# The virtual reality work. Beside, another advanced control approach

The aim of this work is to study the dynamic behavior and control of the triple inverted-pendulum system. A nonlinear dynamic model of the inverted-pendulums fixed on cart, based on CAD model is developed. The Lagrange equation is used to obtain the nonlinear dynamic models of the system. The dynamic model is then linearized around operating point. An augmented dynamic model using the linearized model is also derived. Two control approaches are used to stabilize the pendulums in vertical position.  First approach: State Feedback Control based on the linearized model is used to generate the input force control to stabilize the system. Second approach: Model Predictive Control is designed based on augmented dynamic Model to control the motion of the system. In order to verify the developed model and the chosen controller gains several simulations for different carts’ paths are carried out. Several 3D animations are also presented to verify the usefulness of the designed CAD model and the controllers. As a future work: the 3D model of the triple inverted-pendulum system gives a valuable resource for virtual reality work. Beside, another advanced control approach can be applied on the derived dynamic model. Balancing of an inverted pendulum is an ideal problem in the field of control theory which considerably interests in many fields such as mechanics, physics, applied mathematics and etc., because the inverted pendulum system has the characteristics of non-linear, unstable and non-minimum phase dynamics Brisilla and Sank 2015. Due to nonlinearity and instability challenging control objectives; Inverted pendulum systems have used as excellent test-rigs for control theories. This system has two equilibrium points, one of which is stable while the other is unstable. The stable equilibrium point represents the point where the system swings about with decreasing amplitude until it comes to rest if it is released from any position other than precisely vertical. Therefore, the stable equilibrium point needs no control action to be achieved, thus it is uninteresting from a control point of view. The unstable state of the system in which the pendulum points strictly vertical and requires a control input force in order to keep this position. The inverted pendulum system is used effectively to illustrate several ideas in linear control theory as stabilization of unstable systems Wang 2011. Furthermore, the inverted pendulum systems and due to their nonlinear nature are used to illustrate many ideas in the nonlinear control field such as: variable control structure, back-stepping control, nonlinear state observers, task oriented control, swinging up control, and nonlinear model reduction Roose et. al. 2017. The planar inverted pendulum carried on a cart is the main control idea of the inverted pendulum system, and has several types such as single-inverted pendulum, double-inverted pendulum and triple-inverted pendulum. The control problem of the pendulum system can be divided in general into two problems such as stabilizing and swing-up.The controllability of the non-zero friction joint (damped) of triple-inverted pendulum is investigated in Su and Woodham 2003. The work is aimed to canceling the uncontrollable poles and zeros which can unstabilize the transfer function of the system. A symbolic manipulation for the dynamic model is done in order to calculate the transfer function and a numerical result is also presented.  Gmiterko and Grossman 2009 presented an automatized procedure to derive n-link inverted pendulum dynamic model. The procedure is incorporating the Maple software for deriving the dynamic model symbolically. A numerical example is also addressed and the LQR control approach is applied for double-inverted pendulum resulted from the numerical example.  The motion control of a triple-inverted pendulum on a cart is presented in Glück et. al. 2013, where two degrees-of-freedom controller scheme is used in order to accomplish the swing-up problem. The scheme contains a nonlinear feedforward controller and an optimal feedback controller. The proposed controller is validated using an experimental test-rig. Zhang et. al. 2015 proposed a cloud genetic algorithm + PID neural network to control triple-inverted pendulum. Furthermore, a 3D animation model based on Solidworks MATLAB and LabVIEW is also illustrated. The CGA-PIDNN controller validated using the simulation results and 3D animation model. An alternative representation of fractional Lagrangian for modeling of triple-inverted pendulum is presented in Escamilla et. al. 2016. The dynamic model of the system is derived based on the fractional derivative definitions in order to simplify the modeling process. A numerical simulation using the representations is done.  In this paper, the design and control of triple-inverted pendulum system is presented. The objectives of the present work are: 3D design of triple-inverted pendulum system, mathematical modeling of the system, design of appropriate controller. The CAD system is used to design the system in order to simplify the modeling process. The dynamic model of triple-inverted pendulum under study is derived using the Lagrange equations by implementing the parameters produced from the CAD design model. The model linearized around upward operating point for the sake of controller design. The linearized model is then discretized and augmented in order to design a Model Predictive Controller (MPC). The Laguerre function is used in the MPC to generate the required input force to control the motion of the system. The graphical programming LabVIEW Software is used to design the controllers’ gains for each of the controller type, and also to simulate the controlled motion. The rest of the paper is organized as follow: in the next section Triple-Inverted Pendulum model description is presented, after that the nonlinear dynamic model and the linearized model is addressed. Later controller design is explained, the State Feedback method and the Model Predictive Control method are detailed. The simulation results are illustrated before the final section. Conclusions and future work suggestions are written in the last section.The triple inverted-pendulum system is a nonlinear under actuated system, furthermore triple inverted- pendulum on a cart represent a challenging control design problem. In this section the dynamic modeling of triple inverted-pendulum on a cart is derived. The system under study is graphically shown in Fig. 1.