Target Capturing in an Ellipsoidal Region for a Swarm of Double Integrator Agents

Antonio Bono,, Luigi D’Alfonso, Giuseppe Fedele,, and Veysel Gazi, Senior

Abstract—In this paper we focus on the target capturing problem for a swarm of agents modelled as double integrators in any finite space dimension. Each agent knows the relative position of the target and has only an estimation of its velocity and acceleration. Given that the estimation errors are bounded by some known values, it is possible to design a control law that ensures that agents enter a user-defined ellipsoidal ring around the moving target. Agents know the relative position of the other members whose distance is smaller than a common detection radius. Finally, in the case of no uncertainty about target data and homogeneous agents, we show how the swarm can reach a static configuration around the moving target. Some simulations are reported to show the effectiveness of the proposed strategy.

I. INTRODUCTION

STUDIES on natural and engineering swarms have attracted many researchers in recent years. The collective dynamics of swarm systems is an interesting natural phenomenon, but also has wide potential engineering applications [1], [2]. As reported in [3], in many scenarios such as surveillance, security, rescue and so forth, it is required that the swarm members enclose a target object or a prescribed region by using neighbor information.

Some relevant literature contributions are here summarized.One common approach to this task involves a cyclic-pursuit where agentisimply pursues agenti+1 modulonwhile moving around the target. The paper in [4], for example,considers the target capturing where each agent is modelled as single integrator model and it is able to sense the distance from the target and from the agent to be pursued. More recently, in [5], the cyclic-pursuit approach is managed to be applied in the underwater domain, using a sliding mode control under the hypothesis that one designated agent knows the position of the target object in the inertial frame and broadcasts the target coordinates information to the remaining members of the formation. Other solutions are based on artificial potential functions that constitute the subject of many studies (see [6] and references therein). The method is exploited in [7] by considering multiple circular trajectories around the known moving target. In [8], a control law based on a gradient descent method has been developed using some local data among the agents and the target position in order to encircle and grasp the target. Similarly, in [9] a potential functions and sliding mode control based strategy is developed for capturing/enclosing a moving target by a swarm composed of agents with fully actuated dynamics containing model uncertainties. In the above works, all vehicles require the information of the target in terms of its position or trajectory. Moreover, in those based on the cyclic pursuit, the network topology among the vehicles is limited to cycle graphs and then enclosing the target objects cannot be achieved with other graph configurations. The problem of the partial knowledge on the target has been faced in [10] and[11] where at least one vehicle can observe the target which is included in the connection graph topology. Reference [10]solves the consensus of the agents on a circle around the target in the case of mobile vehicles in Cartesian coordinates. The target position together with its velocity is exactly known by at least one vehicle. These constraints are partially relaxed in[11] where a fleet of UAVs and the evader flying at a constant altitude is considered. In particular the target-centric formation problem is addressed where each UAV should maintain a constant distance from the target at a constant angle. The main limitation of this strategy is the use of the dynamic characteristics of the unmanned aerial vehicles,which makes it difficult to generalise the method in other frameworks. Indeed the uncertainty is managed by using a sliding mode controller to mitigate the partial information available to the capturing vehicles.

In this paper, we consider a swarm of agents modelled by double integrator dynamics and propose a control law that ensures the enclosing of a moving uncertain target without imposing a consensus on prescribed positions around it. The choice of modelling agents as double integrator systems also stems from the fact that overall dynamics of many different agents such as fully actuated robots, drones, or even manipulators can be represented by double integrator models.For example, a general form for a fully-actuated holonomic dynamics model can be represented as [12]

Similarly, the overall dynamics of other types of agents can be captured by the double integrator model. This is one of the reasons for extensive utilization of this model in the literature.Moreover, although in other kinds of cooperative task more complex agents dynamics have been considered [13]–[16], the single or double integrator models can serve as high-level path generator for systems with particular vehicle dynamics [2],[17]. In the simulation section, an example of this approach is described for a swarm of differential drive mobile robots.

In order to enclose the moving uncertain target, the agents arrange themselves in an ellipsoidal ring that is defined by two regions centered on the target: a distancing region that is forbidden to the agents and represents the free space allowed to the target and a containment region that agents enter in steady-state and they cannot leave. Ellipsoid is a shape which requires different level attractions/repulsions on different semi-axes. It is worth mentioning that, differently from many other approaches, the ellipsoidal ring constitutes a non-trivial geometric constraint to address because of its non-convexity.

The swarm is assumed to know the target position but to have only uncertain information about its velocity and acceleration. The estimation errors are bounded by some valuesa prioriknown. Moreover, while agents have a continuous, even if uncertain, knowledge of the target, they can sense the other agents through local proximity sensors (for collision avoidance). Consequently, the topology of connections is time variant.

