# Dynamic Stability of an Electric Monowheel System Using LQG-Based Adaptive Control
## 1 Introduction  
The invention of the wheel was a path-breaking event in the history of humankind. Since then, man has used the concept of the wheel to construct various vehicles that have revolutionized transportation. Over the years, transport vehicles have evolved from having multiple wheels to four wheels, to the most widely used domestic transport mode of the two-wheeled vehicle. In this paper, we have decided to take this evolution a step further by physically constructing, from first principles, a self-balancing single-wheeled vehicle, also known as the monowheel, as shown in Figure 1. The electric vehicle concept is emerging because of increased awareness about ongoing pollution and the energy shortage crisis. Today, automobiles and motorcycles are no longer the best forms of transportation. With the ever-increasing prices of petroleum products, there is a need for a form of transport that is cheaper and more efficient. Saving energy to mitigate the problem of fuel depletion is of prime importance. Such demands are causing a surge in research on eco-friendlytransportation. Electrical vehicle technology is a step towards fulfilling these goals, and so the idea of an electric unicycle or monowheel is attractive [1]. Many cities are now allowing personal mobility vehicles to mix with pedestrian traffic to de-congest modern roads. The electric monowheel is ideal for such a situation owing to its compactness in structure. It is a transitional vehicle that goes to most places where a person can walk or ride a bike, and it is easy to carry around due to its light weight. Typically, in conventional two-wheeled transportation, one wheel provides the force for speed control, while the other wheel deals with changes in direction and steering. However, in a monowheel, both the speed as well as direction are controlled through the single physical machinery. The electric monowheel has attracted a great deal of attention from both academia as well as the industry [2; 3; 4]. These vehicles are available as fun vehicles for children, people who are style-conscious, and creative workers in many domains. The monowheel is derived initially from the unicycle concept, upon which various research works have been conducted. [PERSON] et al. presented an optimal minimum energy control solution for the same through Jacobi Elliptic Functions [5]. [PERSON] et al. [6] proposed a schematic design of a unicycle. Similar to a Segway device, it comes with a handling rod for maneuvering and a supportive seat for the rider. They then conducted a dynamic analysis and proposed a robust control law for their schematic design. [PERSON] et al. [7] developed and conceptualized the ergonomic study of a typical unicycle. Based on a dynamic model, [PERSON] et al. [8] proposed a steering control method for the torque applied on an underactuated unicycle robot. Many researchers believe that the monowheel can be a serious vehicle for transportation in the future. It is a pollution-free vehicle that needs only one DC motor to drive it. A few researchers developed an electric unicycle made from inexpensive components, such as an DC brush motor, with an inexpensive gearbox, an accelerator, an eight-bit micro-controller, a rubber wheel, a rate gyro, a PWM motor driver, and nickel-metal hydride (NiMH) batteries. Also, it is important to ensure an economical, eco-friendly means of short-distance transportation due to rising fuel prices. The electric unicycle has not been a commercial success. However, we trust that it can become a popular single-wheeled vehicle that satisfies the parameters of being low cost, economical, and ecologically friendly. The design and implementation of a practical and logistically safe self-balancing monowheel have garnered attention in modern times. Since the late 1800s, many different kinds of single-wheeled vehicles have been proposed with a similar design, drive mechanism, and power source. In some cases, propellers were the prime movers, while in others gas engines or electric motors were used.