Reinforcement learning based adaptive control for uncertain mechanical systems with asymptotic tracking

This paper mainly focuses on the development of a learning-based controller for a class of uncertain mechanical systems modeled by the Euler-Lagrange formulation.The considered system can depict the behavior of a large class of engineering systems,such as vehicular systems,robot manipulators and satellites.All these systems are often characterized by highly nonlinear characteristics,heavy modeling uncertainties and unknown perturbations,therefore,accurate-model-based nonlinear control approaches become unavailable.Motivated by the challenge,a reinforcement learning(RL)adaptive control methodology based on the actor-critic framework is investigated to compensate the uncertain mechanical dynamics.The approximation inaccuracies caused by RL and the exogenous unknown disturbances are circumvented via a continuous robust integral of the sign of the error(RISE)control approach.Different from a classical RISE control law,a tanh(·)function is utilized instead of a sign(·)function to acquire a more smooth control signal.The developed controller requires very little prior knowledge of the dynamic model,is robust to unknown dynamics and exogenous disturbances,and can achieve asymptotic output tracking.Eventually,co-simulations through ADAMS and MATLAB/Simulink on a three degrees-of-freedom(3-DOF)manipulator and experiments on a real-time electromechanical servo system are performed to verify the performance of the proposed approach.

Prediction and Output Estimation of Pattern Moving in Non-Newtonian Mechanical Systems Based on Probability Density Evolution

A prediction framework based on the evolution of pattern motion probability density is proposed for the output prediction and estimation problem of non-Newtonian mechanical systems,assuming that the system satisfies the generalized Lipschitz condition.As a complex nonlinear system primarily governed by statistical laws rather than Newtonian mechanics,the output of non-Newtonian mechanics systems is difficult to describe through deterministic variables such as state variables,which poses difficulties in predicting and estimating the system’s output.In this article,the temporal variation of the system is described by constructing pattern category variables,which are non-deterministic variables.Since pattern category variables have statistical attributes but not operational attributes,operational attributes are assigned to them by posterior probability density,and a method for analyzing their motion laws using probability density evolution is proposed.Furthermore,a data-driven form of pattern motion probabilistic density evolution prediction method is designed by combining pseudo partial derivative(PPD),achieving prediction of the probability density satisfying the system’s output uncertainty.Based on this,the final prediction estimation of the system’s output value is realized by minimum variance unbiased estimation.Finally,a corresponding PPD estimation algorithm is designed using an extended state observer(ESO)to estimate the parameters to be estimated in the proposed prediction method.The effectiveness of the parameter estimation algorithm and prediction method is demonstrated through theoretical analysis,and the accuracy of the algorithm is verified by two numerical simulation examples.

Loss Factors and their Effect on Resonance Peaks in Mechanical Systems

The loss factors and their effects on the magnitude and frequency of resonance peaks in various mechanical sys-tems are reviewed for acoustic,vibration,and vibration fatigue applications.The main trends and relationships were obtained for linear mechanical models with hysteresis damping.The well-known features(complex module of elasticity,total loss factor,etc.)are clarified for practical engineers and students,and new results are presented(in particular,for 2-DOF in-series models with hysteresis friction).The results are of both educational and prac-tical interest and may be applied for NVH analysis and testing,mechanical and aeromechanical design,and noise and vibration control in buildings.

Interface Mechanical Behavior of Flexible Piles Under Lateral Loads in OWT Systems

This paper investigates the interface mechanical behavior of flexible piles with L_p/D>10 under lateral load and an overturning moment in monotonic loading conditions.To modify the beam-on-Winkler-foundation model of piles in offshore wind farms,the energy-based variational method is used.The soil is treated as a multi-layered elastic continuum with the assumption of three-dimensional displacements,the pile modeled as an Euler-Bernoulli beam.A series of cases using MATLAB programming was conducted to investigate the simplified equations of initial stiffness.The results indicated that the interaction between soil layers and the applied force position should be taken into account in calculating the horizontal soil resistance.Additionally,the distributed moment had a limiting effect on the lateral capacity of a flexible pile.Moreover,to account for the more realistic conditions of OWT systems,field data from the Donghai Bridge offshore wind farm were used.

