3.5 Organisation State
3.5.2 Organisation Properties
The active set of agents, i.e., the coalition structure of the organisation, has to be analysed dynamically in order to evaluate the selected properties efficacy, efficiency and safety. Chap-ter2.1has introduced these properties, which serve as optimisation objectives for the planning approach outlined in Chapter4.
Efficacy Efficacy in the context of MoreOrgdescribes the ability of a reconfigurable multi-robot system to provide a particular functionality. To measure an organisation’s efficacy an objective has to been given as a set of required functionalities. The identification of efficacy leads only to a binary result: either the organisation supports the given functionality or not.
The mapping between structure and functionality as outlined in Section 3.3.2is the basis to quantify the efficacy of an organisation, since it allows to verify whether an active set of agents supports a set of functionality. The efficacy of an agent with respect to a required set of functionalities is defined based upon a general agent’ssupportfor a set of functionalitiesF:
efficacy(GA,F) =
⎧
⎪⎪
⎨
⎪⎪
⎩
1 support(GA,ˆ F)≥1
0 otherwise (3.29)
Likewise, the efficacy of a coalition structure can be accordingly defined as:
efficacy(CS,F) = min
GA∈CSsupport(GA,ˆ F). (3.30)
Efficiency Efficiency describes the cost of performing a task, here measured through the re-sources ’energy’. To analyse efficiency MoreOrg estimates the operation cost for all agents, which is accounted for as consumed energy. The consumed energy depends upon the time of
operation and the power consumption of each agent. The time of operation depends upon esti-mated travel cost, time for the requested operation and time for reconfiguration. This approach extends an approach used by Wurm et al. (2013) to estimate the cost based on the travel time between two locations.MoreOrgexpects the nominal speedvnomas default property for atomic agents, so that based on this information the duration estimate can be computed. As long as no better estimation is available, the line-of-sight distance between two locations is the basis for the cost computation. Typically, robotic systems consume electrical energy andMoreOrg additionally expects a definition of all agents nominal power consumption. Robotic agents can have a different power consumption. Operation time is therefore not an accurate cost measure.
Hence,MoreOrguses the total energy consumption of a reconfigurable multi-robot system as cost measure. Power consumption can still vary over time with the type of activity. MoreOrg assumes a constant power consumption of all operative atomic agents and leaves a more so-phisticated estimation as future enhancement. The total required energy of an organisation represented by the atomic agent setAto perform a missionM(see Chapter4for the complete definition) represents the efficiency of an organisation. It is defined as:
E(A,M) =∑
a∈A
op(a,M)·pw( ˆa) , (3.31)
whereop(a,M) defines the operation time of an agenta∈Ain the missionM.
Reconfiguration of an organisation comes at a cost, and the operation time is influenced by transitions between coalition structures. The time to transition from one coalition structure CSiA to another CSjA is therefore estimated with a heuristic function. The heuristic firstly assumes basic cost for the number of atomic agents which are involved in the reconfiguration.
Secondly, additional and significantly higher cost arise from the need to coordinate multiple agents to exchange atomic agents or to merge. The reconfiguration cost function to form a single agent from an existing coalition structure is:
ρ(GA, CS) =ta· |GA|+ ∑
GA′∈CS
tb· |GA′∩GA| , (3.32) where ta and tb are heuristic time constants. Robotic experiments are required to establish a realistic estimate for the magnitude of these parameters. The evaluation in Chapter6gives an indication for these parameters for small teams. To model the cost of reconfiguration for the reference system in this thesis, the default setting ofta= 100sandtb= 600sapplies. The values are estimates which consider time for additional error handling. The overall reconfiguration cost to transition from a coalition structureCSiAto anotherCSjAis defined as:
ρ(CSiA, CSjA) = ∑
GA∈CSjA
ρ(GA, CSiA) (3.33)
The reconfiguration cost heuristic does not account for relocation cost. Instead, the following assumption holds.
Assumption 3.2 (Precondition for Reconfiguration). All agents which take part in a reconfiguration process to form a single agent operate in direct proximity.
Furthermore, real reconfigurable multi-robot systems come with limitations regarding their reconfigurability. For instance, payload items of the reference system cannot self-reconfigure.
They can neither relocate nor have a degree of freedom. Thus, they require external support to attach to other agents.
Assumption 3.3 (Agent self-reconfigurability). Mobile agents can self-reconfigure, i.e. attach and detach other atomic agents, and permit the reconfiguration of other agents.
In effect, a transition fromCSiAtoCSjAis considered feasible, when all agents operate in direct proximity and at least one mobile agent is present.
Safety InMoreOrg the computation of safety of an agent is based on resource redundancy.
Fostering highly redundant agents in an organisation ultimately leads to a single monolithic agent (if that agent is feasible). For the search for an optimal organisation it has to be con-sidered, that due to a high degree of redundancy a safer organisation might be less efficient.
Therefore, any optimisation has to trade safety and efficiency against each other. A measure for redundancy is the central part of the safety heuristic and it is based on the standard mod-elling approach for parallel and serial component-based systems (Rausand and Høyland2009, pp. 118-125). Each resource can be associated with a probability of survival, so that an over-all probability of survival can be computed using a function decomposition tree approach.
