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Edited by: Fabio Bonsignorio, The BioRobotics Institute Scuola Superiore Sant'Anna, Italy

Reviewed by: Charalampos P. Bechlioulis, National Technical University of Athens, Greece; Zhouhua Peng, Dalian Maritime University, China

This article was submitted to Robotic Control 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 presents results on recent developments pertaining to the coordinated motion control of a fleet of marine robotic vehicles. Specifically, we address the Cooperative Moving Path Following (CMPF) motion control problem, that consists of steering the robotic vehicles along a priori specified geometric paths that jointly move according to a target frame, while achieving a pre-defined coordination objective. To this end, each vehicle will need to communicate with their neighbors in order to cooperatively solve the CMPF task. Two distinct robust Moving Path Following motion control strategies for achieving robustness on the moving path following tasks are proposed. Experimental results demonstrating the application of CMPF to marine vehicles in the context of source localization and tracking of underwater targets are presented backed with stability and convergence guarantees.

The motion control problem for underactuated robotic vehicles is a relatively mature area of research, with important works addressing trajectory tracking and path following schemes. In the path following (PF) problem, the vehicle is tasked to follow a

A generalization of the path following problem is termed the Moving Path Following (MPF) motion control problem, which consists of steering the robotic vehicle along an a priori specified geometric path expressed with respect to a

In path following literature, the problem of robustness has been addressed for example, in Dagci et al. (_{∞} robust controller for ground vehicles is proposed to achieve path following in the presence of disturbances caused by delays and data packet dropouts. All of the above schemes consider robustness for the path following problem. From the best of the authors knowledge, the only work concerning the problem of robustness in MPF literature is Reis et al. (

A further extension of the MPF framework for multi-robot applications and formation control is the Cooperative Moving Path Following (CMPF) control problem, which consists in steering

This paper extends the results obtained for the MPF controllers proposed by Reis et al. (

Consider an inertial frame of reference {_{i}} attached to its center of mass. Define the set of ^{n} with

where _{i}} to an inertial frame {_{i}) ∈ 𝔰𝔬(_{i}.

Finally,

where the body linear velocity _{i} is defined as _{f, i} ∈ ℝ and the body angular velocity _{v, i} and _{ω, i}.

In the CMPF control problem, the vehicles must follow a priori specified paths expressed with respect to a moving target whose position can be accurately estimated, while also maintaining some coordination objective. Define the target frame {

Let _{i} ∈ ℝ. As illustrated by _{i} and time _{d, i}(γ_{i},

where _{i} is the derivative with respect to γ_{i}.

Coordinate frames and vector notation for

Assumption 2.1.

Note that Assumption 2.1 is already needed in order to compute (4) from (3). Suppose we wish to control the position of the nose of the _{i}}. Then, define the MPF error associated to the

The objective of the MPF control problem is to design a control law _{i} such that the origin _{i} ≡ 0 is stable and _{i} → 0 as _{d, i}(γ_{i},

In order to control the progression of the virtual points _{d, i}(_{i}) along the moving paths, the dynamics of the path variable _{i} as

where the scalar _{d} is the desired nominal speed of the path variable and ϑ_{i} is a bounded control signal, designed to achieve CMPF objectives such as: (i) consensus over the path variables of the robotic vehicles to achieve a desired formation along the moving path and (ii) faster convergence to the moving path. To move along the geometric paths with the desired velocity, the vehicles must satisfy the desired speed assignments

Assume that the _{i} (_{r, i}(_{i} − γ_{j}| converges to zero _{i} in (6) is decomposed as

where _{r, i}(_{e, i}(_{ω, i}(_{e, i}(_{ω, i}(

In this section, we consider the kinematic controller proposed in Jain et al. (

Using model (1) with control signal (2) and MPF error (5), the error dynamics can be rewritten as

where Δ is a constant matrix that can take the forms

for the planar (

Remark 3.1. _{i} _{i}∥ ≤ ∥_{v, i}∥ + ∥_{ω, i}∥∥

Assumption 3.1. _{i}

Theorem 1 (

_{i} = ^{†} _{i}

where we have used the fact that _{i} + _{ω, i}) is skew-symmetric. Substituting control law (10) in (13) yields

where _{p} > 0, the first term is negative definite and bounded by _{i}∥ ≥ ϵ_{w} or ∥_{i}∥ < ϵ_{w}.

