Lagrangian Systems Research Articles

Impulsive tracking synchronization control of networked robotic manipulator systems in the task space

This study investigates the impulsive tracking synchronization control of networked robotic manipulator systems based on Lagrangian systems. In the task space, the end-effector positions of networked robotic manipulators can synchronize and track a desired trajectory by designing an impulsive controller. Based on impulsive control, some algebraic synchronization criteria are derived. As a direct application of the theoretical results, tracking synchronization of six double-link robotic manipulators in the task space is discussed in detail. Numerical simulations are given to demonstrate the effectiveness and feasibility of the proposed control strategy.

Safety Flocking of Networked Lagrangian Systems With Event-Triggered Communication

Safety Flocking of Networked Lagrangian Systems With Event-Triggered Communication

Adaptive Reinforcement Learning Strategy-Based Sliding Mode Control of Uncertain Euler–Lagrange Systems With Prescribed Performance Guarantees: Autonomous Underwater Vehicles-Based Verification

Adaptive Reinforcement Learning Strategy-Based Sliding Mode Control of Uncertain Euler–Lagrange Systems With Prescribed Performance Guarantees: Autonomous Underwater Vehicles-Based Verification

Adaptive Saturated Obstacle Avoidance Trajectory Tracking Control for Euler–Lagrange Systems With Velocity Constraints

ABSTRACTThis article pays attention to the obstacle avoidance trajectory tracking control problem for uncertain Euler–Lagrange (EL) systems subject to velocity constraints and input saturation. The existing obstacle avoidance results do not consider velocity constraints under input saturation, which means that an EL system may not be able to obtain sufficient control inputs to avoid a collision with an obstacle if it has a high speed when approaching the obstacle. Therefore, the velocity constraints in the obstacle avoidance tracking control are considered in this paper. A novel velocity constraint function that depends on the distance between the system and the obstacle is proposed. Integral‐multiplicative Lyapunov‐barrier functions (LBFs) are constructed and incorporated into the backstepping procedure to design an adaptive fuzzy obstacle avoidance tracking control scheme. Moreover, an auxiliary dynamic system is designed by constructing a bounded nonlinear vector related to an auxiliary variable to compensate for the effects of saturation. Through the Lyapunov method and boundedness analysis for the barrier function, it is shown that the protocol achieves obstacle avoidance for the EL system without violating the velocity constraints inside the obstacle detection region, while also guaranteeing the ultimate uniform boundedness of all the closed‐loop signals. Numerical simulations are presented to demonstrate the efficacy of the proposed control strategy.

Synchronization of Euler-Lagrangian networks: a periodic intermittent dynamic event-triggered control strategy

In this paper, the synchronization problem of Euler–Lagrange networks is studied by designing periodic intermittent dynamic event-triggered control strategy. The synchronization of all agents is successfully realized by injecting negative comments into the discontinuous interval of the agents described by the Euler–Lagrange systems. Different from the traditional intermittent event-triggered control strategy, the control strategy proposed in this study introduces internal dynamic variables to expand the sampling interval of the event-triggered mechanism, thereby improving resource utilization efficiency. In addition, the event trigger instants of each agent are asynchronous. By using graph theory and Lyapunov method, the synchronization criteria of Euler–Lagrange networks are derived. The effectiveness of the proposed control scheme is verified by simulations for the Euler-Lagrangian network composed of six two-link rotating manipulators.

In search of hidden symmetries

Abstract This paper exemplifies the importance of finding hidden symmetries of differential equations that are models of physical phenomena. The hidden symmetries (Lie symmetries) may be determined by either linking together different equations for certain values of their parameters or transforming the original model into another equivalent system of equations that may have more symmetries. Therefore, hidden symmetries may help to solve the original model or yield its hidden properties, e.g. linearity and conservation laws. Moreover Noether symmetries are shown to be preserved by going from classical to quantum mechanics, namely from Lagrangian systems to the corresponding time-dependent Schrödinger equation.

A Weierstrass approach to the analysis of rarefaction solitary waves in tensegrity mass-spring systems

Abstract The Weierstrass‘ theory of one-dimensional Lagrangian systems and a quasi-continuum approach are employed to study the propagation of solitary waves in tensegrity mass-spring chains, which exhibit softening-type elastic response in the large displacement regime and are subject to external pre-compression. The presented study analytically derives the shape of the traveling rarefaction pulses, and limiting values of the speeds of such pulses. Use is made of a tensegrity-like interaction potential that captures the main features of the real force-displacement response of the examined units. The Weierstrass approach is validated through numerical applications that establish comparisons between the theory developed in the present work and previous results available in literature.

