Initial Condition Deviation Research Articles

Complex dynamics, sensitivity analysis and soliton solutions in the (2+1)-dimensional nonlinear Zoomeron model

This paper delves deeply into the investigation of the (2+1)-dimensional nonlinear Zoomeron model. The primary focus of the study revolves around comprehending the dynamic behaviors inherent to this model. This is achieved through a thorough exploration of bifurcations occurring at equilibrium points. The paper also effectively demonstrates the model’s propensity for chaotic behavior by employing principles rooted in chaos theory. Furthermore, the paper conducts a meticulous sensitivity analysis of the dynamical system. This analysis utilizes the RK4 method to establish that even minor deviations in initial conditions exert minimal influence on the overall stability of the solution. Additionally, the study employs the comprehensive discrimination system of the polynomial method. This is done systematically to construct individual traveling wave solutions for the governing model. The culmination of these findings contributes to the establishment of a robust and dynamic mathematical framework. This framework can be effectively employed to address a wide spectrum of nonlinear wave phenomena.

Adaptive robust control of moving-target tracking for marching tank based on constraint following

This paper is devoted to the constraint-following scheme for the moving-target tracking control problem of tank on move. The mission of tank on the battlefield is to find and shoot the armored vehicle, both conditions are required to accomplish this task: complete the process from finding a moving target (time-varying constraints) to pointing to it; keep the barrel stable under highly nonlinear disturbance (which is caused by the battlefield environment). Considering modeling uncertainty and initial condition deviation, an adaptive robust strategy based on Udwadia-Kalaba scheme is presented to solve the matters of target tracking and stable following. Considering the limitation of the analytical model, a tracking system model and a target movement model are built in virtual prototyping environment, complicated road condition, and real target motion state are restored by this method. The model-based control system and the three-dimensional model are combined to verify the feasibility of the control algorithm by the method of RecurDyn/Matlab. By this way, the barrel responds and follows the movement of the target stably within [Formula: see text] s under the action of the stabilization system, and the constraints are approximately satisfied under complex perturbations.

Adaptive robust control for lower limb rehabilitation robot with uncertainty based on Udwadia–Kalaba approach

ABSTRACT This paper aims to address the trajectory control of a lower limb rehabilitation robot which works under passive mode. Udwadia–Kalaba approach is a novel and concise method to address the control of constrained systems with holonomic and nonholonomic constraints. According to Udwadia–Kalaba approach, the fundamental equation of lower limb rehabilitation robot considering the resistance of human muscles is first obtained, then a closed-form expression of the joint control torques is obtained. In practice, it is quite difficult to meet the control target due to various uncertainties such as the initial condition deviation from the constraints, the uncertainty of model, and the external disturbance. To deal with this condition, an adaptive control on the basis of Udwadia–Kalaba approach is presented. The uniform boundedness and the uniform ultimate boundedness of the proposed control are verified by the Lyapunov method. Experiments and simulations are performed to show the simplicity, efficiency and accuracy of the proposed control. Specifically, the tracking trajectory of the robot is designed based on the data collected from the experiments. The simulation results show that the proposed control can be used in the lower limb rehabilitation robot.

Robust approximate constraint-following control design for permanent magnet linear motor and experimental validation

A robust approximate constraint-following control method based on the fundamental equation of Udwadia–Kalaba approach is designed to control the permanent magnet linear motor. The control scheme consists of three parts: the nominal part to suppress any tendency of deviating from the constraints, the second part to deal with any possible initial condition deviation from the constraint manifold and drive the system toward the constraints, and the robust part to compensate for the effect due to uncertainty. Theoretical proof, numerical simulation, and experimental validation have been done to verify the effectiveness of this control method.

