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Edited by: Ioannis Rekleitis, University of South Carolina, United States

Reviewed by: Jawhar Ghommam, École de Technologie Supérieure (ÉTS), Canada; Dongbin (Don) Lee, Oregon Institute of Technology, United States

This article was submitted to Multi-Robot Systems, a section of the journal Frontiers in Robotics and AI

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

This paper addresses the distance-based formation control problem for multiple Autonomous Underwater Vehicles (AUVs) in a leader-follower architecture. The leading AUV is assigned a task to track a desired trajectory and the following AUVs try to set up a predefined formation structure by attaining specific distances among their neighboring AUVs, while avoiding collisions and enabling at the same time relative localization. More specifically, a decentralized control protocol of minimal complexity is proposed that achieves prescribed, arbitrarily fast and accurate formation establishment. The control signal of each vehicle is calculated based on the relative position of its neighbors and its own velocity only, which can be easily acquired by the onboard sensors without necessitating for explicit network communication. Finally, a realistic simulation study with five AUVs performing seabed scanning was conducted to clarify the approach and verify the theoretical findings of this work.

The use of autonomous underwater vehicles has steadily grown during the last 20 years. Several activities related to the offshore industry, such as surveillance and inspection of underwater facilities, oceanography, seabed map building, search and rescue, marine resource exploitation and so on, have been enabled by underwater robotic vehicles (Griffiths,

As an alternative solution, the deployment of multiple underwater vehicles in various formation schemes has emerged (see

Examples of multiple underwater vehicles in cooperative missions.

Another key feature of multi-agent systems is the upgrade of persistent autonomy via fault tolerance (Longhi et al.,

Formation control, one of the most significant missions in underwater multi-agent systems, is a cooperative task in which multiple AUVs are deployed to achieve a specific geometric structure and move coordinately, so that a global mission is satisfied. Particularly in

Despite the recent progress in marine technology, the most significant challenge in underwater cooperation is imposed by the strict communication constraints, owing to the limited bandwidth and update rate of underwater acoustic devices. Furthermore, as the number of cooperating vehicles increases, communication protocols require complex design to deal with the crowded bandwidth (Stilwell and Bishop, ^{1}

The rest of the manuscript is organized as follows: section 3 introduces the problem, describes the system model and reviews preliminary results in rigid graphs. The control protocol along with the stability analysis are presented in section 4. Section 5 validates our approach via a simulated paradigm. Finally, section 6 concludes the paper and discusses future research directions.

This section describes the dynamic model of the AUVs and introduces a rigorous formulation of the distance based formation control problem that will be tackled herein.

Let us define a body-fixed frame

where:

_{ui}, τ_{vi}, τ_{wi} and torques τ_{pi}, τ_{qi}, τ_{ri} generated by the actuators) applied on the vehicle and expressed in

_{i} ≜ _{RBi} + _{Ai}, where

_{i}(ν_{i}) ≜ _{RBi}(ν_{i}) + _{Ai}(ν_{i}) , where

_{i}(ν_{i}) ≜ _{qi}(ν_{i}) + _{li}, where

Inertial and body-fixed frames for an AUV model.

This work examines the coordination problem of

Moreover, ^{3}. The pair

This definition implies that in a rigid framework, keeping the edge length and at the same time moving one or a set of vertices of the graph does not affect the distances between the other vertices. Moreover, we define the ^{3(N + 1)} → ℜ^{l × 3(N + 1)} of

Hence, given a sequence of edges in

Clearly, the rigidity matrix depends only on the relative positions, so it can be written as

^{3}

It follows from the aforementioned definition that ^{3} if the corresponding graph has at least 3(

^{3}

If a framework is infinitesimally rigid and its underlying graph has exactly 3(

^{2} ^{2} ^{2}.

Several frameworks on ℜ^{2}:

In the leader-follower architecture that is adopted in this work, there is one global plan, i.e., a reference trajectory _{d} (

Let us define the distance errors for each edge of the rigid graph, as:

A critical issue that has to be considered in multi-agent systems concerns collision avoidance among interacting agents. In this respect, the distance of the agents should be kept greater than a safety zone

for all _{ij}_{ij}(_{∞} correspond to the decaying rate and the maximum error at the steady state, respectively.

The proposed control protocol is first derived at the kinematic level assuming that the control signals are the linear body velocities. Subsequently, the kinematic controller is extended to the dynamic model, considering the actual control input signals, i.e., body forces. Hence, given a smooth and bounded desired trajectory

to incorporate via the appropriate selection of ρ_{∞} and λ the desired performance specifications regarding the steady state error and the speed of convergence.

where ^{l × 3(N + 1)} denotes the rigidity matrix defined in (3),

and Ξ_{E} denotes the diagonal matrix of the derivatives of the modulated errors with respect to the actual distance errors:

where

and select the corresponding, exponential decaying, velocity performance functions

where _{vi} denotes the vector of modulated velocity errors defined as:

and Ξ_{Evi} is the diagonal matrix of the derivatives of the modulated velocity errors with respect to the actual velocity errors:

_{vi}_{E}, _{V} affects the control input characteristics as well. Decreasing the gain values leads to increased oscillatory behavior within the performance envelopes, which can be suppressed when adopting higher gain values enlarging however the control effort both in magnitude and rate. Apparently, fine tuning might be needed during real-time implementation to meet certain input constraints that affect severely the AUVs dynamics

The following theorem summarizes the main results of this work.

_{f}) with _{f}). Therefore, the modulated distance and velocity error vectors _{vi}, _{f}). Subsequently, let us assume that τ_{f} ≠ ∞ (otherwise the problem would be trivially solved, since the inequalities would hold for all time). In the sequel, following standard Lyapunov arguments, we shall prove that, for all _{f}), the distance and velocity errors will evolve strictly within the corresponding upper and lower bounds dictated by the performance functions irrespectively of τ_{f}. Thus, invoking Proposition C.3.6 (pp. 481) in Sontag (_{f} = ∞.

