1. Ames Research Center Planning with Uncertainty in Continuous Domains Richard Dearden No fixed abode Joint work with: Zhengzhu Feng U. Mass Amherst Nicolas Meuleau, Dave Smith NASA Ames Richard Washington Google

2. Ames Research Center Motivation Panorama Image rockImage Rock Dig Trench ? Problem: Scientists are interested in many potential targets. How to decide which to pursue?

3. Ames Research Center Motivation Panorama Image rock Image Rock Dig Trench ? Time? Power? Likelihood of Success? Different value targets

4. Ames Research Center Outline  Introduction  Problem Definition  A Classical Planning Approach  The Markov Decision Problem approach  Final Comments

5. Ames Research Center Problem Definition  Aim: To select a “plan” that “maximises” long-term expected reward received given: Limited resources (time, power, memory capacity). Uncertainty about the resources required to carry out each action (“how long will it take to drive to that rock?”). Hard safety constraints over action applicability (must keep enough reserve power to maintain the rover). Uncertain action outcomes (some targets may be unreachable, instruments may be impossible to place).  Difficulties: Continuous resources. Actions have uncertain continuous outcomes. Goal selection and optimization Also possibly concurrency, …

6. Ames Research Center Possible Approaches  Contingency Planning: Generate a single plan, but with branches. Branch based on the actual outcome of the actions performed so far in the plan.  Policy-based Planning: A plan is now a policy: a mapping from states to actions. There’s something to do no matter what the outcome of the actions so far. More general, but harder to compute. Power > 5Ah Power  5 Ah

7. Ames Research Center An Example Problem Drive (-2)Dig(60)Visual servo (.2, -.15)NIR Lo resRock finderNIR E >.1 Ah  =.05 Ah  =.02 Ah E >.6 Ah  =.2 Ah  =.2 Ah  = 40s  = 20s  = 60s  = 1s E > 10 Ah  = 5 Ah  = 2.5 Ah  = 1000s  = 500s V = 100 t  [9:00, 16:00]  = 5s  = 1s E >.02 Ah  =.01 Ah  = 0 Ah  = 120s  = 20s E >.12 Ah  =.1 Ah  =.01 Ah V = 50 HiRes V = 10 E > 3 Ah  = 2 Ah  =.5 Ah t  [10:00, 13:50]  = 600s  = 60s t  [10:00, 14:00]  = 600s  = 60s E > 3 Ah  = 2 Ah  =.5 Ah t  [9:00, 14:30]  = 5s  = 1s E >.02 Ah  =.01 Ah  = 0 Ah V = 5

8. Ames Research Center Value Function Expected Value Power Start time 10 15 20 5 13:20 14:40 14:20 14:00 13:40

9. Ames Research Center Value Function Power Start time 10 15 20 5 13:20 14:40 14:20 14:00 13:40 Drive (-2)Dig(60)Visual servo (.2, -.15)NIR Lo resRock finderNIR E >.1 Ah  =.05 Ah  =.02 Ah E >.6 Ah  =.2 Ah  =.2 Ah  = 40s  = 20s  = 60s  = 1s E > 10 Ah  = 5 Ah  = 2.5 Ah  = 1000s  = 500s V = 100 t  [9:00, 16:00]  = 5s  = 1s E >.02 Ah  =.01 Ah  = 0 Ah  = 120s  = 20s E >.12 Ah  =.1 Ah  =.01 Ah V = 50 HiRes V = 10 E > 3 Ah  = 2 Ah  =.5 Ah t  [10:00, 13:50]  = 600s  = 60s t  [10:00, 14:00]  = 600s  = 60s E > 3 Ah  = 2 Ah  =.5 Ah t  [9:00, 14:30]  = 5s  = 1s E >.02 Ah  =.01 Ah  = 0 Ah V = 5

10. Ames Research Center Plans Drive (-2)Dig(60)Visual servo (.2, -.15)NIR Lo resRock finderNIR Time > 13:40 or Power < 10  Contingency Planning:  Policy-based Planning: Regions of state space have corresponding actions. VisualServo Lo-Res Hi-Res Time 10 : VisualServo Time > 14:15 and Time 10 : Hi-Res …

11. Ames Research Center Contingency Planning 1. Seed plan 2.Identify best branch point 3.Generate a contingency branch 4.Evaluate & integrate the branch ? ? ? ? r VbVb VmVm Construct plangraph Back-propagate value tables Compute gain

12. Ames Research Center Construct Plangraph g1g1 g2g2 g3g3 g4g4

13. Ames Research Center Add Resource Usages and Values g1g1 g2g2 g3g3 g4g4 V1V1 V2V2 V3V3 V4V4

14. Ames Research Center Value Graphs g1g1 g2g2 g3g3 g4g4 V1V1 V2V2 V3V3 V4V4 r r r r

15. Ames Research Center Propagate Value Graphs g1g1 g2g2 g3g3 g4g4 V1V1 V2V2 V3V3 V4V4 r r r r v r v r v r

16. Ames Research Center p r 515.1 V p r 510.2 v r v r 515 v r 10 25 Simple Back-propagation

17. Ames Research Center p r 515.1 V p r 510.2 v r v r 515 v r 10 25 r > 15 Constraints

18. Ames Research Center p r 515.1 V p r 510.2 v r 515 v r 10 25 p q t s v r 515 v r {t} p r 5 {q} v r 10 20 {q} {t} Conjunctions

19. Ames Research Center p r 515.1 V p r 510.2 v r 10 25 p q t s p r 5 v r 10 20 {q} {t} v r 10 25 v r 10 20 Back-propagating Conditions