For ∥_{i}∥ ≥ ϵ_{w} in (14), we have

where the Cauchy-Schwarz inequality was employed on term _{i} such that (12) is satisfied. Therefore, by Remark 3.1, choosing ρ_{i} ≥ ∥_{i}∥ renders the second term on the right-hand side negative definite, which estabilishes that the trajectory _{i}(

When the trajectories are inside _{i}∥ < ϵ_{w}, and (14) gets

where 0 < θ < 1. Then, using the inequality above, one can write:

Note that μ_{i} ≤ ϵ_{w} for all 0 < θ < 1, which means that the trajectory of the closed-loop system _{i}(

This establishes that the trajectories are globally ultimately uniformly bounded, since _{i}(_{w} → 0. □

In the presence of large amplitude disturbances, it may be difficult to tune the parameters ρ_{i} and ϵ_{w} so as to satisfy (12). In these situations, an observer can be designed to provide an estimate of the disturbance. Furthermore, this estimate can be used in the control law to compensate the real disturbance directly.

Without loss of generality, consider the planar problem. Consider that the vehicle pose _{i} ∈ ℝ, such that

Then, the disturbance observer for the translational disturbance is defined as

where the estimation errors are defined as _{i}, _{v, i} (Aguiar and Pascoal,

Similarly, observers for the rotational disturbances _{ω, i} ∈ ℝ can be designed as:

where the estimation errors are defined as _{i} are measured. Again, for positive scalars _{ω1}, _{ω2} ∈ ℝ_{>0}, the dynamics of the estimation errors _{ω, i} (Aguiar and Pascoal,

Theorem 2 (

_{i} = ^{†}

The term _{i} is defined by (11), with scalars ρ_{i} satisfying

where

Note that (20) is similar to (14), but with disturbance _{i}. Therefore, using the same arguments for the proof of Theorem 1 with Assumption 2.2 and condition (19), one can conclude that the trajectories of the MPF error are globally uniformly ultimately bounded and _{i}(_{w} → 0. □

Remark 3.2. _{i}, _{i}

Remark 3.3. _{i} _{i} _{i},

This section provides a proper design for function ϑ_{i} in (7). First, the design of the error correction term _{e, i}(_{ω, i}(_{r, i}(

The term _{e, i}(

and then choosing a gradient descent law _{e, i} = − _{e, i}_{e, i}) with _{e, i} > 0. The saturation function guarantees the boundedness for the correction term. Its effect is to effectively delay the evolution of the virtual point along the path by explicitly avoiding the evolution of γ_{i} if the MPF error norm is too large.

The term _{ω, i}(_{d, i} in a such a way that minimizes the effect of the target rotational motion, which is evident from the term _{t}, the virtual point could move faster than the _{ω, i} such that

If the target angular velocity is known, the minimum can be achieved by the least squares solution

with minimum given by

Remark 3.4.

Assumption 3.2.

From (23) and Assumption 3.2, the error correction term is bounded by.

Consider the distributed consensus law (Aguiar,

where _{c, i} > 0 are consensus gains and

Assumption 3.3. _{i}

Define the vectors

where _{c} = _{c, 1}, _{c, 2}, …, _{c, N}) is a positive definite matrix of consensus gains, ^{N × N} is the _{ij}], with _{ij} = 1 if _{ij} = 0 otherwise. Vector

Theorem 3 (_{i} ≡

^{1}_{e, i} _{ω, i},

_{N}, with

Note that the consensus condition

Next, define the ISS Lyapunov function candidate

Taking its time-derivative and using (25), yields

with

Applying Young's inequality to the last three terms in (27), we have

with a scalar _{>0}. Choosing any _{e, i}, _{ω, i} establishes that the disagreement vector _{e, i} and _{ω, i}, for all

The experiments were performed on Porto de Leixões (Porto, Portugal) using three Light Autonomous Underwater Vehicles (LAUVs) from the Underwater Systems and Technology Laboratory (LSTS) at the Faculty of Engineering of the University of Porto (FEUP) (

The vehicles operate under the DUNE/Neptus environments, which are part of a software toolchain (Pinto et al.,

A target vehicle was

where _{1} = 0 _{2} = 2π/3 _{3} = − 2π/3

For the construction of the MPF errors _{i}, the value ^{T} was used. The controller gain matrices and error correction gains were chosen as _{p, i} = _{d} = 1

Remark 4.1.

