Globally rigid formation of n-link doubly nonholonomic mobile manipulators

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Abstract

This paper provides a new framework for the collective motion control of a team of n-link doubly nonholonomic mobile manipulators in a constrained environment. A continuous decentralized motion planner is proposed. It guarantees the establishment and strict maintenance of a team formation using the Lyapunov-based control scheme (LbCS), which takes into account all the practical limitations, and the constraints due to fixed obstacles, nonholonomy and globally rigid formation requirements. The control scheme inherently utilizes artificial potential fields within an overarching leader–follower framework to mobilize the prescribed globally rigid formation. The designated leader of the team is modeled as a moving reference point, referred to as the virtual leader. With its protective polygonal region, it directs the motion of the team and manipulate the geometry of the formation. It is the authors’ belief that this navigation and globally rigid formation control problem of n-link doubly nonholonomic mobile manipulators is treated for the first time with the use of continuous time-invariant control laws within the framework of LbCS and a variant of the leader–follower scheme. The effectiveness of the motion planner and the resulting acceleration-based control laws are demonstrated via computer simulations.

Introduction

In recent years, the concept of formation control of multi-robot systems has received an unprecedented uptake and attention from researchers all over, in both theoretical research and real-world applications. Formation control is an integrated branch of motion planning and control of robots, which is the coordinated and collision free movement of multiple robots, and completion of tasks [[1], [2], [3]].

While copious methodologies appear in literature that address control of multi-robot systems, formation control of such systems operating in difficult, dull, dirty and constrained environments is been favored because of its relevancy to various real-life applications [[4], [5], [6], [7], [8], [9]]. As emphasized in the literature the primary objective of formation control is to control the posture of a team of robots while normally maintaining constant their relative locations or the prescribed geometrical structure and allowing them to travel to their desired destinations [[10], [8]]. Its applications include surveillance, transportation, health-care, reconnaissance, save and rescue, pursuit-evasion, tunnel passing, surveying, troop hunting, and explorations, in various environments [[9], [11], [6]].

The idea of formation by design is inspired by collective behaviors seen in nature [12], such as ant swarming, bird flocking, fish schooling and animal herding. Accordingly, to mimic these group behaviors, formation control can be divided into four layers: formation shape, formation type, formation tracking, and robot roles [[6], [7]]. In general, the literature harbors numerous approaches to address the problem of formation control in robotic applications. These approaches can be roughly categorized into five generic approaches [13], namely, leader–follower, virtual structure, generalized coordinates, behavior-based, and social potential fields. A brief review of these approaches is presented in [[14], [13], [7], [15], [3]] and the references therein. Nonetheless, incorporating leader–follower schemes via an artificial potential fields (APFs) method is gaining popularity for formation control due to their seamless formulation and scalability [[16], [17], [18], [9]].

One of the adaptive challenges of formation control is the maintenance of a degree of rigidity of the pattern formed by the multi-agents. Using the nomenclature from [[19], [20]]; while one end of the formation spectrum allows for ungoverned changes to the formation, essentially known as the split/rejoin or minimally rigid formation, the other end requires a strict maintenance of the formation which is classified as the globally rigid formation. Finally, there is the locally rigid formation which allows for temporary distortions to the formation upon encountering obstacles and restrictions.

Recent applications include the split/rejoin maneuvers of flocks of nonholonomic vehicles in [[9], [20]], and the establishment and translation of locally rigid formations of swarms of planar 2-link nonholonomic mobile manipulators in [8], to name a few. However, while the aforementioned research proved stability, the proposed algorithms are unable to guarantee globally rigid formation of nonholonomic systems in obstacle-ridden environments. In situations such as cooperative payload transport or in distributed sensor deployment applications, strict maintenance of the geometric formation is critical throughout the motion [[21], [22], [23], [9], [8]]. The authors note that the problem of motion planning and control of a globally rigid formation of nonholonomic vehicles is still an open problem.

In this paper, we present a unique framework for formation control of a team of n-link doubly nonholonomic mobile manipulators in a constrained environment using an integration of the Lyapunov-based control scheme (LbCS) and the leader–follower scheme containing a virtual leader. While in practice, the leader–follower approach is not robust to communication delay or loss, and external disturbances, the principal benefits of a virtual leader is that the formation control problem of a multi-robot system reduces to a single robot control from which the other robots derive their control laws with the requirement of communication of coordination information [12], and it offers a high degree of robustness since the multi-robot system does not depend on any of its robots, rather a virtual object. As such, we design a motion planner derived from the LbCS, that guarantees the establishment and maintenance of a globally rigid formation of a team of doubly nonholonomic mobile manipulators, considering all practical limitations, and constraints due to fixed obstacles, nonholonomy of the mobile manipulators, and the formation constraints within an overarching leader–follower framework adopted from [9].

While literature is inundated with research on motion planning and control of an array of robotic systems, this paper adds a unique algorithm to establish and mobilize a globally rigid formation of a team of n-link doubly nonholonomic mobile manipulators within an obstacle cluttered environment. It is the authors’ belief that this navigation and formation control problem of n-link doubly nonholonomic mobile manipulators is treated for the first time with the use of continuous time-invariant nonlinear control laws within the framework of LbCS and a variant of the leader–follower scheme.

Within this variant leader–follower scheme, we consider a novel technique of using a virtual guide and its protective polygonal region to prevent possible distortions or splits to the prescribed globally rigid formation, which is mobilized through the concept of ghost targets.