Hence, consider a candidate Lyapunov function of the modulated distance errors

Differentiating with respect to time, we obtain:

Employing the fact that _{ρij}, _{ij}(_{v}

where _{vi} denote the modulated velocity errors, we obtain:

It should be noted that: (a) the rigidity matrix _{f}. Hence, invoking the positive definiteness of the square matrix ^{T} by Lemma 2, it is easy to deduce the existence of a positive constant Ē, which depends of the aforementioned upper bound, such that _{f}) (see Remark 4). Moreover, since _{I} and consequently _{d} and its derivative remain bounded as well. Similarly, invoking the aforementioned boundedness properties and observing the proportional and derivative terms of the leader's control law in (10), we also establish accurate tracking of the reference trajectory by the leader for all _{f}) and for a sufficiently high gain _{P}. Finally, what remains to be proved is that _{f}). Hence, we follow the aforementioned line of proof for the velocity modulated errors _{vi}, 0, 1, …,

Similarly to the previous step, notice that all terms in the right parenthesis are bounded by construction or by assumption; hence invoking the positive definiteness of the inertia matrix _{i}, it is easy to conclude the boundedness of all elements of the modulated velocity errors _{vi}, from which it is straightforward to deduce that _{f}). Moreover, since _{vi} was proven bounded then the control signals τ_{ui}, τ_{vi} and τ_{wi} remain bounded, which completes the proof.

_{vi} _{P}, _{E}, and _{V}. In the same spirit, large model uncertainty and external disturbances involved in the vehicle non-linear model (1) can be compensated, as they affect only the size of the ultimate bound of _{vi}, _{vi},

We consider a leader-follower scheme composed of five identical underwater robotic vehicles. The model that was used for simulation is a 4 DoFs Seabotix LBV (LBV150,

The dynamic parameters of the Seabotix LBV150.

9.7532 | Mass and added mass along surge axis | |

8.6636 | Mass and added mass along sway axis | |

_{z} = _{ẇ} |
10.898 | Mass and added mass along heave axis |

_{z} |
0.1589 | Inertia about yaw axis |

_{u} |
−8.6040 | Linear damping term along surge axis |

_{v} |
−18.1106 | Linear damping term along sway axis |

_{w} |
−17.1828 | Linear damping term along heave axis |

_{r} |
−1.4146 | Linear damping term about yaw axis |

_{|u|u} |
−17.8534 | Quadratic damping term along surge axis |

_{|v|v} |
−1.0594 | Quadratic damping term along sway axis |

_{|w|w} |
−3.6482 | Quadratic damping term along heave axis |

_{|r|r} |
−10.3483 | Quadratic damping term about yaw axis |

−1.1881 | Vehicle weight minus buoyancy |

The task of scanning the sea-bed was assigned to the multi-agent system. The scan was performed according to a pattern, used by divers for object recovery, called “compass box search pattern” (see

Compass box search pattern. In this work we selected

The adopted undirected graph with ^{2}.

The collision avoidance and connectivity specifications were selected as: (a) sensing range _{E} = 1, _{P} = 5, _{V} = 100.

The simulation results of the aforementioned study are illustrated below. The evolution of the sea-bed scanning, which is displayed in the

The evolution of the seabed scanning for six consecutive time instances.

Distance errors—Leader.

Distance errors—Follower 1.

Distance errors—Follower 2.

Distance errors—Follower 3.

Distance errors—Follower 4.

Velocity tracking—X axis.

Velocity tracking—Y axis.

Velocity tracking–Z axis.

Control input signals.

This paper proposed a solution to the formation control problem for multiple AUVs in a leader-follower architecture. The derived control protocol guarantees formation establishment with prescribed transient and steady state performance while avoiding collisions and connectivity breaks and despite the presence of external disturbances and dynamic model uncertainty. Moreover, no explicit communication among the fleet is needed. Furthermore, for each AUV the control signal is calculated based only on the relative position of the neighboring vehicles and its own velocity, which both can be easily acquired by the onboard sensors. Additionally, the proposed decentralized control protocol is of low complexity. Finally, realistic simulation results clarified and verified the proposed approach.

Future research directions will be devoted toward studying the effect of: (i) underactuated translational dynamics (i.e., AUVs unactuated in the sway or heave degrees of freedom), (ii) input uncertainties (i.e., mapping uncertainties between desired body forces/torques and actuator commands) and constraints (e.g., the thruster limitations) as well as (iii) sensor filtering (underwater relative localization is plagued with noise, intermittent failures and latency) on the closed loop stability, in order to increase the applicability of the proposed control methodology in open sea scenarios. Extra attention should also be pledged on studying how the achieved transient and steady state performance specifications could encapsulate further configuration constraints that may arise owing to extra sector-based (not only range-based) limited capabilities of the on board sensors (e.g., the limited field of view of cameras or sonars) adopted to acquire relative localization. Finally, the increasingly challenging mission scenarios in the field of marine robotics, call for inexpensive and robust control solutions for obstacle avoidance.

CB took over the technical writing of the manuscript. FG and GK conducted the simulation study. Finally, KK supervised the work.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The Supplementary Material for this article can be found online at:

Scanning the seabed via the compass box search pattern.

^{1}Although underwater robots are equipped with acoustic modems to communicate with the surface control station, avoiding the use of intense explicit inter-robot communication is clearly motivated by the limited bandwidth of underwater acoustic devices and issues related to time delays and packet drops, which are critical in such communication medium.