ankle joint as a function of the delayed state vector (tilt rotation angle + tilt rotational speed) of a Virtual Inverted Pendulum (VIP), namely a pendulum that links the ankle to the CoM, embedded in the real DIP. Simulations of such DIP/VIP model, with the hybrid control mechanism, show that it can reproduce the in-phase/anti-phase interaction patterns of the two joints described by several experimental studies. Moreover, the simulations demonstrate that the DIP/VIP model can also reproduce the measured minimization of the CoM acceleration, as an indirect biomechanical consequence of the dynamic interaction between the active control of the ankle joint and the passive control of the hip joint. We suggest that although the SIP model is literally false, because it ignores the ankle-hip coordination, it is functionally correct and practically acceptable for experimental studies that focus on the postural oscillations of the CoM.

Interval estimation and optimization for motion trajectory of overhead crane under uncertainty

The parameter uncertainty has an important effect on the motion planning of overhead cranes, especially in relation to its industrial safety of production activities. Thus, a novel uncertain estimation-and-optimization strategy is proposed for motion planning of overhead cranes with uncertainty in this paper. The main work of this paper includes the following aspects. First, the overhead crane is simplified as a double pendulum model and the corresponding motion planning is described as an optimal control problem with uncertainty. Second, uncertainties are expressed as interval parameters where only the upper and lower bounds are required without probability information and a bounds estimation problem for optimal control with uncertainty is established; the solution contains all possible values. Third, the bounds estimation problem is solved by a surrogate model-based method, the motion trajectory intervals of overhead cranes are obtained. Fourth, in order to reduce the influence of uncertainty on the motion planning of overhead cranes, an optimization method is introduced to reduce the sensitivity to uncertainty. Finally, the numerical examples show that high accurate interval estimation results are obtained with a reasonable computational cost, and the sensitivity of motion trajectory to uncertainty is reduced obviously with the help of optimization. The proposed strategy provides a guidance for uncertain analysis and online controller design of overhead cranes.

Decentralized event-triggered output feedback control for a class of interconnected large-scale systems

This paper is concerned with the problem of decentralized event-triggered dynamic output feedback control for large-scale systems with unknown interconnections. By using a modified cyclic-small-gain condition and introducing a free-matrix-based integral inequality, a novel sufficient condition is derived to ensure that the overall closed-loop system is asymptotically stable with a prescribed H∞ performance. Based on this condition, a convex condition is presented to design the controller gains and the event-triggered parameters simultaneously. In contrast with the existing event-triggered results, the derived stability criterion can be applied to the large-scale systems with unmatched unknown interconnections. The effectiveness of the proposed control strategy is demonstrated by a double-inverted pendulum model.