Control of a 2-DOF Omnidirectional Mobile Inverted Pendulum

Abstract

Nowadays, simple and executable controllers that can control many complex processes are more popular, remarkable and justifiable. Studying and designing such controllers for an inverted pendulum system is a suitable way to prove the controllers performance. The reason to study the pendulum intensively relies on the fact that many important engineering systems can be approximately modeled as a pendulum. For example, in thrust vectored rocket control, the pitch dynamics of a rocket can be approximated by a simple pendulum. In robot systems, their dynamics can be modeled as the inverted pendulum systems. In biomechanics, the pendulum is used to model bipedal dynamic walking. The pendulums are also used in the study of wheeled motion and balancing mechanisms. Many challenging control algorithms have been tested with the inverted pendulum system. These controllers are from classical control theories to intelligent control algorithms. An 2-degrees-of-freedom (2-DOF) omnidirectional mobile inverted pendulum (OM-IP) is made of a 2-DOF inverted pendulum (IP) based on omnidirectional mobile platform (OMP) to take advantage of the convenient characteristics of omnidirectional wheels to balance the IP which is mounted on the OMP. Until now, there are little researches about this system with disturbance and friction. It is difficult to derive the dynamic modeling of an OMP with disturbance and friction and control it. Moreover, it is also difficult to derive the dynamic modeling of a 2-DOF OM-IP with an assumption that the distance from the center of gravity of the inverted pendulum to the rotary point on the OMP is unknown and how to control to keep the IP balance. So obtaining modeling and motion control of the 2-DOF OM-IP are deeply needed. Objective of this dissertation is to present modeling and controlling the balance of the IP mounted on the OMP with disturbance and friction by using adaptive backstepping technique. The OM-IP is separated into two subsystems, the OMP and the 2-DOF inverted pendulum. To control the motion of the OMP, two controllers are designed for the OMP to track a desired path with a desired velocity, and an adaptive backstepping controller to balance the IP which is mounted on the OMP. Therefore, the following tasks are implemented in this dissertation. First, the structure of the OM-IP used for experiment is proposed. Hardware configuration of the proposed system is implemented. A control system is developed based on PIC18F452 microcontroller technology. One PIC18F452 is used as a master unit, and other PIC18F452s are used as slave units. The master unit is connected with other slave units through I2C communication. The master unit receives data from the sensors that are used for main controller, and then it sends the commands to the slave controllers via I2C communication, respectively. The slave controller integrates PIC18F452 with motor drivers, LMD18200, for the DC motor control. Second, an OMP with three driving omnidirectional wheels in the presence of disturbance and friction is presented. A kinematic modeling of the OMP is presented, and a dynamic modeling of the OMP with disturbance and friction is derived based on the Newton’s second law of motion. Based on the modeling of the system, controllers are designed to control the OMP to track a desired path. Third, dynamic modeling for the 2-DOF OM-IP is presented. The IP is a rod that rotating around a universal joint structure with 2-DOF. The motions of the OMP are separated into two independent motions: motion in plane and motion in plane, so the 2-DOF inverted pendulum is decoupled into two subsystems of 1-DOF inverted pendulum. By applying the Newton’s second law at the center of gravity of the pendulum, the nonlinear mathematical modeling equations of the 1-DOF inverted pendulum in plane and plane are obtained. Fourth, based on the dynamic modeling, a backstepping controller is designed to stabilize the OMP to follow a desired path. The controller is designed based on a backstepping control theory. It includes two steps: firstly, a virtual state and a stability function are introduced. Secondly, Lyapunov functions for the system are chosen and an equation for the virtual control that makes the system stabilisable is obtained. The system stability is guaranteed by the Lyapunov stability theory. The simulation and experimental results are presented to illustrate the effectiveness of the proposed tracking controller. Fifth, based on the dynamic modeling of the OMP system with disturbance and friction, an adaptive backstepping controller is designed to stabilize the OMP with parameter variations and uncertainties caused by friction and slip to follow a desired path. The controller and update laws are designed based on an adaptive backstepping control theory. The system stability is guaranteed by the Lyapunov stability theory. The simulation and experimental results are presented to demonstrate the effectiveness of the proposed controller and update laws. After that, the controllers in chapter 4 and chapter 5 are compared by simulation and experimental results in the same conditions. The concepts of two controllers in Chapter 4 and 5 are applied to control the OM-IP in Chapter 6, respectively. Finally, from the dynamic equation of the OM-IP system, an adaptive backstepping control method is proposed to keep the inverted pendulum balance with an assumption that the distance from the center of gravity of the rod to the rotary point on the OMP is unknown. The stability and the convergence properties of OM-IP is guarantee by Lyapunov function. The effectivenesses of the proposed system are shown through simulation and experimental results. So the system can be applicable and implemented in the applications. Keywords: Inverted Pendulum Omnidirectional Mobile (OM-IP), 2 degrees of freedom (2-DOF), Omnidirectional Mobile Platform (OMP), Inverted Pendulum (IP), Lyapunov Function Candidate, Backstepping Method (BC), Adaptive Backstepping Controller (ABC).