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Journal of Vibration Testing and System Dynamics 8(2) (2024) 207--234 | DOI:10.5890/JVTSD.2024.06.005

Hassène Gritli$^{1,2}$, Sahar Jenhani$^{1}$

$^{1}$ Laboratory of Robotics, Informatics and Complex Systems (RISC Lab, LR16ES07), National Engineering School of Tunis, University of Tunis El Manar, BP. 37, Le Belvédère, 1002 Tunis, Tunisia

$^{2}$ Higher Institute of Information and Communication Technologies, University of Carthage, 1164 Borj Cedria, Tunis, Tunisia

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Abstract

One of the frequent tasks in the robotic research field is to control the position of the robot and then change its current position to the intended state. This study focuses on the position feedback control through a state-feedback control law of an underactuated Lagrangian-type robotic system, called the inertia wheel inverted pendulum (IWIP), to its unstable upright state. Furthermore, using a rescaled dynamic model that describes the difference between the nonlinear dynamics and its approximate linear model, and based on the S-procedure, the Young inequality and the Schur complement lemma, we develop conditions on the feedback gain for the stabilization using two different methodologies. These designed methodologies are realized based on the Linear Matrix Inequality (LMI) techniques. We show that an initially obtained bilinear matrix inequality is converted into an LMI via some mathematical tools. Moreover, we introduce some further LMIs in order to minimize the feedback gain's size. Finally, we show some numerical results to illustrate the effectiveness of the proposed state-feedback control law for stabilizing the underactuated IWIP.} [\hfill Underactuated robotic system \par \hfill Inertia wheel inverted pendulum \par \hfill Position control \par \hfill State-feedback controller \par \hfill LMI optimization\par][J. O. Maaita][5 February 2023][6 May 2023][1 April 2024][2024 L\&H Scientific Publishing, LLC. All rights reserved.] \maketitle %\thispagestyle{fancy} \thispagestyle{firstpage} \renewcommand{\baselinestretch}{1} \normalsize \section{Introduction} \label{Sec:1} Mechanical systems appear in many real-world applications such as industrial robots, domestic robots, military robots, medical robots, service robots and security robots \cite{Jablonski2016,Wang2021,Barros2021,Chai2021,Gonzalez-Aguirre2021,Tipary2021,Kalita2021}.~Different types of robotic systems need to be able to move and manipulate in order to accomplish the desired task \cite{Chai2021,Biswal2021,Wang2021}. Such robots are intended to carry out repetitive jobs that are harmful or painful for humans. According to their degree of actuation, the robotic systems may be divided into three major classes. The first is that the underactuated robotic systems \cite{seifried2014dynamics,Choukchou2014,Zhang2022}, which have fewer actuators than the degrees of freedom (DoFs), such as the crane system, the rotating pendulum, the beam-and-ball system, the inertia wheel pendulum, the surface vessel, the cart-pole system \cite{Liu2013,Krafes2018,Zhang2022}, as well as the monopod robot, the two-link and three-link bipedal robots, the pendubot and the non-holonomic robotic systems \cite{Added2022a,Rehman2018,Xing2022,Added2021a,Jenhani2022DATA}. The second category is the fully-actuated robotic systems \cite{garofalo2012walking,gu2018exponential} that have the same number of actuators as the DoFs. The third category is the over-actuated robotic system. It is a system in which there are more control inputs than the number of DoFs, such as the over-actuated hover capable autonomous underwater vehicle (AUV) \cite{steenson2012experimental}, the over-actuated hexapod robot \cite{bjelonic2017autonomous} and the monopod robot mounted to a vertical slider \cite{Xing2022}. In order to control these different types of robots, a simple and appropriate controller should be designed and hence applied. Different techniques for controlling robotic systems have been proposed in the literature, to regulate a robotic system's position or trajectory \cite{Kelly2005,Choukchou2014,Krafes2018,Liu2013,Liu2020,Gu2013,Spong2020,Kurdila2019,Abbas2021,Mobayen2017,Gritli2020g,Adheem2021,Perrusquia2020,Vrabel2021,Jenhani2022CHTA}. These controllers can be divided into two main families: (1) the linear controllers \cite{Narayan2021,Chawla2021,Singla2017,Kelly2005,Gritli2020g}, like the PD control law, the LQR control law and the PID control law, and (2) the nonlinear controllers \cite{Kelly2005,Mobayen2017,Hasan2021,Jiang2020,Perrusquia2020,Nho2003,Jenhani2022CHTA}, such as the PD plus gravity compensation controller, the augmented PD controller, the computed torque control law, the sliding mode control law, the PD plus feedforward controller, the PD plus desired gravity compensation controller, among others. The underactuated mechanical systems are found to be one of the most significant type of robotic systems \cite{Choukchou2014,Rudra2017}. The design of control laws for the underactuated robotic systems is more difficult compared to the fully-actuated and over-actuated robotic systems \cite{Choukchou2014,Krafes2018,Liu2013,Liu2020,Jenhani2022g}. Then, stabilizing the underactuated robots