STRATA: A Unified Framework for Task Assignments in Large Teams of Heterogeneous Robots

03/12/2019 ∙ by Harish Ravichandar, et al. ∙ 0

Large teams of robots have the potential to solve complex multi-task problems that are intractable for a single robot working independently. However, solving complex multi-task problems requires leveraging the relative strengths of different robots in the team. We present Stochastic TRAit-based Task Assignment (STRATA), a unified framework that models large teams of heterogeneous robots and performs optimal task assignments. Specifically, given information on which traits (capabilities) are required for various tasks, STRATA computes the optimal assignments of robots to tasks such that the task-trait requirements are achieved. Inspired by prior work in robot swarms and biodiversity, we categorize robots into different species (groups) based on their traits. We model each trait as a continuous variable and differentiate between traits that can and cannot be aggregated from different robots. STRATA is capable of reasoning about both species-level and robot-level differences in traits. Further, we define measures of diversity for any given team based on the team's continuous-space trait model. We illustrate the necessity and effectiveness of STRATA using detailed simulations and a capture the flag game environment.

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I Introduction

The study of multi-robot systems has produced significant insights into the process of engineering collaborative behavior in groups of robots [22, 5]. These insights have resulted in large teams of robots capable of accomplishing complex tasks that are intractable for a single robot, with applications including environmental monitoring [28], agriculture [29], warehouse automation [33], construction [32], defense [3], and targeted drug delivery [17]. Efficient solutions to the above problems typically rely on a wide range of capabilities. The multi-robot task assignment (MRTA) problem [9, 16, 15] addresses the above challenges by seeking to optimally assign robots to tasks.

In this work, we focus on an instance of the MRTA problem with an emphasis on large heterogeneous teams. Teams of heterogeneous robots are particularly well suited for performing complex tasks that require a variety of skills, since they can leverage the relative advantages of the different robots and their capabilities. We present Stochastic TRAit-based Task Assignment (STRATA), a task assignment algorithm that enables a heterogeneous team of robots to optimally divide the various tasks among its members. STRATA models the topology of tasks as a densely connected graph, with each node representing a task and or a physical location and the edges indicating the possibility of switching between any two tasks. We assume that the optimal robot-to-task associations are unknown and that the task requirements are specified in terms of the various traits (capabilities) required for each task. Thus, in order to effectively perform the tasks, the robots must reason about their combined capabilities and the limited resources of the team. To enable this reasoning, we take inspiration from prior work in robot swarms [25] and biodiversity [24], and propose a group modeling approach [1] to model the capabilities of the team. Specifically, we assume that each robot in the team belongs to a particular species (group). Further, each species is defined based on the traits possessed by its members. Assuming that the robots are initially sub-optimally assigned to tasks on the task graph, STRATA computes optimal assignments such that the robots can reorganize themselves to collectively aggregate the traits necessary to meet the task requirements as quickly as possible.

Fig. 1: Top row: STRATA defines the effects of task-species distribution and the species-trait model on the task-trait distribution. Bottom row: Given a team defined by the species-trait model, we aim to perform optimal task assignments such that the desired task-trait distribution is achieved.

In Fig. 1, we illustrate the basic building blocks of STRATA and the task assignment problem. As seen in the top row, STRATA models the effects of task assignments (task-species distribution ) and the species’ traits (species-trait model ) on how the traits are aggregated for each task (task-trait distribution

). Thus, for any distribution of robots on the graph, we can compute the corresponding trait distribution across the graph. We also derive a closed-form expression to quantify the effect of the variance of the robots’ traits on the achieved task-trait distribution. The task assignment problem, as seen in the bottom row, involves computing the optimal task assignments, given a desired distribution of traits across task and a team of heterogeneous robots.

Using the above model, STRATA allows for the optimization of two separate task assignment goals: (1) exact matching and (2) minimum matching. In exact matching, the algorithm aims to distribute the robots such that the achieved trait distribution is as close to the desired as possible. In minimum matching, the algorithm aims to distribute the robots such that the achieved trait distribution is higher than or equal to the desired as possible, i.e., over-provisioning is not penalized.

The STRATA representation of both task and species traits is inspired by [25], which considered binary instantiations of traits. However, binary models fail to capture the nuances in the scales and natural variations of the robots’ traits. For instance, consider an unmanned aerial vehicle and a bipedal robot. While both robots share the mobility trait (the ability to move), their speeds are likely to be considerably different. To address these challenges, in STRATA we have extended the representation to model traits in the continuous space. Additionally, STRATA also captures robot-level differences within each species by using a stochastic trait model.

