Reducing Network Load via Message Utility Estimation for Decentralized Multirobot Teams

Multirobot teams have shown promise for effectively exploring and accurately estimating quantile values to aid understanding of natural environments [1]. When studying spatially varying fields, quantile estimation is useful for describing varied spreads of interest, and entails using measurements from the environment to estimate the values of desired quantiles (e.g., the 25th percentile is equivalent to the 0.25 quantile) [2]. We are interested in using a team of communication-enabled mobile robots to collaboratively explore a natural environment such that we achieve an accurate estimate of quantile values in that environment. However, realistic multirobot deployments must consider limitations in communication capabilities; here, we focus on reducing the network load as measured by the number of attempted transmitted messages with the goal of sacrificing minimal quantile estimation accuracy. We address the question of how to decide if a message is worth sending. Specifically, we:

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  Propose methods to reduce network load for bandwidth- and resource-constrained decentralized multirobot teams that leverage utility-based message evaluation, with no guarantees on information from teammates;

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  Describe a decision-theoretic approach to choosing whether to transmit a message to the network rooted in the utility-based evaluation and a cost of transmission;

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  Present results showing significant reductions in network load are achievable with minor corresponding drops in performance on the problem task of estimating quantile values, even with probabilistic transmission failure and a distance-based transmission probability, on a dataset from a real aquatic environment;

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  Discuss results in the context of real-world applications.

We assume a decentralized team where each robot plans independently and is not guaranteed to receive information from others. However, we assume a communication protocol such that a robot can, at each step, decide to broadcast its newly acquired measurements to the network. The network has limited bandwidth, so robots are incentivized to only send those that will be most beneficial to the others. We encode the cost of loading the network with a message through a threshold value, where utilities higher than the threshold have a higher benefit than cost. With these assumptions in place, the question of how to evaluate a potential message for its utility emerges as a key consideration to this problem; in this work, we propose several approaches to solve it. A general overview can be seen in Figure 1.

There is an extensive literature in utility-based communication, and our methods share similarities with previous work. Work that has considered the problem of deciding what or when to communicate is largely motivated by the same desire to respect bandwidth limitations that real networks must face and to reduce communication costs. However, particular methods differ in how they approach solutions and in what context the problem is situated. Some of the varied contexts in which utility-based decision-making for inter-robot communication has been investigated include months-long data collection to be routed to a base station [3], disaster response [4], non-cooperative multi-agent path finding [5], and those with request-reply protocols [5, 6].

One common approach has been to use message category, type, or intrinsic data features to determine which messages are most valuable to send [3, 4, 7]. These approaches typically leverage domain knowledge for their particular application. Previous work aimed at the problem of multirobot localization has applied directed acyclic graphs built by forward-simulating the robot team under both transmission and non-transmission of each message [8]. The goal then is to reach the best group localization accuracy while minimizing the cost (number of communications required in the DAG).

The expected effect of the message on robot actions has also been used to determine utility [5, 6]. One approach for restricting the number of communications when robots are tasked with selecting goal waypoints to visit is to use a heuristic based on limiting the new path’s length, and only request information from others if the path would be within a certain factor length of the current path [9]. Monte Carlo Tree Search (MCTS) has been used within a general middleware in order to make decisions on whether a message should be sent to the network [7]. Their approach takes into account the temporal aspect of message utility potentially changing over time, and the tree built at each decision point includes all previous and future messages. However, it does not explicitly encode network constraints and assumes that robots are always within communication range of each other.

Another class of solutions propose metrics to determine utility based on the difference in expected reward after incorporating a message [10, 11, 12]. In [13], the authors proposed a utility function computed as the sum of two competing factors: maximizing utility gained by the entire team and minimizing the number of times a message is sent. There, the network is point-to-point, and thus at each step, a decision is made whether to forward the received message to a random recipient or to cease forwarding. [7] and [8] also include a measure of utility based on expected reward, and the teams are decentralized in terms of planning and reward calculation. A major difference between these works and ours is that our end goal is to explore the environment in whatever way results in the best quantile estimates, while those define a static goal location for the robots to reach. We also exclusively choose to plan using a greedy algorithm for computational efficiency and tractability, motivated by resource-constrained systems, and do not forward-simulate the models of other robots.

Significantly, we make the (more realistic) assumption of stochastic communication on top of the communication decision; just because a robot decides to send a message does not guarantee reception by all or any of the recipients. We do not assume perfect measurement sensing. Despite our relatively simple planning and modeling methods and more failure-prone communication assumptions, we are able to achieve a significant decrease in network loading while maintaining good task performance.

