Resilient Task Allocation in Heterogeneous Multi-Robot Systems
Siddharth Mayya, Diego S. D'antonio, David Saldaña, Vijay Kumar
RResilient Task Allocation in Heterogeneous Multi-Robot Systems
Siddharth Mayya , David Salda˜na , Vijay Kumar Abstract — For a multi-robot system equipped with heteroge-neous capabilities, this paper presents a mechanism to allocaterobots to tasks in a resilient manner when anomalous environ-mental conditions such as weather events or adversarial attacksaffect the performance of robots within the tasks. Our primaryobjective is to ensure that each task is assigned the requisitelevel of resources, measured as the aggregated capabilitiesof the robots allocated to the task. By keeping track oftask performance deviations under external perturbations, ourframework quantifies the extent to which robot capabilities (e.g.,visual sensing or aerial mobility) are affected by environmentalconditions. This enables an optimization-based framework toflexibly reallocate robots to tasks based on the most degradedcapabilities within each task. In the face of resource limitationsand adverse environmental conditions, our algorithm minimallyrelaxes the resource constraints corresponding to some tasks,thus exhibiting a graceful degradation of performance. Simu-lated experiments in a multi-robot coverage and target trackingscenario demonstrate the efficacy of the proposed approach.
I. INTRODUCTIONIn recent years, heterogeneous multi-robot systems havedemonstrated a potential to achieve complex real-worldobjectives due to their versatility in accomplishing special-ized tasks which might require collaboration among differ-ent types of robots, e.g. [1], [2], [3]. A crucial step to-wards achieving such behaviors is multi-robot task allocation(MRTA), which concerns itself with allocating robots totasks in such a way that the resources required to executethe tasks successfully are made available (see [4], [5], [6]for a taxonomy and survey of the topic). For instance, apossible approach is to classify the robots according to theirheterogeneous capabilities (e.g., speed, sensor range, batterylife, etc), and then assign aggregated capabilities to each task,based on given specifications [7], [8].For heterogeneous multi-robot systems operating in dy-namic and complex environments, the diversity in the capa-bilities of the robots presents another advantage— resilience :the ability to continuously operate and recover from failureswith limited resources, e.g. [9], [10]. In our context, whena multi-robot system experiences difficulties in executingtasks due to changing environmental conditions or certaintypes of adversarial attacks, reallocating robots to tasks cansignificantly improve their performance as a team, e.g., [11].Such a reallocation can take different forms, based on the
This research was sponsored by the Army Research Lab through ARLDCIST CRA W911NF-17-2-0181. S. Mayya and V. Kumar are with the GRASP Laboratory,University of Pennsylvania, Philadelphia, PA, USA { mayya,kumar } @seas.upenn.edu. D. Salda˜na is with the Autonomous and Intelligent RoboticsLab (AIRLab) at Lehigh University, Bethlehem, PA, USA [email protected]. type of failure that has occurred. For instance, if a teamof ground robots tasked with surveilling an area encountersslippery terrain, a reallocation of aerial robots to the taskmight be desirable. However, if an adversarial attack wereto reduce the effective communication range of the groundrobots, supplying additional robots of the same kind toact as intermediate communication links might be a bettersolution. Note that, in these scenarios, specific capabilitiesof the robots were affected by disturbances, i.e., groundmobility and communication range, respectively. Hence, away to facilitate effective and resilient task allocation isby: i) identifying the extent to which the robot capabilities within each task are affected; and ii) performing a suitablereallocation which ensures progress in each of the tasks.In this paper, we propose a novel heterogeneous multi-robot task allocation framework which explicitly quanti-fies the extent to which robot capabilities—pertaining torelevant aspects of the robots’ operation such as groundspeed or sensor coverage—are degraded by environmentaldisturbances. The primary objective of our optimization-based formulation is to allocate a team of robots to a setof given tasks in a deterministic manner such that con-straints on the minimum aggregate capability requirementsfor each task are satisfied. Distinct from previous worksin the literature [7], [8], [12], we impart resilience to ourframework in two ways. First, we explicitly model the factthat, a given task can be accomplished via multiple possiblecombinations of robot capabilities—one of which can beselected based on the extent to which robot capabilities havebeen degraded by environmental disturbances. This achieves resilience via reconfiguration —by allowing the algorithm tomove robots to tasks where they can contribute