Adaptive Event-Triggered Time-Varying Output Group Formation Containment Control of Heterogeneous Multiagent Systems

In this paper,a class of time-varying output group formation containment control problem of general linear hetero-geneous multiagent systems(MASs)is investigated under directed topology.The MAS is composed of a number of tracking leaders,formation leaders and followers,where two different types of leaders are used to provide reference trajectories for movement and to achieve certain formations,respectively.Firstly,compen-sators are designed whose states are estimations of tracking lead-ers,based on which,a controller is developed for each formation leader to accomplish the expected formation.Secondly,two event-triggered compensators are proposed for each follower to evalu-ate the state and formation information of the formation leaders in the same group,respectively.Subsequently,a control protocol is designed for each follower,utilizing the output information,to guide the output towards the convex hull generated by the forma-tion leaders within the group.Next,the triggering sequence in this paper is decomposed into two sequences,and the inter-event intervals of these two triggering conditions are provided to rule out the Zeno behavior.Finally,a numerical simulation is intro-duced to confirm the validity of the proposed results.

The Jaffa Transform for Hessian Matrix Systems and the Laplace Equation

Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of linear second-order partial differential equations. This paper discusses the Hessian and its applications in optimization, and then proceeds to introduce and derive the notion of the Jaffa Transform, a new linear operator that directly maps a Hessian square matrix space to the initial corresponding scalar field in nth dimensional Euclidean space. The Jaffa Transform is examined, including the properties of the operator, the transform of notable matrices, and the existence of an inverse Jaffa Transform, which is, by definition, the Hessian matrix operator. The Laplace equation is then noted and investigated, particularly, the relation of the Laplace equation to Poisson’s equation, and the theoretical applications and correlations of harmonic functions to Hessian matrices. The paper concludes by introducing and explicating the Jaffa Theorem, a principle that declares the existence of harmonic Jaffa Transforms, which are, essentially, Jaffa Transform solutions to the Laplace partial differential equation.

Lie-form invariance of nonholonomic mechanical systems

The Lie-form invariance of a nonholonomic mechanaical system is studied. The definition and criterion of the Lie-form invariance of the nonholonomic mechaaical system are given. The Hojman conserved quantity and a new type of conserved quantity are obtained from the Lie-form invariance. An example is givea to illustrate the application of the results.

A Novel Method for Vibration Mitigation of Complex Mechanical Systems

Taking the complex mechanical systems as the research project,a theoretical multi-degree-of-freedom(MDOF)model was established.Based on the vibration characteristics analysis of this system,a novel method of vibration mitigation was proposed,which can be applied to most of the complex mechanical systems.Through this method,limited grounding stiffness was made use of and added to certain degree of freedom(DOF)discretely.Thus,the root-meansquare(RMS)of the systems amplitude can be reduced to ideal level.The MATLAB code based on this method was attached,which was tested on the theoretical model.Consider that complex mechanical systems are nonlinear and uncertain,theoretically the optimal solution of vibration mitigation is inaccessible.However,this method can always provide a relatively effective solution.

MOTION CONTROL OF A CLASS OF UNDERACTUATED MECHANICAL SYSTEMS: THE ACROBOT EXAMPLE

Presents a control strategy for underactuated mechanical system: the acrobot example, which combines fuzzy control and linear quadratic control. The fuzzy controller designed for the upswing ensures that the energy of the acrobot increases with each swing. After the acrobot enters a neighborhood of the unstable straight up equilibrium position, a linear quadratic regulator is designed to balance it.

Two kinds of generalized gradient representations for holonomic mechanical systems

Two kinds of generalized gradient systems are proposed and the characteristics of the two systems are studied. The conditions under which a holonomic mechanical system can be considered as one of the two generalized gradient systems are obtained. The characteristics of the generalized gradient systems can be used to study the stability of the holonomic system. Some examples are given to illustrate the application of the results.

Multi-rate sensor fusion-based adaptive discrete finite-time synergetic control for flexible-joint mechanical systems

This paper proposes an adaptive discrete finite-time synergetic control （ADFTSC） scheme based on a multi-rate sensor fusion estimator for flexible-joint mechanical systems in the presence of unmeasured states and dynamic uncertainties. Multi-rate sensors are employed to observe the system states which cannot be directly obtained by encoders due to the existence of joint flexibilities. By using an extended Kalman filter （EKF）, the finite-time synergetic controller is designed based on a sensor fusion estimator which estimates states and parameters of the mechanical system with multi-rate measurements. The proposed controller can guarantee the finite-time convergence of tracking errors by the theoretical derivation. Simulation and experimental studies are included to validate the effectiveness of the proposed approach.

