The number of robots used in everyday activities is steadily increasing and is expected to keep growing. This will undoubtedly occur in industrial settings where the next generation of robots will be crucial in meeting the dynamic needs of collaborative and intelligent manufacturing that characterize the so-called Industry 4.0 and Industrial Internet of Things [1]. However, international market analyses anticipate widespread use of robots also by the general public [2]. In different scenarios, different types of robots with different capabilities and marketed by different companies will need to work together, and possibly with humans, to assess the current situation [3,4] and achieve shared goals. Therefore, "teammates" will likely require to trust each other for efficient teamwork. In multi-agent systems, it is possible to define trust as "the subjective probability by which an agent (the trustor) expects that another agent (the trustee) succeeds in performing a given action on which its welfare depends" [5].
Robotic literature has primarily investigated trust in the context of human-robot interaction [6,7,8]. However, not all situations in which trust is involved require the trustee or the trustor to be human. Both in industrial and service scenarios, there are situations where autonomous robots must cooperate without having perfect knowledge about each other's capabilities. In this context, the trust that robots have in their own abilities versus other robots' capabilities may affect the selection of the most suitable candidate to perform the task.
In this work, we propose a novel framework for auction-based task assignment that can operate in a distributed open environment. The framework may handle heterogeneous robotic agents, with different perceptual, reasoning, and actuation capabilities, that are periodically assigned tasks to be performed to achieve a shared objective. The system takes inspiration from popular models in the literature, adapting them to an open environment in which robots may dynamically enter or exit, execute assigned tasks, or verify the correct execution of tasks by other robots. The purpose of such a framework is to allow agents to model trust in each other regarding the capability to accomplish an assigned task and use this model to evaluate possible candidates for task assignments whenever needed. In other words, agents may use the framework to outsource tasks they need to complete to achieve a particular goal.
The general structure of the framework may be summarized as follows. First, using a portfolio of trust-related metrics, agents dynamically gather data about the other agents' capability to (i) perform actions; (ii) verify the outcomes of actions performed by other agents. Then, they will iteratively use and update these metrics to evaluate the trustworthiness of other agents, including themselves, during auctions, thus ultimately taking trustworthiness into account when taking a new decision for task assignment.
Given that, each agent in the framework should be able to:
• start a plan to manage one or more events;
• auction an action in its plan or bid on an action auctioned by another agent;
• execute or verify the execution of one or more actions;
• gather data from other agents about the success or failure of a given action;
• update its trust in other agents.
Concerning the latter, and considering a set of 𝑁 𝑒 events ℰ = {𝐸 𝑖 } (which may be triggered by an explicit command from a user, an alarm, or any other external stimulus that is processed by a robot and whose effect is a specific plan), a set of 𝑁 𝑎 actions 𝒜 = {𝐴 𝑖 }, and a set of 𝑁 𝑔 agents 𝒢 = {𝐺 𝑖 }, which can communicate with other agents, auction or bid for actions, we have defined the following context-independent metrics [7]:
• Reliability, one of the attributes of Logical Trust [6], which is the estimate, made by agent 𝐺 𝑖 , of the success rate of agent 𝐺 𝑗 in executing action 𝐴 𝑘 ; • Verification Trustworthiness, which measures the degree of the trustworthiness of a verifying agent depending on the consensus it has around its judgment skills. Formally, the Verification Trustworthiness measures how much the verification made by agent 𝐺 𝑗 about the success of action 𝐴 𝑘 is considered trustworthy according to an agent 𝐺 𝑖 and requires counting the number of opinions that are concordant or discordant with the judgment of 𝐺 𝑗 ; • Perceived Competence, which is computed, in its general formulation, as a function of the Reliability of 𝐺 𝑗 estimated by all agents {𝐺 1 . . . 𝐺 𝑛 } participating to the auctions as well as their own Verification Trustworthiness.
Regarding Reliability, it is worth noticing that it may be correctly computed only when a reasonably large sample of observations is available. To handle the transitory, i.e., when the agent 𝐺 𝑗 has not yet executed the action 𝐴 𝑘 a sufficiently large number of times, we have identified three possible strategies:
• a Boot mode, which requires the definition of a "boot phase" length, expressed as the number of 𝐴 𝑘 's auctions to which an agent 𝐺 𝑗 needs to participate before an auctioneer 𝐺 𝑖 starts using its Reliability as a measure of 𝐺 𝑗 's trustworthiness. In the boot phase, each auctioneer 𝐺 𝑖 shall trust 𝐺 𝑗 's declared Reliability, or rely on the outcomes of the other actions. • a Window mode, similar to the Boot mode, but taking into account only a "memory window", instead of considering the full sequence of observations; • a BCI mode, where each agent computes and shares not only the average Reliability of other agents, but also the binomial confidence interval (BCI) around such estimate that converges to zero as the number of executions increases.