the most.Second, in situations where the capabilities of the robotsare too degraded to satisfy the requirements for all thetasks, we allow the algorithm to minimally relax the capa-bility requirement constraints for some tasks, to ensure thatconstraints corresponding to higher priority tasks continueto be met. Such a graceful degradation of performance ensures that infeasible task allocation specifications in theface of significant environmental disturbances are handledeffectively.Leveraging robot heterogeneity in MRTA problems hasclassically been approached by scoring the ability of eachrobot to perform different tasks [12], [13], [14], and byexplicitly enumerating the various task-related capabilitiesof the robots [7], [15]. The above discussed features of theproposed task allocation algorithm are owed to a quantifi-able understanding of how different robot capabilities are degraded due to changing environmental conditions. Towards a r X i v : . [ c s . M A ] S e p his end, we restrict our framework to tasks whose executioncan be encoded as the minimization of a non-negative scalarcost function, such as in distributed coordinated multi-robottasks [16], [17], [18]. At every point in time, we alloweach robot to measure the discrepancy between the expectedand measured progress that it makes towards minimizing itsportion of the overall cost function. A similar approach ispresented in [11], where the real-time performance of robotsat tasks is used to modify the suitability of robots towardsdifferent tasks. In this paper, we instead leverage the hetero-geneity model to identify which capabilities are primarilyresponsible for the observed performance deviations—thusallowing the algorithm to make more expressive reallocationdecisions.The capability degradation metrics are then leveragedby a centralized mixed-integer quadratic program (MIQP)which i) selects a capability configuration that is best suitedfor each task, ii) generates the robot-to-task allocationsto meet the requirements set by the chosen configuration,and iii) minimally violates the constraints corresponding tothe allocation requirements of some tasks, if required. Ourframework deploys robots in a resource-aware manner, byminimizing the team size and allowing the mission designerto specify a cost of deployment for each type of robot.Similarly, the mission designer can also specify which tasksare less critical to the mission than others (and hence shouldbe degraded in quality first). Lastly, robots experiencinghigh performance degradation—based on a user definedthreshold—are automatically excluded from the allocationprocess.To circumvent the computationally intensive nature ofsolving MIQPs frequently, we present an event-triggered exe-cution framework, where the MIQP is solved only when theestimated capability degradations change beyond a certainthreshold. Figure 1 illustrates the system architecture for theresilient task allocation paradigm presented in this paper. V i denotes the task cost associated with robot i , which isused to compute the difference between the measured andpredicted performance of the robots. This information isused to compute degradation metrics for the different robotcapabilities by the mission evaluation block which decidesif a reallocation of robots to tasks is warranted or not.II. TASK PERFORMANCE EVALUATIONIn this section, we first characterize the heterogeneitywithin the robot team in terms of the different types of robotsavailable, and the capabilities possessed by each type ofrobot. This framework is then coupled with a task executionmodel to quantify the extent to which robot capabilities areaffected by environmental disturbances within each of thetasks. A. Robot Heterogeneity
We consider a team of N heterogeneous robots, indexedby the set R = { , . . . , N } . Let U denote the total number ofunique task-related capabilities available to the robot team,e.g., perception, ground mobility, aerial mobility, object Fig. 1. Architecture diagram for the resilient task allocation frameworkpresented in this paper. The task performance discrepancies computed bythe robots (using the task costs V i ) are converted into capability degradationscores in a centralized fashion by the mission evaluation block. If asufficiently large change in performance is detected, the resilient taskallocation algorithm is invoked to redistribute robots among tasks whiletaking into account their degraded capabilities. manipulation, etc. Individual robots can exhibit differentcombinations of capabilities, depending on their size, power,and cost constraints. In the literature, robots with identicalsets of capabilities are often said to belong to the same species [19]. Let S denote the total number of species in theteam. Let Q ∈ R S × U ≥ denote the capability matrix , whichspecifies the capabilities available to each robot species: Q = (cid:2) q (1) q (2) . . . q ( S ) (cid:3) T , (1)where q ( s ) = [ q ( s )1 , . . . , q ( s ) U ] T ∈ R U ≥ is a vector describingthe capabilities available to species s . Section IV and var-ious examples throughout the paper will demonstrate howphysically meaningful values can be assigned to the robotcapabilities. Let Q ∈ { , } S × U denote the binary versionof Q , where, Q su = 1 if and only if q ( s ) u > . Similarly, let P ∈ { , } S × N denote the robot-species mapping matrix ,whose binary-valued element P si = 1 if and only if robot i belongs to species s . B. Task Execution
