Cognitive Robotics 2016/2017

Transcription

1 based on Manuela M. Veloso lectures on PLNNING, EXEUTION ND LERNING ognitive Robotics 2016/2017 Planning:, ctions and Goal Representation Matteo Matteucci rtificial Intelligence and Robotics Lab - Politecnico di Milano

2 Recall «Think hard, act later»? Planning is about «thinking» Given the actions available in a task domain. Given a problem specified as: an initial state of the world a goal statement (set of goals) to be achieved Find a solution to the problem Plan: a way, in terms of a sequence of actions, to transform the initial state into a new state of the world where It s all about states, the goal statement is true. actions, and plans! Newell and Simon

3 The lock World The lock World is a useful abstraction to introduce s, ctions and Plans locks are on the Table, or on top of each other. There is an rm the rm can be empty or holding one block. The table is always clear. 3

4 The lock World: s Objects locks:,, Table: Table Some predicates might be redundant Predicates On(, ), On(, Table) lear(), Handempty, Holding() On-table(), On(,), Top(), s onjunctive On(,) and On(,) and lear() and Handempty On-Table(), On-Table(), On(,), lear(), lear(), Handempty On-Table(), On-Table(), lear(), lear(), Holding() 4

5 The lock World: ssumptions/limitations The lock World models lassical Deterministic Planning... There is a single initial state The description is complete The plan is deterministic What is not true in the state is false The basic operators perform queries on states On(,) returns true or false On(,x) returns x=table or x= On-table (x) returns x= and x= W: losed World ssumption 5

6 The lock World: Description -on- -on-table -on- -on-table Holding- Holding- Handempty lear- lear- -on- -on-table -on- -on-table Holding- Holding- -on- -on- 2^4 Possible states ll these define the Space -on-x {, table, } -on-x {, table, } 3^2 Possible s 6

7 The lock World: Planning as -Space Search 7

8 Models for Spaces Different models for states exist... tomic identification of states (s1, s2,...) Symbolic feature based states Symbolic predicate based states together with different ways of combining them onjunctive observable Probabilistic approximate Incremental on-demand Temporal dynamic Predicates, conjunctive, complete, correct, deterministic 8

9 Goal Specification We can specify a Goal according to different levels of generality: Goal ompletely specified state Goal ment Partially specified state Objective function Defines good or optimal plan Increased Generality Goal ment example: Initial: -on-x = Table; -on-x = ; -on-x = Table Goal: -on-x = 9

10 What is an ction? Plan: a way, in terms of a sequence of actions, to transform the initial state into a new state of the world where the goal statement is true. Newell and Simon 1956 ction: a transition from one (partial) state to another May be applicable only in particular states Generates new state Deterministic: t det : S x S Non-deterministic: t non-det : S x 2 S Probabilistic: t prob : S x <2 S, r> Explicit ction Representation 10

11 The lock World Dynamics: ctions How do these transform a state into another? locks are on the Table, or on top of each other locks are picked up and put down by the arm block can be picked up only if it is clear, i.e., without a block on top The arm can pick up a block only if the arm is empty, i.e., if it is not holding another block, i.e., the arm can pick up only one block at a time The arm can put down blocks on blocks or on the table The table is always clear 11

12 STRIPS ction Representation STRIPS (Stanford Research Institute Problem Solver) was the planner used by Shakley, it was developed at SRI International by Richard Fikes and Nils Nilsson in Explicit action a representation {preconds(a), effects (a), effects + (a)} effects (a) effects + (a) = (S, a) = {S effects (a) effects + (a)}, where S 2 S Example in the lock World Pickup_from_table(?b) Pre:... dd:... Delete:... Let s try this out together! 12

13 ctions in the lock World In the lock World: n action a is applicable in s if all its preconditions are satisfied by s. RESULT(s,a) = (s Del (a)) U dd (a) No explicit mention of time o The precondition always refers to time t o The effect always refers to time t+1 13

14 The lock World: ctions Pickup_from_table(b) Pre: lock(b), Handempty lear(b), On(b, Table) dd: Holding(b) Delete: Handempty, On(b, Table) lear(b) Pickup_from_block(b1, b2) Pre: lock(b1),lock(b2), Handempty lear(b1), On(b1,b2) dd: Holding(b1), lear(b2) Delete: Handempty, On(b1,b2) lear (b1) Putdown_on_table(b) Pre: lock(b), Holding(b) dd: Handempty, On(b, Table) Delete: Holding(b) Putdown_on_block(b1, b2) Pre: lock(b1), Holding(b1) lock(b2), lear(b2), b1 b2 dd: Handempty, On(b1, b2) Delete: Holding(b1), lear(b2) 14

15 More Realistic ctions Representations onditional Effects Pickup (b) Pre: lock(b), Handempty, lear(b), On(b, x) dd: Holding(b) if (lock(x)) then lear(x) Delete: Handempty, On(b, x) Quantified Effects Move (o, x) Pre: t(o, y), t(robot, y) dd: t(o, x), t(robot, x) forall (Object(u)) [ if (In(u, o)) then t(u, y)] Delete: t(o, y), t(robot, y), forall (Object(u)) [ if (In(u, o)) then t(u, y)] Disjunctive and Negated Preconditions Holding(x) Or Not[Lighter_Than_ir(x)] ll these extensions can be emulated adding actions! 15

