Probabilistic Reasoning for Robust Plan Execution Steve Schaffer, Brad Clement, Steve Chien Artificial Intelligence Group Jet Propulsion Laboratory California Institute of Technology

Handling Uncertainty Why? –Better account for competing requirements of safety, science, and engineering concerns –Sequence generation and validation rely on the accuracy of models –Current process limits the capabilities by improperly evaluating alternative courses of action restricting the time horizon of operations between command uplinks –Fully exploit the capabilities of a spacecraft and enable greater autonomous operation Capabilities –Avoid constraint violations –Identify sources of greatest risk –Balance risk and efficiency –Allow for varying user risk preferences –Make execution more reliable Approaches –Conservative estimates of effects –Bound resource capacities –Slack padding –Re-planning –Handle in execution system –Contingency planning

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 PessimisticSingle peak Chebyshev consumable35s2s 30s25s consumable 2x std dev 35s2s 35s25s non- consumable 3s 400s5s600s 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

Result: Time Behavior

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 values from -∞ 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 with same 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 to derive 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 given temporal constraints (e.g. back-to-back), there should be no probability of overlap. 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.