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Planning with Resources at Multiple Levels of Abstraction. Ed Durfee Artificial Intelligence Lab University of Michigan durfee@umich.edu. Brad Clement, Tony Barrett, Gregg Rabideau Artificial Intelligence Group Jet Propulsion Laboratory California Institute of Technology

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planning with resources at multiple levels of abstraction

Planning with Resources at Multiple Levels of Abstraction

Ed Durfee

Artificial Intelligence Lab

University of Michigan

durfee@umich.edu

Brad Clement, Tony Barrett, Gregg RabideauArtificial Intelligence GroupJet Propulsion Laboratory

California Institute of Technology

{bclement, barrett}@aig.jpl.nasa.gov

Paper to appear in proceedings of ECP-01

http://ai.eecs.umich.edu/people/bradc

motivation
Motivation
  • Hierarchical refinement search can reduce the search space exponentially when there is no need to backtrack up the hierarchy. (Korf ’87, Knoblock ’91)
  • Often planning problems have interacting goals such that backtracking is necessary.
  • How can this performance advantage be exploited for such problems?
  • By summarizing the constraints of an abstract task’s refinement, the planner can reduce the complexity exponentially. (Clement & Durfee 2000)
  • How can this work be applied to the use of metric resources?
contributions
Contributions
  • Algorithm summarizing metric resource usage for abstract activities based on their decompositions
  • Complexity analyses showing that scheduling is exponentially cheaper at higher levels of abstraction using summary information
  • Experiments in a multi-rover domain that support the analyses
  • Search techniques for directing the refinement of activities in an iterative repair planner that further improve performance
resource usage
Resource Usage

interval of task

  • Depletable resource
    • usage carries over after end of task
    • gas = gas - 5
  • Non-depletable
    • usage is only local
    • zero after end of task
    • machines = machines - 2
  • Replenishing a resource
    • negative usage
    • gas = gas + 10
    • can be depletable or non-depletable
mars rover
Mars Rover

3

D

A

B

C

1

2

F

E

morning activities

sunbathe

move(A, B)

soak raysuse –4W

20 min

soak raysuse –6W

20 min

low path

high path

soak raysuse –5W

20 min

middle path

go(2,B)

use 12W

20 min

go(A,1)

use 6W

10 min

go(A,B)

use 8W

50 min

go(A,3)

use 8W

15 min

go(3,B)

use 12W

25 min

go(1,2)

use 6W

10 min

summarizing resource usage

morning activities

sunbathe

move(A, B)

soak raysuse –4W

20 min

soak raysuse –6W

20 min

low path

high path

soak raysuse –5W

20 min

middle path

go(2,B)

use 12W

20 min

go(A,1)

use 6W

10 min

go(A,B)

use 8W

50 min

go(A,3)

use 8W

15 min

go(3,B)

use 12W

25 min

go(1,2)

use 6W

10 min

Summarizing Resource Usage

soak rays

soak rays

soak rays

soak rays

soak rays

soak rays

soak rays

soak rays

soak rays

-4

-5

-6

-4

-5

-6

-4

-5

-6

low

medium

high

6

6

12

8

8

12

-4

2

2

1

7

6

-6

0

-4

4

3

2

-6

0

-4

4

3

7

6

-6

0

Battery power usage for three possible decompositions

summarizing resource usage7
Summarizing Resource Usage

summarized resource usage 

< local_min_range, local_max_range, persist_range >

Captures uncertainty of decomposition choices and

temporal uncertainty of partially ordered actions

40

30

20

10

0

-7

-20

< [-7, -20],[30, 40],[10, 20] >

resource summarization algorithm
Resource Summarization Algorithm
  • Can be run offline for a domain model
  • Run separately for each resource
  • Recursive from leaves up hierarchy
  • Summarizes parent from summarizations of immediate children
  • Considers all legal orderings of children
  • Considers all subintervals where upper and lower bounds of children’s resource usage may be reached
  • Exponential with number of immediate children, so summarization is really constant for one resource and O(r) for r resources
resource summarization algorithm9
Resource Summarization Algorithm

<[0,5][3,6][0,6]>

OR

<[2,3][3,4][3,4]>

<[0,5][3,6][0,6]>

slide10

Resource Summarization Algorithm

Serial AND

<[0,5][3,6][0,6]>

<[2,3][3,4][3,4]>

<[0,5][3,10][3,10]>

slide11

Resource Summarization Algorithm

Parallel AND

<[0,5][3,6][0,6]>

<[2,3][3,4][3,4]>

<[2,9][5,6][3,10]>

resource summarization algorithm12
Resource Summarization Algorithm
  • for each consistent ordering of endpoints
    • for each subtask/subinterval summary usage assignment
      • use Parallel-AND to combine subtask/subinterval usages by subinterval
      • use Serial-AND over the chain of subintervals
  • use OR computation to combine profiles