When reasoning about the collective strengths of the team, attention must be paid to the fact that not all capabilities are improved in quantity by aggregation of individual robots’ abilities. For instance, a coalition of any number of slow robots does not compensate for a faster robot. Taking this observation into account, we consider two types of traits: cumulative and non-cumulative. We consider a trait to be (non) cumulative if it can (not) be aggregated from different robots in order to achieve certain task requirements.

Finally, we extend the diversity measures introduced in [25] to the continuous space. We derive two separate diversity measures, one for each goal function. The diversity measures provide insights about the trait-based heterogeneity of the team. Specifically, the diversity measures help define a a minimum subset of the species that can collectively compensate for the rest of the team.

In summary, the key contributions of our work include a unified framework for optimal task assignment of large heterogeneous teams that:

  1. incorporates a stochastic trait model that captures both between-species and within-species variations,

  2. optimally assigns tasks to robots with respect to two separate goals: exact matching and minimum matching, and

  3. computes measures of diversity in teams with continuous trait models.

We evaluate STRATA using detailed simulations and a capture-the-flag game environment. Our results demonstrate the necessity and effectiveness of STRATA in terms of optimal task assignment and improved team performance, when compared to a baseline that only considers binary traits.

Finally, we note that heterogeneous teams can be composed of both autonomous and human agents. Human traits are vastly different from and complementary to those of robots [7]. For instance, when compared to humans, robots can carry heavier payloads, move faster, and be immune to fatigue. On the other hand, humans’ abilities to assimilate and maintain situational awareness, process noisy information, and adapt to highly unstructured environments are unmatched by the abilities of their robotic counterparts. Further, individual differences are considerable in teams involving humans (see [7] and references therein). Although not yet evaluated with human-robot teams, STRATA’s ability to characterize humans as one or more separate species (e.g., soldiers, pilots, medics) possessing stochastic traits makes it a promising representation for modeling human-robot teaming.

Ii Related Work

Significant efforts have been focused on problems in multi-robot systems task assignment (MRTA) [9, 16, 15]. Broadly, the problems are categorized based on three binary characteristics: (1) Task type (single-robot [SR] vs multi-robot [MR]), (2) robot type (single-task [ST] vs multi-task [MT], and (3) assignment type (instantaneous [IA] vs time-extended [TA]) [9]. While task type indicates the number of robots required to complete each task, robot type indicates whether the robots are capable of simultaneously performing a single task or multiple tasks. The assignment type is used to differentiate between tasks that involve scheduling constraints and those tat do not. Indeed, numerous approaches related to the different variants of MRTA, including assignments involving single-robot tasks, are available in the literature. However, we limit our coverage of related work to variants of MRTA that involve multi-robot tasks - tasks that involve the coordination of several robots. We refer readers to [9, 16] for comprehensive categorizations and examples of all approaches pertaining to task assignment.

Several methods for task assignment with homogeneous robots have been proposed. A graph-theoretic framework, named SCRAM, is proposed in [20]. SCRAM maps robots to target locations while simultaneously avoiding collisions and minimizing the time required to reach target locations. The work in [19] presents a hierarchical algorithm that is correct, complete, and optimal for simultaneously task assignment and path finding. A fast bounded-suboptimal algorithm, that automatically generates highways for a team of homogeneous robots to reach their target locations, is introduced in [6]. Notably, the methods in [20, 19, 6] emphasize optimal path finding for each robot and collision avoidance in order to assign each robot to a single task (reaching a target location). However, these methods assume that all the robots in the team are interchangeable, and thus are not suitable for multi-task scenarios that involve several heterogeneous robots.

Approaches involving single-task robots solve the assignment problem by assuming that each robot is specialized and can only perform one task. The method proposed in [26] addresses a transportation task involving multiple single-task robots. Some of the items to be transported can be transported by a single robot, while others need coordinated efforts from several robots. [26] use a greedy set-partitioning algorithm to form coalitions of robots required to perform the tasks. Potential coalitions are iteratively computed for each task involved. The coalition formation algorithm introduced in [26] was later extended in [31]. The extended algorithm in [31] reduces the communication effort, balances the coalitions, and constrains the requirements to specify if and when all the required traits must be possessed by a single robot. These approaches, however, require the listing of all potential coalitions and thus are not suitable for problems involving large number of possible coalitions. Indeed, the number of possible coalitions is a factor of both the number of robots in the team and the inherent diversity of the team. Specifically, as the number of robots in the team and their similarities increase, so does the number of possible coalitions. STRATA, on the other hand, is scalable with the number of robots as it does not list all possible coalitions.

