# Inverse Optimal Design for Position Control of a Quadrotor
## 1 Introduction  
Over the past few years, the field of unmanned aerial vehicles (UAVs) has rapidly expanded. In addition, because of the advantage of UAVs, there has been an increase in their applications, such as aerial reconnaissance, where life-saving operations are risky for pilots, and disaster surveillance and monitoring. In particular the quadrotor, one type of UAV, has many advantages. It has the advantage of easy implementation compared to other aerial vehicles. In addition, it has vertical take-off and landing (VTOL) ability, high agility and maneuverability. Recently, there has been significant interest in the application of quadrotors, so there has been much attention on the studies that focus on the control of a quadrotor. In the field of a quadrotors, the problem of control design has focused primarily on typical quadrotor missions including attitude stabilization and movement from one position to another position. Linear control methods such as proportional-integral-differential (PID) control and the linear quadratic regulation (LQR) control methods are proposed in [1, 2]. However, linear control methods simplify the model and focus only on the local behavior of the system. In practice, quadrotors have complicated dynamics with coupled states that cause nonlinearities. Furthermore, the dynamics of a quadrotor are not only nonlinear, but they are also difficult to characterize because of the complexity of the aerodynamic properties. For these reasons, various nonlinear control methods have been proposed including nested saturations [3], dynamic inversion control approach [4], sliding mode control methods [5, 6], feedback linearization method [7], backstepping control approaches [8, 9, 10], integral predictive nonlinear \(H_{\infty}\)control [11], and neural network based output feedback control method [12]. Meanwhile, using optimal control techniques, research for battery operation and missions to consider the optimality have been performed [13; 14]. In other words, researchers are interested in realizing the optimality of controllers other than the stability. The optimal control problem for nonlinear systems is usually attributed to the solvability of the Hamilton-Jacobi-Bellman (HJB) equation. Because the HJB equation involves partial differential equations, optimal control design is generally too difficult to be solved analytically, with the exception of a few special cases. An alternative method is the so-called inverse optimal control method, which does not require a solution to the HJB equation [15]. In this case, the minimization of the performance index is determined _a posteriori_ according to the given controller, so the aforementioned difficulty can be alleviated. In [16], the author proposed an inverse optimal control of a quadrotor by employing the backstepping technique. However, it involves controlling only the attitude of a quadrotor. Moreover, to the best of the authors' knowledge, there have been no reports on the application of inverse optimal based position control for quadrotor systems. Most of the research into the position control of quadrotors use the small-angle assumption to simplify the position of the dynamic equation of a quadrotor [1; 3; 17], and they derive the reference roll and pitch angles that are outputs from the position controller. Unlike the aforementioned methods, in this paper, we derive the reference roll and pitch angle caused by the modified dynamics of the position of a quadrotor using the state transformation technique without any assumption. Furthermore, we employed the state transformation technique to simplify the design process of the inverse optimal controller. In addition, it can make the proof of inverse optimality simple. In this paper, we propose an inverse optimal position control of a quadrotor. First, we derive the dynamics of the quadrotor using the Newton-Euler formulation. Then, we present the state transformation technique to derive the position dynamics from the kinematic and dynamic models of a quadrotor.