# Experimental Evaluation of a Takagi-Sugeno Fuzzy Controller for an EV3 Ballbot System +
## 1 Introduction  
An underactuated robot is one for which the number of generalized coordinates is greater than the number of control inputs. Fully actuated robots possess strong properties that facilitates the control design. In fact, fully actuated manipulator robots are globally feedback-linearizable. This generally is not true for most underactuated systems. The control problem for underactuated systems often requires the development of new tools for controller design. There is a large class of underactuated robots for which the control problem is more difficult to solve than for fully-actuated robots [1]. A ballbot is an omnidirectional underactuated robotic system driven through a spherical wheel and whose dynamics are based on the spherical inverted pendulum system [2, 3]. The NXT Ballbot Robotic System (NXTBRS) is a small version of a huge robotic platform introduced in [3] but built using the robotics LEGO Mindsorms NXT kit plus two gyro sensors [4]. Due to the benefit that this robotics kit is reconfigurable and reprogrammable, in addition to its low cost, it has been used in education and research activities [5]. The robotics kit comprises Direct Current (DC) motors with encoders, sensors, and the NXT brick, as well as structural elements like wheels, gears, bars, and balls [6]. The principalparts of the NXTBRS are two rubber tires attached to servomotors, one spherical wheel, two gyro sensors, and the NXT brick (See Figure 1). Because some DC motor parameters cannot be accurately measured, namely the moment of inertia and friction coefficient, the main control problem for a ballbot is the balancing of its body for the upright position at all times due to the uncertainties when considering the actuator's dynamics. Some authors have used different values for the physical parameters of the servomotors based on experimental tests [4; 6; 7; 8; 9]. In [4], a type-1 servo system as described in [10], also known as an integral control structure [11], was employed in the control of an NXTBRS where feedback gains were calculated through the Linear Quadratic Regulator (LQR) design approach. The performance of the servo system was validated via simulation and experimental results. In [6], an LQR was designed to control an NXTBRS whose performance was also validated through numerical simulation and experiments. In [2], following a similar procedure as that for the Takagi-Sugeno Fuzzy Controller (TSFC) design methodology, a robust controller was designed under the focus of Polytopic Linear Differential Inclusion (PLDI) models and Linear Matrix Inequalities (LMIs). A Polytopic Model (PM) for the NXTBRS was proposed, where the gains for an LQR were designed via the LMIs technique. In this work, the parameters corresponding to both the moment of inertia and the friction coefficient from the NXT servomotors were taken from [4; 6; 7; 8; 9]. So the NXTBRS's PM was constituted by four linear subsystems, the so-called vertices, in such a way that all possible minimum and maximum values for the parameters from the NXT servomotors may be considered in the same PM. The Polytopic Controller (PC) (control law) was then given by state feedback plus an integral action, namely, in the form of a type-1 servo system via the LQR design approach. In their proposal, it is not clear how the interpolation between both PM, i.e., linear subsystem models, and PC, i.e., linear local controllers, is given. However, it could be considered that it was carried out in the form of a gain scheduling scheme. Simulation and experimental results validated the performance of the PC in contrast with those from the classical LQRs tuned for each vertex from the PM, showing that for some vertices, the classical LQR was not able to balance the NXTBRS; i.e., the PC was robust for the widespread range of parameters from the servomotors. In [12], Discrete-Time LQRs (DTLQRs) were designed for the control of an NXT ballbot considering the physical parameters from each one of the systems given in [4; 6; 7; 8; 9].