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1、Nonlinear Feedback Control Design for Two-link Robot ArmReport for the CourseNonlinear Control System 专业：控制理论与控制工程 Problem DescriptionDynamic control model of a two-link robot arm in a vertical plane shown as follows：Fig.1 Two-link robot armPosition equation is given by ( 1 ) ( 2 )Where is ( 3 )The
2、dynamic equation of matrix which relates the total joint torque vector to the joint angle vector is given by ( 4 )Where is the inertia matrix, is the coriolis centrifugal torque vector and is gravity loading vector. And , are defined as following. ( 5 )is the inertia matrix; ( 6 )is the Coriolis cen
3、trifugal torque vector; ( 7 )is gravity loading vector with following parameters: ,.Solution.1. System linearizationRewrite the system into the form of affine nonlinear system Define ,. Let , then the state space equation can be written as ( 8 )The general task of output is given by Linearize the sy
4、stem using feedback linearization method1） Calculating the values of the parameters in formulas（5）,（6） and （7）：，、。2）We check the nonlinear system to see if it an be totally transformation, it requires computation of Lie derivatives and decoupling matrix . The same，for ， ( 9 )Matrix is nonsingular，So
5、 the relative degree of the system is r=2.3）Choose a coordinate transformation .Let , namely, ( 10 )then ( 11 )4）The normal form is（12）And the output is（13）5）If we define ，letWhere Further obtainingnamely（14）let2. Tracking controller designIn this section, controllers are designed to show the simula
6、tion results.Optimal controlThe actual output of the system become as much as possible close to the ideal output by Using LQR method for solving to determine the optimal control law. This problem belongs to the tracking problem.For the optimal control problem, the performance index is （15）where is t
7、he error vector, denotes the desired output and is the actual output.If (A,B) is controllable and (A,C) is observable, then the optimal control law is ,is a symmetric positive-definitematrix that satisfies the following Riccati equation .letThen 3. Simulation resultsOptimal control caseThe simulatio
8、n model based on the optimal control method in Matlab/Simulink is shown in Fig.2. Some functions are defined to represent the process of the nonlinear feedback .Fig.2 Simulation model for optimal controllerGive the desired output, which means the arm moves around a oval of which the semi-major axis
9、is 0.7 and the short half shaft is 0.3. Controller parameters are chosen as.The tracking trajectories are shown in Fig.3 from which we can see that the designed optimal controller can guarantee the tracking performance.Fig.3 Desired trajectory and actual outputIn order to have a good view of the tra
10、cking errors both in Px and Py, they are shown in Fig.4.Fig.4 Tracking errors Because of the limit of the robot arm length, tracking length must be less than. In this paper, the input curve is a oval of which the semi-major axis is 0.7m and the short half shaft is 0.3m. From figure 3, we can see tha
11、t the tracking curve has the same shape with the input curve, so the control method works.It can be seen from figure 4 that the tracking errors are limited in0.1, which shows a good performance of tracking. And there is a large error at the beginning, it is because that the optimal control chosen is
12、 infinite time control and the output of the system is zero, the system needs some time to track the reference input. 4. ConclusionIn this project, the nonlinear system of two-link robot arm is first linearized by feedback linearization, then an optimal controller are designed for the tracking problem. Simulation results show that the optimal controller is effective for tracking control. Lastly, through the work of this project, I get a better understanding of the knowledge of nonlinear control than I once learned from class.