At first, there seems to be a contradiction in the ability of the agent to know the position of the target at each time instant and instead the position of other agents only when they are within its detection range. However, it is worth mentioning that in the robotic context there are usually different types of sensors on board. In the scenario presented,for example, the perception of the leader and of the neighbors agents could be provided by two different modules: one, based on computer vision and therefore capable of detecting objects at a considerable distance, devoted to the perception of the leader; another one, based on proximity sensors (ultrasonic,infrared, etc.), devoted to the perception of the closest agents.The benefit of such a setup would be to use computer vision,which is computationally expensive, only for the leader and to use the simpler and faster proximity sensors for the neighbours agents. Regardless of the possible scenarios, the assumption of target position knowledge is not limiting. In the early stages of the tracking, in fact, we can always use an estimate of the target’s position in place of the real data. This estimate will be replaced as soon as the agent is able to locate the target thanks to its perceptive abilities. Different strategies have been proposed in this regard. If at least one vehicle in the swarm has target information also with uncertainties (only the upper and lower bounds on the exact target motion are available), and the corresponding communication graph is connected, then a target-centric formation can be maintained using [11]. Many systems, however, have some kind of sensors to measure the relative position and/or velocity of the target with possibly bounded measuring errors. For example,[18] uses the two-step weighted least squares as well as time of arrival measurements to estimate the position of moving targets. These targets only passively receive signals from the multi-agent network, which then broadcasts the related information to the agents. An estimator of the target location is proposed in [19] based on the position of the agents and the bearing angle to the target. The papers in [20], [21] extend the results in [19] in the case of multiple targets and provide a method for elliptical circumnavigation around them. A bearing-based coordinated circumnavigation control scheme is also proposed in [22], based on the networked agents’coordinated estimation algorithm for the velocity of the target as well as the distance towards the target. In particular, an estimation algorithm is designed to estimate the target’s velocity and the distance between the agent and the target,based on the local velocity and bearing information.

The proposed strategy relies on the works in [23]–[26]where multi-agent systems based on kinematic models have been studied in the context of finite-time reference tracking and rotational aggregation around a given reference. Although in all these previous works the swarm model benefits of similar interaction terms among the agents, here the control law usesad hocattractive and repulsion terms that ensure agents to enclose the target without colliding with it.

Finally, the main contributions of the paper can be summarized as follows:

i) Cooperative enclosing of a moving target, with uncertain information about velocity and acceleration available, is achieved by spreading the follower agents in an ellipsoidal ring.

ii) The ring is formed by two ellipsoidal regions centered on the target: a containment region where agents converge and remain inside during the tracking and a distancing region that is forbidden to the swarm.

iii) In case of exact information, at steady-state, the agents can achieve a static configuration such that the moving target lies in the convex-hull formed by their positions.

iv) The proposed strategy holds for any finite space dimension (d≥2).

The remainder of this paper is organized as follows:Section II defines the capturing problem that the paper aims at solving; the proposed model and its main properties are described in Sections III and IV, respectively; Section V shows the applicability of the strategy through examples and a software in the loop (SIL) simulation; Section VI concludes the paper by summarizing the achieved goals and discussing future research directions.

Remark 1:Apparently the definition of a containment region seems a trivial consequence of the stability and cohesiveness of the swarm. However, it is worth noting that,in general, the parameterc, i.e., the width of the containment region where agents should remain in order to solve the TCPUD, depends on the estimation errors bounds (bv,ba). In Section IV this dependency is clearly analyzed and it is also shown that the value ofccan be arbitrarily chosen in two cases: i) the relative velocity of each agent with respect to the target, namelywi(t), is properly constrained; ii) no uncertainty on the target data exists.

As can be easily seen from above, both the distancing and the containment regions are described by the same type of ellipsoid with the only difference that semi-axes of the containment region are multiplied by the factorc, which depends on the accuracy of the target’s velocity and acceleration estimates. Thus, to meet these requirements,agents have to remain in an ellipsoidal ring around the target as the one shown in Fig. 1.

Fig. 1. An example of the ellipsoidal ring where agents have to remain in order to solve the TCPUD. The ellipse drawn with the dashed line represents the limit on the smallest containment region that the user can choose in this example according to the uncertainty on the target’s velocity and acceleration.

III. SWARM MODEL

In order to solve the TCPUD, the control protocol of an individual agent is chosen as

IV. ANALYSIS OF AGENT’S DYNAMICS

A. Boundedness of Agent’s Velocities

B. Solution of the TCPUD

Following these indications, which respect the formal conditions required by Theorem 1, it is always possible to find a set of parameters that solve the TCPUD.

C. Stationary Behavior

This section shows some characteristics of the swarm model in the case of known target. It proves that asymptotically the individuals will converge to a static position w.r.t. the target and therefore to a constant relative configuration. In particular it is shown (see Theorem 3) that, thanks to the interaction term, when the agents are in the ellipsoidal ring and, at steadystate at least two agents are connected then, they organize so that the target is a convex combination of the agents’positions. With that objective, let us consider the case in which:

i)No uncertaintyaffects the data about the target, i.e.,

ii) All the agents have thesame mass, i.e.,

Note that assuming the same mass for all agents does not imply an additional restriction since the effect of the mass can easily be compensated by appropriately scaling the agents’control input. In terms of the perfect knowledge of the velocity and acceleration, the analysis here can also represent the case in which the estimation errors on the velocity and acceleration of the target vanish as time progresses. For example, it could be the case that a tool such as a high gain observer [29] is utilized to estimate the velocity and acceleration of the target from its position data guaranteeing asymptotic convergence of the estimation errors to zero.