Decentralized approaches for task assignment are introduced in [13, 4, 12, 21]. A game-theoretic task assignment strategy is introduced in [13] to assign tasks to a team of homogeneous robots with social inhibition. In [4], multiple tasks are assigned to a team of homogeneous robots. The authors develop of a continuous abstraction of the team by modeling population fractions and defining the task allocation problem as the selection of rates of robot switching from and to each task. In [12], the authors extend the method in [4] with a wireless communication-free quorum sensing mechanism in order to reduce task assignment time. In [21]

, a decentralized approach for heterogeneous robot swarms is introduced. The approach computes optimal rates at which the robots must switch between the different tasks. These rates, in turn, are used to compute probabilities that determine stochastic control policies of each robot. However, a common shortcoming of these decentralized approaches is that they assume that the desired behavior is specified as a function of the distribution of robots across the tasks.

Auction or market-based methods also provide solutions to the MRTA problem involving single-task robots [10, 18, 30, 8]. In [10], the robot responsible for any given task is the robot who discovers the task. Once discovered, the robot holds an auction to recruit other robots into a coalition. [18] introduce combinatorial biding to form coalitions and provides explicit cooperation mechanism for robots to form coalitions and bid for tasks. A homogeneous task assignment algorithm for robot soccer is presented in [30]. Sensed information from the robots are shared to compute a shared potential function that would help the robots move in a coordinated manner. We refer readers to [8] for a survey of market-based approaches applied to multi-robot coordination. A common of trait of auction or market-based methods is that they require extensive communication for biding and scale poorly with the number of robots in the team. Further, the methods discussed thus far are limited to either single-robot tasks or single-task robots. In contrast, STRATA considers robots capable of performing tasks that require coordination between multiple robots.

Our work falls under the category of Single-Task Robots Multi-Robot Tasks Instantaneous Assignment (ST-MR-IA) problem, also known as the coalition formation problem [9]

. In other words, we are interested in assigning a team of robots to several tasks, each requiring several robots. The assignment type is instantaneous since our task assignment does not reason about future task assignments or scheduling constraints. The ST-MR-IA is an instance of the set-partitioning problem in combinatorial optimization and is known to be strongly NP-Hard

[9]. Albeit not developed for MRTA, a few efficient approximate solutions have been proposed for the set partitioning problem [2, 11]. Based on prior work in set partitioning problems, centralized and distributed algorithms to solve a ST-MR-IA problem have been proposed in [26, 27]. The method in [27] has also been adapted to be more efficient by reducing the required communication and discouraging imbalanced coalitions [31]. A method for coalition formation is introduced in [23] by building a solution to a task by dynamically connecting a network of behaviors associated with individual robots.

A limitation of most of the existing approaches is that the desired behavior is assumed to be specified in terms of optimal robot distribution. A notable exception to this generalization is the framework introduced in [25], which is capable of receiving the task requirements provided in the form a desired trait distribution cross tasks. We take a position similar to [25], and do not assume that the desired distribution of robots is known. Another similarity between STRATA and [25] is being suitable for a decentralized implementation. Thus, both approaches are scalable in the number of robots and their capabilities, and are robust to changes in the robot population.

While STRATA shares several similarities with [25], there are a number of notable relative advantages. First, our species-trait model is continuous, while [25] uses a binary model. Second, we differentiate between cumulative and non-cumulative traits. Third, the framework in [25] utilizes a deterministic model of traits. In contrast, we consider the inherent randomness in the robots’ traits, thereby capturing the variations at both species and robot levels. Finally, while the diversity measures introduced in [25] are limited to binary trait models, our measures are compatible with continuous-space models.

Iii Modeling framework

In this section, we introduce the various elements of STRATA that enables task assignments in large heterogeneous teams. Assigning tasks to the different robots in the team requires reasoning about their complementary traits and the limited resources of the team. STRATA handles this challenge using (1) a stochastic trait model that describes the capabilities of each species in the team along with the corresponding variance, (2) a task model that explains the combinations of capabilities currently available at each task and how robots are allowed to switch between the tasks, and (3) a model that describes the dynamics of robots traversing the task graph.

Based on the above mentioned models, we formulate and solve a constrained optimization problem to distribute the robots across the different tasks to satisfy certain trait-based task requirements. Specifically, we compute the optimal transition rates on the task graph which in turn dictate how task assignments vary as a function of time such that the desired trait distribution is achieved and maintained as quickly as possible. Further, our optimization explicitly reasons about the expected variance of the trait distribution.