turned out to be a challenge for the robotics community. Various controllers have been developed in the literature for this purpose, such as the energy and passivity-based control, the hybrid and switched controllers, the sliding-mode control, the backstepping and forwarding controllers, the intelligent and fuzzy controllers, the adaptive control, and others \cite{Moreno2018,Choukchou2014,Krafes2018,Liu2020,Idrees2019,Gnucci2021,Huang2015,Rudra2017}. Recently, the control of underactuated robotic systems has received a lot of attention due to its complexity and applicability. Various underactuated systems, including the rotating inverted pendulum, the wheeled inverted pendulum, the double inverted pendulum, the flywheel inverted pendulum, and the Furuta pendulum, are described in the literature \cite{Krafes2018,Liu2020,siciliano1998control,li2010robust}. Inverted pendulums are often excellent examples of several concepts in the automated control of nonlinear systems. The Inertia Wheel Inverted Pendulum (IWIP) robotic system is a unique underactuated device made up of a real pendulum and a symmetrical disc. It has a long history and has been extensively applied to test, illustrate, and contrast novel control theories and concepts \cite{Khraief2018,Gritli2018g,Andary2009a,Gritli2017self,Olivares2014,Gritli2021b}. It is still a topic of ongoing research today. For the stability of the IWIP, in \cite{Khraief2018}, an adaptive control approach was used with the IDA-PBC (Interconnection Damping Assignment, Passivity Based Control) controller. Indeed, in \cite{chen2019stabilization}, a fuzzy-based hybrid control was used in order to securely balance the IWIP system around the upright position. The tracking strategy of a self-generated stable limit cycle for the underactuated mechanical system, the IWIP, was also covered in \cite{Gritli2017self}. The work in \cite{Andary2009a} presented the control strategy for the stable limit cycle production of the underactuated inertia wheel inverted pendulum. In this study, we are concerned with the control of the underactuated IWIP robotic system, which has 2 DoFs with only one joint to be controlled to stabilize the whole robot at its upper unstable equilibrium. Thus, to resolve the position control issue of this underactuated robotic system, the IWIP must be controlled to and hence stabilized at its unstable upward state. Then, using the dynamic model, which defines the difference between the nonlinear dynamic model and its approximate linearized one, we will adopt a state feedback controller.~Moreover, relying on the S-procedure lemma, the Schur complement, and the Young inequality, we will design the LMI conditions on the feedback gain of the proposed controller through two methodologies to ensure stabilization at the desired state. Finally, several numerical and graphical simulations were presented to verify the accuracy of the developed conditions and the effectiveness of the proposed controller. The remainder of this article is structured as follows: The dynamic model of the IWIP and the problematic in this study are described in Section~\ref{Sec:2}. The rescaled dynamic model and the proposed state-feedback control law are presented in Section~\ref{Sec:3}. The synthesis of LMI conditions via two approaches for developing the gain of the adopted controller is described in Section~\ref{Sec:4}. The improved LMI conditions are given in Section~\ref{Sec:6}. The simulation results for these two design methodologies are shown in Section~\ref{Sec:7.1} and Section~\ref{Sec:7.2} to demonstrate the stabilization issue of the IWIP. A discussion about the results is given in Section~\ref{Sec:7.3}. The last section, Section~\ref{Sec:Con}, will draw a conclusion and some future works. \section{Dynamic model of the underactuated IWIP and problem formulation} \label{Sec:2} \subsection{The underactuated IWIP} \label{Sec:2.1} The inertia wheel inverted pendulum (IWIP), which has two degrees of freedom (DoF) and only the rotating inertia wheel is regulated, is the underactuated robotic system that we use in this study. It is shown in Fig.\ref{paper2_ITSISfig:1Ad} \cite{Gritli2018g,Gritli2021b}. The underactuated IWIP is composed of a vertical inverted pendulum that has an entirely passive joint and where it is free swinging motion is constrained by two symmetric mechanical stops. At the upper extremity of the pendulum, an inertia wheel is mounted, and it is controller via the input $u$. Such wheel and then its corresponding controller input $u$ are employed to control and stabilize the motion of the complete IWIP robotic system. Additional details on such underactuated robotic system can be found in \cite{Andary2009a,Touati2013,Khraief2018,Gritli2017self}. Let us consider the following state constraint that limits the free motion of the inverted pendulum: \begin{equation} -\sigma\leq\theta_{1}\leq \sigma \label{Eq:00} \end{equation} where $\sigma=10^{\circ}$. This specific value of the maximum/minimum angle that defines the symmetric constraint on the swinging motion of the underactuated IWIP can be found in the previous works like in \cite{Gritli2018g,Gritli2021b,Andary2009a,Touati2013,Khraief2018,Gritli2017self}. \begin{figure

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