Skew-gradient representations of constrained mechanical systems

The characteristics of stationary and non-stationary skew-gradient systems are studied. The skew-gradient representations of holonomic systems, Birkhoffian systems, generalized Birkhoffian systems, and generalized Hamiltonian systems are given. The characteristics of skew-gradient systems are used to study integration and stability of the solution of constrained mechanical systems. Examples are given to illustrate applications of the result.

Gradient systems and mechanical systems

All types of gradient systems and their properties are discussed. Two problems connected with gradient systems and mechanical systems are studied. One is the direct problem of transforming a mechanical system into a gradient system, and the other is the inverse problem, which is transforming a gradient system into a mechanical system.

Unified symmetry of nonholonomic mechanical systems with variable mass

Based on the total time derivative along the trajectory of the system the definition and the criterion for a unified symmetry of nonholonomic mechanical system with variable mass are presented in this paper. A new conserved quantity, as well as the Noether conserved quantity and the Hojman conserved quantity, deduced from the unified symmetry, are also obtained, An example is given to illustrate the application of the results.

Udwadia–Kalaba Equation for Constrained Mechanical Systems:Formulation and Applications

There are many achievements in the field of analytical mechanics, such as Lagrange Equation, Hamilton’s Principle, Kane’s Equation. Compared to Newton–Euler mechanics, analytical mechanics have a wider range of applications and the formulation procedures are more mathematical. However, all existing methods of analytical mechanics were proposed based on some auxiliary variables. In this review, a novel analytical mechanics approach without the aid of Lagrange’s multiplier, projection, or any quasi or auxiliary variables is introduced for the central problem of mechanical systems. Since this approach was firstly proposed by Udwadia and Kalaba, it was called Udwadia–Kalaba Equation. It is a representation for the explicit expression of the equations of motion for constrained mechanical systems. It can be derived via the Gauss’ s principle, d’Alembert’s principle or extended d’Alembert’s principle. It is applicable to both holonomic and nonholonomic equality constraints, as long as they are linear with respect to the accelerations or reducible to be that form. As a result, the Udwadia–Kalaba Equation can be applied to a very broad class of mechani?cal systems. This review starts with introducing the background by a brief review of the history of mechanics. After that, the formulation procedure of Udwadia–Kalaba Equation is given. Furthermore, the comparisons of Udwadia–Kalaba Equation with Newton–Euler Equation, Lagrange Equation and Kane’s Equation are made, respectively. At last, three di erent types of examples are given for demonstrations.

Perturbation to Mei symmetry and Mei adiabatic invariants for mechanical systems in phase space

For a perturbed mechanical system in phase space, considering d/dt in the structure equation and process of proof including infinitesimal parameter ε obviously, this paper studies the perturbation to Mei symmetry and adiabatic invariants. Firstly, the exact invariant induced directly from the Mei symmetry of the system without perturbation is given. Secondly, based on the concept of high-order adiabatic invariant, the determining equations of the perturbation to Mei symmetry are established, the condition of existence of the Mei adiabatic invariant led by the perturbation to Mei symmetry is obtained, and its form is presented. Lastly, an example is given to illustrate the application of the results.

Adaptive Control of MIMO Mechanical Systems with Unknown Actuator Nonlinearities Based on the Nussbaum Gain Approach

This paper investigates MIMO mechanical systems with unknown actuator nonlinearities. A novel Nussbaum analysis tool for MIMO systems is established such that unknown time-varying control coefficients are tackled. In contrast to existing literatures on continuous-time systems, the newly-developed Nussbaum tool focuses on extending the traditional Nussbaum result from one dimensional case to the multiple one. Specifically,not only the multiple unknown input coefficients are extended to the time-varying, but also the limitation of the prior knowledge of coefficients' upper and lower bounds is removed. Furthermore,an adaptive robust controller associated with the proposed tool is presented. The asymptotic tracking of MIMO mechanical systems is guaranteed with the help of the Lyapunov Theorem. Finally,a simulation example is provided to examine the validity of the proposed scheme.

Perturbation and Adiabatic Invariants of Mei Symmetry for Nonholonomic Mechanical Systems

基于这个概念断热不变，不安和为 nonholonomic 的 Mei 对称的断热的 invariants 机械系统被学习。为没有不安的系统的 Mei 对称的准确 invariants 被给。到 Meisymmetry 的不安被讨论，为使不安的系统的 Mei 对称的断热的 invariants 被获得。