We now introduce a model for the execution of differenttasks by the multi-robot team. The following sections willleverage this model to allow each robot to evaluate itsperformance at a given task. We let x i ∈ R p denote the stateof robot i ∈ R , and u i ∈ R q denote the control input, whichmodifies the state according to the following control-affinedynamics: ˙ x i = f ( x i ) + g ( x i ) u i . (2)Let M denote the total number of tasks among whichthe robots must be allocated. Subsequently, let T m ⊆ R represent the index set of robots that are currently allocatedto task m ∈ { , . . . , M } := M . We assume that robots canonly contribute to one task at a time, so T m ∩ T n = ∅ , ∀ m (cid:54) = n ∈ M . Let x m ∈ R p |T m | represent the stacked ensembletate of robots allocated to task m , where | · | denotes the setcardinality operator.In this paper, we encode the execution of tasks as theminimization of a non-negative scalar cost function. A largeclass of robotic tasks can be encoded via such a formu-lation, e.g., when robots modify their states according tothe gradient flow of a cost functional [20], which mightrepresent planning, mapping, or target tracking objectives ofthe robots [21]. To this end, let V m : R p |T m | → R denote thecost function corresponding to task m ∈ M . We assume thatthis cost can be expressed as the sum of robot-wise costs, V m ( x m ) = (cid:88) i ∈T m V ( i ) m ( x m ) . (3)Note that, the individual robot cost function V ( i ) m in (3) candepend on the states of other robots in the task, as is commonin coordinated control multi-robot tasks [20]. C. Task Performance Discrepancy
As discussed in Section I, we would like to endow therobots with an ability to evaluate their performance in thetasks, with the aim of quantifying the extent of degradation ofdifferent robot capabilities within each task. Ultimately, thesemetrics will be leveraged in Section III to design a resilienttask allocation algorithm. Towards this end, we allow eachrobot in task m , i ∈ T m , to compute a predicted cost function pred V ( i ) m which represents the value of the cost function atthe next time step as predicted by robot i . More specifically,we consider discrete time intervals, indexed by t ∈ N , andevenly spaced by a small time interval ∆ t , at which thepredicted cost function is computed as, pred V ( i ) m [ t + 1] = V ( i ) m ( x m [ t ]) + ∆ t dV ( i ) m ( x m [ t ]) dt (4)where, dV ( i ) m ( x m [ t ]) dt = ∂V ( i ) m ( x m [ t ]) ∂x i ˙ x i + (cid:88) r ∈N i ∂V ( i ) m ( x m [ t ]) ∂x r ˙ x r . (5)Here, N i represents the neighborhood set of robot i , andcan be described using a graph embedding—for example,representing physical proximity among the robots [20]. Someexamples of multi-robot tasks described in this fashioninclude coverage control [16], formation control [18], ren-dezvous [22], and target tracking [17].At discrete time t , robot i can then use (4), to computethe predicted cost at time t + 1 . Comparing this against themeasured cost function at the next time step allows the robotto evaluate its task performance as discussed next. Definition 1 (Task Performance Discrepancy) . Let ∆ V ( i ) [ t +1] denote the discrepancy associated with the task perfor- mance of robot i at time t + 1 , given as, ∆ V ( i ) [ t + 1] =min (cid:32) max (cid:32) − V ( i ) m ( x m [ t + 1]) − V ( i ) m ( x m [ t ]) pred V ( i ) m [ t + 1] − V ( i ) m ( x m [ t ]) , (cid:33) , (cid:33) . (6) For a small time interval ∆ t , the task-performance discrep-ancy ∆ V ( i ) encodes the fractional deviation between howmuch progress the robot made towards modifying its costfunction (encoded in the numerator) and how much progressit expected to make in the same time interval (encoded inthe denominator). As seen in Definition 1, if the robot did not experience anydisturbance, the predicted cost pred V ( i ) m [ t + 1] and the actualmeasured cost function V ( i ) m ( x m [ t + 1]) would be equal,implying that the task performance discrepancy ∆ V ( i ) [ t + 1] would be equal to 0. Similarly, a discrepancy value of implies that the robot did not make any progress towardsthe task execution. As seen in (6), we cap values of ∆ V ( i ) which are less than or greater than , which correspondto situations where the robot did better than expected, orits actions resulted in an unexpected direction of changeof the cost function, respectively. In the following example,we demonstrate how the task performance discrepancy canquantify the real-time disturbances experienced by a multi-robot system. Example 1.
