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Edited by: Savvas Loizou, Cyprus University of Technology, Cyprus

Reviewed by: Saptarshi Bandyopadhyay, NASA Jet Propulsion Laboratory (JPL), United States; George C. Karras, National Technical University of Athens, Greece

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.

We develop a synchronous rendezvous strategy for a network of minimally actuated mobile sensors or

There is much interest in using networked distributed robotic systems for large-scale environmental monitoring applications, such as coastal surveillance, scientific data collection, and surveying for ocean mining (Yuh et al.,

In this work, we consider the teams of networked minimally actuated drifters or similarly power-constrained mobile sensors that must leverage the dynamics of the ocean flow in order to minimize consumption during navigation. These

Motion plans and control strategies for robots that are part of a mobile sensor network needs to capture the interplay between sensing, communication, and mobility. Existing work has mostly focused on enabling robots to efficiently harvest and transport data from stationary sensors deployed across large geographical regions (Bhadauria et al.,

In this work, we observe that synchronous rendezvous between agents in the ocean-like flows is a variant of the non-linear oscillator synchronization problem. However, since robot motions are dictated by the geophysical fluid dynamics, the synchronized arrival of these mobile sensors must rely on motion plans and control strategies that are ^{1}

Simulation of a contaminant spill in a time-varying wind-driven double-gyre flow. The LCS boundaries are shown as red curves and the red

While the model shown in ^{2}

Phase portrait of the wind-driven double-gyre model at

Snapshot (August 2005) of visualization of ocean surface currents for June 2005 through December 2007 generated using NASA/JPLs Estimating the Circulation and Climate of the Ocean, Phase II (ECCO2) ocean model.

Leveraging our understanding of LCS, we assume that the workspace can be modeled as a collection of LCS bounded regions exhibiting gyre-like flows. Decomposing the workspace along LCS boundaries allows mobile sensors to leverage the surrounding fluid dynamics for navigation, thus enabling an energy aware control strategy (Kularatne et al.,

This paper builds upon our preliminary work (Wei et al.,

The rest of the paper is organized as follows: section 2 offers a more complete problem statement, while section 3 presents the analysis of the synchronous rendezvous conditions for a single pair of agents, and section 4 provides the synthesis of the short-range coordination strategies. Section 5 analyzed the effect of disturbance in the input. Section 6 presents simulation results. Conclusions and final thoughts close the paper in section 7.

Let the workspace ^{2} indexed by _{i}(_{i}(_{i} = θ_{i}(0) denote the initial phase, then the single vehicle dynamics is given by

where ω_{i} denotes the natural frequency where agents move along the orbit and _{i}(_{i}(_{i}. The agent can maintain a desired fixed period _{i} if there exists a mapping of _{i} = _{i}). Notice that any flow that allows an agent to travel on a closed curve can be modeled as such circular orbits.

Two orbits are tangent to each other if the two gyres share an LCS boundary. For a tangent pair _{i.j} or the _{i,j} as shown in

where _{i,j} is a radius pre-selected according to the communication range of both agents _{i,j} as Ψ_{i,j}, and the phases of it entering and exiting Γ_{i,j} as _{i} ∈ Γ_{i,j}∋_{j} is therefore equivalent to

The details of a rendezvous zone Γ_{i,j} with the entering and the exiting phases

Agents are only aware of the existence of other neighboring agents when they are both within the rendezvous zones. We call the _{i}, using the exchanged information when they are in the rendezvous zones. Once agents leave the rendezvous zones, they continue executing the same control input which is not updated until the next time they enter the rendezvous zone and exchange information with a neighboring agent. Without loss of generality, we assume _{i}(

The layout of seven agents on their orbits, and the abstraction to a graph.

In this work, we are interested in the

For a team of agents indexed as

_{i}(_{i}(_{j}(_{j}(_{0}, we test whether

If there exists ^{rend} satisfying the conditions above, we consider agent

For any pair _{i/j} is a random variable. Let _{i/j} the same as the solution of

_{i,j} is a random variable on a bounded interval [−η_{s}, η_{s}], we solve for η_{s}, such that the rendezvous will still be guaranteed to take place.

_{i,j} is a Gaussian white noise ^{2}.

Whether a pair of oscillators would achieve rendezvous spontaneously has been studied in Wei et al. (_{i}, ω_{j}, ϕ_{i}, ϕ_{j}) → {1, 0} to determine the possibility of a spontaneous rendezvous based on the initial phases and natural frequencies of both parties.