The Distributed Adaptive Bipartite Consensus Tracking Control of Networked Euler–Lagrange Systems with an Application to Quadrotor Drone Groups

Actuator faults and external disturbances, which are inevitable due to material fatigue, operational wear and tear, and unforeseen environmental impacts, cause significant threats to the control reliability and performance of networked systems. Therefore, this paper primarily focuses on the distributed adaptive bipartite consensus tracking control problem of networked Euler–Lagrange systems (ELSs) subject to actuator faults and external disturbances. A robust distributed control scheme is developed by combining the adaptive distributed observer and neural-network-based tracking controller. On the one hand, a new positive definite diagonal matrix associated with an asymmetric Laplacian matrix is constructed in the distributed observer, which can be used to estimate the leader’s information. On the other hand, neural networks are adopted to approximate the lumped uncertainties composed of unknown matrices and external disturbances in the follower model. The adaptive update laws are designed for the unknown parameters in neural networks and the actuator fault factors to ensure the boundedness of estimation errors. Finally, the proposed control scheme’s effectiveness is validated through numerical simulations using two types of typical ELS models: two-link robot manipulators and quadrotor drones. The simulation results demonstrate the robustness and reliability of the proposed control approach in the presence of actuator faults and external disturbances.

Periodic solutions of Lagrangian systems under small perturbations

In this paper, we consider the existence of periodic solutions of Lagrangian systems of the form [Formula: see text], where [Formula: see text] is a [Formula: see text]-function in the sense of Trudinger, [Formula: see text] are [Formula: see text]-smooth, [Formula: see text]-periodic in the time variable [Formula: see text] and [Formula: see text] is a real parameter. We prove the existence of a [Formula: see text]-periodic solution [Formula: see text] for any sufficiently small [Formula: see text], and show that the found solutions converge to a [Formula: see text]-periodic solution of the unperturbed system if [Formula: see text] tends to [Formula: see text]. Let us stress that our theorem only makes a natural assumption on the potential [Formula: see text] and no additional assumption on [Formula: see text] except that it is continuously differentiable. Its proof requires to work in a rather unusual (mixed) Orlicz–Sobolev space setting, which bears several challenges.

Physics-informed deep Koopman operator for Lagrangian dynamic systems

Physics-informed deep Koopman operator for Lagrangian dynamic systems

Neural network-based distributed consensus tracking control for uncertain Euler–Lagrange systems over directed topologies

This paper proposes a neural network-based robust control approach for driving disturbed Euler–Lagrange systems (e.g., robot manipulator) in a directed network to perform consensus tracking of a heterogeneous leader’s output signal. The parameter matrices and external disturbances in each follower’s Euler–Lagrange dynamics are unmeasurable, thus we unify them into one lumped uncertainty term. Neural networks are then employed to online estimate these uncertainties, with adaptive update laws ensuring the boundedness of the estimation errors. Subsequently, a set of distributed observers are developed to adaptively estimate the leader’s states as well as the unidentified parameter vector and output matrix in the leader’s model. With their help, the robust cooperative controller is designed. Its efficacy on directed networks is theoretically justified by designing a Lyapunov function with a special structure. This function produces symmetric positive definite matrices when its first derivative terms interact with the specified parameters in the observer and Laplacian matrices for directed graphs. Finally, numerical simulations based on two-link robot manipulators are carried out to verify that the tracking errors converge to zero when applying the neural network-based robust controller.

Practical output regulation of uncertain Euler–Lagrange systems

This paper deals with the problem of robust output regulation of uncertain Euler–Lagrange systems. An internal model controller is synthesized by a systematic framework that considers polynomial mappings of the steady-state trajectory. The closed-loop stabilization is guaranteed by using a descriptor differential–algebraic representation of the system. This methodology allows the controller design problem to be cast as a convex optimization problem subject to linear matrix inequalities. Simulations considering a two-link robotic manipulator illustrate the method.

Bipartite tracking for Euler–Lagrange systems via prescribed‐time hierarchical control based on the matrix‐weighted signed digraphs

AbstractThe goal of this article is to investigate the bipartite tracking control problem using a prescribed‐time convergence method for Euler–Lagrange systems (ELSs) with external disturbances. Agent interactions are represented by a matrix‐weighted signed directed graph. There are not only cooperative but also adversarial interactions. An essential component of the system, the prescribed‐time hierarchical control (PTHC) algorithm, including a more general time‐varying function, is newly developed to guarantee that all agents converge to the same leader state but with opposite signs within a prescribed time. The salient feature of the introduced method lies in the fact that the convergence time of the control objective can be prespecified by the user. The state transformation, the property of the matrix‐weighted Laplacian, and the generalized Lyapunov stability argument are employed to theoretically validate the proposed algorithms. Finally, the effectiveness of the algorithm is confirmed through the execution of numerical examples.