Throughout the paper, we illustrate STRATA using a running example of a task assignment problem. We will progressively build the example as we introduce the different parts of the framework.

Iii-a Trait Model

Base model: Consider a heterogeneous team of robots. We assume that each robot is a member of a particular species. The number of species is finite, and the number of robots in the th species is denoted by . Thus, the total number of robots in the team is given by We define each species by its abilities (traits). Specifically, the traits of each species are defined as follows

(1)

where is the trait of the species, and is the number of traits. Thus, the traits of the team is defined by a species-trait matrix , with each row corresponding to one species.

Stochastic traits: To capture the natural variation found in each species, we maintain a stochastic summary of each species’ traits. Specifically, each element of

is assumed to be an independent Gaussian random variable:

. Thus, expected value of the species-trait matrix and the corresponding matrix summarizing the variance of each of its elements are given by

(2)
(3)

Using a stochastic model allows us to automatically find clusters in the trait space and thus aids in automatically identifying the different species and their expected traits along with their observed variance. STRATA learns both the means and variances directly from data. Given the trait values of each robot in the team and the number of species

, a Gaussian mixture model (GMM) is used to approximate the joint distribution over the traits of the team. The parameters of the GMM with

Gaussian kernels and diagonal covariance matrices are learned using the Expectation Maximization algorithm. Each Gaussian cluster in the mixture is assumed to represent a single species using the mean and variance values for each trait.

[colback=blue!5!white, enhanced jigsaw, breakable] Example: Consider an example scenario in which the team is made up of 4 species, each consisting of 25 robots. Each species is categorized based on the following 4 traits: viewing distance (miles), speed (mph), number of health packs, and ammunition. Let the expected value of the species-trait matrix and the corresponding matrix of variances for our example team be given by

(4)
(5)

Note that STRATA allows for modeling traits of different orders of magnitude. Further, for the same trait, the variation observed in each species is different. For instance, consider the ammunition trait (th columns of and ). The distribution of this trait is considerably different in each species. Specifically, while Species 1 has the largest average units of ammunition (), it also has the smallest variance (). On the other hand, Species 3 has considerably lower units of ammunition () while its variance () is considerably higher than that of Species 1. Encoding these aspects of each species enables STRATA to reason about the various trade-offs when recruiting robots to meet the task requirements.

Cumulative traits: STRATA explicitly differentiates between cumulative and non cumulative traits. While examples of cumulative traits include ammunition, equipment, and coverage area, those of non-cumulative traits include speed, special skills, and training. We model the cumulative traits as continuous variables (i.e.,

), and the non-commutative traits as binary variables (i.e.,

), where is the set of indices corresponding to cumulative traits. In the case of non-cumulative traits, the binary values are assigned based on the following rule

(6)

where is a user-defined minimum acceptable value for the th trait. The binary representation of non-cumulative traits captures information about whether the robots of each species posses the minimum required capabilities. Further, when aggregated (Section III-D), the binary representation provides the total number of robots meeting the minimum requirements, as opposed to aggregating the trait values.

[colback=blue!5!white, enhanced jigsaw, breakable] Example: In our example, the first two traits (viewing distance and speed) are non-cumulative since they can not be aggregated. Let the minimum acceptable values for the traits be miles, and mph. Thus, the modified expected value of the species-trait matrix is given by

(7)

Note that the average viewing distance of Species 1 ( miles) is lower than the minimum requirement of miles. Thus, Species 1 is considered as not meeting the requirements for viewing distance and is assigned a zero for the same trait. Similarly, Species 2 and 3 are assigned zeros for speed.

Iii-B Task Model

Given the trait model from the previous section, we require the team to accomplish tasks. We model the topology of the tasks using a densely connected graph . The vertices represent the tasks, and the edges connect adjacent tasks and each edge represents the possibility of a robot to switch between the corresponding two adjacent tasks. For each species, we aim to identify the optimal transition rate for every edge in , such that . The transition rate defines the rate which which a robot from species currently performing task switches to task . The transition rates dictate how all the robots are distributed around the graph as time evolves.

[colback=blue!5!white, enhanced jigsaw, breakable] Example: Our example problem involves 5 tasks and the task graph is shown in Fig. 2. Note that the graph is not fully connected. This reflects the restrictions on how the robots can switch between tasks. For instance, let each task be carried out in a different physical location. The presence (absence) of an edge between any two tasks implies that it is (not) possible for the robots to move between the two tasks. STRATA explicitly takes these restrictions into consideration.