In this section we analyze the rendezvous condition for a more general case that, by knowing the current phases at _{0} and future periodic control schemes (for _{0}) for both agents _{i/j} = ω_{i/j} + _{i/j}, the time before either agent's first entrance of the rendezvous zone from now on is denoted as Δ_{i,j} (or Δ_{j,i}) and satisfies

and the time either agent enters the rendezvous zone for the _{i} and _{j} are the periods of both agents. For a pair of agents with no rendezvous before, we take θ_{i/j}(_{0}) = ϕ_{i/j} and _{i/j} ≡ 0. Then

The time either agent spends to travel through the rendezvous zone is denoted as δ_{i,j} (or δ_{j,i}) such that

For a pair of agents with no rendezvous before,

Therefore the time that agent

respectively. The rendezvous occurs when _{i}, _{j} ∈ ℕ satisfying the following inequalities (as shown in

Time schedule for a pair of agents. Shaded parts indicate the time the agents spent in the rendezvous zone. A rendezvous will occur if and only if there is an overlap between shaded parts on both axis.

For Equation (5) to hold, the time set for

By rearranging the inequalities we get

Lemma 3.1.

Proof: See Wei et al. (

The solution space

Corollary 3.1.1.

Proof: Corollary 3.1.1 follow directly the proof of Lemma 3.1.

Corollary 3.1.1 is a weak sufficient condition. However, further analysis on Lemma 3.1 reveals much tighter results.

Corollary 3.1.2. _{i} ≤ _{j}.

Proof: Corollary 3.1.2 follows directly the proof of Lemma 3.1.

Notice that Equation (7) holds only when δ_{j,i} − _{i} < −δ_{i,j}. If there is δ_{j,i} + δ_{i,j} ≥ _{i}, Equation (7) cannot hold and

We now analyze Equation (7) in two categories: (i)

Lemma 3.2.

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Proof: Follow the steps for (Wei et al.,

Notice that Lemma 3.2 suggests a similar but stronger result as in (Wei et al.,

Lemma 3.2 suggests that after a time period of length _{j}_{i} = _{i}_{j}, both agents would have completed integer multiples of rounds and appears at the same locations of _{0}. The rendezvous must happen before this _{0} + _{rend} that both θ_{i}(_{rend}) and θ_{j}(_{rend}) are inside the rendezvous zone, θ_{i}(_{rend} + _{j}(_{rend} +

When

Lemma 3.3. _{i,j} and δ_{j,i} is non-zero. Then

Proof: See

Together with previous lemmas, the cases that agents _{i} and _{j} with some certain combinations of rendezvous zones and current phases.

Theorem 3.4. _{i} ≤ _{j}. The time set that agent _{i}_{j} and k_{j}_{i}

Corollary 3.4.1.

Proof: Theorem 3.4 and its corollary follow directly the proof of previous lemmas and corollaries.

Since the set of rational numbers has a Lebesgue measure of zero, we consider it is almost impossible for a pair of agents to maintain frequencies with an exact rational ratio. Practically, for agents actuated to maintain such frequencies intentionally, noises and disturbance always arise to deviate the agents and result in (most likely) irrational ratios. According to our analysis, any tangent pair of agents are almost always able to discover each other and are re-united in the rendezvous region. The sensitivity analysis is provided in section 5.

Although a pair of agents are able to reach rendezvous relying solely on their frequencies holding an irrational ratio, such rendezvous cannot happen periodically, and the next rendezvous may not occur until after a long interval. To synchronize a pair into periodic rendezvous, the agents' motion will be actuated to yield desired frequencies. For agents _{i,j}, ideally, either agent shall travel at a constant angular velocity outside of the rendezvous zone, such that it is able to return to the zone after completing integer multiples of periods _{i/j}. A periodic rendezvous occurs only when _{i} periods in approximately the same amount of time of _{j} periods, where _{i}, _{j} ∈ ℕ. The rendezvous period _{i} and _{j} that satisfies

When the pair are both in the rendezvous zone, a coordinating controller can be applied to actuate them such that a new pair of angular velocities with a desired rational ratio will be reached and maintained before either party exiting the rendezvous zone. Meanwhile, this controller is tasked with maximizing the rendezvous duration, which can be realized by regulating their motions to align them before exiting the rendezvous zone. In this section we show an example of designing a time-optimal controller to adjust both agents' angular velocities to the mean value, and align the agents to hit the tangent point at the same time, which satisfies

The synchronization task is accomplished in a split way. WLOG agent

The controller letting agent

with the error dynamics as follows

The control input in Equation (9), _{i,j}, is the difference between _{i} and _{j}. As _{j} is determined by Equation (8), _{i} can be acquired straightforwardly by designing a bang-bang controller following Athans and Falb (

After rendezvous was initiated between a pair of agents, they synchronize themselves to a common rendezvous period that should become invariant. Such synchronization can be extended to a connected network of multiple agents. Wei et al. (

Wei et al. (

Wei et al. (

Theorem 4.1.