Lyapunov Based Robust Control for Tracking Control of Lower Limb Rehabilitation Robot with Uncertainty

This study designs a control of two-degree of freedom lower limb rehabilitation robot (LLRR) for the patient who needs the proper physical therapy after a spinal cord injury (SCI), stroke, or a surgical operation. The robot manipulator can perform specified passive exercises as well as copy exercise motions and perform them without the physiotherapist. Specifically, the uncertainties including the model uncertainty, initial condition deviation and the external disturbance are also considered. Firstly, a unilateral man-robot dynamical model is proposed based on Lagrange method. Then, we propose a Lyapunov based robust control to suppress the effect of uncertainties. The control algorithm consists of a PD feedback component and a piecewise function component. Theoretical analysis is provided to demonstrate that the controller can guarantee the uniform boundedness and uniform ultimate boundedness of the system. Moreover, the joint angle trajectory of a healthy person is explicitly obtained by the experimental platform and used as the pre-specified trajectory of the LLRR. Finally, numerical simulation is presented to illustrate the effectiveness and the trajectory-tracking control performance of the control.

Adaptive robust control for a lower limbs rehabilitation robot running under passive training mode

This paper focuses on the problem of the adaptive robust control of a lower limbs rehabilitation robot (LLRR) that is a nonlinear system running under passive training mode. In reality, uncertainties including modeling error, initial condition deviation, friction force and other unknown external disturbances always exist in a LLRR system. So, it is necessary to consider the uncertainties in the unilateral man-machine dynamical model of the LLRR we described. In the dynamical model, uncertainties are (possibly fast) time-varying and bounded. However, the bounds are unknown. Based on the dynamical model, we design an adaptive robust control with an adaptive law that is leakage-type based and on the framework of Udwadia-Kalaba theory to compensate for the uncertainties and to realize tracking control of the LLRR. Furthermore, the effectiveness of designed control is shown with numerical simulations.

A novel adaptive robust control approach for underactuated mobile robot

Underactuated mobile robot (UMR) is a typical nonlinear underactuated system with nonholonomic and holonomic constraints. Based on the model of UMR, we propose a novel adaptive robust control to control the UMR and compensate the uncertainties from the view of constraint-following. The uncertainties, which are (possibly fast) time-varying and bounded, include modeling error, initial condition deviation, friction force and other external disturbances. However, the bounds are unknown. To estimate the bounds of the uncertainties, we design an adaptive law which is of leakage type. The uniform boundedness and the uniform ultimate boundedness of the proposed control are verified by Lyapunov method. Furthermore, the effectiveness of the control is shown via numerical simulation of a case.

Udwadia–Kalaba theory for the control of bulldozer link lever

We propose to apply Udwadia–Kalaba theory to the bulldozer dynamics analysis. The bulldozer system is divided into several subsystems by this approach, which simplifies the modeling process for multi-link mechanism. Based on this approach, the constraints are classified into structure constraints and performance constraints. Structure constraints are used to set up dynamic model without regard for trajectory. Performance constraints are the desired trajectory. Then, according to the equation of motion of the unconstrained system established by Lagrange approach and system constraints which include structure and performance constraints, an explicit, closed-form analytical expression for the control force can be obtained by solving Udwadia–Kalaba equation. We demonstrate that this approach does not need to solve Lagrange multiplier, which is always difficult to obtain. However, for bulldozer link lever system, the initial conditions are difficult to satisfy the constraints in the actual situation. Thus, the problem of initial condition deviation is taken into consideration. In the end, the numerical simulations are done to prove that the trajectory of the bulldozer satisfies the designed one and the real-time forces are conveniently acquired.

Adaptive robust control methodology for active roll control system with uncertainty

The active roll control system (ARCS) can impose anti-roll moment quickly to prevent the vehicle rolling when the vehicle generates the roll tendency and effectively enhance the vehicle dynamic performance without sacrificing the ride comfort. In the dynamic model of the ARCS, the sprung mass of the vehicle is considered to be the uncertain parameter, which is (possibly) fast-varying. However, what we know about the uncertainty is just that it is bounded. Furthermore, the bound is unknown. The target roll angle is regarded as the constraint when the vehicle equipped with the ARCS is running under a given case. Taking the parameter uncertainty and possible initial condition deviation from the constraint into account, an adaptive robust control scheme based on the Udwadia and Kalaba’s approach is proposed to drive the ARCS to follow the pre-specified constraint approximately. The adaptive law is of leakage type which can adjust itself based on the tracking error. Numerical simulation shows that by using the adaptive robust control scheme, the error between the actual roll angle and the desired roll angle converges to zero quickly in 0.3 s from initial error 0.287 deg, and the final error is of the order of $$10^{-7}$$ . Thus, the control design renders the ARCS practically stable and achieves constraints following maneuvering.