Fig. 2: The task graph of our example task assignment problem.

Iii-C Robot Model

With the capabilities and the tasks of the team defined, the modeling of individual robots and their assignments remains. The distribution of robots from species across the tasks at time is defined by . Thus the distribution of the whole team across the tasks at time can be described using a abstract state information matrix .

[colback=blue!5!white, enhanced jigsaw, breakable] Example: Let us assume that the initial distribution of robots, perhaps a result of earlier task requirements, is given by

(8)

Thus, initially, all the robots from Species 1 are assigned to Task 1, all robots from Species 2 are to Task 2, and so on. Further, no robots are assigned to Task 5. Note that each column adds up to the number of robots in the corresponding species.

The dynamics of the number of robots from Species at Task is give by

(9)

and thus the dynamics of each species’ abstract state information can be computed as

(10)

where is the rate matrix of species , defined as follows

(11)

The solution of the dynamics in (10) for each species is given by

(12)

Thus, the solution to the dynamics of the abstract state information is given by

(13)

where , denotes a

-dimensional vector of ones, and

is the -dimensional unit vector with its th element equal to one.

Iii-D Trait Aggregation and Distribution

Finally, we represent the trait distribution across the tasks by the trait distribution matrix and is computed as

(14)

Thus, is computed by aggregating the traits of all the robots assigned to a particular task. For cumulative traits, each column of represents the aggregated amounts of the corresponding trait available at each task, and for non-cumulative traits, each column of represents the total number of robots (who meet the minimum requirements for the corresponding trait) assigned to each task.

Note that since the stochastic nature of results in the elements of being random variables. The expected value of can be computed as follows

(15)

and since the elements of are independent random variables, the variance of each element of is given by

(16)

where denotes the Hadamard (entry-wise) product. Furthermore, the covariance between any two elements of is given by

(17)

[colback=blue!5!white, enhanced jigsaw, breakable] Example: The expected value of the species-trait matrix and the initial abstract state information of our example problem are defined in (7) and (8), respectively. Thus, the corresponding initial trait distribution is given by

(18)

Given that the robots are distributed as given by (8), the above matrix explains how the team’s aggregate capabilities are distributed across the different tasks.

Iv Problem Formulation

Based on the modeling framework described in Section III, this section considers the problem of task assignment that achieves a desired trait distribution across tasks. Specifically, we wish to find the transition rates for each species such that the trait distribution over tasks , defined in (14), reaches the desired trait distribution as quickly as possible.

We express the problem as the following optimization problem

(19)
(20)

where , named the goal function, is a function that defined the set of admissible trait distribution matrices . As in [25], we consider two goal functions:

  1. Exact matching:

  2. Minimum matching:

where denotes the element-wise less-than-or-equal-to operator. While goal function requires achieving the exact desired trait distribution, goal function requires the trait distribution be greater than or equal to the desired trait distribution. In other words, does not allow any deviation from the desired trait distribution, and allows for over-provisioning.

[colback=blue!5!white, enhanced jigsaw, breakable] Example: Let the desired trait distribution for our example be given by

(21)

Note that the initial trait distribution is defined in (18). Thus, the task assignment algorithm is required to find the optimal transition rates such that the team’s trait distribution changes from to as quickly as possible and continuous to remain at .

V Diversity Measures

Large heterogeneous teams with multiple species might result in capabilities that are complementary and or redundant. Inspired by [25], we study the properties of the average species-trait matrix to understand the similarities and variations among the species of a given team. Measures of team diversity were defined in [25] for species defined by binary traits. In this section, we extend and define diversity measures for species defined by continuous traits. We define two measures of trait diversity for a given team, one for each of the two goal functions defined in Section IV. To this end, we utilize the following definitions.

Definition V.1.

Minspecies: In a team described by an average species-trait matrix , a minspecies set is a subset of rows of with minimal cardinality, such that the system can still achieve the goal .

Definition V.2.

Minspecies cardinality: The cardinality of the minspecies set is defined as the Minspecies cardinality and is represented by the function that takes the average species-trait matrix as the input and returns the minimum number of rows to achieve the goal .

V-a Eigenspecies

First, we define a diversity measure related to the exact matching goal, .

Proposition V.1.

The cardinality of eigenspecies (the minspecies corresponding to goal function ) is computed as follows

(22)
(23)

where is a subset of all the species in the team, denotes the cardinality, and is the set of all non-negative integers.