_{f, i} <

_{f, i} _{C}, ω_{C} ∈ ℝ _{C} = _{C} =

The first experiment shows the results of the CMPF controller with velocity compensation, ρ_{i} = 0 and no disturbance compensation (

Vehicle trajectories for the CMPF controller with velocity compensation.

Experimental results for the CMPF controller with velocity compensation.

The second experiment shows the results of the robust CMPF controller with velocity compensation and Sliding Mode term, with MPF control law given by (10) with ρ_{i} = 0.2 for the three vehicles and ϵ_{w} = 0.5_{i} ≡ 0. That means that the controller is able to achieve better performance than the previous one, given that ϵ_{w} can be designed to be arbitrarily small. However, from (11), small values of ϵ_{w} can result in higher gains for _{i}, which can potentially saturate the control inputs. In fact, sometimes the control saturation limits are reached after the transient, as shown in _{w}, and moments of occasional increase in the target velocity. Even so, performance is slightly better than in the previous case, and the amount of control chattering is acceptable.

Vehicle trajectories for the robust CMPF controller.

Results for the robust CMPF controller.

The consensus law, error correction signals and rotation correction signals are omitted, but are similar to those observed in

The third and last experiment shows the results of the robust CMPF controller with velocity compensation, Sliding Mode term and direct disturbance compensation using a linear observer. The control law is given by (17) with ρ_{i} = 0.2 and ϵ_{w} = 0.5

As seen from _{1} = 0 and ϕ_{3} = π_{i} ≡ 0, except during the instants where the control inputs are saturated (

Vehicle trajectories for the robust CMPF controller with disturbance observer.

Results for the robust CMPF controller with disturbance observer.

Results obtained with the disturbance estimator.

This work addressed the robust cooperative MPF problem for marine vehicles. We demonstrated that the origin of the MPF errors associated to the vehicles are stable with the two proposed robust CMPF control schemes in the presence of bounded disturbances acting on the vehicles. Furthermore, it was theoretically demonstrated that the cooperative control scheme is ISS with respect to the path variable estimation errors and to two other bounded, auxiliary input variables, named error correction term and rotation correction terms. The proposed robust controllers (10, 17) guarantee that the MPF error is globally uniformly bounded to a small neighborhood of the origin while maintaining acceptable control chattering. The narrow linear region of the actuators imposes limits on how small ϵ_{w} can be designed in practice. Lastly, we conclude that control law (17) actually improved the control chattering in practice, corroborating the theoretical insight of Remark 3.2.

Some of the future works are: (i) to investigate how to extend the proposed controllers to the case of unknown bounds for the disturbances (ii) to take the existence of actuator saturation limits in the control design and (iii) to incorporate obstacle avoidance techniques into the cooperative MPF approach to prevent vehicle collision during the cooperation tasks.

The datasets generated for this study are available on request to the corresponding author.

MR has written the manuscript, implemented the algorithms in C++ code, and performed the experiments using the LSTS vehicles. He also proposed the sliding mode based scheme for adding robustness to the moving path following controllers. RJ has proposed the cooperative control scheme using a consensus law, and contributed significantly to the stability proof of the cooperative controller. He also helped by suggesting important changes on the code and with the organization of the manuscript. AA contributed with the proposition of the disturbance compensation method for improving the performance of the first controller (Theorem 2), and also strongly contributed to the stability proofs and overall organization of the paper. JS made the experiments possible by setting up the mission at Porto de Leixões and has contributed by suggesting some changes on the code.

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 authors would like to acknowledge the support of the LSTS staff in conducting the experiments on Porto de Leixões. The code used in the experiments was written in C++ and can be accessed from the

_{∞}path following control for autonomous ground vehicles with delay and data dropout

^{1}Khalil (_{0}) and any bounded input _{0} and satisfies