Probabilistic Reasoningfor Robust Plan Execution

Steve Schaffer, Brad Clement, Steve Chien{first_name.last_name}@jpl.nasa.gov

Artificial Intelligence GroupJet Propulsion Laboratory

California Institute of Technology

Main Idea

• Represent uncertainty of action effects and durations as parametric, continuous probability distribution functions

• Propagate distributions through plan to project states/resources

• Score plan based on risk– risk = probability outside limits

• Plan to reduce risk

“Full Probabilistic” Modeling

Durations and resource usage normally distributed

Modeling Approximations

• Full probabilistic

• Means only– Track only the expected value (mean)– Same as non-probabilistic risk-ignorant

• Pessimistic– Track only “worst” possible value– “Worst” depends on domain / resource

• Single peak– For time-dependent multimodals– Track only one “average” Gaussian

• Chebyshev bound– Distribution-free limit on probability density– Only track the mean and standard deviation

single peak

Evaluation Domains

• Abstract testbed– one resource– various consumers, replenishers– schedule within time horizon– conflicts resolvable

• Orbiter domain– image planet, process, downlink– ~10 resources, ~10 actions– schedule goals within horizon– conflicts not all resolvable – must minimize

Evaluation Methodology

• Generate plan (batch mode)– Use different approximations– Planner is not allowed to remove goals

• Run plan on stochastic simulator

• Score execution by # errors caused– error = resource oversubscribed

Planning in ASPEN

Start (if conflicts exist and user time-limit not exceeded)

...Select probable conflict

Select a repair method ...move

...

...

Select an activity

Select a start time

Results: Abstract Domain

– Full probabilistic performs best

– Single peak performs well when variance low

– Chebyshev worst

Results: Runtime

Full probabilistic

Means only

Pessimistic Single peak

Chebyshev

consumable 35s 2s 2s 30s 25s

consumable 2x std dev

35s 2s 2s 35s 25s

non-consumable

3s 3s 400s 5s 600s

Mean run times on the abstract domain problems

Runs were terminated after 2000 iterations

Results: Orbiter Domain

– Full probabilistic performs best

– Single peak performs well

– Chebyshev worst

Results: Problem Size

– Means only worse on average for larger problems

Results: Problem Difficulty

Results: User Risk Metric

– Means only worse on average for low risk tolerance

Conclusions

• Alternative methods for handling uncertain continuous variables

• Full probabilistic reasoning is most robust– superior plans

• fewer errors• tailored to user risk attitudes

– but requires modeling overhead– and computationally expensive– suited for high risk-averseness / cost of failure

• Naive approximations do almost as well

Future Directions

• Direct temporal constraints

• Domain-specific pessimistic approximation

• Need to also evaluate– bounded distributions– particle filter representation

• Integration with execution system– observations update distributions– dynamic replanning

Overcoming Normal Representation Inaccuracies

• Normals give probability to valuesfrom -∞ to ∞

• Variable domain inaccuracy– duration must be greater than zero– usage is either > 0 or < 0 (if replenishing)– more problematic with small means and high variance

• Timeline domain inaccuracy– Resources often have only one bound of conflict

(e.g. can’t have an overfull battery)– Becomes a problem for mixture of consumers and replenishers

μ0

• Solution to variable domain inaccuracy– redistribute impossible value probability into

normal

• Timeline domain inaccuracy– move impossible value probability

into a spike withsame integral

Overcoming Normal Representation Inaccuracies

μ0

μ0

μ0

μ0

Particle Filter Representation

• Commonly used for robot localization

• A Monte Carlo simulation draws sample values (particles) from source random variables toderive likelihoods of alternative states

• In a planner, the particles approximate state/resource projections; the more particles, the more precise the estimate

• Gets around exponential peak computations of normal representation by trading precision and time

Handling Temporal Constraints

• A good execution system can issue a command when a preceding activity finishes.

• When activities are giventemporal constraints (e.g. back-to-back), thereshould be no probability ofoverlap.

• To handle this, a Bayes net can be constructed based on temporal constraints to calculate the resource distribution resulting from different possible usage contribution combinations.

Prob.

Non.