Path Following of Autonomous Agents under the Effect of Noise Krishna Raghuwaiya, Bibhya Sharma and Jito Vanualailai School of Comp, Info and Mathematical Sciences University of the South Pacific Suva, Fiji {raghuwaiya k, sharma b, vanualailai}@usp.ac.fj Parma Nand School of Computing and Mathematical Sciences Auckland University of Technology Auckland, New Zealand parma.nand@aut.ac.nz Abstract—In this paper, we adopt the architecture of the Lyapunov-based Control Scheme (LbCS) and design attractive and repulsive potential field functions to ensure a collision- free path following strategy of a group of mobile car-like robots using a leader-follower approach. A robot is assigned the responsibility of a leader, while the follower robots position itself relative to the leader so that the path of the leader robot is followed with arbitrary desired clearance by the follower robot avoiding any inter-robot collision while navigating in a terrain with obstacles. The set of artificial potential field functions is proposed using the Direct Method of Lyapunov for the avoidance of obstacles and attraction to their designated targets. The effectiveness of the proposed nonlinear acceleration control laws is demonstrated through computer simulations which prove the efficiency of our control technique and also demonstrates scalability for a large group of robots. Index Terms—Lyapunov, nonholonomic mobile robots, path following. I. I NTRODUCTION Motion control problems in mechanical systems with non- holonomic constraints have been widely addressed in litera- ture can be roughly classified into three groups: point stabi- lization, trajectory tracking and path following [1]. Formation control algorithms that enable groups of autonomous agents to follow designated paths can be useful in the planning and execution of various assignments. This could be in a variety of problem domains where robots are used such as search and rescue, space exploration, environmental surveillance to name a few [2]. From a control science point of view, the accuracy and performance of wheeled mobile robots trajectory tracking are subject to nonholonomic constraints and usually difficult to achieve stabilized tracking of trajectory points using linear feedback laws [3]. To deal with these, many researchers have proposed controllers that utilized discontinuous control laws, piecewise continuous control laws, smooth time varying con- trol laws or hybrid controllers [4]. Path following problems are more flexible than trajectory tracking and is primarily concerned with the design of control laws when manoeuvring objects (robot arm, mobile robots, ships, aircraft etc) to reach and follow a geometric path without strict temporal specifications [5], [6]. following, the control laws consider the distance from the vehicle to the reference path and the angle between the vehicles’s velocity vector and the tangent to the path [7]. For multi robot systems, coordinated path following entails making each robot approach a preassigned path and once on the path, the robots are required to coordinate. This could mean getting into formation, maintaining a desired intervehicle formation, or getting its path variables [8]. In this paper, we adopt the architecture of the LbCS and design attractive and repulsive potential field functions to ensure a collision-free path following strategy of a group of mobile car-like robots using a leader-follower approach. A robot is assigned the responsibility of a leader, while the follower robots position itself relative to the leader so that the leader robot is followed with arbitrary desired clearance by the follower robot. The scheme uses Cartesian coordinate’s representation as proposed in [9]. Based on artificial potential fields, the LbCS is then used to derive continuous acceleration-based controllers which render our system stable. The control algorithm used merges together the problems of path following and obstacle collision avoidance as a single motion control algorithm. The remainder of this chapter is structured as follows: in Section II, the robot model is defined; in Section III, the artificial potential field functions are defined under the influence of kinodynamic constraints; in Section IV, the acceleration-based control laws are derived and stability anal- ysis of the robotic system is also carried out; in Section VI, we demonstrate the effectiveness of the proposed controllers via computer simulations which guide the follower robot to follow the leaders reference path with an error; and finally, Section VII closes with a discussion on its contributions. II. VEHICLE MODEL Consider the vehicle model of N i for i =1,...,n in the Euclidean plane. Without loss of generalization, we let N 1 represent the leader and N i , for i =2,...,n take the role of followers. With reference to Fig. 1 and for i =1,...,n,