On The Nonlinear Control Algorithm Of Underactuated Ball And Plate System

Nonlinear Ball and Plate System is an underactuated system which has four DOF (degree of freedom), but it has only two actuators. The four DOF have to be stabilized by just two control inputs. And it is a nonlinear dynamics system of multi-variable, strong coupling, uncertainty. The controlled object is the ball which is unrestricted, that is it only touch the surface of the plate and free from other constraints. The movement of the ball depended on the inclination of the plate. So if the control system wanted to regulate the position of the ball, it could to regulate the controller to control the inclination degree of the plate.Because of no sensor for the ball, it can not recognize the outside world; and because of the existed friction which is a nonlinear factors, etc. This is a very difficult problem to control the controlled objected tracking the expected trajectory. The systematic study of ball and plate system in the nonlinear control theory and practical engineering applications have important meaning, and the paradigm of nonlinear underactuated dynamic systems can be used as a standard experimental platform for testing various control methods. At present, there are a number of universities take the system as the device for control theory and control engineering research and development of a comprehensive experimental study and research platform for a variety of control and experimental study abroad.Although the trajectory control problem of ball and plate system have something in common with the research of robot, but there also existed a lot of difference. The control objects as manipulator or mobile robot, bounded by mechanical agencies, if they can not work in the absence of external effects of instruction. The control for the ball and plate system usually required to ensure the accuracy of the position of the ball, and expected the consumed time for the task as short as possible, that is the velocity of the ball on the plate would be high. At fact, the existence of uncertainty factors, this would be a challenging task.The problems with nonlinear control algorithm of nonlinear system are investigated on the ball and plate equipment BPVS-JLU II which is independently developed by our laboratory. The work is supported by College Doctor Special Scientific Research Fund of China under Grant 20060183006.The main study objectives of this paper are as follows: (1) Analysis the neglected factors of the ball and plate system in the former modeling, and set up the system more accurate mathematical model;(2) Use the relevant knowledge of differential geometry and nonlinear controllability of to prove the controllability of the ball and plate system.(3) Apply nonlinear control methor to carry out the stabilization control and tracking control for the expected trajectory.The study contents of this paper are included:(1) Modeling: considered the difference of the inertia for the X-axis and Y-axis of the plate is neglected in the former modeling; this paper substituted the two different values into the dynamic equation of ball and plate system. At last, apply the Lagrange equation to modify the state equation.(2) Controllability: the ball and plate system is nonlinear underactuated system, whose the number of control variables are less than the number of degrees of freedom, its controllability has not been well studied. At first, this paper studied knowledge of differential geometry and the controllability theory of the affine nonlinear systems. This knowledge can prove controllability of the underactuated system which has some especial nature. The ball and plate system can also be proved by the application of this knowledge.(3) Design: firstly, use the nonlinear algorithm, backstepping, to design the controller for the inclination degree of the plate; then, in order to improve the precise of control, this paper designed the torque position controller, although it is not applied to the actual system, in the future it can also improve the precise for tracking.(4) Simulation and Experiment: analysis and research the capability of the designed nonlinear controller by simulation and experiment. The simulation and experiment results show that the stabilization and tracking capability of the designed controller, and can verify the performance of anti-disturbance.Summarily, this paper studied a lot of theoretical and experimental research on the controllability and nonlinear control algorithm for underactuated ball and plate system, and basically achieves its goals. For the controller that is designed for the inclination of the plate, this paper not only did some simulation on the simulation platform and some experiment on the actual platforms. The other controllers are verified on the simulation platform.