Proof: See

The discussion in section 3 points out it is extremely rare for a neighboring pair of agents

Section 3 also points out that one key factor to make sure a pair of agents always rendezvous in the future is that noise and disturbance exist in the system which deviate the frequency ratio to a most likely irrational number. It is also worth noticing that, for an already synchronized pair of agents, the achieved (rational) frequency ratio may also be deviated by the inevitable noise and disturbance. The resulting irrational ratio may not cause a total loss of future rendezvous, but is still able to result in a much longer rendezvous period that is not applicable for certain real world cases. In section 5 we will analyze the effect of noise and disturbance.

The analysis in section 3 provides a theoretical basis for a promised future rendezvous for any pair of agents on neighboring circular orbits. However, in practice, there are at least two types of factors that may cause a loss of a synchronized rendezvous scheme: (i) There is usually an upper bound of the rendezvous period due to the requirements of the specific application, for example (but not limited to) the recharging of an agent, uploading data from an agent's limited storage, or a regular recalibration of an agent; and (ii) the disturbance and noise that accumulated in the agents' dynamics, especially when an agent is not in rendezvous, which deviate the agents' angular velocities from the desired value.

Definition 1. If agents _{i}._{j} ∈ ℕ, and _{i}, _{j} co-prime, we say that agents _{i}-_{j} rendezvous scheme. A synchronized pair fails to maintain its _{i}-_{j} rendezvous scheme is said to be

In this section we analyze in what conditions the effect of the disturbance and noise will cause a synchronized pair of agents fail to rendezvous in their pre-selected scheme and, when such desynchronization happens, whether they would be able to synchronize themselves into another periodic rendezvous scheme. Consider agents _{i}-_{j} scheme, where WLOG _{i} < _{j}, and

The agent dynamics contains certain noise that results in a disturbance on the control input. The dynamics shown in Equation (1) is therefore rewritten as

where η_{i/j} is the disturbance associated with agent _{i/j} is an independent random variable. The actual period of either agent,

We discuss two typical types of η, (i) that η_{i/j} is a random variable on a bounded interval [−η_{s}, η_{s}]; and (ii) that η_{i/j} is unbounded, but a Gaussian white noise with a mean of μ = 0 and a standard deviation σ. In the first case that η is bounded, we have

and

Let _{i,j} of a circle while _{i,j} is bounded by

Take the time that _{i}-th circle is _{j}-th circle is

Lemma 5.1. _{s}, η_{s}_{i} and _{j}, such that _{i}_{i} = _{j}_{j}, with _{i}, _{j}_{i}-_{j} scheme under the effect of disturbance iff

Proof: Given the agents are aligned such that _{i}-_{j} periodic scheme.

Since in practice, both

Theorem 5.2. _{s}, η_{s}_{i} and _{j}, such that _{i}_{i} = _{j}_{j}, with _{i}, _{j}_{i}-_{j} scheme under the effect of disturbance if and only if

Proof: See

For disturbance greater than the limit provided in Corollary 5.2, agents are not able to rendezvous in the pre-selected _{i}-_{j} scheme. However, it is still possible that the pair may fall into another scheme. We denote

Lemma 5.3. _{i}-_{j} scheme if and only if

Proof: Lemma 5.3 follow directly the proof of Lemma 5.1.

If the requirements of the specific application arise that the pair needs to rendezvous before

Lemma 5.4. _{j} periods if and only if

Proof: This lemma holds directly following Lemma 5.3.

For any given pair of _{i} and _{j}, the range of a valid α_{i,j} is a neighborhood around _{j}, with the increase of _{i}, the distribution of _{i} is great enough, all valid ranges of α_{i,j} overlap with each other and form a continuous range.

Theorem 5.5.

Proof: See

Corollary 5.5.1. _{s}, η_{s}_{i} and _{j}, such that _{i}_{i} = _{j}_{j}, with _{i}, _{j}

Notice that Theorem 5.5 and Corollary 5.5.1 suggest that there exists some gap between the valid ranges of α_{i,j}, such that for _{j} ≤ _{j}, _{j} is _{j} can go infinitely large such that all rational numbers are included, the gaps are narrowed to only some points upon certain rational numbers.

