# Dynamic Stability of an Electric Monowheel System Using LQG-Based Adaptive Control
## 1 Introduction
In 1869, the first monowheel was constructed, which used a mechanism analogous to a bike to pedal the inner wheel, which became the standard structure for the many designs of unicycles that were to follow. A pedal force was used to drive an outer wheel directly using a gear in an unicycle patented in 1893. After that, a gear-driven chain coupe was used to give traction to the unicycle. The system is nonlinear and highly unstable, thus requiring a complicated and robust control design. [PERSON] et al. Figure 1: Basic design sketch of the monowheel. designed an observer-based cascade stabilizer, and a fuzzy controller for a similar system [9]. Various techniques were proposed by [PERSON] et al. for the PD control of an electric unicycle [10]. A modeling approach was undertaken by [PERSON] et al., along with a controller for self-balance [11]. Many developed system designs and nonlinear electric unicycle control methods have been proposed, such as those in [12; 13]. However, none of these have been implemented in the real world, and they do not account for friction. The LQR method, or the Linear Quadratic Method, has been a control branch for various such model-based systems, which over time minimizes an integral function as well as giving a quadratic measure of the states along with the control actions' quadratic measures. The LQR solution finally produces a set of constants, which at each point in time gives a control action that is optimal when multiplied by each of the states. LQR provides robustness in performance. Based on the sub-optimal approach for an AC servo motor, [PERSON] et al. develope a controller design scheme based on the discrete sliding mode [14]. [PERSON] et al. implemented this method to design a robust PI controller for a one-link robot by integrating it with a reduced parameter sensitivity technique [15]. This control method has been used in single-wheeled mobile robots [16], as well as to develop a state feedback controller, as in [17]. For establishing self-balancing, an adaptive friction compensation method was integrated by [PERSON] et al. along with this LQR method. It also included the speed tracking of the system while considering variable rider weights along with varying friction coefficients [18]. [PERSON] presented a working self-balancing unicycle, while using an existing control mechanism [19]. Bombardier worked on the monowheel as a sports vehicle powered by a hydrogen cell [20]. Ryno motors [21] demonstrated a prototype of the Ryno bike monowheel. [PERSON] et al. introduced a motorized transport vehicle for pedestrians, and they filed a US patent for the vehicle. A \"self-propelled unicycle engaged with vehicle\" was patented by Ford Motor Company [22], which was used only for independent use. This included a hub and a coupled wheel [23]. [PERSON] et al. thought of a compact folding bicycle which would lead to easy parking and transportation. The mechanism of folding could vary, with each offering various combinations of folding speed and folding ease, durability, compactness of frame, weight, ride, and variations of prices with changes in the internal material [24]. [PERSON] et al. designed a self-balancing mobile robot on two wheels, which primarily used a control moment gyroscope module. The robot is capable of achieving improved rotation in more confined spaces and has more speed than regular legged robots, Because of this, it is mostly used as a mobile robot platform [25]. [PERSON] et al. proposed the Microycle system, involving an analysis of the components of the system. The automatic steering mechanism helps the rider to balance in the rolling direction.This was followed by developing the dynamics of the unicycle in the pitch direction through the Lagrangian formulation. The plant was stabilized in the pitch direction using a linear PD controller, and the system was simulated on MATLAB Simulink. The controller helped in the implementation of the simulated and physical system, and data helped to quantitatively assess the control system [10].