# Experimental Evaluation of a Takagi-Sugeno Fuzzy Controller for an EV3 Ballbot System +
## 1 Introduction
Numerical simulation results validated the performance of the DTLQRs when comparing their response versus that from their respective continuous-time versions. In [13], an extended Kalman filter was used to estimate, under position measurements, the state vector of the NXTBRS. The estimated state vector is used by an LQR to achieve the control objective. That work showed that the closed-loop system is locally asymptotically stable. The viability of this proposal was validated via numerical simulation. In [14], a dual TSFC was proposed for the control of a Ball Robot System (BRS) to operate independently without coupling. The fuzzy model for the BRS comprises two fuzzy rules whose local  
Figure 1: EV3 BRS’s generalized coordinates. linear models were obtained from linearizing the nonlinear model from zero to five degrees with respect to the vertical unstable equilibrium position. Membership functions of the Gaussian type were used. The BRS has the physical configuration of an inverted pendulum but is supported by a bowling ball and driven through two 24 V DC motors orthogonally placed. The BRS is about 40 cm in height and 17 kg in weight. Simulation and experimental results validated the proposal. In [15], from the dynamic model for a ballbot given in terms of the Euler-Lagrange equations, an H-infinity (nonlinear optimal) controller was designed, whose performance is evaluated via simulation results. The nonlinear model is linearized around a temporary operating point which is updated at each time step of the control algorithm. Feedback control gains are computed through the solution of an algebraic Riccati equation solved at each time step. Closed-loop stability is shown via the Lyapunov approach. Mobile robots have been playing a key role in the areas related to transportation, logistics and healthcare. Single-wheel robots, two-wheel robots, and omnidirectional-wheel (ODW) robots have been designed in order to gain an increase in mobility. A ballt segway (BS), a ballbot-type mobile robot, was introduced in [16], where a PID two-loop controller is proposed to control ODWs for both balancing and transferring of the BS whose performance was evaluated experimentally. In [17], a double-loop controller is proposed to control the BS and whose performance was validated via simulation and experimental results. This BS has the capability to transport a single human passenger. It was shown that this BS can maintain its balance, perform trajectory tracking and rotate around its own vertical axis. The double-loop controller comprises an LQR outer-loop controller and a PI-plus-feedforward inner-loop controller. A PD controller was added to control the rotation of the BS around its vertical axis. The driving mechanism for the BS consists of three ODWs. Its governing equations are given by a second-order multi-input multi-output (MIMO) system with five degrees of freedom (DOF). The system dynamics can be decoupled into three planar planes under certain conditions. So, the whole system can be operated through the separate control of these decoupled dynamics. Robust controllers must be designed to face modeling uncertainties due to backlash between mechanical components, location center of mass, mass moment of inertia, nonlinearities from the motors, and friction between the ball and the road surface. In [18; 19; 20; 21], a hierarchical sliding mode control (HSMC) approach was used to address the balancing and movement problem for the BS. In [22], an adaptive version of the HSMC approach to the control of the BS was proposed. Energy/passivity methods, partial feedback linearization, and switching control have shown to be a good choice in the control of underactuated systems. A passivity-based nonlinear controller for a BS was suggested in [23]. In [24], a control scheme based on radial basis function networks was addressed, in conjunction with the dynamic surface control approach for balancing and trajectory tracking of a BS.