4.2 Robotic Missions
4.2.1 Mission Specification & Representation
The overall mission planning problem which arises from the use of reconfigurable multi-robot systems has been cast into a constraint-based mission planning approach. This approach builds upon the definitions for reconfigurable multi-robot systems developed in Chapter2.4and the organisation model as outlined in Chapter3. The definition of a mission combines the domain and problem representation in a temporal database which allows to define the evolution of a mission using a set of constraints. This means, that the initial, intermediate and final state of a mission can be encoded to represent the planning problem. The same representation can be used to describe partial and full solutions.
A human operator has the opportunity to detail a mission for individual atomic agents by defining constraints, e.g., such as on partial paths. Alternatively, a mission can be defined only through functionality requirements; requirements are defined as a set of functionalities which are expanded to a set of suitable agents. The mission specification only accounts for agent roles, i.e., anonymous instances of agent types. Therefore, after a solution to a mission planning problem has been computed, all involved agent roles have to be mapped to the set of real atomic agents. A mission specification is based on so-called spatio-temporal requirements which reflect the use of embodied agents in combination with time:
Definition 4.1 (Spatio-temporal requirement). A spatio-temporal requirement is represented as a spatio-temporally qualified expression (stqe) s, which describes the functional requirements and agent instance requirements for a time-interval and a location:
s= (F,GAˆr)@(l,[ts, te]),
where F is a set of functionality constants, GAˆr is the general agent type representing the required agent type cardinalities,l∈Lis a location variable, and ts, te∈T are temporal variables describing a temporal interval with the implicit constraintts< te.
Each spatio-temporal requirement represents a persistence constraint, i.e., the requirements have to hold throughout the time interval. The mission specification allows to relate spatio-temporal requirements to the organisation model which defines agent types and functionalities along with the associated properties (cf. Chapter3.5).
t0 t1 t2 t3 . . . l1
l0 [ (∅,ˆC0)]
[ (F0,{( ˆa0,3)})] [ (∅,Cˆ1)]
Figure 4.5: A mission specification example consisting of three spatio-temporal re-quirements: (∅,ˆC0)@(l0,[t0, t1]) and (∅,Cˆ1)@(l1,[t0, t1]) representing the initial state and (F0,{( ˆa0,3)})@(l1,[t2, t3]), wherel0, l1are location variables andt0, . . . , t3are timepoint variables
.
Definition 4.2 (Mission). A robotic mission is a tupleM=⟨GA, ST R,ˆ X,OM, T , L⟩, where the general agent typeGAˆ describes the available agent types,ST Ris a set of spatio-temporally qualified expressions,X is a set of constraints,OM represents the organisation model, T is the set of timepoints and L the set of locations.
Constraints inX can refer tostqesas described in Section4.2.1. The initial state of a mission is defined by the earliest timepoint and binds available agents to their starting depot. The earliest timepoint ist0 ∈T and ∀t∈T , t ,t0 :t > t0. Note that this neither requires a single starting location for all agents nor a single final destination, e.g., in contrast to VRP1-M-1 variants using a single depot.
Graph-based Representation Graphs serve as an intuitive representation for manyVRP re-lated problems (cf. (Drexl2013) and (Ahuja, Magnanti, and Orlin1993)). A graph representa-tion of the mission specificarepresenta-tion is also basis for the planning approach. Each vertex represents a space-time requirement, and each edge a set of agent roles. Figure4.5illustrates a mission specification, where the starting state is defined by two composite agentsCˆ0andCˆ1which are assigned to the start depots: l0 andl1. A single goal requirement demands a functionality set F′ and at least three instances of an agent type ˆa0at locationl1andt0< t1< t2< t3. Figure4.6 shows an exemplary solution for this mission, whereF0defines a transport provider function-ality, i.e., a mobile agent type with transport capacity, and three instances of an agent type
’payload’ are required at locationl1 within the timeframe oft2 tot3.
Mission Constraints
A mission can be detailed by providing constraints in the constraint setX. The only initially required constraints are temporal ones to describe the starting timepoint and state. Other constraints are optional, but can be added to the mission specification as part of the planning process to detail the mission and resolve flaws in infeasible solutions. Human operators can use the additional constraints to further restrict solutions characteristics and manually explore new solutions.
Reasoning with agent types instead of agent instances is a first means to reduce the combina-torial explosion,O(|A|)≤O(|θ(A)|). Still, to fully control a mission, agent roles, i.e., instances,
Figure 4.6: A mission solution representation: each vertex in the network represents a space-time tuple describing the agent type assignments. Fillbars on the capacitated edges indicate the total consumed capacity of the transitioning agent. Colour-coded boxes represent unique agent roles: green for fulfilled requirements, grey for an agent role presence without requirement.
have to be considered for timeline construction. Only this timeline construction allows to con-trol the path of individual agents, e.g., such that the same agent performs a given set of tasks.
This requirement exists in VRPs as a client delivery preference, for instance when a driver should (re)visit a regular set of clients. For robotic space exploration a sample return mission serves as application example. After sampling a payload cannot be substituted by any other (unused or used) sampling module. Hence, the action execution effectively changes the type of the used sample payload.
