The Study On Control And Stability Of Robot Arms Vibrating System

This thesis studies the modeling and stability analysis of robot ?exiblearms vibrating control system described by partial di?erential equations. Wemainly discuss the design of the controller and the dynamic characteristicbehavior including stability. Secondly, the tree-shape networks of string systemand the laser propagation networks are considered.The main body of the thesis is composed of five parts.In the first part, we give the model of single robot arm vibrating controlsystem described by partial di?erential equations and design the controllerof ?exible arm. Firstly, we prove the well-posedness of the system by usingC0 semigroup theory. Secondly, by a detailed asymptotic spectrum analysisof the system operator, we conclude that there is a sequence of generalizedeigenfunctions of the system which forms a Riesz basis with parenthesis forthe state space, hence the system satisfies the spectrum determined growthassumption. Lastly, we obtain the system is exponentially stable under somecondition.In the second part, we give the model of robot cooperative vibratingcontrol system and design the controller. By the technique of asymptoticanalysis, we obtain the eigenfrequency (the spectrum of the system) and theRiesz property. Lastly, by the Matlab simulation, we show the system isexponentially stable by suitable choice of the system parameter.In the three part, the reliability of a periodically repairable robot systemwith safety device is discussed. Using the method of functional analysis, thefinite expansion of the solution is obtained by the analysis of the essential spec-trum to the system and the change of essential spectrum radius of semigroupunder perturbation. Furthermore we prove that 0 is the strictly dominanteigenvalue of the system and there are two nonzero eigenvalues of the systemat most. Therefore we show the solution to the system converge exponentially. In the four part, we study the variable coe?cients di?erential equation ona network with small time-delay in nodal feedbacks. By Lax-Milgram theoremand C0-semigroup theory, we obtain the well-posedness of the system. By adetailed analysis of the system operator, we show that the spectrum of thesystem distributes in a strip parallel to the imaginary axis. As a consequence,we conclude that there is a sequence of generalized eigenfunctions of the systemwhich forms a Riesz basis with parenthesis for the state space. Moreover, weprove the exponential stabilization of the system under some condition.In the five part, we mainly discuss the laser propagation in the laserdevice, the laser travel is viewed as the wave propagation connected Y-shapenetworks. Firstly, we give the mathematical modeling of the laser propagation,herein we consider the viscous damping in the nodes. By using semigrouptheory we prove the existence and asymptotic stability of the wave system.In theoretical, we obtain the energy is dissipative during the wave propagatealthough the viscoelasticity is very small. Finally, we give the transmissionformula of laser on network.