# Formation Control and Distributed Goal Assignment for Multi-Agent Non-Holonomic Systems
## 1 Introduction
The second of these papers proposes a solution to the problem of collision avoidance with static and dynamic obstacles present in the environment. In [21], collision avoidance is obtained by applying safety constraints in optimal trajectory generation. The robots do not have non-holonomic constraints (they are quadrotors) and they have to change the shape of the formation using goal assignment that minimizes the total distance travelled by the robots. [PERSON] et al. [22] present concurrent assignment and planning of trajectories (CAPT) algorithms. The authors propose two variations of the algorithm: centralized and decentralized, and test them on a group of holonomic mobile robots moving in a three-dimensional space. The solution is based on the Hungarian assignment algorithm. The above-mentioned works do not deal with the problem of closed-loop control. For this reason stability issues were not considered there. In comparison to the above two approaches, here the user or the higher level controller determines the locations of desired trajectories according to the needs (this can be considered as an expected feature in many applications). In addition, trajectory generation is decoupled from the closed-loop control (the system is modular). Such a solution is considered as a design pattern in robotics. The algorithm is responsible for tracking these trajectories and reacting to the risk of collisions between agents at the same time. Even if initial states of individual robots are far from the desired ones, the collision avoidance module works correctly. Furthermore, the algorithm characterizes conceptual and computational simplicity. The paper considers preserving data integrity during the goal exchange as it requires a simultaneous change of states in remote systems. This subject is omitted in the literature on goal assignment in multi-robot systems. To the best of the author's knowledge, no work has been published so far that proposes closed-loop control for multiple non-holonomic mobile robots combined with target assignment with analysis based on a Lyapunov-like function for both the tracking algorithm and target assignment. [PERSON] et al. [23] propose a similar method, but it assumes that agents are fully-actuated (modeled using integrators), and the analysis is based on multiple Lyapunov-like functions. This algorithm also uses a different criterion for goal exchange. The algorithm proposed in this paper is applicable mainly to the homogeneous formations of non-holonomic mobile robots but also in the scenarios when two or more robots of the same type are involved in task execution. For a high number of applications of multiple mobile robots this situation occurs, e.g., exploration, mapping, safety, and surveillance. In Section 2 a control algorithm for the formation of non-holonomic mobile robots is described. Section 3 analysis stability. The simulation results are presented in Section 4. The goal exchange algorithm is introduced in Section 5. Section 6 provides stability analysis of extended algorithm. Section 7 details distributed implementation. Some generalization of the proposed goal exchange rule is given in Section 8. Section 9 discusses the problem of maintaining data integrity in the process of the goal exchange. Section 10 offers simulation results for the goal exchange algorithm. Section 11 presents simulation results for limited wheel velocity controls. In the last Section concluding remarks are provided.