# Backstepping Based Formation Control of Quadrotors with the State Transformation Technique
## 1 Introduction  
Unmanned aerial vehicles (UAVs) have received much attention in the past few years due to their great potential applications in exploration, transportation, reconnaissance and surveillance. Among all the UAVs, the quadrotor is the most widely used because of its easy implementation compared to other aerial vehicles, its vertical take-off and landing (VTOL) ability and its high agility and maneuverability. Recently, there has been significant research interest in quadrotor applications especially those using multiple quadrotors. By operating multiple quadrotors, tasks that are impossible with a single quadrotor can be performed, and single quadrotor tasks can be performed faster and more efficiently. Therefore, there has been considerable attention on studies focusing on quadrotor formation control [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14]. There are several quadrotor formation control methodologies including behavior based [1], leader-follower [2; 3; 4; 5; 6; 7; 8], virtual structures [9; 10] and graph theory [11; 12; 13; 14]. Among these methodologies for the formation control, the leader-follower approach is well recognized as the most popular and intuitive approach [15]. In the leader-follower approach, one of the agents in the formation is designated as a leader, and the other members are defined as the followers. The leader transmits its state information to the followers, but it does not receiving any information from the followers. However, the followers can transmit and receive information. The formation control methodologies mentioned above commonly require precise position control of each quadrotor for precise formation control of multiple quadrotors. However, since the quadrotor system is an underactuated system in which the number of outputs is greater than the number of inputs, the precise position control of a quadrotor is not easy to achieve since the \(x,y\) and \(z\) position system of a quadrotor is connected to only one altitude input unlike the attitude system of a quadrotor. Therefore, the \(x\) and \(y\) positions of the quadrotor are not directly controlled by the inputs. This is why the precise position control of a quadrotor is not easy. Position control based on linear control methods such as proportional-integral-differential (PID) and the linear quadratic regulator (LQR) has been proposed [2; 3; 5; 16; 17], but it has limited performance because it was only focused on the local behavior of the system. Considering the nonlinear dynamics of a quadrotor, various nonlinear control methods have been proposed. Many researchers applied the backstepping control method to solve the problem of the position control mentioned above [4; 8; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28]. The position control of a quadrotor using the backstepping method can be roughly classified in two ways. The first one is to derive the reference attitude angles for the position control and formation control [4; 18; 20; 21; 22; 23; 24; 27; 28]. The other one is to derive the final inputs of a quadrotor by connecting the position to the attitude dynamics of the quadrotor without the use of reference attitude angles [19; 25; 26]. However, this approach has the disadvantage that it is difficult to actually implement because the final input formula is complicated and there is much computational burden with respect to the input. Therefore, most researchers commonly use the method to derive the reference attitude angles for the position and formation controls and the position and formation controls achieved through attitude tracking control of the reference attitude angles [29; 30]. However, it is not easy to derive the reference attitude angles from the position dynamics of a quadrotor consisting of several coupled trigonometric functions. Therefore, in most studies, the reference attitude angles for the backstepping-based position control are obtained through simplified quadrotor dynamics or linearization methods such as the small angle assumption (SAA) [4; 18; 21; 22; 23; 28].