Finally, the way in which individual Reliability and Verification Trustworthiness contribute to the Perceived Competence will depend on:
• the behaviour of agents towards the community: individualistic or collectivistic. Individualistic agents compute the Perceived Competence only based on their own observations; on the contrary, collectivistic agents take into account other agents' opinions by calculating a weighted average mediated by their Verification Trustworthiness (Weighted Reliability). • the disposition of agents towards other agents: optimistic, pessimistic, realistic. The optimistic/pessimistic/realistic disposition of an auctioneer may play a key role to evaluate a bidder's trustworthiness. For example, in BCI mode, an optimistic auctioneer will consider the upper bound of the BCI to estimate the agent's Perceived Competence, therefore being more prone to forgive and give a second chance to an agent that failed the first attempts; a pessimistic agent will use the lower bound, thus being very conservative and cautious when it encounters a new bidder about which it has little data.
The framework has been tested in a real-world scenario comprising two SoftBank Pepper robots and one NAO robot, which we repeatedly explored for human-robot interaction in socially assistive contexts [9,10,11,12,13]. In order to simplify the execution of actions and their verifications, we have decided to associate actions 𝐴 1 and 𝐴 2 with sentences to be pronounced by the robots. Please notice that, while the robot that won the auction and pronounced the sentence always considers its action as a success, the other agents may judge outcomes differently.
For instance, the distance between the speakers of a robot and the microphones of another robot, or the fact that NAO's and Pepper's microphones are placed over the robots' heads, may lead the speech recognition systems to fail.
Based on the considerations above, we performed experiments by placing NAO (agent 𝐺 3 in the following) on a table in front of the two Pepper robots (𝐺 1 and 𝐺 2 ). NAO's speakers point towards Pepper microphones from above (Figure 1). The speech volume of the three robots is set to 80% of the maximum value. In this configuration, we expect that 𝐺 1 and 𝐺 2 can correctly verify the actions performed by all other robots, while 𝐺 3 will not be very performing in verifying the actions of 𝐺 1 and 𝐺 2 (given that its microphone is not oriented toward the two Pepper robots' speakers).
Robots work in BCI mode, with an optimistic disposition, and tests have been performed with robots showing both individualistic and collectivistic behaviours. Finally, please notice that event 𝐸 1 can only be handled by 𝐺 1 , that will then auction 𝐴 1 (pronouncing the sentence "Take the medicine"); event 𝐸 2 can only be handled by 𝐺 2 , that will then auction 𝐴 2 (pronouncing the sentence "Drink some water"). All agents can execute and observe both actions 𝐴 1 and 𝐴 2 .
Results are shown in Figure 2, which reports, for each agent 𝐺 𝑖 how the estimated Reliability and Verification Trustworthiness of the three agents in the framework evolve as the number of auctions increase. In all plots, Reliability estimates are plotted with a continuous line, whereas Verification Trustworthiness estimates are plotted with a dashed line. A small circle means that an action has been assigned to the corresponding agent at that time.
When robots exhibit a collectivistic behaviour (Figure 2 (a)) it can be observed that 𝐺 1 and 𝐺 2 can correctly recognize the actions performed by all other robots (even if in some cases the actions performed by 𝐺 2 are wrongly evaluated by 𝐺 1 ), while 𝐺 3 is not capable of correctly verifying the performance of 𝐺 1 and 𝐺 2 . As a result, after a few iterations, the collectivistic auctioneers 𝐺 1 and 𝐺 2 tend to assign all actions to 𝐺 3 , since it is considered the most reliable agent in terms of Weighted Reliability (i.e., it's the only agent whose actions are perceived as correctly executed by 𝐺 1 , 𝐺 2 , and 𝐺 3 itself). When the robots show an individualistic behaviour (Figure 2 (b)), the actions are more uniformly distributed among all agents. Indeed, the two auctioneers 𝐺 1 and 𝐺 2 judge all agents equally trustworthy, and they only rely on their Perceived Competence. It may be also observed that, due to possible errors in speech-to-text conversion, the auctioneers may sometimes judge that an action has not been executed correctly. See for instance the trust metrics computed by G1: 𝐴 1 performed by 𝐺 2 during auction 4 is negatively evaluated by 𝐺 1 therefore producing a decrement in 𝐺 2 's Reliability. When this happens, the probability that the same action will be assigned again to a robot that has possibly failed decreases, since each auctioneer relies only on its own opinion. For the same reason, even if other agents judge that an auctioneer has failed an action, the auctioneer will ignore their opinions since all agents consider themselves as perfectly capable to execute actions. As a consequence, it can be observed in Figure 2 (b) that 𝐺 1 repeatedly assigns 𝐴 1 to itself even if it is considered very unreliable by 𝐺 2 and 𝐺 3 .
Finally, it is worth noticing how the Verification Trustworthiness of the three robots is much lower in the second experiment. When robots are collectivistic, most of the auctions are won by 𝐺 3 , whose actions are judged as correctly performed by all robots, but when robots are individualistic, actions are shared among all agents, and, concerning their verification, usually 𝐺 3 disagrees with 𝐺 1 and 𝐺 2 about the outcome of the actions.
These results, although preliminary, confirm the potentials and the functionalities of the presented framework, and its applicability to a real scenario.