Consider a multi-robot team composed of tworobots: a ground “leader” robot r and an aerial “follower”robot r . The ground robot is tasked with tracking a movinggoal and the aerial robot is tasked with maintaining a pre-specified distance with respect to the ground robot. Therobot-wise cost functions, whose minimization encodes theseobjectives, are given as, V (1) = 0 . (cid:107) x − g (cid:107) (7) V (2) = 0 . (cid:107) x − x (cid:107) − d ) , (8) where g represents the location of the goal and d representsthe desired following distance for the aerial robot. Forsimplicity, we model the motion of both robots using singleintegrator dynamics: ˙ x = u . At time t = 0 . s , gusts ofhead wind affect the motion of the aerial robot, but notthe ground robot. We model this disturbance as a multiplierto the control input applied by the robot. More specifically,for robot r , ˙ x = (1 − w ) u where w gradually increasesfrom to . . Figure 2a shows how the task performancediscrepancy corresponding to both the robots, evolves. Thehigher values of discrepancy for robot r capture the factthat, the robot is making a smaller amount of progresstowards minimizing its cost function than it expects, ascomputed by (6) .D. Capability Degradation Metrics While (6) gives us the robot-wise task performance dis-crepancies, it does not tell us which robot capabilities are
Time . . . . . V (1) ∆ V (2) (a) Time . . . . . C a p a b ili t y D i s t u r b a n ce S c o r e perceptionground mobilityaerial mobility (b)Fig. 2. Task-performance discrepancy and capability degradation metricsfor a goal tracking task executed by a ground and aerial robot, r and r ,respectively. As described in Example 1, at time t = 0 . s , the aerialrobot experiences a simulated wind disturbance. Figure 2a illustrates acorresponding increase in the task performance discrepancy computed bythe robot, according to (6). As explained in Section II-D and in Example 2,this causes an increase in the capability degradation of aerial mobility (seeFig. 2b). affected by environmental disturbances. Towards that end, weassemble the task performance discrepancies of the robots intask m ∈ M into a vector denoted as ∆V m ∈ [0 , |T m | . Inthe following definition, we use the heterogeneous mappingsdescribed in Section II-A to compute a capability degradationmetric for the robots in each task. Definition 2.
Let d ∗ m [ t ] ∈ [0 , U denote the extent to whicheach capability is degraded within task m at time t . Thehigher the score, the more ineffectual the robots havingthis capability are, at executing task m . We compute thiscapability degradation metric based on the task performancediscrepancy values computed in (6) , d ∗ m [ t ] = Q TS m , − P S m , T m ∆V m [ t ] , (9) where P S m , T m denotes a submatrix of P which containsonly the rows and columns corresponding to the species andindices of robots currently present in task m , respectively. Q S m , − contains the rows corresponding to the species ofrobots in task m along with all columns. The rows of P S m , T m and columns of Q S m , − are normalized to preserve the value of the disturbances between 0 and 1. Note that (9) represents the instantaneous capability degra-dation at time t based on the task performance discrepancies ∆V m [ t ] . We introduce the following update law to capture atime-averaged version of the capability degradation metrics, d m [ t + 1] = d m [ t ] + ∆ t Θ m [ t ] (cid:16) d ∗ m [ t ] − d m [ t ] (cid:17) , (10)where d m [ t ] now represents the time-averaged capabilitydegradation at discrete time t . Here, Θ m is a binary diag-onal matrix, whose u th diagonal element indicates whethercapability u is currently available on any robot allocated totask m , defined as, Θ m [ t ] = diag (cid:0) T Q S m , − (cid:1) (11)where for g ∈ R n , diag( g ) = G ∈ R n × n and the columnsof Q S m , − are normalized as before. The introduction of Θ m allows us to update only the degradation values for thecapabilities which are currently deployed in task m , and keepthe other values constant. The following example continuesthe scenario presented in Example 1 to illustrate how theupdate law presented in (10) can be leveraged. Example 2.
