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The Uniform-Velocity-Walking LQR Controller Research and Design for Linear Inverted Pendulum SystemHuang Chengyu(North China Institute of Science and Technology Information and Control Technology Research Institute, Yanjiao East, Beijing, 101601) Abstract: The problem about the control of the self-balancing mechanical systems uniform motion has great research value and strong reality application background, and the inverted pendulum has become the typical experiment equipment for the analysis and design of the control theory because of its nonlinearity. In allusion to the problem about the uniform motion of the linear inverted pendulum system, we set up the mathematical model first, next we put up the modules of the system one by one in the Matlab/Silumink environment and compiled lots of functions which are related to the hardware communication and so on, then we designed the LQR controller adopting the optimum control theory by using linear quadratic form and realized the control of the linear single or double inverted pendulum systems uniform motion both in simulation and physical environment, during which the controller has shown good anti-jamming capability for external disturbance.Keywords: inverted pendulum, uniform-velocity walking, LQR, Matlab/SimulinkChinese classified number: The literature identification code: Article Numbers:IntroductionAs a typical application example of control theory, Inverted Pendulum has many characters such as high-order times, multivariate, instability, nonlinearity and strong coupling, and it is an absolutely unstable system. There are many researches on Linear Inverted Pendulum currently, but they generally set the angle control or the position control as control goals and adopted various control strategies such as PID, LQR, and fuzzy control. Robot control is the hot research area nowadays, and the control task about the two-rounds self-balancing robot requires not only the balancement or the assured position of its location, but also the ability to walk. In view of the actuality that the research on the walking control of this kind of self-balancing mechanical system is deficient at present, this paper sets speed control as the goal, considering to design a uniform-velocity-walking controller. As the two-rounds self-balancing robots prototype is the double inverted pendulum system, this paper regards the linear one or two stage inverted pendulum system as the research objects, and researches on the design of LQR controller which is about the objects uniform-velocity walking.1. The mathematical modeling of linear one or two stage inverted pendulumThe 2-stage inverted pendulum is composed by four particles, whose mathematical model is shown in figure 1.To make the system simplified, we ignored the resistance of the air during our modeling process, and regarded the swinging rod as a kind of rigid body. Inverted pendulums parameters were defined as follows: “M” for the cars quality, “m1” for swinging rod 1s quality, “m2” for swinging rod 2s quality, “m3” for the quality of the plastid between swinging rod 1 and 2, “l1” for the distance from rotation center to swinging rod 1s centroid, “l2”for the distance from rotation center to swinging rod 2s centroid, “1 ” for the angle between swinging rod 1 and vertical upward direction, “2” for the angle between swinging rod 2 and vertical upward direction, “b” for the friction coefficients between the car and the guide rail. Applied the second category Lagrange equation: (s=1, 2, 3) (1) In the upper equation, “T” is the kinetic energy of the system, and “Qs” is the generalized force of the system. At the same time, we selected the generalized coordinates as follows: ，.The cars barycenter coordinates: (2)Swinging rod 1s barycenter coordinates: (3)Swinging rod 2s barycenter coordinates: (4)The plastids barycenter coordinates: (5) Active force: Gained the generalized force: (6) (7) (8)Kinetic energy of the system: (9)“”in formula (9) means taking 2-norm of . (10)“”in formula (10) means replacing “”in formula (3) with “”.Similarly, (11) (12)The kinetic energy in Inverted pendulum system: (13)Substituting the T and， above into Lagrange equation, we obtained the dynamic equations of the model, which were non-linear equations and needed to be linear approximated in the inverted equilibrium point. Regarding the acceleration as input, namely, ，we obtained the linear equation as follows : (14)Of which: A=。 (15)As the linear 2-stage pendulum would become a linear 1-stage pendulum once it loses its second-stage pendulum and the plastid, we obtained the linear 1-stage pendulums linear equation approximately from the upper equation: (16)2. The design of the Uniform-Velocity-Walking LQR Controller for Linear Inverted Pendulum For linear system, if we take the state variable and the integration of the control variables quadratic form as the performance indexes, this kind of problems about optimizing the dynamic system can be called optimal control problem of linear systems quadratic form performance indexes, which is abbreviated to linear quadratic form problem, also known as LQR problem. Over the years, people have studied LQR controllers structure, nature and design methods from various aspects, and LQR control has become the most fruitful part of modern control theory and its applications. In this paper, the LQR controller was used in inverted pendulums uniform-walking control, and the performance indexes were as follows: (17)Thereunto, “X” is the systems state variable,;“”is the input variable for the system, which is also known as the acceleration of the car in the one or two inverted pendulum system; “Q” is positive definite (or semi-definite) Hermitian matrix, which determines the relative importance of decay rate of state; “R” is a first-order positive definite Hermitian matrix, which determines the relative importance of the energy loss. As the system is single-input, we may assume that R = 1 and Q is in following form: (18)Thereinto, “n” is the number of systems state variables, which fetch 4 for a 1-stage pendulum system while fetch 6 for a 2-stage one. By adjusting the diagonal elements of the upper matrix on the value, we obtained different values for Q. Matlab Software Control System Toolbox provides lqr, lqry and some other functions