For the case that η is not bounded but a Gaussian white noise _{i} + _{i} and ω_{j} + _{j}. As a necessary and sufficient condition on η that yields certain rendezvous schemes with a good confidence is hard to solve analytically, a sufficient condition is still relatively easy to obtain. The probability that a normal deviate lies in the range between (μ −

where

The probability of α_{i,j} falls between the bounds

is simply (^{2}. Thus we have

Theorem 5.6. _{i,j} is _{i} and _{j}, such that _{i}_{i} = _{j}_{j}, with _{i}, _{j}_{i}-_{j} scheme with a confidence level of at least^{2}

Theorem 5.7. _{i,j} is _{i} and _{j}, such that _{i}_{i} = _{j}_{j}, with _{i}, _{j}^{2}

Proof: Both theorems follow Theorem 5.2 and Corollary 5.5.1 directly.

In this section we show simulations of the synchronization and desynchronization of a network of multiple agents. We first show seven agents deployed as shown in

Seven agents converge to the same frequency; the black dash line shows how Δ(

Seventeen agents converge to the same frequency.

Two-hundred agents converge to four subgroups.

Now we show the effect of the noise and disturbance on the synchronized system. We take the synchronized groups formed by seven agents, which is shown as in _{s}, the network almost always maintains its current configuration. We ran the simulation for a 50,000 s time window.

The rendezvous network formed by the agents synchronized following the second approach. An edge means that the pair is able to rendezvous periodically with a period no longer than

The comparison between the rendezvous events in the beginning and near the end of the simulation. Every bar has two readings, indicating that the two agents are in rendezvous at this time. The width of each bar is the rendezvous duration.

While the disturbance is set to be a normal distribution of zero mean, and 2σ = η_{s}, the network is more likely to be desynchronized.

The rendezvous network after this group of agents has been desynchronized due to the existence of a disturbance. Agent 3 is disconnected from all the neighbors, and the network was split into two sub-graphs.

The comparison between the rendezvous events in the beginning and near the end of the simulation. The bar indicating the rendezvous between agent 1 and 3 disappeared below the axis (after 400 s of simulation).

The control input to align an agent's phases with any companion it is in rendezvous with. Agent 3 failed to rendezvous with any of its neighbors since ~ 240 s, and therefore no control input was generated after that time.

This paper addressed a synchronous rendezvous problem for a network of mobile sensors monitoring large-scale ocean regions bounded by LCS. It approximated the coherent structures as circular orbits tangential to each other, and assuming that the agents flowing along these orbits can only interact while in close proximity, this formulation gave rise to a graph of intermittently interacting 2-D oscillators. Conditions under which a pair of oscillators can rendezvous solely relying on flow dynamics were presented, controllers were designed to lock them into subsequent periodic rendezvous, and sensitivity analysis was provided for two typical types of disturbance on the control input.

The results in this paper can also find use in other fields, such as perimeter surveillance or space docking. In this work, agents are assumed to travel only along their own circular orbits. Future directions include allowing agents to drive off the orbits to explore the inner circle of the bounded region and to optimally plan its trajectory subject to the ocean environment.

XY and CW developed the theoretical formalism under supervision of MH and HT. XY performed the simulation under supervision of MH. XY and MH wrote the manuscript with support from CW and HT.

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.

We gratefully acknowledge the support of ONR Award No. N00014-17-1-2690 and ARL DCIST CRA W911NF-17-2-0181.

Lemma 3.3

Proof: Let

Without loss of generality, take the time point when agent _{i}(0). Before _{+}. The corresponding phases of

An irrational ^{+} such that

Theorem 4.1

Proof: Let

Pick randomly any pair (^{rend}, and the minimum time

If ϑ_{i}(_{j}(

Thus note that Θ(

While the network of oscillators has not yet reached consensus on their frequencies, there is bound to be at least one agent with angular velocity greater than

Theorem 5.2

Proof: Rearranging (16) in section 5 yields

For a valid _{i,j} in the range of (15) in section 5.

Rearranging (25) provides us with

Since (26) holds for all α_{i,j} exists in (15) in section 5, it is clear that we should have

As

and

Theorem 5.5

Proof: For any

there is

therefore the valid ranges of α_{i,j} to realize _{i}-1 scheme and (_{i} + 1)-1 scheme overlap for _{i} ≥ _{i}. Any α_{i,j} satisfying (21) in section 5 falls into some rendezvous scheme as long as both agents can finish at least one round within

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