In the described case, we still can apply the results of Lemma 2 and Theorem 1 to solve the TCPUD. The resulting control law is

V. SIMULATIONS

In this section we will use three simulations in order to better show the behavior of the swarm described in the previous results. The first two are numerical simulations of the proposed protocol while the last is based on the SIL methodology.

A. Example 1

TABLE I INITIAL POSITIONS OF THE AGENTS IN THE FIRST TWO EXAMPLES

Fig. 2. Example 1. Agents’ paths in coloured lines along the target trajectory in red. Circles are the initial positions while crosses are the final ones. In the magnified view, the final agents’ positions inside the ellipsoidal ring are shown.

B. Example 2

Fig. 3. Example 2. The evolution of velocities of the agents w.r.t. the target,i.e., wi(t). The magnified view helps showing how the relative velocities tend to zero according to Theorem 2. After about 60 s, the agents reach a static configuration around the target (see Fig. 4). The dashed line, finally, is the common upper bound w=3m/s provided by Lemma 2 (see (36)).

C. Software in the Loop Simulation

This last simulation aims to show how the proposed control protocol can be used in a realistic scenario where the TCPUD is assigned to a team of differential drive UGVs.

For this purpose, we have chosen the Simulink-Gazebo cosimulation environment [33]. The advantage of such a tool is to quickly implement the control logic in Simulink and, at the same time, to exploit the high fidelity ODE physics engine of Gazebo for the dynamics simulation. This relationship can be appreciated in Fig. 5, where a scheme with the main components of the simulation is shown. Here the proposed protocol is not used directly as control input for the single robot. The actual controller, in fact, is the classic non linear tracker for differential drive robots reported in [12] followed by a proportional controller that computes the torque commands for the wheels. Since this algorithm is based on the differential flat property of the unicycle model, a reference trajectory and its derivative up to the second order must be provided. The evolution of the error (6) and the control input(8) are sufficient to generate these required data if we assume that the coordinate bases of the different agents are directionally aligned (this is commonly achieved by using compasses or landmark data). To better show the behavior of the team due to the control protocol (8) we also assume that each robot receives from Gazebo the relative positions of the target and all the other robots inside its detection radius. In this way we avoided the influence of all the issues related to the measurements and localization processes. The exchange of data between Simulink and Gazebo occurs at each sampling timeTs=0.01s, that is also used to compute an Euler-forward discretized version of the proposed protocol (8) and the tracking algorithm [12].

Fig. 4. Example 2. Agents’ configuration at the final instant t =60s.

Fig. 5. SIL simulation. The input/output relationships among the main components of the simulation.

With this setup, we have simulated six Pioneer 2DX [34]mobile robots whose mass ismi=5.67kg,i=1,2,...,6 and equipped with a detection radiusr=10m. The target robot is of the same kind and tracks the following figure-eight trajectory:

While all robots know the target relative position (zi(t)=xi(t)?xT(t)), the estimates of its velocity and acceleration are affected by the same kind of error reported in the first simulation. The chosen distancing region is an ellipse defined by the matrixP?=diag(1/4,9/16). By tracking the references given by the protocol, however, there is no guarantee for a real robot not to enter the distancing region. This is due to two factors: the real size of the robot that is not taken into account in the protocol and the performance of the tracking algorithm.Such a drawback can be easily overcome by using a larger region in the protocol than the one actually desired. In this simulation, for example, we have enlarged both the axes of a factor 1.5 obtainingP=diag(1/9,1/4). The velocities matching coefficient is γ =25 and the other control parameters have been chosen according to Theorem 1. In particular,c=1.5,βi=2, and αi=73.42,i=1,2,...,6. To define thehi(t) term the sameg(//q(t)//) function of the first simulation has been chosen (see (51)). The simulation run with these parameters gives satisfactory results as confirmed by the tracking errors reported in Fig. 6 and by the convergence in the containment region that occurs att=11.58 s. A snapshot of the simulation at the final instantt=60 s, reported in Fig. 7, shows that all the agents are in fact inside the ellipsoidal ring. A video of the entire experiment is also available at the following link: https://www.youtube.com/watch?v=nS3vMuO8j3g.

Fig. 6. SIL simulation. The norm of the tracking error for the relative position reference: ei, i=1,...,6, represents the norm of the tracking error between the position of the robot ith and its reference trajectory.

Fig. 7. SIL simulation. A snapshot from the simulation video focusing on the formation around the target robot. The yellow ellipse is the distancing region and the blue is the containment one.

VI. CONCLUSIONS

In this paper a strategy for the uncertain target capturing problem has been proposed. It has been shown that the swarm enters an ellipsoidal ring around the target and remains therein, indefinitely. Moreover, steady-state properties of the swarm have been investigated particularly stressing the capability of the agents to organize such that the target is a convex combination of the agents’ positions. Future developments will be devoted to consider more general agent’s dynamics and removing hypotheses on the communication facilities, e.g., send/receive operations amongst the agents may be subject to latency phenomena.