to solve optimal control problem of linear quadratic form, so we can obtain the systems feedback matrix from the state equations and Q, R matrices.3. Simulation and physical experimentsEstablishing a linear inverted pendulum system under the environment of Matlab / Simulink, whose simulation block diagram of LQR control experiment was shown in Figure 2 . Considering the travel restrictions of the actual inverted pendulum system, we adopt the speed turnaround technology when it reached to the journey boundary, that was, the system would reverse its speed direction immediately after walking in a target speed for a period of time, so that the inverted pendulum would be able to keep track of the positive and negative values within a certain speed, which leads to its back and forth movement. In figure 2, Pulse Generator is a square wave signal whose amplitude is 2;Gain module is the target speed of the inverted pendulums uniform walking, which is setting in advance; LQR controller module is the LQR controller part; L1ip model module is the linear approximate equation of the linear 1-stage inverted pendulum system, which is as follows: (19)Took =0.25，,and selected =1000，=10，=200，=10, we got LQR feedback control matrix K=-31.62, -21.33, 78.74, 14.63.When it comes to simulating, set the systems initial state as follows: =0，=0.2，=0，=0,took the simulation step as 0.005s and Solver as ode1 (Euler) algorithm.Figure2 Linear 1-stage inverted pendulum uniform-walking LQRs simulation block diagramWhen the setted speed value of the Gain module in figure 2 was changed, the system could reach the asymptotic steady state, and had a good anti-interference capability, which was also reflected by the response curve of the system while changing the direction of the speed. In addition, for the same Q-matrix, even when the setted value of the target speed changed largely, the system could get a good response curve. Figure 3 shows the LQR simulation results of the linear 1-stage inverted pendulum system when the target speed is setted as 0.05m / s .Figure3 The simulation results of the 1-stage inverted pendulum when the target speed is 0.05m/sUsing the same method, we did LQR simulation control experiments on the linear 2-stage inverted pendulum system, and the state equation was as follows: (20)Where the actual parameters were calculated as follows:，=0.05Kg，=0.14Kg，=0.24Kg，=0.075m，=0.275m.At the same time, we selected =1000，=10，=200，=10，=200，=10.Then we got the LQR feedback control matrix K = 17.32, 129.16, -228.47, 19.89, 3.68, -39.01.When it comes to simulation, we set the systems initial state as this: =0，=0.2，=0.1，=0，=0，=0; Figure 4 gives the LQR simulation results of the linear 2-stage inverted pendulum system when the target speed is 0.02m/s.Figure4 The simulation results of the 2-stage inverted pendulum when the target speed is 0.02m/sAfter a series of simulation experiments, we found that 1-and 2-stage inverted pendulum systems uniform-walking LQR control experiments yielded very good results. Adjusting the parameters in the matrix Q, we could gain different control effects, and if the adjusting was proper, it could do good to improving systems dynamic properties such as transition time and overshoot and so on.Experimental inverted pendulums model is GLIP2001 and GLIP2002, which is produced by GuGao Technology Company and the motion control card is GT-400-SV-EDU. We constructed the physical experiment module in the Matlab / Simulink environment. Using the acceleration input mode, we draw up a series of S functions, such as the initialization, detection and control functions of the motion control card. Then we carried out uniform-walking LQR control experiments on linear 1-and 2-stage inverted pendulum system, whose results showed that the 1-stage inverted pendulums control effect is good. Figure 5 shows the physical-control experimental results of the linear 1-stage inverted pendulum system while the target speed is 0.25m/s and an external disturbance is applied at the time of 2.85s,from which we can see, the system has almost met the requirement to achieve a uniform motion during the whole progress except the speed direction-changing phase, and the smaller the value of the target speed was set, the higher-powered the uniform nature and immunity of the system would be. Compared to the 1-stage inverted pendulum system, the 2-stage inverted pendulum system was relatively large buffeting during the experiment, and the probability of failure was large, which has shown that it was more difficult to do the physical experiment of the 2-stage inverted pendulum system than to do it of the 1-stage one, though there might be some errors of the inverted pendulum system itself.Figure5 The physical-control experimental results of the 1-stage inverted pendulum system while the target speed is 0.25m/s4. ConclusionIn this paper, we established the mathematical model for the one or two inverted pendulum system firstly by using Lagrange method, then we went along a local linearization around the inverted equilibrium point, and get the state equation of the system. Next, we designed the linear quadratic optimal controller and carried out the uniform-walking experiments on the one and two inverted pendulum system, which included simulation and physical experiments and achieved good control effect. From the study results we can get that the inverted pendulum control system is an ideal experimental device to verify varieties of control methods, which also plays an important role for the study on control theory, and the success of the uniform-walking LQR control experiment for inverted pendulum system has laid a certain foundation for this kind of experiment of other more complex self-balancing mechanical systems.References1 Wu Liqiang, Han Jingqing. Linear inverted pendulum disturbance rejection control design J. Drawn Theory and Applications, 2004, 21 (5): 665-669, 688.2 Yang Shiyong, Xu Li Ping, Wang Pei Jin. Single-stage inverted pendulum PID control study J. Drawn Engineering, 2007, 14 (S0): 23-24, 53.3 Liao Road competition, Zhang Minglian. A nonlinear robust PID controller design method J. Journal of Beijing University of Aeronautics in 2005, 31 (12): 1355-1357.4 Luo Cheng, Hu Dewen, etc. Based on LQR and fuzzy interpolation of five inverted pendulum control J. Control and Decision, 2005, 20 (4):