Information about the probability of survival of components has to be part of an initial sys-tem identification and has to be augmented with performance information from real syssys-tems.
Using redundancy as safety measure follows an assumption regarding failing components.
Assumption 3.4 (Component substitution). To maintain the functionality of an agent, one component can replace another if it is an instance of the other’s class, which also includes instances of subclasses.
This seems like a strong assumption, since even if components are instances of the same con-cept (e.g., a camera) it might not be possible to substitute one with the other without loos-ing functionality. However, this is a matter of modellloos-ing equivalence classes in the ontology.
Hence, according to Assumption2.5, MoreOrgconsiders a shared use of resources in a com-posite agent.
The reliabilityRf (also referred to as probability of survival) of a single functionalityf can be computed by accounting for parallel components, i.e., resources that are not strictly required but which can serve as replacement:
Rf(t) =
⎧
⎪⎪
⎨
⎪⎪
⎩ 1−∏n
i=1(1−pi(t)) parallel system
∏n
i=1pi(t) serial system , (3.34)
where pi(t) is the time-dependant probability of survival with 0≤pi(t)≤1. Component de-grading can be one reason for a change of the probability of survival.MoreOrgleaves the use of time-dependence as future improvement and instead uses a static probability of survival with t= 0.
Definition 3.9 (Functional reliability). R(F,GA) denotes theˆ reliability of a set of required functionalitiesF which is provided by an agentGA.ˆ
Figure 3.15:Schematic of a system composition consisting of three resource types: a,b,c, where the ratio from required to available is for a 1:3, for b 1:1, and for c 2:8.
The computation ofR(F,GA) is based on the functional decomposition of the agent typeˆ GAˆ into atomic resources. For each resource a redundancy at component level (cf. (Rausand and Høyland 2009, p. 129)) is assumed. As a heuristic the redundancy is computed based on a type partitioning considering all resources which have no further dependencies. All resources of the same type are modelled as subsystems, which again form a serial system. Figure3.15 illustrates this modelling approach.
For each subsystem which is composed of a single resource type the redundancy is computed forrrequired instances,navailable instances and the probability of survivalpfor the resource type:
rsub(r, n, p) =
⎧
⎪⎪
⎨
⎪⎪
⎩
1−[(1−p)r]nr , wheren≥r
0 otherwise (3.35)
The functionREQmaps a set of functionalitiesF to the required number of instances for each resource type:
REQ(F) ={req1, . . . , req|REQ(F)|} , (3.36) wherereqi represents the minimum cardinality of a resource typei to fulfil all functionalities f ∈ F.
The function AV L maps an agent type to the number of maximum available resources with respect to a functionality setF. Only resources that can contribute to the provision ofF need to be considered:
AV L(F,GA) =ˆ {avl1, . . . , avl|REQ(F)|} , (3.37) where avli represents the maximum cardinality of a resource type i available in the general agent typeGA.ˆ
Resources lead to a heuristic system structure as shown in Figure3.15using serial and parallel systems. Based on this structure, an agent’s reliability is defined as:
R(F,GA) =ˆ
|REQ(F)|
∏
i=1
rsub(reqi, avli, pi) , (3.38) wherepi represents the probability of survival for a resource typei.
Table 3.8:Resources of the agent typeSherpaT T which are relevant to provision the function-alityLocationImageProvider.
Resource reqi avli pi Localization 1 1 0.95
Locomotion 1 1 0.95
Mapping 1 1 0.95
PowerSource 1 1 0.95
Camera 1 2 0.95
Example A functionalityLocationImageProviderdepends on a functionalityImageProvider and a functionalityMoveTo. The requirements for theImageProviderare one of each resource:
CameraandPowerSource. MoveTorequires one of each resource: Localization,Locomotion, Mapping,PowerSource.
Table3.8lists the cardinalities and probabilities of survival for an atomic agent typeSherpaT T with relevant resources. This agent type provides the functionalityLocationImageProvider with a redundant camera system, and otherwise a series system. According to Equation3.38 the probability of survival for the functionalityLocationImageProviderisP = (1−(1−0.95)2)· 0.954 ≈ 0.81. A composite agent which has an additional atomic agent P ayloadBattery can increase the redundancy of the resource PowerSource by 1. This leads to an increase of the probability of survival for the functionality LocationImageProvider, since now two redun-dant subsystems exist:P = (1−(1−0.95)2)2·0.953≈0.85.
Goal Dependant Reliability An agent’s reliability is based on information about required functionalities and relevant subsystems, i.e., reliability can be computed with respect to a functionality requirement. The overall goal and objective of an organisation lies in support-ing activities of multiple agents. These agents can operate in parallel at different physical locations. The objective for an active coalition structureCS ={GA1, . . . , GA|CS|} of an organi-sation is described by a corresponding set of functionality sets denotedFS={F1, . . .F|CS|}, and a2f :CS→FSallows to map each operative agentGAito the required functionality setFi. The current redundancy of the organisation is then the minimum achievable level of redundancy:
R(FS, CS) = min
GA∈CSR(a2f(GA),GA)ˆ (3.39)
This redundancy computation is used as safety metric for the planning approach in Chapter4.