For the heterogeneous multi-robot team pre-sented in Example 1, we first specify the capability matrix Q ,which consists of three capabilities—perception (measuredin terms of the area that the robot can sense around it),ground mobility, and aerial mobility (both measured in termsof speed): Q = (cid:20) m m/sec m/sec m m/sec m/sec (cid:21) . (12) The robot-species mapping is simply: P = I , and Θ = I since all three capabilities are present in the task. For thesame scenario presented in Example 1, Fig. 2b plots eachelement of the capability degradation metric d computed ac-cording to the update law presented in (10) (note that the taskindex is hidden). As seen, the degradation metric for aerialmobility increases as the task performance discrepancy forthe aerial robot is mapped to the capabilities it possesses us-ing (9) . The degradation metric for the perception capabilityalso increases, where the lower magnitude is explained by thefact that it represents the average degradation experiencedin this capability by the ground and the aerial robot (theformer of which is unaffected by the wind). III. RESILIENT TASK ALLOCATIONIn this section, we develop an optimization-based taskallocation framework which meets the resilience objectivesdescribed in Section I. Towards this end, we take into accountthe fact that, tasks can often be accomplished with one of multiple possible capability configurations . For example, asurveillance task over a large region could be accomplishedby slow moving ground robots with large perception ranges,or fast moving aerial robots with smaller perception ranges.This notion is formalized in the definition below. efinition 3 (Task Requirement Matrix) . Let K m denote thenumber of possible alternative configurations of capabilitieswhich can support the accomplishment of a given task m ∈ M . We denote Y ∗ m : R K m × U ≥ as the requirementmatrix for task m , which specifies the aggregated capabilitiesrequired to effectively execute the task in each of the differentconfigurations. In other words, each row of Y ∗ m specifiesa possible combination of minimum aggregated capabilitieswhich need to be assigned to task m . In this paper, we are interested in generating an allocationmatrix , A ∈ { , } M × N , whose element A ji = 1 if andonly if robot i is allocated to task j . For each task m ∈ M and candidate allocation A , let c m ∈ R U + denote the totalaggregated capabilities assigned to the task (computed in asimilar manner to [8]), given as, c m = (cid:0) A m, − P T Q (cid:1) T , (13)where A m, − denotes the m th row of A . However, asdiscussed in Section II, the performance of different robotspecies will be different in the tasks, due to environmentaldisturbances. To explicitly account for these variations in theallocation process, we introduce the effective total capabili-ties assigned to a given task, which leverages the capabilitydegradation metrics computed in (10). Thus, the effectiveaggregated capabilities in task m can be given as, ˆc m = c m − ( d m (cid:12) c m ) , (14)where (cid:12) is the Hadamard product. Using (14), the followingdefinition outlines the conditions which would ensure that asufficient amount of aggregated capabilities are assigned tothe tasks. Definition 4 (Effective Task Execution) . The capabilityrequirements for a given task m ∈ M are met, when theeffective aggregated capabilities of the robots allocated to itare greater than those specified by one or more of the con-figurations in the task requirements matrix (see Definition 3).This is encoded by the following two conditions: ˆc m − (cid:0) ι Tm Y ∗ m (cid:1) T = δ m (15) δ m ≥ (16) where ≥ is interpreted element-wise, and ι m ∈ { , } K m is an indicator matrix specifying which of the K m possibleconfigurations is selected. In particular, the condition ι Tm =1 ensures that only one configuration is selected at a givenpoint in time. δ m ∈ R U then represents the aggregatedcapability margin, which is the difference between the totalavailable capabilities assigned to the task and the require-ments of the task. However, environmental conditions might force a situationwhere it is impossible to meet the requirements for everytask, i.e. constraint (16) might not be satisfied ∀ m ∈ M .To impart a second