Mission constraints augment the set of spatio-temporal requirements and they can be divided into four different categories: (i) temporal, (ii) model, (iii) functionality, and (iv) property. The following paragraphs describe and motivate these constraints.
Temporal Constraints Temporal planning allows to synchronise activities and also allows to perform activities in parallel. While time can be represented qualitatively and quantitatively, the presented mission planning approach uses a combination of a qualitative and quantitative representation of time. Spatio-temporal requirements only require qualitative timepoints to define synchronisation constraints. However, a quantification of time is basis for computing the cost of a mission; the transition of agents between two locations is lower bound by the minimum travel time. A quantification of the temporal network is part of the solution pro-cess, where qualitative as well and quantitative constraints are considered. Table4.4lists the point-algebra and duration constraints which apply to the qualitative and quantitative time representations respectively.
Model Constraints Model constraints set requirements for agent types and agent roles. They allow bounding the cardinality of agent types so that the combinatorial search problem can be
Table 4.4:Temporal constraints for a missionM=⟨A, ST R,ˆ X,OM, T , L⟩.
Name Syntax Description
temporal relation ⟨tn, REL, tm⟩ tnandtmare qualitative timepoints andRELis the set of permitted relations, so thatREL⊆ {<, >,=}(Dechter2003) min duration minDuration(tn, tm, d) sets a lower bound for the duration of a time interval:tn− tm ≥d, wheretn andtm are two qualitative timepoints d∈R+; implies the qualitative relationshiptn> tm max duration maxDuration(tn, tm, d) sets an upper bound for the duration of a time interval:
tn−tm≤d, wheretnandtmare two qualitative timepoints d∈R+; implies the qualitative relationshiptn> tm
limited according to a least-commitment principle. Equality constraints allow to restrict agent routes partially or even completely. Requiring the minimum equality of a single agent type over the full mission defines the full route for a single atomic agent of this type. Modelling constraints allow to detail a mission to a high degree. In general, constraints have to be con-sidered which apply to the dimensions space, time, agent types, and roles. TemPlimplements only a subset of feasible (meta-)constraints and Table4.5lists the available model constraints.
Table 4.5:Model constraints, whereS⊆ST RandAasˆrepresents the general agent type require-ment ofs∈S.
Name Syntax Description
min cardinality minCard(S,a, c)ˆ Minimum cardinality constraint ∀s ∈ S : γ
GAˆs( ˆa) ≥ c, wherec≥0
max cardinality maxCard(S,a, c)ˆ maximum cardinality constraint corresponding to minCardso that∀s∈S:γ
Aˆs≤c, wherec≥0 all distinct allDistinct(S,a)ˆ ∀s∈S:⋂
Aasˆ=∅ min distinct minDistinct(S,a, n)ˆ ∀si, sj∈S, i,j:
⏐
⏐
⏐
⏐
|Aasˆi| − |Aasˆj|
⏐
⏐
⏐
⏐
≥n, wheren >0
max distinct maxDistinct(S,a, n)ˆ the equivalent maximum constraint to minDistinct, so that∀si, sj∈S, i,j:
⏐
⏐
⏐
⏐
|Aasˆi| − |Aasˆj|
⏐
⏐
⏐
⏐
≤n, wheren≥0 min equal minEqual(S, Ar) minimum existence of the same agent roles so thatAeq=
⋂
s∈Sr(As) andAr⊂Aeq, whereAr ⊂r(A),Ais the avail-able agent pool for a mission, andAsis the agent pool that fulfilss∈S
max equal maxEqual(S, Ar) maximum existence of the same agent roles so thatAeq=
⋂
s∈Sr(As) andAeq⊂Ar, whereAr ⊂r(A),Ais the avail-able agent pool for a mission, andAsis the agent pool that fulfilss∈S
all equal allEqual(S, Ar) the constraint conjunction: minEqual(S, Ar) ∧ maxEqual(S, Ar)
Functionality & Property Constraints Agents either comprise a functionality or they do not.
In effect, functionality is requested with a maximum cardinality of one, which makes the in-troduction of a maximum function constraint unnecessary. Note that this is a limitation of the current modelling approach, which is discussed in Chapter4.5. However, the property of a agent providing a particular functionality can be of importance, and the use of property con-straints allows to narrow applicable agents. The so-calledTransportProvider functionality servers as an examples. This functionality characterises all mobile agents, which offer a
trans-Table 4.6:Functionality and property constraints.
Name Syntax Description
min function minFunc(s, f) functionalityf to be available atspatio-temporal require-ment (str)s∈ST R, so thatf ∈ Fs, whereF represents the functionality requirements associated withs
min property minP rop(s, f , p, n) constrain the numeric propertypf of a functionalityf to bepf ≥n, wheren∈RandminP rop(s, f , p, n) implies that minFunc(s, f) holds
max property maxP rop(s, f , p, n) equivalent maximum property value constraint to minP rop(s, f , p, n), so that propertypf ≤n,n∈mathbbR
port capacity. A particular demand can only be handled by systems with enough transport capacity, and a property constraint allows to filter these out, here by requiring a minimum transport capacity. Table4.6lists the constraints for functionalities and properties.