20. Ames Research Center p r 515.1 V p r 510.2 v r 10 25 p q t s p r 5 v r 10 20 {q} {t} r 30 v 15 30 v 15 v r 10 25 v r 10 20 Back-propagating Conditions

21. Ames Research Center B D A C CDAB CABD CADB ACBD ACDB ABCD Which Orderings

22. Ames Research Center v2v2 r p r 1020.1 p r 510.2 v1v1 r 510 v2v2 r 20 r v1v1 v2v2 r 1020 v1v1 p r 5 v2v2 r 1020 v1v1 Max Combining Tables

23. Ames Research Center v2v2 r p r 1020.1 p r 510.2 v1v1 r 510 v2v2 r 20 r v1v1 p r 5 v2v2 r 1020 v1v1 v 1 + v 2 30 v2v2 r 1020 v1v1 v 1 + v 2 30 Achieving Both Goals

24. Ames Research Center V1V1 V2V2 V3V3 V4V4 V r V r V r V r Max Estimating Branch Value

25. Ames Research Center r V1V1 V2V2 V3V3 V4V4 r P r plan value function resource probability VmVm VbVb Estimating Branch Value

26. Ames Research Center r V1V1 V2V2 V3V3 V4V4 VbVb r P r Gain = ∫ P(r) max{0,V b (r) - V m (r)} dr ∞ 0 VmVm VbVb Expected Branch Gain

27. Ames Research Center Heuristic Guidance  Plangraphs generally used as heuristics – the plans they produce may not be executable: Not all orderings considered. All the usual plangraph limitations: –Delete lists generally not considered. –No mutual-exclusion representation. Discrete outcomes not (currently) handled. –Action uncertainty is only in resource usage, not resulting state.  Output used as heuristic guidance for classical planner: Start state Goal(s) to achieve  Result is an executable plan of high value! Drive (-1)Dig(5)Visual servo (.2, -.15)Hi res Lo resRock finderNIR

28. Ames Research Center Expected Value Power Start time 10 15 20 5 13:20 14:40 14:20 14:00 13:40 Evaluating the final plan  Plangraph gives a heuristic estimate of the value of the plan.  Better estimate can be computed using Monte-Carlo techniques, but these are quite slow for a multi-dimensional continuous problem.  Figure required 500 samples per point, 4000x2000 points, so simulation of every branch of the plan 4 thousand million times. Slow!

29. Ames Research Center Outline  Introduction  Problem Definition  A Classical Planning Approach  The Markov Decision Problem approach  Final Comments

30. Ames Research Center MDP Approach: Motivation Expected Value Power Start time 10 15 20 5 13:20 14:40 14:20 14:00 13:40 Constant value function throughout region. Wouldn’t it be nice to only compute the value once! Approach: Exploit the structure in the problem to find constant (or linear regions).

31. Ames Research Center Continuous MDPs  States: X = {X 1,X 2,...,X n }  Actions: A = {a 1, a 2,..., a m }  Transition: P a (X 0 |X)  Reward: R a (X)  Dynamic programming (Bellman Backup):  Can’t be computed in general without discretization

32. Ames Research Center Symbolic Dynamic Programming  Special representation of transition, reward and value using MTBDDs for discrete variables, kd-trees for continuous.  Representation makes problem structure (if any) explicit.  Dynamic programming on both the value function and the structured representation.  Idea is to do all operations of Bellman equation in MTBDD/kd-tree form.

33. Ames Research Center  Requires rectangular transition, reward functions: Continuous State Abstraction  Transition probabilities remain constant (relative to current value) over region.  Transition function is discrete: approximate continuous functions by discretizing. Required so family of value functions is closed under the Bellman Equation.

34. Ames Research Center  Requires rectangular transition, reward functions: Continuous State Abstraction  Reward function piecewise constant or linear over region.  This, along with discrete transition function, ensures all value functions computed using Bellman equation are also piecewise constant or linear.  Approach is to compute exact solution to approximate model.

35. Ames Research Center Value Iteration  Theorem: If V n is rectangular PWC (PWL), then V n+1 is rectangular PWC (PWL). P a V n V n+1  Represent rectangular partitions using kd-trees.

36. Ames Research Center Partitioning

37. Ames Research Center Performance: 2 Continuous Variables

38. Ames Research Center Performance: 3 Continuous Variables  For naïve, we just discretize everything at the given input resolution.  For the others, we discretize the transition functions at that resolution, but the algorithm may increase the resolution to accurately represent that final value function. This means that the value function is actually more accurate than for the naïve algorithm.

39. Ames Research Center Final Remarks  Plangraph–based approach: Produces “plans” - easy for people to interpret. Fast heuristic estimate of the value of a plan/plan fragment. Need an effective way to evaluate actual values to really know a branch is worthwhile. Efficient representation for problems with many goals. Still missing discrete action outcomes  MDP-based approach: Produces optimal policies – the best you could possibly do. Faster, more accurate value fn. computation (if there’s structure). Hard to represent some problems effectively (e.g. fact that goals are worth something only before you reach them). Policies are hard to interpret by humans.  Can be combined: Use MDP approach to evaluate quality of plans/plan fragments.

40. Ames Research Center Future Work  We approximate by building an approximate model, then solving it exactly. One could also approximately solve the exact model.  The plangraph approach takes advantage of the current system state when planning to narrow the search. The MDP policy probably includes value computation for many unreachable states.  Preference elicitation is very important here. With many goals we need good estimates of their value.  This is part of a greater whole—rover planning problems. Is the policy sufficiently efficiently encoded to transmit to the rover? How much more complex does the executive need to be to carry out a contingent plan?