layer of resilience to our framework,we introduce a task relaxation matrix φ ∈ { , } M which indicates whether the requirements for each task will be metor not, φ m = (cid:40) , task m requirements are being met , otherwise . (17)Using φ , we can modify the requirement in (16) as follows: δ m ≥ − φ m δ max , where δ max ∈ R represents the maxi-mum extent to which the task requirements constraints canbe violated for all capabilities.We now present the mixed-integer quadratic program(MIQP) which can be solved to generate a resilient taskallocation for the multi-robot team. Resilient and Resource-Aware Task allocation (18) minimize ι , ι ,..., ι M , A , φ T ( AP T ) W s + w Tt φ + l (cid:107) D (cid:107) + (cid:107) T T ( A − A p ) (cid:107) (18a) subject to ˆc m − (cid:0) ι Tm Y ∗ m (cid:1) T = δ m , (18b) D ≥ − φ T δ max (18c) T ( AP T ) ≤ λ T (18d) T A ≤ + (∆ V thresh − ∆V ) (18e) ι Tm = (18f) ∀ m ∈ M , where D = [ δ , δ , . . . , δ M ] T ∈ R M × U and theinequality in constraint (18c) holds elementwise. We willnow define the symbols and the roles played by various termsin the above defined optimization problem. First, W s ∈ R S × S + is a diagonal weight matrix which represents the cost-of-deployment associated with robots of different species (forinstance, a robot with an expensive LIDAR might have ahigher cost of deployment associated with it). Furthermore, w t ∈ R M + represents the relative importance among thevarious tasks, which is taken into account when consideringwhich task constraint to relax first. For example, when con-sidering the mission objective of defending a perimeter [23],it might be better to relax the constraints of the patrol task(which detects new intruders) than the defense task (whichintercepts them) if both cannot be achieved simultaneously.The third term in the cost function (18a), (cid:107) D (cid:107) , scaledby a positive constant l , serves two purposes: it penalizesexcessive allocation of capabilities to a given task andalso ensures that in case the constraints corresponding toa given task are relaxed due to significant environmentaldisturbances, they are done so to a minimal extent. Thefinal term in the cost function (18a), (cid:107) T T ( A − A p ) (cid:107) ,represents the cost of transitioning robots between the tasks.In this regard, A p simply represents the current allocationof the multi-robot team to tasks (computed as the solutionof the MIQP in the previous iteration, see Algorithm 1). Thetransition cost matrix, T ∈ R M × M + is a diagonal matrix,where | T i,i − T j,j | represents the cost incurred by eachrobot when it transitions from task i to task j , or vice versa.Similarly, T i,i simply indicates the cost associated with andle (unallocated) robot being assigned to task i ∈ M . Forinstance, these costs can be assigned by the mission designerbased on the distances that robots have to traverse whentransitioning between tasks.The vector λ ∈ N S represents the total number ofrobots of each species available for allocation and thus,constraint (18d) ensures that the resource constraints of theoverall team are accounted for by the allocation algorithm.Along a similar vein, constraint (18e) ensures that each robotis allocated to only one task at most. Here, ∆V ∈ { , } N represents the stacked task-discrepancies corresponding tothe entire team (see Definition 1), and ∆ V thresh ∈ { , } represents the maximum acceptable task performance dis-crepancy that a given robot can have, for it to be eligible forallocation by the algorithm. Thus, if the following conditionholds for robot i ∈ R , ∆ V thresh − ∆ V ( i ) < , (19)then robot i will not be allocated to any task, since it isdeemed unfit to perform any task. For instance, a groundrobot stuck in a crevice might not be able to perform anytask in the environment, and will not be considered in theallocation.As discussed earlier, the capability degradation metric d m is updated every ∆ t seconds, which is then incorporated intothe resilient task allocation optimization problem (18) viaconstraint (18b). However, if we assume that environmentaldisturbances affect the multi-robot team at time scales muchlarger than ∆ t , it is clear that the MIQP described by (18)need not be solved every ∆ t seconds. This idea is