Proof.

The species-trait matrix can be factorized as where and . Now, where . Thus, there exists a robot distribution that can achieve the goal with only a subset of the species, defined using the minimal species-trait matrix . ∎

Thus, contains the minimal set of species that can exactly match any desired trait distribution without recruiting robots from species not in , and denotes the number of species that form . Note that weighting factors of the sum are restricted to be natural numbers. The motivation behind this restriction is that, when aggregating traits, the weighting factor corresponds to the number of robots we are considering when aggregating traits.

[colback=blue!5!white, enhanced jigsaw, breakable] Example: For the team in our example, the average species-trait matrix is defined in (7). Note that the sum of the first row rows is equal to the last row. Further, no other rows is equal to the weighted (by natural numbers) sum of the remaining rows. Specifically, . The average species-trait matrix can thus be factorized as , where

(25)

Thus, and consequently . In other words, the traits of only one species (Species 4) can be exactly matched by aggregating the traits of other species (Species 1 and 2).

V-B Coverspecies

Next, we define a diversity measure related to the minimum matching goal, .

Proposition V.2.

The cardinality of coverspecies (the minspecies corresponding to goal function ) is computed as follows

(26)
(27)
Proof.

The species-trait matrix can be factorized into two matrices and such that , , and . Now, . Thus, there exists a robot distribution that can achieve the goal with only a subset of the species, defined using the species-trait matrix . ∎

Thus, contains the minimal set of species that can satisfy (with potential over-provision) any desired trait distribution without recruiting robots from species not in , and is the number of species that form such a minimal set.

[colback=blue!5!white, enhanced jigsaw, breakable] Example: For the team in our example, the average species-trait matrix is defined in (7). Note that each element of the last row is larger or equal to the corresponding elements in every other row. The average species-trait matrix can thus be factorized as , where

(28)

Thus, and consequently . In other words, the traits of three species (Species 1,2, and 3) can be minimum matched by the traits of one species (Species 4).

Vi Solution Approach

This section details the proposed solution to the optimization problem defined in (19)-(20). Our solution finds the optimal transition rates without assuming the knowledge of the optimal robot distribution .

Vi-a Goal Constraints

We begin by considering the time evolution of average trait distribution over the tasks. To this end, we combine (15) and (12), yielding

(29)

In order to satisfy the goal function constraint, as defined in (20), we impose constraints on two error functions [25]. The first error function computes the trait distribution error and is defined separately for each goal function as follows:

(30)
(31)

where denotes the Frobenius norm of a matrix. Note that we have omitted the dependence of on the transition rates and initial conditions for brevity. The second error function measures the deviation from the steady state robot trait distribution and is defined as follows for both goal functions:

(32)

The first error function (for both goal functions) penalizes the system when the trait distribution at time does not satisfy the appropriate goal, and the error function penalizes the system if its trait distribution does not reach steady state at time . Thus, enforcing upper bounds on these error functions guarantees a certain minimum level of performance.

Vi-B Optimization Problem

Based on the definitions in Sections VI-A, we reformulate the optimization problem in (19)-(20) for goal as follows

(33)
(34)
(35)
(36)
(37)
(38)

where , , and are user-defined positive scalars. Note that the optimization problem for goal is identical except that we replace the constraint in (34) with .

Note that the solution to the optimization problem in (33)-(38) guarantees minimum levels of performance, both in terms of achieving and maintaining the appropriate goal, as defined by the arbitrary constants and , respectively. The constraint in (36) helps ensure that the expected variance of the achieved trait distribution is below a desired threshold. Thus, for each task, the constraint in (36) encourages the system to recruit robots who possess traits (required for the task) with relatively low variance, there by minimizing the variation in the actual trait distribution.

Vi-C Analytical Gradients

To efficiently solve the the optimization problem in (33)-(38), we derive and utilize the analytical gradients of all the constraints with respect to the decision variables. In this section we define the analytical gradients of constraints defined in (34)-(38) with respect to the unknowns and . We refer the readers to [25] for closed-form expressions of the derivatives of , and with respect to both and .

Fig. 3: The four error measures used to quantify the proportion of trait mismatch.
Fig. 4: The initial (left) and desired (right) team configurations from an exemplary run with eight tasks (nodes) and four traits. The bar plots denote the trait distribution at each task and the edges represent the possibility of switching between the corresponding tasks.

