Velocity controllers for a swarm of unmanned aerial vehicles

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Abstract

The biological concept of a swarm’s emergent behavior resulting from the self-organization of the individuals in a swarm is an important piece of information that can be integrated into industrial manufacturing of unmanned ground or aerial vehicles. In this paper, we present a Lyapunov-based path planner that organizes the individuals, governed by a simple rigid-body model, of a swarm into formations which ensure a safe collision-free path for the swarm to its target in obstacle-cluttered environments. Via the Direct Method of Lyapunov that establishes the swarm system’s stability, we propose the instantaneous velocity function for each individual. The velocity functions could be easily integrated into industrial designs for the individuals of a swarm of unmanned ground or aerial vehicles. As an application, we consider the planar formation of a swarm of unmanned aerial vehicles (UAVs) using their kinematic models for simplicity. Via computer simulations, we illustrate several self-organization such as elliptic and linear formations, split/rejoin, and tunneling maneuvers for obstacle avoidance and helical trajectory for milling. In particular, the linear formations have been proposed as suitable for the surveillance of large areas such as the Exclusive Economic Zones.

Introduction

The ability of a swarm to take several formations, which is the emergent behavior of the swarm, is mesmerizing. The integration of this ability into the industry is of importance as swarm formations play an increasingly important role in diverse fields such as robotics, computer science, surveillance, military, economics, biology, and industrial automation [1]. The principle of swarming from which emergent patterns arise has a various wide range of applications or possible applications in the industry. For instance, the possible use of swarm robots for carrying out deep mining operations in hazardous environments [2], the use of swarm unmanned aerial vehicles (UAVs) in the monitoring of; air pollution caused by the gases released due to industries [3], large farms for precision agriculture [4], and exclusive economic zone (EEZ) [5], and using swarms of robots for cooperative transportation and geological surveys [6]. Swarms of robots are also utilized to monitor defects in civil infrastructure by the construction industry. In the energy production industry, swarms of UAVs or unmanned ground vehicles (UGVs) are also used for monitoring power lines, oil and gas pipes, to name a few.

Emergent formations arise naturally and ostensively in a swarm due to the self-organization of individuals which use simple local rules to govern their action [5], [7], [8], [9], [10], [11], [12], [13]. Self-organization of swarm individuals is a process in which patterns at the global level emerge solely from numerous interactions among the individuals in a local context through direct communication or environmental observations without reference to the global pattern. The emergent behavior of a swarm emerges by the meaningful collaboration of swarm individuals exhibiting the overall system’s capabilities beyond the capabilities of a single member of the swarm [14], [15]. Operational principles of swarming can be a foundation for developing distributed cooperative control, formation control, coordination and learning tactics for autonomous multi-agent systems such as UAV and UGVs [10], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31]. These have gained the attention of many researchers to understand the mechanisms of swarming and its industrial applications such as in aircraft industry [32], [33].

Over the last twenty years, the attempts of researchers to comprehend biological swarming can be categorized into two different modeling approaches: the Eulerian and the Lagrangian approaches [5], [34], [35], [36], [37], [38], [39], [40]. In the Eulerian approach, the swarm is considered a continuum described by its density in one-, two- or three-dimensional space. The time evolution of swarm density is described by partial differential equations. In the Lagrangian approach, the state (position, instantaneous velocity or instantaneous acceleration) of each individual and its relationship with other individuals in the swarm is studied; it is an individual-based approach, in which the velocity and acceleration can be influenced by spatial coordinates of the individual. The time evolution of the state is described by ordinary or stochastic differential equations. Comprehensive reviews of these approaches and their advantages and disadvantages can also be found in Gazi (2004) [41] and Merrifield (2006) [42].

A fundamental problem in robotics is to identify a continuous path that allows a robot, or a part of it, to reach its destination without colliding with obstacles that may exist in the workspace [43], [44]. This is the so-called findpath problem [21], [22], [23], [45]. In this paper, we want to solve the findpath problem for the individuals of a set of autonomous UAVs with the requirement that the UAVs are heading in the direction of their goal in a priori known dynamic environment. We begin by developing a system representing multiple rigid bodies that will be easily transferable to a system of UAVs, given that UAVs are examples of rigid bodies which also require an angular coordinate to fully describe its configuration in planar space. The navigation of the swarm rigid bodies are based on Reynolds rules [46] which are (1) collision avoidance with neighbors, (2) matching velocity of the neighbors, and (3) staying close to the neighbors. Since the current research involves the state (position and instantaneous velocity) space of each individual and its relationship with other individuals in the swarm, a Lagrangian swarm model that guides the rigid bodies to their target have been developed. The swarm model is based on the hypothesis that swarming is an interplay between long-range attraction and short-range repulsion between the individuals in the swarm [47], [48]. Our interest lies in utilizing the key component of the Lagrangian approach, and that is, the use of attraction and repulsion functions to model the swarming behavior in which there is a long-range attraction and a short-range repulsion between individuals in the swarm [37], [41]. This behavior leads to aggregation and formation, which are important for the survival of the members of the swarm [49], [50], [51]. If constructed appropriately, these attraction and repulsive functions can be expressed as a gradient of some artificial or social potential function [52], [53], [54]. This means that there is a Lyapunov function, a minimum of which corresponds to a stable [55] stationary state of the individual-based Lagrangian system [37]. As noted in the publications [41], [56], the use of a gradient system ensures there is an element of distribution of tasks among the members of the swarm and that the swarm members are performing distributed optimization. Indeed, because of the existence of the Lyapunov function, each individual in the swarm is individually and optimally searching a minimum. The stability conditions provided by the Lyapunov function can also provide the cohesiveness of the swarm in which the distances between individuals in the swarm are bounded from above [40].