As an application, we imitate a real-life scenario wherein a snowplough removes or clears snow and ice from existing routes and possibly creating newer and safer routes. The snowplough reveals an unobstructed path for vehicles to travel on. Acting like a leader, it provides a prescribed formation of movement for all vehicles following it. The novelty of the paper resides with the use of the snowplough system to translate a globally rigid formation within the framework of LbCS and the leader–follower scheme.

Section snippets

Doubly nonholonomic mobile manipulator

We adopt a homogeneous team of N (NN) mobile robots and associated nomenclature from [[24], [20]], where the ith robot is an under-actuated n-link doubly nonholonomic mobile manipulator (nDNM) denoted as Ai.

Definition 1

The set Ai consists of nonholonomic car-like wheeled platforms with a n-link nonholonomic manipulator mounted on the mid-front axle of the platform. For each Ai, the wheeled platform is a disk with radius r0 and is positioned at the center (xi0,yi0). The wheeled platform

Problem formulation

Assumption 2

The prescribed globally rigid formation is referenced with respect to Cartesian positions of end-effectors of the mobile manipulators.

In this research, we establish and maintain rigid distances between the end-effectors of the mobile manipulators fixed in a globally rigid formation with the aid of a virtual leader. Essentially, we maintain a strict and continued cohesion of the team of nDNMs once the prescribed globally rigid formation is generated, and guarantee

Lyapunov-based control scheme

We deploy an artificial potential field method known as the Lyapunov-based Control Scheme (LbCS) from [[28], [24]] for this research. The scheme has been recently applied by the authors to motion planning and control of various robotic systems, including ones tagged with holonomic and/or nonholonomic constraints [[9], [29], [30], [28], [24], [20]]. The main idea behind LbSC is to design attractive and repulsive potential functions which are summed to form a suitable Lyapunov function that acts

Artificial potential field function

In accordance with the LbCS we first construct attractive and repulsive potential functions required to treat each subtask integrated to the control objective.

Assumption 3

For the target attraction, and the obstacle and collision avoidance functions that follow, we have i{1,2,,N}, NN, for Ai and m{0,1,,n}, for the mth articulated body of Ai.

Design of controllers

The nonlinear control laws for system (2) will be designed using the LbCS. To ensure that the controllers vanish at the target configuration, auxiliary functions, denoted as F0 and Fi, are designed for the virtual leader and Ai, respectively F0(x)=12(x0p01)2+(y0p02)2,Fi(x)=12(xipi1)2+(yipi2)2+ρi0(θi0pi3)2+k=1nρik(θikpik+3)2.Note that pi3 and pik+3 are the final orientations of the platform and Link k , respectively, of Ai. Here, ρi0 and ρik are the angle-gain parameters, which will be

Stability analysis

We utilize the Second Method of Lyapunov to provide a mathematical proof of the stability of system (2).

Theorem 2

If the fixed points xi=pi1,pi2,pi3,pi4,,pin+3,0,0,0,0Rn+7(i=1,2,,N) and x0=p01,p02,0,0R4 are equilibrium points of the end-effector of Ai and the virtual leader, respectively, then xe=(x0,x1,x2,,xN)D(V  (2)(x))D is, at least, a stable equilibrium state of system (2) .

Proof

One can easily verify the following, for i{1,2,,N} and m{0,1,2,,n}

Simulation results

This section provides computer simulations of two traffic situations for a team of 3DNMs (n=3) navigating in a workspace WS cluttered with obstacles, under the guidance of a virtual leader. In both scenarios, dimensions, maximum translational and rotational velocities, and the maximum steering angle of each 3DNM are kept the same. We note that for the two scenarios the obstacle avoidance rules tagged to the solid objects fixed within the boundaries of WS are applied to the virtual leader only

Concluding remarks

In this paper, we have presented a set of decentralized acceleration control laws that successfully tackle the problem of rigid formation control of a team of n-link doubly nonholonomic mobile manipulators (nDNMs) operating within a constrained environment. The advancement of the prescribed formation was facilitated by moving ghost targets, which is a variant of the leader–follower scheme. To the author’s knowledge this is the first time in literature that a rigid formation of nDNMs is

Dr. Bibhya Sharma (Ph.D.) is an Associate Professor of Mathematics at the University of The South Pacific (USP). He has also been the Associate Dean for Learning and Teaching within the Faculty of Science, Technology and Environment at USP since 2010. He is a member of a number of professional mathematics societies and unions and has published more than 70 articles and book chapters in the fields of mathematics education, robotics, biologically inspired processes and the TLPs of higher

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    Dr. Bibhya Sharma (Ph.D.) is an Associate Professor of Mathematics at the University of The South Pacific (USP). He has also been the Associate Dean for Learning and Teaching within the Faculty of Science, Technology and Environment at USP since 2010. He is a member of a number of professional mathematics societies and unions and has published more than 70 articles and book chapters in the fields of mathematics education, robotics, biologically inspired processes and the TLPs of higher education. He champions adaptive works and innovations in the areas of eLearning and mLearning for higher education. More information regarding his work and achievements can be found at http://scims.usp.ac.fj/~bibhyasharma/.

    Mr. Shonal Singh received his M.Sc. in Applied Mathematics from The University of the South Pacific, Fiji, in 2012. He is currently a Ph.D. candidate in Mathematics at The University of Sydney. His main fields of interest are integrable systems, with a particular focus on geometry of higher-order Painlevé equations, and mathematical modeling and simulations.

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    Dr. Avinesh Prasad received his Master’s degree in applied mathematics from the University of the South Pacific, Fiji, in 2008. He completed his PhD degree in applied mathematics from the same university. He is currently a Senior Lecturer in Mathematics with the School of Computing, Information & Mathematical Sciences. His area of research interest are dynamics of nonlinear systems and robotics.

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