We adapt the closed-form expressions for derivatives of the matrix exponential provided in [14]

, and use the chain rule to derive the derivatives of

with respect and as follows 111we drop the arguments of for brevity

(39)
(40)

where

(41)

Thus, the closed form expressions of the derivatives are given by

(42)
(43)

where

is the eigenvalue decomposition of

, is the diagonal matrix with the eigenvalues of , is a matrix defined as

(44)

and is a matrix with its th element is given by

(45)

Vii Experimental Evaluation

We evaluate STRATA using two sets of experiments. In the following experiments we compare STRATA’s performance with that of a baseline. Our baseline method is a bootstrapped version of the binary-trait-based method introduced in [25]. Since the baseline requires the desired trait distribution to be specified in the binary trait space, we make the following modifications to the baseline. We define a binary species-trait matrix to be , where is the signum operator applied to each element of its matrix argument. We also define a modified desired trait distribution for the baseline: , where is the floor function applied to each element of a matrix, refers to Hadamard (element-wise) division, , is the mean value of the th trait computed across all species, and is a -dimensional vector of ones.

The task assignment performance of each method is evaluated in terms of four measures of percentage trait mismatch, as defined in Fig. 3. In these measures, is the deterministic trait distribution achieved by the algorithm when the species-trait matrix is assumed to be (ignores variance), and is the stochastic trait distribution achieved when considering the stochastic species-trait matrix. As illustrated in Fig. 3, the metrics measure performance in two scenarios: when the actual traits of the robots are (1) assumed to be known and equal to the average of the corresponding trait, and (2) assumed to be unknown and sampled based on the trait distribution. Additionally, in each scenario, the performance is measured in terms of both exact minimum trait matching, irrespective of the optimization goal.

Fig. 5: Comparison of the performances of STRATA and the baseline [25] (binary) framework when optimizing for exact matching (). The performance of each framework is quantified in terms of four measures of percentage trait mismatch.

Vii-a Simulation

In the first set of experiments, we study the performances of STRATA and the baseline in terms of matching the desired trait requirements for a large heterogeneous team. To this end, we simulate a task assignment problem with nodes (tasks), traits (3 cumulative and 2 non cumulative traits), and species (each with 200 robots). We present the results computed from 100 independent simulation runs.

In each run, we make the following design choices. The task graph along with its connections is randomly generated. The initial and final robot distributions, and , are uniformly randomly generated. Based on the obtained , a desired trait distribution is computed for each run. The expected value of the species-trait matrix is chosen to be , where each element of

is sampled from a uniform distribution:

, and each element of -dimensional is sampled from a discrete uniform distribution: . Each element of is sampled from a uniform distribution: . An example initial and desired trait distribution is illustrated in Fig. 4

To ensure a fair comparison, we limit both STRATA and the baseline framework to a maximum of 20 meta iterations of the basin hopping algorithm during each run. In order to measure and for each run, 10 samples of the trait-species matrix are generated and used to compute .

Fig. 6: Comparison of the performances of STRATA and the baseline [25] (binary) framework when optimizing for minimum trait matching (). The performance of each framework is quantified in terms of four measures of percentage trait mismatch.

Exact Trait Matching: First, we compute the optimal transition rates according to both STRATA and the binary trait framework [25] with respect to the function Goal . STRATA is found to converge during 79 of the 100 simulation runs and the binary trait framework during 10 runs. In Fig 5, we present the performances of both frameworks by plotting the errors (defined in Fig. 3) as functions of time. Note that the error plots for each method reflect the error measures computed only across the converged runs.

As shown in Fig. 5(a) and (b), STRATA consistently performs better than the baseline in terms of deterministic performance, as measured by both and . Further, as shown in Fig. 5(c) and (d), when the robots’ traits are randomly sampled, STRATA performs better than the baseline on average. The stochastic nature of the species-trait matrix forces the errors to be larger than 0.

Fig. 7: Number of converged simulation runs.

Minimum Trait Matching: Next, we compute the optimal transition rates according to both STRATA and the binary trait model [25] with respect to the function Goal . STRATA is found to converge during 85 of the 100 simulation runs and the binary trait framework during 16 runs. In Fig 6, we present the performances of both frameworks by plotting the errors (defined in Fig. 3) as functions of time.

STRATA consistently performs better than the baseline when optimizing to satisfy minimum trait distribution, as measured by . On average, STRATA performs better than the baseline when considering stochastic species-trait matrix, as measured by . These assertions are supported by the plots in Fig. 6(b) and (d). In Fig. 6(a) and (c), the baseline exhibits high error and variance in terms of both and . This implies that when optimizes for , the binary trait model, unlike STRATA, results in a high level of over-provisioning.