furtherreinforced by the fact that, the MIQP must be solved ina centralized manner, and is not amenable to real-timesolutions due to its NP-complete nature [24].Indeed, a reallocation of robots to tasks is warranted onlywhen there are significant changes in the capability degrada-tion metrics associated with any of the tasks. Let t l denotethe time index when the MIQP was most recently solved.We introduce a binary variable β [ t ] , which determines if theMIQP should be solved at time t , β [ t ] = (cid:40) , if ∃ m ∈ M , s.t. max ( d m [ t ] − d m [ t l ]) ≥ ∆ , , otherwise (20)where ∆ is a user defined threshold on the change inany capability degradation value. Algorithm 1 outlines theoperations of the resilient task allocation framework. Step 6computes the task performance discrepancy values ∆ V ( i ) based on the environmental disturbances experienced bythe robots. Following this, step 7 computes the capabilitydegradation metrics d m for each task m ∈ M and uses thisto compute β using (20). If β = 1 , the task allocation MIQPpresented in (18) is solved to generate the allocation matrix A which subsequently results in an rearrangement of robotsamong the tasks. The next section illustrates the salientfeatures of the proposed framework in a heterogeneous multi-robot coverage control and target tracking scenario. Algorithm 1
Resilient task allocation via online task perfor-mance evaluation
Require:
Robot team heterogeneity specifications Q , S , P , λ Task Specifications Y ∗ m , T Parameters W s , w t , ∆ V thresh , l , δ max , ∆ Initialize: t = t l = 0 , A p = M × M Compute A and transmit to robots. (cid:46) (18) Set A p = A while true do Execute tasks m ∈ M Compute ∆ V ( i ) , ∀ i ∈ R (cid:46) (6) Update capability degradation d m , ∀ m ∈ M (cid:46) (10) Compute reallocation trigger β (cid:46) (20) if β = 1 then Compute A and transmit to robots (cid:46) (18) Set A p = A and t l = t end if t = t + 1 end while IV. ENVIRONMENT COVERAGE AND TARGETTRACKING: AN APPLICATIONWe consider a team of aerial and ground robots whichneed to be allocated among three tasks: tracking target 1(task 1), tracking target 2 (task 2), and monitoring of theenvironment (task 3). In particular, we use the coveragecontrol algorithm [16] to execute the monitoring task, withthe importance density function chosen as a zero-centeredGaussian function. The robots performing tracking tasks and are also required to maintain a certain quality ofsurveillance on the target, which is modeled as a functionof both the distance to the target and a scalar state e i denoting the environmental effects on the sensing. Thesetask objectives are encoded into the following cost functionswhose minimization represents the execution of the tasks, V k = 12 (cid:88) i ∈T k ( (cid:107) x i − γ k (cid:107) − d k ) + e i (cid:107) x i − γ k (cid:107) + (21) (cid:88) j ∈T k \ i (cid:107) x i − x j (cid:107) − d , k = 1 , ,V = 12 (cid:88) i ∈T (cid:107) x i − c i (cid:107) , (22)where γ k denotes the locations of target k , d k denotes thedesired distance to be maintained between the robots andtarget k , d determines the minimum distance maintainedbetween the robots in the task, and c i denotes the centroidof the Voronoi cell corresponding to robot i [16].The simulated experiment considers two species of robots:an aerial and a ground platform. The heterogeneity amongthe robots is characterized via five capabilities: perception(m ), sensing resolution (m), air speed (m/sec), ground speed(m/sec), and communication rate (mb/sec). These specifica-ions are captured by the robot capability matrix: Q = (cid:20) m m m/sec m/sec Mb/sec m m m/sec m/sec Mb/sec (cid:21) . (23)The task requirements matrix (see Definition 3) is given as, Y ∗ = Y ∗ = (cid:20) . . . . (cid:21) (24) Y ∗ = (cid:2)