The main advantage of our method is the simplicity in the design of continuous and distributed velocity based controllers. It is also relatively easier to capture the mechanical constraints in the control laws derived from our method, compared to the other motion and control schemes reported in literature [21], [22], [23], [45], [57]. Though we have this element of distributed optimization, our methodology does not ensure scalability. Since every individual knows the position of every other individuals in the swarm, any increase in swarm size results in a greater demand for computing resources, and in applied situations, for sensing capabilities. However, the exponentially increasing processing power, memory, and storage of the computing devices accompanied by decreasing prices of these devices, and the increasing use of the Global Positioning System to offset sensing limitations will play a major role in diminishing the impact of scaling. Indeed, already several existing approaches have extensively exploited global communication capabilities together with distributed control principles in a series of case studies in collective robotics, namely, aggregation and segregation, foraging, collaborative stick pulling, cooperative transportation, flocking and navigation in formation, the odor source localization, cooperative mapping, and soccer tournaments [58], [59].

The major contribution of this paper is the development of a new Lagrangian swarm model that shows several swarming characteristics or self-organization patterns like split-and-rejoin and tunneling maneuvers, elliptic and linear formations, and helical trajectory while avoiding multiple obstacles in a dynamic environment.

The remainder of the paper is organized as follows: Section 2 describes the generic swarm model. In Section 3, the velocity control laws are derived for the swarm model in the presence of stationary and dynamic obstacles from a Lyapunov function. In Section 4, the properties of the Lyapunov function is discussed. The velocity control laws derived in Section 3 are applied to unmanned aerial vehicles in Section 5. In Section 6, simulation studies are presented.

Section snippets

A two-dimensional swarm model

Consider a swarm of nN individuals that we shall treat as rigid bodies. In two-dimensional space, the positions of the individuals of the swarm can be described by their translational components and their rotational component, yaw, about the vertical axis of the body-frame reference. Let the position of the ith individual at time t0 be xi=(xi(t),yi(t)) with yaw angle ψi=ψi(t), for all i{1,2,3,,n}, with (xi(t0),yi(t0))(xi0,yi0) and ψi(t0)=ψi0 as initial conditions.

Definition 2.1

The ith individual is a

Velocity-based controllers in presence of stationary and dynamic obstacles

Consider the configuration space of system (4) clustered with mN stationary and qN dynamic obstacles.

Roles of the parameters in the Lyapunov function

In this section, we provide an overview of the roles of the parameters. This is a modified approach from work carried out in [13], [21], [23], [61]. Consider again our Lyapunov function (20) L(x)=αT(x)+i=1nT(x)γiRi(x)+j=1,jinβijQij(x)+ξijEij+k=1mλikWik(x)+i=1np=1qϖipT(x)Sip(x). The parameter α>0 can be considered as a measurement of the strength of attraction between the swarm centroid (xC,yC) and the swarm target. The smaller the parameter is, the slower the convergence of the swarm

Application: Planar formation of a swarm of autonomous UAVs

The UAV that will be used is a miniature sized quadrotor helicopter-type analyzed in Vaualailai (2013) [11]. At the onset, the reader is referred to papers such as Beard (2008) [62] as further reading of the complicated dynamics of quad-rotor UAVs and the resulting kinematic and dynamic models based on the Newtonian equations of motion. In contrast, the present paper treats the kinematic model of the quadrotor UAVs as a geometric problem, rendering the kinematic model simpler to derive, which

Simulation results for the UAV system

Simulations were generated using Wolfram Mathematica 8 software. To achieve the desired results a number of sequential Mathematica commands were executed. The positions of the obstacles were randomly generated. The radius of the obstacles were randomized between 4 and 5. The target for the centroid of the swarm was also randomly generated. The arm length l of all the UAVs in all the simulations is 2.5. We numerically simulated system (27) using RK4 method (Runge–Kutta Method). At t=0, the

Discussion

In literature for a swarm to navigate in unconstrained and constrained environments:

  • there are different algorithms such as the ones present in [10], and

  • algorithms such as the one present in [9] is not able to show many emerging behaviors and the stability of such system is not shown.

Hence, integration of such algorithms into industry for manufacturing of UAVs would be expensive as different algorithm will serve different purpose. However, this paper has a single system that shows the emergent

Conclusion

A Lagrangian swarm model was developed, based on the assumption that swarming is an interplay of long-range attraction and short-range repulsion between individuals in a swarm. It is a theoretical exposition wherein a simple rigid-body model of individuals in a swarm is extended to the kinematic equations governing the planar movement of a UAV. The swarm exhibited planar movements such as elliptic formation, and split-and-rejoin maneuver and tunneling abilities for the obstacle avoidance.

CRediT authorship contribution statement

Sandeep A. Kumar: Conceptualization, Methodology, Writing - original draft. J. Vanualailai: Methodology, Writing - review & editing. B. Sharma: Methodology, Validation, Writing - review & editing. A. Prasad: Validation, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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