This chapter describes a mathematical model and design structure of five-link inverted pendulum on cart. The system comprises of five rigid pendulums or links mounted on a mutable cart. The objective is to control all the five links at vertical upright position when cart is stationary at particular location. The study considered free-body-diagram (FBD) analysis of proposed system and applied Newton's second law of motion for deriving a mathematical model of proposed system. The derived governing equations of motion can be further used by researchers for developing a Matlab-Simulink model of five-link inverted pendulum system. The developed model can be further used for deriving equations of motions for n-link cart and pendulum system. Researchers can further apply various control techniques for control of proposed system.
TopIntroduction
Inverted Pendulum system belongs to a category of highly non-linear, multi-variable and dynamic system which acts as a testing bed for validating various control algorithms (Lee & Jung, 2008; Lobas, 2005). Some of the popular control algorithms which are being widely tested using inverted pendulum system include Proportional-integral-derivative (PID) controllers (Prasad et al., 2014), neural networks (Noh et al., 2010), fuzzy logic controllers (Elsayed et al., 2006), adaptive neuro fuzzy inference system (ANFIS), genetic algorithms (Dadios et al., 2006), particle swarm optimsation algorithm etc. The system comprises of a rigid/flexible pendulum mounted over a movable cart. The system has got its center of mass above its pivot point which makes the behaviour of complete system highly nonlinear. The objective is to control the pendulum at vertically upright orientation when cart is being stabilised at particular location under influence of a control force (Soumelids et al., 1997). A traditional pendulum system employed in wall clocks is inherently stable when it is positioning vertically downwards whereas an inverted pendulum needs a counter force for its balancing (Ananevskii & Anokhin, 2014). The dynamics of inverted pendulum is related to the behaviour of missile or rocket guidance system, ship yaw motion control, aircraft control in turbulence, human standing etc (Gawthrop et al., 2014). There exist various variants of conventional single-stage cart and pendulum system. These variants may include a multi-link cart and pendulum system (Bowden et al., 2012), an elastic/flexible inverted pendulum (Chao & Yu, 2004), cart and pole system climbing on slope (Dai et al., 2014), ball and beam system, rotary inverted pendulum etc (Yavin, 1999). These systems are widely employed in various robotics and industrial applications in order to satisfy futuristic needs of mankind (Pomales & Gonzalez, 1996). This chapter focuses on deriving a mathematical model of five-link cart and pendulum system. The system comprises of five rigid links attached to each other and mounted on a movable cart as shown in Figure 1. A five-link cart and pendulum system comprises of five identical rigid links of masses (m1, m2…..m5) and lengths (l1, l2…..l5) respectively attached one above another. These links or pendulums are inclined at an angle of (θ1, θ2…..θ5) respectively from the vertical axis. The bottom most link is pivoted to a movable cart of mass (m) which can move freely along horizontal direction under action of frictional force (b). The cart is supplied an input force (F) to drag it on the surface. The overall objective is to balance all the pendulums in vertical upright orientation while the cart is stationary at the desired location. The configuration of proposed system can be effectively used for analysing behaviour of flexible machine structures and humanoid robots (Lee et al., 2012). The governing equations of motion for five-link inverted pendulum system have been derived applying Newton’s second law of motion. The complete system has been divided into separate cart and pendulum sub-systems for better free-body-diagram (FBD) analysis.
Figure 1. A schematic view of Five-link cart and pendulum system