Formation path following control of unicycle-type mobile robots

Abstract

This paper presents a control strategy for the coordination of multiple mobile robots. A combination of the virtual structure and path following approaches is used to derive the formation architecture. A formation controller is proposed for the kinematic model of two-degree-of-freedom unicycle-type mobile robots. The approach is then extended to consider the formation controller by taking into account the physical dimensions and dynamics of the robots. The controller is designed in such a way that the path derivative is left as a free input to synchronize the robot’s motion. Simulation results with three robots are included to show the performance of our control system. Finally, the theoretical results are experimentally validated on a multi-robot platform.

Introduction

During recent years, efforts have been made to give autonomy to single mobile robots by using different sensors, actuators and advanced control algorithms. This was mainly motivated by the necessity to develop complex tasks in an autonomous way, as demanded by service or production applications. In some applications, a valid alternative (or even the mandatory solution) is the use of multiple simple robots which, operating in a coordinated way, can develop complex tasks [1], [2], [3]. This alternative offers additional advantages, in terms of flexibility in operating the group of robots and failure tolerance due to redundancy in available mobile robots [4]. In the literature, there have been three main methods to formation control of multiple robots: leader following, behavioral and virtual structure. Each approach has its own advantages and disadvantages.

In the leader following approach, some vehicles are considered as leaders, whilst the rest of robots in the group act as followers[1], [5]. The leaders track predefined reference trajectories, and the followers track transformed versions of the states of their nearest neighbors according to given schemes. An advantage of the leader following approach is that it is easy to understand and implement. In addition, the formation can still be maintained even if the leader is perturbed by some disturbances. However, the disadvantage related to this approach is that there is no explicit feedback to the formation; that is, there is no explicit feedback from the followers to the leader in this case.

The behavioral approach prescribes a set of desired behaviors for each member in the group, and weighs them such that desirable group behavior emerges without an explicit model of the subsystems or the environment. Possible behaviors include trajectory and neighbor tracking, collision and obstacle avoidance, and formation keeping. In [6], the behavioral approach for multi-robot teams is described where formation behaviors are implemented with other navigational behaviors to derive control strategies for goal seeking, collision avoidance and formation maintenance. The advantage is that it is natural to derive control strategies when vehicles have multiple competing objectives, and an explicit feedback is included through communication between neighbors. The disadvantages are that the group behavior cannot be explicitly defined, and it is difficult to analyze the approach mathematically and guarantee the group stability.

In the virtual structure approach, the entire formation is treated as a single, virtual, structure and it acts as a single rigid body. The control law for a single vehicle is derived by defining the dynamics of the virtual structure and it then translates the motion of the virtual structure into the desired motion for each vehicle [7], [8], [9]. In [10], virtual structures have been achieved by having all members of the formation tracking assigned nodes which move into desired configuration. A formation feedback has been used to prevent members leaving the group. Each member of the formation tracks a virtual element, while the motion of the elements is governed by a formation function that specifies the desired geometry of the formation. The main advantages of the virtual structure approach is that it is fairly easy to prescribe the coordinated behavior for the group, and the formation can be maintained very well during the maneuvers; that is, the virtual structure can evolve as a whole in a given direction with some given orientation and maintain a rigid geometric relationship among multiple vehicles. However, if the formation has to maintain the exact same virtual structure all the time, the potential applications are limited, especially when the formation shape is time varying or needs to be frequently reconfigured.

In addition to the references mentioned so far, there have been many results on the mathematical analysis of formation control. In [11], the authors analyze the stability of the behavior of the first-order vehicles in a formation that relied on the Graph Laplacian associated with a given communication topology. Consensus algorithms applied to a team represented by a second-order dynamics are presented in [12] to guarantee attitude alignment and velocities in a group of vehicles using undirected communication information. New results on formation control based on second-order consensus protocols was given in [13]. In [14], new general potential functions are constructed to design formation controllers that yield semi-global asymptotic convergence of a group of mobile agents to a desired formation, and guarantee no collisions among the agents. However, most of those works model the agents as point robots and consider only the problem of position control of the robots. In numerous applications the orientation of the agents plays an important role, which means that the agents must be modeled as rigid bodies. Taking into account the position and orientation of the rigid bodies in a formation of a team of mobile robots, Breivik et al. [15] proposed a guided formation control scheme based on a modular design procedure that makes the design completely decentralized in the sense that no variables need to be communicated between the formation members. In [16], the problem of coordinated path following of multiple wheeled robots was solved by resorting to linearization and gain scheduling techniques. Even though the solution given to the problem is simple, it lacks global results; that is, convergence of the vehicles to their paths and to the desired formation pattern is only guaranteed locally.

The challenge we are tackling in this paper is to design a nonlinear formation control law based on a virtual structure approach for the coordination of a group of $N$ mobile robots [17]. This control law should force the robots’ relative positions with respect to the center of the virtual structure during the motion. The control system is derived in four stages. First, the dynamic of the virtual structure is defined. Second, the desired motion for each mobile robot is defined by translating by a given distance the motion of the virtual structure center. Third, the path following problem for each mobile robot is solved individually, by introducing a virtual target that propagates along the path. We interpret the path following errors in a triangular form, for which the backstepping technique can be applied to control the rate of progression of the virtual target along the path, and a local control law that makes the mobile robot track the virtual target. Finally, coordination is achieved by synchronizing the parametrization states that capture the positions of the virtual targets with the parametrization states of the virtual structure’s center. The formation of a simplified kinematic model of robots is first considered to clarify the design philosophy. The proposed technique is then extended to the dynamics of mobile robots with nonholonomic constraints. The experimental controller testing was performed on EtsRo vehicles, available at the GREPCI, Robotics Lab of the École de Technologie Supérieure of Montreal.

The reminder of this paper is organized as follows. The problem statement is presented in Section 2. Section 3 is devoted to modeling of the path following configuration for a wheeled mobile robot, while Sections 4 Control design, 5 Extension of formation control to the dynamic model of the mobile robot are the main part of this paper focussing on the study of the coordinated control for, respectively, a team of kinematic wheeled mobile robots and dynamic mobile robots. Simulation and experimental results are presented in Section 6. A conclusion is given in Section 7.