Summary: As demonstrated by the results in this section, reasoning about stochastic traits is essential to consistently satisfy complex task requirements. Firstly, continuous trait distributions might not be achievable when reasoning over binary traits due to incompatibility with the continuous trait space. Further, to construct the modified trait distribution , the binary trait method is required to consider, at minimum, the average value of each trait in the team. In the process, however, the binary trait model ignores all variations both at the species and individual levels. The advantages of considering these variations are reflected in the results above. Particularly, as seen in Fig. 7, STRATA successfully converged in significantly () more runs than the binary trait framework for both exact trait matching () and minimum trait matching ().

# Species
# Agents
per Species
# Tasks # Traits Role Assignment
Team A 4 3 3 4 STRATA
Team B 4 3 3 4 Baseline [25]
Team C 4 3 3 4 Random
TABLE I: Specifications of the teams implemented in the capture the flag environment.

Vii-B Capture the Flag

In this section, we study the question: Does STRATA improve higher-level team performance? To this end, we quantify the effect of STRATA on team performance in a capture the flag (CTF) game. We built the environment using the Unity 3D game engine. We compare the performances of three teams, named A, B, and C. The details pertaining to each team are listed in Table I. The three tasks (roles) in the game are defend, attack, and heal, in that order. The four traits are speed, viewing distance, health, and ammunition, in that order. Indeed, speed and viewing distance are non-cumulative traits while the other traits are cumulative.

Fig. 8: Performances of STRATA and the baseline framework [25], both against random task assignment in the capture-the-flag game. The asterisk indicates the statistically significant () difference between the proportion of wins of STRATA and that of the baseline.

The baseline task assignment strategy, similar to the experiment in Section VII-A, is a bootstrapped version of the binary-trait-based method introduced in [25]. For the random task assignment strategy, each autonomous agent is assigned uniformly randomly to one of the three roles. Note that all the algorithms are provided with identical teams, consisting of 12 agents. Thus, any variation in performance is limited to the task assignment strategy used by each team and the inherent randomness of the game.

The traits of the agents are sampled from the following stochastic species-trait matrix

The minimum acceptable value for the non-cumulative traits are chosen to be as follows: for speed and for viewing distance. The desired trait distribution is designed to be

All games are played with two teams at a time, one versus the other. We compare the performances of both STRATA and the baseline framework against random task assignment. First, we simulate a total of 500 games between Teams A (STRATA) and C (random). Then, we simulate 500 games between Teams B (baseline) and C (random). We consider a team to have won a game it captures the opponent’s flag and brings it back to the starting position. If neither team is able to capture and bring back the opponent’s flag in 120 seconds, then the team with the highest number of active agents is considered the winner. Lastly, if both teams retain the same number of active agents after 120 seconds, the game is considered to have ended in a draw. The performances of STRATA and the baseline framework in terms of number of matches won is given in Fig. 8.

Summary: As shown in Fig. 8, the baseline framework performs slightly better than random task assignment (241/500 vs 203/500). This observation is likely due to the fact that the baseline framework reasons about the average trait values and a corresponding task requirements in the binary trait space. This type of reasoning while limited is still more beneficial than not reasoning about any of the factors that influence team performance. In contrast, STRATA is significantly more likely to win (337/500) than lose (122/500) against a team with random task assignment. Further, based on the z-test, we find that STRATA’s proportion of wins (337/500) is significantly () higher than that of the baseline (241/500). Thus, STRATA’s ability to reason about the traits and task requirements translates to high-level team performance.

Viii Conclusion

We presented STRATA, a unified framework capable of optimal task assignments in large teams of heterogeneous robots. The members of the team are modeled as belonging to different species, each defined by a set of its capabilities. STRATA models capabilities in the continuous space and explicitly takes into account both species-level and robot-level variations. Further, we quantified the diversity of a given team by introducing two separate notions of minspecies, each specifying the minimal subset of species necessary to achieve the corresponding goal. Finally, we illustrated the necessity and effectiveness of STRATA using two sets of experiments. The experimental results demonstrate that STRATA (1) successfully distributes a large heterogeneous team to meet complex task requirements, (2) consistently performs better than the baseline framework that only considers binary traits, and (3) results in improved higher-level team performance in a simulated capture the flag game.

Acknowledgement

This work was supported by the Army Research Lab under Grant W911NF-17-2-0181 (DCIST CRA).

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