Quasioptimal control with feedback for multiorbit low-thrust transfer between noncoplanar elliptic and circular orbits

The problem of synthesizing stable feedback control is considered based on solving the problem of time minimization for a multiorbit transfer between noncoplanar elliptic and circular orbits in a Newtonian gravitational field. The problem is solved using asymptotic properties and symmetries of optimal control in the unperturbed problem. Stability of the obtained control against external perturbations, deviations of initial conditions, and errors in thrust vector realizations is demonstrated. The obtained quasioptimal control with feedback can be used as an onboard algorithm of spacecraft control and when performing design and ballistic analysis.

Features of motion of dynamic systems with disturbances in the neighborhood of manifolds of switchings

The method of computation of control in real time of a linear system with disturbance is suggested. The system of linear algebraic equations is obtained, which links the deviations of phase coordinates to the deviations of initial conditions of the normalized conjugate system and to the deviation of the finite moment. The calculations reduce to the sequence of the solutions of systems of linear algebraic equations and the integration of a matrix differential equation over transfer intervals of the control switching moments and the finite moment of time. The correction of switching moments and the finite moment of control in the accompaniment of the phase trajectory of motion of a controllable object is considered. Simple constructive conditions of the origin of the sliding mode, motions of the representative point over manifolds of switchings, and changes of the control structure in accompanying the phase trajectory of the system motion are obtained. The convergence of the computational method is proved.

Vibrational Echoes: Dephasing, Rephasing, and the Stability of Classical Trajectories

The experimental observables in coherent, multiple pulse infrared spectroscopic measurements can be calculated from the nonlinear response functions describing the nuclear dynamics of molecular and condensed phase systems. Within classical mechanics, these nonlinear response functions can be expressed in terms of the monodromy matrices that quantify the stability of classical trajectories. We use an ensemble of noninteracting, anharmonic oscillators to examine the effects of the divergence in time of the classical stability matrix on the analytic properties of the third-order response function, relevant to vibrational echo spectroscopy. The twopulse echo measurement is designed to rephase a macroscopic variable, that is, to reverse the effects of destructive interference among the dynamics of microscopic systems characterized by a static distribution of energies. Within classical mechanics, this rephasing is shown to preserve the growth with time of the nonlinear response function that is the signature of the divergence of nearby trajectories. For systems with nearly classical nuclear motions, the vibrational echo measurement may then be interpreted as a probe of the stability of atomic trajectories. I. Introduction Observables in nonlinear infrared and Raman measurements 1-7 of nuclear motions may be computed from nonlinear response functions characterizing the material system. The challenges posed by computing nonlinear response functions for large anharmonic systems with time-dependent quantum mechanics has motivated the analysis of these quantities within classical mechanics. 8-17 Nonlinear response functions of a classical mechanical system may not be calculated directly from a conventional, equilibrium molecular dynamics simulation, because their computation requires knowledge of stability matrices, 8 which quantify the effects of small deviations in initial conditions on classical trajectories. An alternative to simulating nonlinear response functions, which obviates the need to compute stability matrices, is to perform a nonequilibrium molecular dynamics simulation of the material system in the presence of an electromagnetic field and to evaluate numerically the appropriate low-field limit. 15-17 The relevance of stability matrices to nonlinear optical measurements is intuitively clear.

Initial Condition Sensitivity Functions and Their Applications

The well-known method of obtaining sensitivity functions is usually restricted by the condition of continuity imposed on the functions of the coordinates in system equations. For the case of discontinuous functions, more elaborate procedures are required to give a good linear approximation. It is shown that sensitivity functions are discontinuous at the discontinuous points. Also, relations between elements of sensitivity functions at these points are shown to be linear for a given nominal solution of the system. The switching time of the desired Bang-Bang control can be estimated if variations in initial conditions are known. The changes in terminal states and cost function due to deviations in initial conditions can be determined, to first order, by the use of sensitivity functions. Bounds on the deviations in initial conditions can be found by a worst case approach so that the desired terminal conditions are satisfied within given tolerances. The fundamental importance of these techniques in a number of areas of application, for instance, guidance and control of aerospace vehicles, is well known.