# Balance Control of a Configurable Inverted Pendulum on an Omni-Directional Wheeled Mobile Robot
## 1 Introduction  
The inverted pendulum model can be found in many engineering applications such as walking robots [1], personal transporter [2], and single-wheeled mobile robot [3]. The dynamics of an inverted pendulum are inherently open loop unstable, nonlinear, underactuated, and non-minimum phase, so balance control of the inverted pendulum is a formidable challenge. It is widely used as a benchmark for designing and validating different control techniques. Due to its importance in control engineering, the inverted pendulum has been often used in control education for many decades. For research and education on the control, several variants of inverted pendulum systems with distinct mechanism constructions and actuation have been proposed and studied, such as rotary inverted pendulum [4], Acrobot [5], Pendubot [6], inertia wheel pendulum [7], cart-pole inverted pendulum [8], double inverted pendulum [9], triple-link inverted pendulum [10], and spherical inverted pendulum [11, 12], etc. The omni-directional wheeled mobile robot [13, 14] is a particularclass of wheeled mobile robots. This robot uses orthogonal omni-directional wheels, which provide traction in the direction normal to the wheel shaft, and the wheel can slide passively in the direction of the wheel shaft. This mobile robot does not have nonholonomic constraints and can simultaneously perform translational movement along an arbitrary trajectory combined with any orientation. In [15], an omni-directional wheeled mobile robot was used to generate the planar control force for balancing a spherical inverted pendulum. This paper uses second-order sliding mode control [16] to stabilize the inverted pendulum systems. Sliding mode control has been known as a robust and nonlinear control technique appropriate for compensating matched uncertainties and disturbances in dynamical systems [16; 17]. Essentially, the state trajectories of the closed-loop system are constrained to a specific manifold in the state space in finite time and to retain their motion to slide along the specific manifold for all successive times when sliding mode control uses switching control actions. The closed-loop system is insensitive to external disturbances and model uncertainties because the motion of the state trajectory is constrained on a predefined sliding manifold. The chattering behavior [17] is the main obstacle for the practical application in the conventional or first-order sliding mode control. The chattering problems can be effectively attenuated using higher-order sliding mode controls [16; 18] with a relative degree two or higher. Another merit is that higher-order sliding mode control extends the relative degree of the sliding variable in the conventional sliding mode control from one to a higher order. Due to its simplicity and low information demand, second-order sliding mode control is the most widely used in practice [19; 20; 21] among the higher-order sliding mode control. This paper considers the problems of the control design and experimental validation of balance control of a configurable inverted pendulum system with an omni-directional wheeled mobile robot. This system can be configured as a rotary inverted pendulum system or a spherical inverted pendulum system. The omni-directional wheeled mobile generates the designed control torque for balancing the rotary inverted pendulum and the designed control planar force for balancing the spherical inverted pendulum. Figure 1 is the schematic diagram of the proposed system, (a) is a rotary inverted pendulum system, and (b) is a spherical inverted pendulum system. For simulation and control design, mathematical models of omni-directional wheeled mobile robot and both inverted pendulums are derived. To enhance robustness to model uncertainties and disturbance, a coupled sliding manifold [22] and second-order sliding mode control are applied to design stabilizing controllers for both inverted pendulum systems.