# Leader-Follower Formation Control of Wheeled Mobile Robots without Attitude Measurements
## 1 Introduction
Some very interesting works where the leader-follower formation is used are, for example [19], in which the formation problem is converted to a trajectory tracking problem, where each follower robot tracks its corresponding generated reference trajectory such that the whole group forms and maintains the desired shape. In this work, some experiments were successfully conducted and reported using a group of four TURTLEBOTs. In [20], the authors tackle the leader-follower formation control problem of non-holonomic mobile robots. In this case, the trajectory tracking control for a single non-holonomic mobile robot is extended to the formation control for two non-holonomic robots in which one is the leader and the second is the follower. The controllers proposed by the authors are based on the PI control technique. Simulation results are presented to demonstrate the good performance of the proposed controller. In this work, we addressed the leader-follower formation control problem for a group of nonholonomic wheeled mobile robots (WMRs). We do not follow the common distance-based approaches where the follower tracks the trajectory generated by the leader with an offset to avoid a collisions. In our approach, the followers track the delayed desired trajectory of the leader. The time delay is not arbitrary, on the contrary, it is obtained as the output of a dynamical system whose inputs are the position of the robots. The aforementioned dynamical system is designed in such way that, when the distance between the robots increases, the magnitude of the delay decreases and vice versa. The proposed methodology allows to achieve a convoy formation or platooning without collisions. Another advantage of the proposed approach is that the followers do not deviate from the leader's path during cornering [21] like the distance-based approaches. Moreover, the distance between robots can be modified by simple tuning the parameters of the delay's dynamical system. To track the desired trajectory, a novel control is proposed that exploits the cascade structure of the robot's kinematic model. On the other hand, one of the most common problems during the implementation of controllers is the lack of state measurements such as velocity, acceleration, orientation, to name a few. This absence of information could be treated by using different sensors to mediate it. However, this would make the system more complex and above all more expensive. Another factor, for example, is that accurate velocity measurements can be difficult, and actually, may be contaminated by the noises in real environments, which can deteriorate the control performance [22]. One of the alternatives to solve the problem of lack of information, is the design of state observers to estimate the measures that the controller needs. There are many works that use observers to estimate information of a system. In [23], the Cartesian position and the kinematic model is employed to design nonlinear observers to estimate the orientation angle and the linear velocity of a mobile robot. On the other hand, in [24] a state-feedback controller for the non-linear error dynamics of the robot is combined with an observer that estimates the orientation error based on available trajectory information and measurement of the position coordinates. Furthermore, in [25] kinematic and dynamic models of the WMR are described, and an output feedback controller is proposed using adaptive sliding mode controller and a high gain observer is designed for velocity estimation to obtain WMR trajectory tracking. In this paper, we assume that only the Cartesian position and its time derivative are available from measurements. To overcome the lack of attitude measurements a nonlinear observer is proposed based on the kinematic model of the robot. The orientation observer is designed directly on \(SO(2)\) and can be used in either open and closed loop schemes. The stability analysis is carried out by means of Lyapunov theory.