Control Dynamics and Simulation of Inclined Cart and Pendulum System

1. Introduction

Cart and pendulum is a highly non-linear and multi-variable systems which are objects of theoretical investigation in area of control engineering, robotics etc (Xu and Yu, 2004). It consists of a rigid pole hinged on a cart moving horizontally. The objective is to control pole vertically upwards while cart is stabilised at desired location (Tripathi et al. 2013). A review of past literature suggests that many control techniques has been successfully applied for control of these systems (Tao et al. 2008; Becerikli et al. 2003; Johnson and Moradi, 2005). Seven et al. (2011) discussed a Zero moment point (ZMP) technique which was capable of generating reference for control of 12 degree of freedom humanoid robot climbing on a slope. The authors employed a fuzzy controller for controlling orientation of the robot feet while climbing. Dong et al. (2012) achieved balancing of first order inverted pendulum using PID and fuzzy controllers. A mathematical model of proposed system has been built. The simulation results showed satisfying performance of both schemes. According to Wang et al. (2012), ANFIS and PID control can be combined to stabilise inverted pendulum system and eliminate tracking error. A mathematical model of proposed system has been elaborated. The results proved superiority of ANFIS-PID controller over linear quadratic regulator (LQR) control. Sugano et al. (2013) introduced a mobile inverted pendulum which was capable of moving on changing slope. The study considered a simulation model which applied a constraint contact formulation technique to determine a contact point between wheel and the ground. Almeshal (2013) discussed a hybrid fuzzy control strategy for two-wheeled robotic vehicle having payload moving in different terrains. The proposed system was designed using Euler-Lagrange approach. Dai et al. (2014) developed a multi degree of freedom two wheel inverted pendulum robot having a movable weight. The proposed system was capable of climbing and descending a slope while maintaining its upright position. Controllers were designed for balance and velocity control of robot using a simplified model derived from Lagrange’s method. In a study by Yusuf and Magaji (2014), PID controller was optimized using Genetic algorithm (GA) for control of pendulum angle. The tuning of PID was achieved manually for obtaining an optimum response. The results showed better performance of GA-PID controller over conventional PID controller. Xu & Lee (2014) proposed a real time sliding mode control of two-wheeled mobile robot. The proposed system was able to climb different terrains with minimal disturbance. The study for the first time introduced an underactuated system which utilized a single actuator for position and balance control of mobile robots. The validity of the proposed control was proved through simulations and experiments. In a research by Ponce at al. (2014), a neural-fuzzy-genetic control of inverted pendulum has been proposed. The study presents conventional controllers for observe implementation problems. The results proved validity of proposed control. Hanafy and Metwally (2014) proposed a concept of qualitative dynamic process modeling using a dynamic adaptive neuro fuzzy systems for control of inverted pendulum. The study also presents an identification method based on combination of linguistic knowledge. In a study by Liang et al. (2016) a dynamical model of two-wheeled self-balancing robot moving on inclined surface has been established. The study examined equilibrium point of the system and presented its dynamics in state space form. The authors further proposed a linear quadratic regulator (LQR) controller for control of proposed system. In this paper, we have focused on offline control of cart and pendulum system moving on an inclined surface. The control was achieved using four different soft-computing control techniques. Initially, a mathematical model of inverted pendulum system has been derived which was controlled using conventional PID controllers. The results of PID controllers were further used to train ANFIS and Neural network controllers. The study also considered a linguistic based fuzzy approach for stabilisation of proposed system. The simulation results are further compared to analyse the superiority and limitations of proposed techniques.