Robust arbitrary trajectory tracking of multiple heterogeneous Euler–Lagrange systems under directed graphs: A fully distributed event‐triggered control

AbstractIn this paper, the robust arbitrary trajectory tracking problem of multiple heterogeneous Euler–Lagrange (EL) systems with disturbances under a directed graph is investigated via a novel fully distributed event‐triggered control (ETC) strategy. Firstly, for an arbitrary tracked trajectory, the Fourier transform technique is employed to approximately obtain its linear form, which represents the leader of multiple EL systems. Since the dynamics and state of the leader are not available to all followers, fully distributed adaptive ETC‐based observers are proposed for each follower to reconstruct the leader's trajectory, where the global information requirements are avoided, and the Zeno behavior is excluded by giving the minimum triggering intervals. Armed with the observed information, a robust adaptive ETC protocol is further developed such that the position‐like state of each follower can track the leader with the intermittent controller updating, where a minimum control update interval is obtained theoretically. Moreover, the adaptive technique is used to eliminate the influence of the parameter uncertainties of multiple EL systems, thus the tracking is achieved asymptotically. Finally, simulation results are illustrated to show the feasibility and effectiveness of the proposed control strategy.

Predefined‐time hierarchical control for bipartite time‐varying formation tracking of networked Euler–Lagrange systems with parametric uncertainties

AbstractThis paper mainly addresses the bipartite time‐varying formation tracking (BTVFT) problem for networked Euler–Lagrange systems (NELSs) with external disturbances and parametric uncertainties, considering a directed signed communication network that encompasses both cooperative and competitive interactions. A novel nonsingular sliding mode control (SMC) method serves as the foundation for a predefined‐time hierarchical control algorithm (PTHCA), introduced to tackle the BTVFT challenge for NELSs in a predefined time. This approach enables the settling time to be predetermined as a control parameter. Moreover, sufficient conditions for establishing the predefined‐time stability of closed‐loop systems are provided based on the Lyapunov argument. Simulations are conducted to demonstrate the effectiveness of the main findings.

A New Class of Separable Lagrangian Systems Generalizing Sawada–Kotera System

Some characteristics of stationary flows of the Sawada–Kotera system lend themselves to generalization, producing a large class of separable Lagrangian systems with two degrees of freedom. All of these systems come in couples that have the same equations of motion, although they are not related by a gauge transform. Some nonpolynomial examples are provided.

Linearly implicit time integration scheme of Lagrangian systems via quadratization of a nonlinear kinetic energy. Application to a rotating flexible piano hammer shank.

This paper presents a general and practical approach for nonlinear energy quadratization based on the Euler-Lagrange formulation of the physical equations. A Scalar Auxiliary Variable -like method based on a phase formulation of the equations is applied. The proposed scheme is linearly implicit, reproduces a discrete equivalent of the power balance. It is applied to a rotating and flexible piano hammer shank. An efficient solving strategy leads to a quasi explicit algorithm which shows quadratic space/time convergence.

Adaptive Robust Control of Uncertain Euler-Lagrange Systems Using Gaussian Processes.

This article proposes a novel adaptive robust control approach based on Gaussian processes (GPs) for the high-precision tracking problem of uncertain Euler-Lagrange (EL) systems with time-varying external disturbances. Given a prior dynamic model, the GP regression (GPR) technique is employed to obtain a nonparametric data-based uncertainty model, including its probabilistic confidence intervals. Based on the adaptive sliding mode control (ASMC) framework, the posterior means of GPs are utilized for dynamic compensation, whereas the posterior variances are applied to adjust the feedback gains. This proposed control strategy is robust against significant system uncertainty with low feedback gains. A novel adaptive law for updating hyperparameters based on tracking error feedback is presented, thereby improving the performance of both tracking control and GP modeling simultaneously. Compared to existing likelihood-based optimization methods, this hyperparameter adaptive law enables data-efficient and fast uncertainty learning for control applications. The proposed control strategy guarantees the semiglobal asymptotic convergence to zero tracking error with a specified probability. Simulations using an underwater robot model demonstrate that the utilization of GPs and hyperparameter adaptive law significantly improves the performance of tracking control and uncertainty learning.

Distributed finite-time optimization for networked Euler–Lagrange systems under a directed graph

In this paper, we investigate the finite-time distributed optimization problem for multiple Euler–Lagrange systems. A new distributed optimization control scheme is presented to achieve the state agreement in finite time while minimizing the sum of each agent’s local cost function. The proposed algorithm has the advantage of being able to achieve distributed finite-time optimization consensus under general unbalanced connected directed communication graphs. By virtue of finite-time Lyapunov theory and convex optimization, the finite-time convergence for the algorithm is analyzed. A numerical example is also presented to illustrate the effectiveness of the obtained results.

Periodic Solutions of Generalized Lagrangian Systems with Small Perturbations

In this paper we study the generalized Lagrangian system with a small perturbation. We assume the main term in the system to have a maximum, but do not suppose any condition for perturbation term. Then we prove the existence of a periodic solution via Ekeland’s principle. Moreover, we prove a convergence theorem for periodic solutions of perturbed systems.