Planning with Resources at Multiple Levels of Abstraction

Presentation Transcript

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

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

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

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

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

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

-4

-5

-6

-4

-5

-6

-4

-5

-6

low

medium

high

6

12

8

12

-4

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 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

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

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

OR

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

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

Serial AND

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

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

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

Parallel AND

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

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

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

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

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 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

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 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

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

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 Domain

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

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

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