# Dynamic Stability of an Electric Monowheel System Using LQG-Based Adaptive Control
## 1 Introduction
The study of the monowheel is relatively recent, with noticeable research work in this field. Although some concepts of self-stabilizing monowheel robots already exist, most of them are structurally quite different from our approach, and their stabilization methods lead to some restrictions which we wish to overcome. [PERSON]'s robot possesses a single wheel with an extended arm, which is stretched out to the left and right to maintain the roll balance. Although not many positive experimental results were obtained, this robot was speculated to be the the first self-balancing unicycle robot [20]. In [26], the authors describe one of the most similar versions to our system. Another such version with a constructed model is known as Gyrover [3], which implements a mechanical gyroscope inside the wheel in order to stabilize the vehicle while stationary or when moving at very low velocities. The gyroscope's axis is altered using a tilt mechanism to enable the turning of the robot through the precision effect. This is similar to the design proposed by [PERSON] [27]. A robotic self-balancing unicycle was proposed by [PERSON], which implements a horizontal flywheel that mimics the twisting of an arm and torso movement to establish balance as well as the motion control of the robot [19]. An LQG controller action was proposed by [PERSON] et al. in 1990 in order to obtain control stability over [PERSON]'s robot [21]. In 2007, the University of California, San Diego developed the UniiBot, which combined the ideas of the \"inertia wheel pendulum\" along with the \"wheel inverted pendulum\" and achieved lateral balance with the help of vertical flywheel control [23]. In 2005, [PERSON] et al. developed a self-balancing unicycle robot that achieved lateral stability with the help of two gyroscopes that acted as actuators [22]. In 2009, [PERSON] et al. developed a dynamic nonlinear model with a vertical flywheel for a unicycle robot and simulated the model [25]. With the sliding mode control of a single-wheel robotic vehicle, [PERSON] et al. developed a similar robot that utilized a flywheel that was vertical, which was then verified through simulations [24]. There are numerous ongoing research works about this system in the fields of robotics and automation. However, what is noticeable is the limited work aimed towards a physical application that considers a rider. Furthermore, while various control systems for this system are available in the literature, a sufficient focus has not been placed on the LQR control of such systems with varying load masses. Constructing laboratory-grade dynamic self-stabilizing robots has been a research topic for a while in the field of automation. We aim to take this a level further so that the monowheel becomes a viable mode of future transport. We wish to build a more realistic system, considering the rider's necessities, that adapts to an extensive range of rider masses and external disturbances such as wind, friction, bumpy roads, etc. Further, it is very much possible that the monowheel may be subjected to sudden parameter changes, such as a change in the rider mass. This is possible if the rider uses the monowheel as a means to deliver goods. There will be an increase in load mass as he picks up or drops the delivery goods at certain points in time. In order to make the monowheel robust to these sudden parameter changes, we also aim to develop an adaptive control strategy that accounts for this. There are many adaptive control techniques available in the literature [28; 29; 30]. In this paper, we have aimed to achieve the following:  
1. Establish the dynamic stability of the system and make it self-balanced in the linear direction (forward and backward movement), with variations in rider input velocity, using the method of LQR control, and for a wide range of rider masses;
2.