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Modelling Temporal Relations with Coexistence Goal

In mission modeling and planning for space systems, having the capability to establish temporal dependencies between events or activities is crucial. Aerie includes the capability to model flexible temporal relations through the use of Coexistence Goals.

A Coexistence Goal can be used to define a causality relation between two activities: the first activity is a directive that should be already included in the plan, while the second is an activity type to be added to the plan under certain conditions. Both activities can be instantaneous or have a duration, and can be related temporally in several ways. The Allen’s Temporal Relations describe 14 different ways in which two time intervals relate to each other. Many of these relations can be easily generalized to relations between timepoints for instantaneous activities.

The following sections introduce the syntax to represent temporal relations in Aerie and provide a number of examples.

Flexible Time Intervals in eDSL​

Consider a time interval as a time range defined by two timepoints that specify the lower and upper bounds of the interval. For example, [lb, ub] represents a closed time interval bounded by the timepoints lb and ub. We define a flexible interval as a time range where its timepoints are specified relative to another timepoint. For example, considering ae as the timepoint at which an activity a ends, we can define a time interval relative to it as follows: [ae + x, ae + y] where x and y represents the delta lower and upper bounds with respect toa's end.

Flexible Time Intervals are represented in Aerie by means of TimingConstraint.bounds, which are composed of two TimingConstraint.singletons to model the lower and upper bound. For example,

TimingConstraint.bounds(TimingConstraint.singleton(WindowProperty.START).plus(Temporal.Duration.from({ minutes : 5})), TimingConstraint.singleton(WindowProperty.START).plus(Temporal.Duration.from({ minutes : 10})))

represents a time interval that ranges between 5 and 10 units of time after the start timepoint of a Window (WindowProperty.START).

bounds represent the core expression to model Allen relations.

Allen's Temporal Relations​

The following paragraphs provide examples for 7 Allen relations, while the other 7 are the opposite relation (e.g. Starts vs Started_by). The examples are based on the Banananation scenario. GrowBanana and DurationParameterActivity are used as examples of durative activities while peelBanana is used whenever instantaneous activities are applicable.

Before​

The relation x BEFORE[lb, ub] y indicates that y must start between [lb, ub] units of time after x completes. Both activities can be instantaneous or durative. In the example below, peelBanana starts between [5, 10] units of time after GrowBanana finishes.

Allen Relation BEFORE
Figure 1: Relation x BEFORE [lb, ub] y
  export default () => Goal.CoexistenceGoal({
    forEach: ActivityExpression.ofType(ActivityTypes.GrowBanana),
    activityTemplate: (span) => ActivityTemplates.PeelBanana({peelDirection: "fromStem"}),
    startsWithin: TimingConstraint.bounds(TimingConstraint.singleton(WindowProperty.END).plus(Temporal.Duration.from({ minutes : 5})), TimingConstraint.singleton(WindowProperty.END).plus(Temporal.Duration.from({ minutes : 10})))
  })

Equals​

The relation x EQUALS y indicates that y starts and finishes at the same time as x. Both activities must be durative. The timepont variant would need to consider the start value. In the example below, DurationParameterActivity starts and finishes at the same time as GrowBanana.

Allen Relation EQUALS
Figure 2: Relation x EQUALS y
  export default () => Goal.CoexistenceGoal({
    forEach: ActivityExpression.ofType(ActivityTypes.GrowBanana),
    activityTemplate: (span) => ActivityTemplates.DurationParameterActivity({duration: Temporal.Duration.from({ hours : 1})}),
    startsAt: TimingConstraint.singleton(WindowProperty.START),
    endsAt: TimingConstraint.singleton(WindowProperty.END)
  })

Meets​

The relation x MEETS y indicates that y starts right after the time x finishes. x must be a durative activity while y could be durative or instantaneous. In the example below, peelBanana takes place at the time when GrowBanana finishes.

Allen Relation MEETS
Figure 3: Relation x MEETS y
  export default () => Goal.CoexistenceGoal({
    forEach: ActivityExpression.ofType(ActivityTypes.GrowBanana),
    activityTemplate: (span) => ActivityTemplates.PeelBanana({peelDirection: "fromStem"}),
    startsAt: TimingConstraint.singleton(WindowProperty.END)
  })

Overlaps​

The relation x OVERLAPS[lb,ub] y indicates that y start time overlaps x end time by an amount of time between lb and ub. Both activities must be durative. In the example below, DurationParameterActivity starts between [5,10] units of time before GrowBanana finishes.

Allen Relation OVERLAPS
Figure 4: Relation x OVERLAPS y
  export default () => Goal.CoexistenceGoal({
    forEach: ActivityExpression.ofType(ActivityTypes.GrowBanana),
    activityTemplate: (span) => ActivityTemplates.DurationParameterActivity({duration: Temporal.Duration.from({ hours : 1})}),
    startsWithin: TimingConstraint.bounds(TimingConstraint.singleton(WindowProperty.END).minus(Temporal.Duration.from({ minutes : 10})), TimingConstraint.singleton(WindowProperty.END).minus(Temporal.Duration.from({minutes : 5})))
  })

Contains​

The relation x CONTAINS[lb1,ub1] [lb2, ub2] y indicates that y starts between [lb1,ub1] units of time after x does, and it ends between [lb2,ub2] before x does. Both activities must be durative. In the example below, DurationParameterActivity starts between [5,10] units of time after GrowBanana starts and finishes between [5,10] units of time before GrowBanana does.

Allen Relation CONTAINS
Figure 5: Relation x CONTAINS y
  export default () => Goal.CoexistenceGoal({
    forEach: ActivityExpression.ofType(ActivityTypes.GrowBanana),
    activityTemplate: (span) => ActivityTemplates.DurationParameterActivity({duration: Temporal.Duration.from({ minutes : 50})}),
    startsWithin: TimingConstraint.bounds(TimingConstraint.singleton(WindowProperty.START).plus(Temporal.Duration.from({ minutes : 5})), TimingConstraint.singleton(WindowProperty.START).plus(Temporal.Duration.from({minutes : 10}))),
    endsWithin: TimingConstraint.bounds(TimingConstraint.singleton(WindowProperty.END).minus(Temporal.Duration.from({ minutes : 10})), TimingConstraint.singleton(WindowProperty.END).minus(Temporal.Duration.from({ minutes : 5})))
  })

Starts​

The relation x STARTS[lb,ub] y indicates that y starts between [lb,ub] after x does. x must be durative. In the example below, DurationParameterActivity starts between [5,10] units of time after GrowBanana does.

Allen Relation STARTS
Figure 5: Relation x STARTS y
  export default () => Goal.CoexistenceGoal({
    forEach: ActivityExpression.ofType(ActivityTypes.GrowBanana),
    activityTemplate: (span) => ActivityTemplates.PeelBanana({peelDirection: "fromStem"}),
    startsWithin: TimingConstraint.bounds(TimingConstraint.singleton(WindowProperty.START).plus(Temporal.Duration.from({ minutes : 5})), TimingConstraint.singleton(WindowProperty.START).plus(Temporal.Duration.from({ minutes : 10})))
  })