20 4 12 0 15 (cid:3) . (25)As seen, the tracking tasks can either be accomplishedusing a team of aerial and ground robots (configuration1) or only aerial robots (configuration 2). The parametersfor the optimization program are chosen as follows: w t =[100 , , , (indicating that the coverage task is less criti-cal to mission success compared to the target tracking tasks), W s = diag ([0 . , . , where diag is the diagonalizationoperator, ∆ V thresh = 0 . , l = 1 . , δ max = 1000 , ∆ = 0 . ,and T = diag ([65 , , .Figure. 3a shows the initial deployment of robots to tasksas generated by solving the task allocation MIQP (18). Inparticular, the red and green dots represent aerial and groundrobots, respectively. The robots in the middle of the domainare executing the monitoring task using coverage control.Two aerial robots remain idle at their starting locations(denoted by purple circles), as all task requirements aremet by the rest of the team. At a certain point in time,target 2 enters a region of low-friction terrain—indicatedby the blue colored area—where, as seen in Fig. 3b, themotion of the ground robot allocated to the task is impededand it cannot track the assigned target anymore. Thanks tothe capability degradation computations in Section II, thisanomaly is accounted for in constraint (18b) of the MIQP. Inparticular, Fig. 4 illustrates how the capability margin corre-sponding to ground mobility for task 2 ( D , ) decreases afterthe failure and becomes negative. Consequently, the eventtriggered MIQP switches task 2 to the second configuration(showcased by ι in the top left corner of Fig. 3b). Thisensures that only aerial robots—unaffected by the slipperyground—are deployed to track target 2. Figure 3b shows thetwo additional aerial robots joining task 2 (highlighted bythe green ellipse). Furthermore, constraint (18e) ensures thatthe stuck ground robot is not assigned to any task.At time seconds, a weather event affects the abilityof the robots in task 2 to maintain an effective trackingquality of the target—signified by an increase in the valueof e i . This is depicted in Fig. 3c as a white shaded areaaround target and a decreasing capability margin D , corresponding to the “resolution” capability in the right tileof Fig. 4. Since there are no more robots available to jointask 2, the algorithm relaxes the constraints corresponding tothe monitoring task, and reallocates one aerial robot to thetracking task ensuring that the overall capability margin staysabove zero, while that for task 3 (depicted by D , ) fallsbelow zero. This demonstrates the ability of our algorithmto gracefully degrade performance when necessary.In order to verify the ability of the proposed allocationalgorithm to deal with large robot teams and varied envi- ronmental conditions, we ran multiple randomized trails ofthe coverage and target tracking mission described abovewith a team of 32 aerial robots and 8 ground robots (witha modified (24) and (25)). The timing corresponding tothe target movement, weather events, as well as the initialpositions of the robots were randomized in each of the trials.Over 20 independent trials, Fig. 5 depicts the minimum(worst case among all runs) capability margins correspondingto two cases: with and without the event-triggered resilienttask allocation algorithm. For the second case, the allocationalgorithm was only executed once at the beginning of eachtrial. As seen, the resilient allocation algorithm (executedon average in . seconds for N = 40 ) ensures that thecapability margins remain close to or at zero, ensuring thatthe tasks can progress successfully, despite the environmentalvariations. V. CONCLUSIONSFor heterogeneous multi-robot systems operating in dy-namic conditions, we present a resilient task allocationframework which explicitly leverages information pertainingto the real-time task performance of robots when generatingrobot-task assignments. We endow the allocation algorithmwith the ability to reconfigure robots among tasks to ensurethat the detrimental effects of environmental disturbancesare mitigated, thus showcasing a degree of resilience. Theresulting framework not only degrades performance of thesystem gradually but presents a flexible mechanism for themission designer to specify important parameters like robotdeployment costs and task priorities.R EFERENCES[1] Luca Iocchi, Daniele Nardi, Maurizio Piaggio, and Antonio Sgorbissa.Distributed coordination in heterogeneous multi-robot systems.
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Time . . . . . . D , D , Disturbance beginsResilient reallcation 12 14 16 18 20
Time − . − . − . − . . . . . D , D , Disturbance beginsResilient reallcation
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