Our formulation extends the setup in [1] to which we refer the reader for details; we briefly mention relevant definitions here. The team consists of $N$ aerial robots, each operating with a fixed planning budget $B_{T}$ at a fixed height in a discretized environment $\mathbold{G^{\#}}\subset\mathbb{R}^{2}$. Robots can take one of four possible actions at each step: $\pm x$ or $\pm y$. Since our focus is on inter-robot communications, we assume a low-level controller, a collision avoidance routine, and perfect localization. The human operator supplies a set of arbitrary quantiles of interest $Q$, and the end goal is to produce accurate estimates $\tilde{V}$ of $Q$ where the true values are defined as $V=\textit{quantiles}(\mathbold{Y^{\#}},Q)$, the set of all possible locations a robot can measure with its sensor are $\mathbold{X^{\#}}$, and the corresponding true values at those locations are $\mathbold{Y^{\#}}$. Each robot has a camera with which it can take noisy measurements $\mathbold{Y}$ representing the concentration of algae at those pixel locations in the environment $\mathbold{X}$. Throughout the deployment, each robot collects measurements $\mathbold{X},\mathbold{Y}$ both firsthand via its camera as well as via any messages received from others through the network. The robots use all their collected measurements to construct and update their internal Gaussian process (GP) model of the environment. Robot $i$’s estimate of the quantile values is given by

where $\mu_{GP^{i}}(\mathbold{X^{\#}})$ is the estimate of all possible locations using the GP conditioned on the measurements $\mathbold{X}^{i}$ and $\mathbold{Y}^{i}$ that $i$ has collected thus far.

In what follows, when viewing the problem from the perspective of a given robot, we will refer to it as $i$ or the ego robot and to any of the other $N$ robots as $j$. Variables superscripted with $j$ such as $GP^{j}$ indicate the GP for $j$ from the perspective of $i$.

Motivated by the computational constraints of resource-constrained systems, we opt to use a locally computable, online, greedy planning policy. At each planning step $t$, $i$ maximizes an objective function $f$ over feasible next measurement locations. We use a modified version of the quantile standard error objective function to evaluate a proposed location for time $t$ [2], which measures the difference in the standard error of the estimated quantile values using two versions of the robot model: the current model $\mu_{GP^{i}_{t-1}}$, and the model updated with the hypothetical new values at the proposed locations $\mu_{GP^{i}_{t}}$. We modify $f$ such that it is conditioned not only on the proposed measurement locations $\mathbold{X}$, but also the values associated with the locations (denoted by $\mathcal{Y}$ generally) and the GP to be used. The reasoning is that $f$ is also used later in the process for computing message utilities, and this allows it to generalize to these cases when $\mathcal{Y}$ and $GP$ may take on different values. When $f$ is used in the planning step by $i$ to select $i$’s next location, $\mathcal{Y}$ are the expected values at the proposed location $\mathbold{X}$: $\mu_{GP^{i}_{t-1}}(\mathbold{X})$ (just like in [2]). However, when $f$ is used by $i$ to determine the utility of $j$ receiving a real measurement that $i$ has already acquired, $\mathcal{Y}$ are those acquired $\mathbold{Y}$, and the GP may be either $GP^{i}$ or $GP^{j}$ (see Section III-C). The final term encourages exploration of high-variance areas.

We now present experimental results for the methods described in Section III. We test performance on 3 sets of quantiles $Q$: Quartiles (0.25, 0.5, 0.75), Median-Extrema (0.5, 0.9, 0.99), and Extrema (0.9, 0.95, 0.99). For each combination of parameters, we run 5 seeds, resulting in 60 total instances. We fix the following parameters: $N=4$; budget $B_{T}=10$ ($40$ steps total); initial spread $\alpha=0.2$; $\eta=0.4$; communication radius $r=15.0$; thresholds $T_{rw}=2.8\times 10^{-4}$ and $T_{ego}=8.3\times 10^{-5}$. $T_{rw}$ is used for Reward and Action, while $T_{ego}$ is used for Ego-reward. We selected these $T$ experimentally by observing the received $u$ values during testing and setting $T$ to the approximate 25th percentile of observed rewards. This can be interpreted as only messages that are in the top 75% of utility being worth the cost of sending to the network, while those in the lower quarter are not.

The robots operate in simulation using a real-world dataset collected with a hyperspectral camera mounted on a drone from a lake in California. The environment is roughly $80\times 60$ meters, discretized to $\mathbold{G^{\#}}=25\times 25$ locations. Each set of measurements is an image of $5\times 5$ pixel intensities for the 400nm channel of the hyperspectral image, normalized to $[0,1]$. To each measurement, we add zero-mean Gaussian noise with a standard deviation $0.05$.

We presented the first study on restricted communication in the context of quantile estimation for environmental analysis. We show that restricting message transmission based on whether the actions that the sender believes other robots will take at the next state would change with the message results in a $42\%$ decrease in network load while also decreasing quantile estimation error by $1.84\%$, when compared to transmitting every message possible. This indicates that targeted communication during quantile estimation can have real benefits to real-world multirobot deployments when robots and their networks are resource-constrained.

In addition to receipt handshaking, there are several interesting directions of future work. We believe allowing the sender to choose the most valuable measurements from its collected data rather than restricting it to the most recent measurements may improve performance. However, this would lead to challenges in scalability as the environment is explored, and could be addressed by applying a rolling window over the previous measurements to mitigate this. Further, we are interested in methods for better approximating the models that a robot maintains for each of its teammates, which could be achieved by augmenting message headers with metadata such as that robot’s current dataset size or current quantile estimates.