Each iterationgenerates a profile

complexity analysis
Complexity Analysis

level

branchingfactor b

0

  • Iterative repair planners (such as ASPEN) heuristically pick conflicts and resolve them by moving activities and choosing alternative decompositions of abstract activities.
  • Moving an activity hierarchy to resolve a conflict is O(vnc2) for v state or resource variables, n hierarchies in the schedule, and c constraints in hierarchy per variable.
  • Summarization can collapse the constraints per variable making c smaller.
  • In the worst case, where no constraints are collapsed because they are over different variables, the complexity of moving and activity hierarchies at different levels of expansion is the same.

1

. . .

d

1

2

n

c constraintsper hierarchy

vvariables

complexity analysis14
Complexity Analysis

level

branchingfactor b

0

1

. . .

d

1

2

n

  • In the other extreme, where constraints are always collapsed when made for the same variable, the number of constraints c is the same as the number of activities and grows bi for b children per activity and depth level i. Thus, the complexity of scheduling operations grows O(vnb2i).
  • Along another dimension, the number of temporal constraints that can cause conflicts during scheduling grows exponentially (O(bi)) with the number of activities as hierarchies are expanded.
  • In addition, by using summary information to prune decomposition choices with greater numbers of conflicts, exponential computation is avoided.
  • Thus, reasoning at abstract levels can resolve conflicts exponentially faster.

c constraintsper hierarchy

vvariables

decomposition strategies
Decomposition Strategies
  • Expand most threats first (EMTF)
    • instead of moving activity to resolve conflict, decompose with some probability (decomposition rate)
    • expands activities involved in greater numbers of conflicts (threats)
  • Level expansion
    • repair conflicts at current level of abstraction until conflicts cannot be further resolved
    • then decompose all activities to next level and begin repairing again
  • Relative performance of two techniques depends decomposition rate selected for EMTF
decomposition strategies16
Decomposition Strategies
  • FTF (fewest-threats-first) heuristic tests each decomposition choice and picks those with fewer conflicts with greater probability.

rover_move

path1

path2

path3

10 conflicts

20 conflicts

15 conflicts

multi rover domain
Multi-Rover Domain
  • 2 to 5 rovers
  • Triangulated field of 9 to 105 waypoints
  • 6 to 30 science locations assigned according to a multiple travelling salesman algorithm
  • Rovers’ plans contain 3 shortest path choices to reach next science location
  • Paths between waypoints have capacities for a certain number of rovers
  • Rovers cannot be at same location at the same time
  • Rovers cannot cannot cross a path in opposite directions at the same time
  • Rovers communicate with the lander over a shared channel for telemetry--different paths require more bandwidth than others
experiments using aspen for a multi rover domain
Experiments using ASPEN for a Multi-Rover Domain

Performance improves greatly when activities share a common resource.

Rarely shared resources (only path variables)

Mix of rarely shared (paths) and often shared(channel) resources

Often shared (channel) resource only

experiments using aspen for a multi rover domain19
Experiments using ASPEN for a Multi-Rover Domain

CPU time required increases dramatically for solutions found at increasing depth levels.

experiments in aspen for a multi rover domain
Experiments in ASPEN for a Multi-Rover Domain
  • Picking branches that result in fewer conflicts (FTF) greatly improves performance.
  • Expanding activities involved in greater numbers of conflictsis better than level-by-level expansion when choosing a proper rate of decomposition
summary
Summary
  • Algorithm summarizing metric resource usage for abstract activities based on their decompositions
  • Complexity analyses showing that scheduling operations are exponentially cheaper at higher levels of abstraction when summarizing activities collapses state/resource constraints and temporal constraints
  • Experiments in multi-rover domain support analyses
  • Reasoning at abstract levels is wasteful when hierarchies must be fully expanded and OR branch selection is not important
  • Search techniques for directing the refinement of activities further improve performance
future work
Future Work
  • How does abstract reasoning affect plan quality using iterative repair?
  • How should complex resources be abstracted?
    • resource usage functions
    • window variables
    • geometrical constraints
    • volatile objects (file system)
  • What is the effect of abstracting over resource classes?
  • What are efficient protocols for coordinating a group of agents’ plans continually?