16 More Realistic ctions Representations Inference Operators / xioms lear(x) iff forall(lock(y))[ Not[On(y, x)]] These extensions make the planning problem significantly harder Functional effects Move (o, x) Pre: t(o, y), t(robot, y), Fuel(f), f Fuel_Needed(y, x) dd: t(o, x), t(robot, x), Fuel(f Fuel_Needed(y, x)), forall (Object(u)) [ if (In(u, o)) then t(u, y)] Delete: t(o, y), t(robot, y), Fuel(f), forall (Object(u)) [ if (In(u, o)) then t(u, y)] Disjunctive Effects Pickup_from_block(b) Pre: lock(b), Handempty, lear(b), On(b, c), lock(c) 1: dd: lear(c), Holding(b); Delete: On(b, c), Handempty 2: dd: lear(c), On(b, Table); Delete: On(b, c) 3: dd: ; Delete: Much harder and you can add probability!!! 16

17 ognitive Robotics Planning: Plan Generation Matteo Matteucci rtificial Intelligence and Robotics Lab - Politecnico di Milano

18 Different Plans... plan can have different degrees of generality Sequence of Instantiated ctions Partial Order of Instantiated ctions Set of Instantiated ctions Policy (a direct mapping from states to actions) Increased Generality and adopt different search stratiegies: Progression, a.k.a. forward state space search, a.k.a. forward chaining Regression, a.k.a. backward state-space search, a.k.a. backward chaining 18

19 Plan Generation acktracking Search Through a Search Space How to conduct the search How to represent the search space How to evaluate the solutions Non-Deterministic hoices Determine acktracking hoice of actions hoice of variable bindings hoice of temporal orderings hoice of subgoals to work on 19

20 Properties of Planning lgorithms Soundness planning algorithm is sound if all solutions are legal plans, i.e., all preconditions, goals, and any additional constraints are satisfied ompleteness planning algorithm is complete if a solution can be found whenever one actually exists planning algorithm is strictly complete if all solutions are included in the search space Optimality planning algorithm is optimal if it maximizes a predefined measure of plan quality 20

21 Linear Planning and Means-ends nalysis Linear Planning Uses a Goal stac and work on one goal until completely solved before moving on to the next goal Mean-ends nalysis Search by reducing the difference between the state and the goals, i.e., What means (operators) are available to achieve the desired ends (goal)? GPS lgorithm (state, goals, plan) If goals state, then return (state,plan) hoose a difference d goals between state and goals hoose an operator o to reduce the difference d If no applicable operators, then return False (state,plan) = GPS (state, preconditions(o), plan) If state, then return GPS (apply (o, state), goals, [plan,o]) Initial call: GPS (initial-state, initial-goals, []) Newell and Simon 60s 21

22 The lock World: GPS at Work Goal 22

23 The lock World: GPS at Work Goal 23

24 The lock World: GPS at Work Goal 24

25 The lock World: GPS at Work Goal 25

26 The lock World: GPS at Work Goal 26

27 The lock World: GPS at Work Goal 27

28 The lock World: GPS at Work Goal 28

29 The lock World: GPS at Work Goal 29

30 The lock World: GPS at Work Sound? Optimal? omplete? Goal 30

31 The Sussman nomaly Pickup(?b) Pre:(handempty) (clear?b) (on-table?b) dd:(holding?b) Delete:(handempty) (on-table?b) (clear?b) Goal Putdown(?b) Pre:(holding?b) dd:(handempty) (on-table?b) Delete:(holding?b) Unstack(?a,?b) Pre:(handempty) (clear?a)(on?a?b) dd:(holding?a)(clear?b) Delete:(handempty) (on?a?b)(clear?a) Stack(?a,?b) Pre:(holding?a) (clear?b) dd:(handempty)(on?a?b) Delete:(holding?a) (clear?b) 31

32 The Sussmann nomaly Linear Solution 1 (on ) Pickup () Stack (, ) Goal 32

33 The Sussmann nomaly Linear Solution 1 (on ) Pickup () Stack (, ) (on ) Unstack (, ) Putdown () Unstack (, ) Putdown () Pickup() Stack (, ) Goal 33

34 The Sussmann nomaly Linear Solution 1 (on ) Pickup () Stack (, ) (on ) Unstack (, ) Putdown () Unstack (, ) Putdown () Pickup() Stack (, ) (on ) Unstack (, ) Putdown () Pickup () Stack (, ) Goal 34

35 The Sussmann nomaly Linear Solution 1 (on ) Pickup () Stack (, ) (on ) Unstack (, ) Putdown () Unstack (, ) Putdown () Pickup() Stack (, ) (on ) Unstack (, ) Putdown () Pickup () Stack (, ) (on ) Pickup () Stack (,) Goal 35

36 The Sussmann nomaly Linear Solution 2 (on ) Unstack (, ) Putdown () Pickup() Stack (, ) Goal 36

37 The Sussmann nomaly Linear Solution 2 (on ) Unstack (, ) Putdown () Pickup() Stack (, ) (on ) Unstack (, ) Putdown () Pickup () Stack (, ) Goal 37

38 The Sussmann nomaly Linear Solution 2 (on ) Unstack (, ) Putdown () Pickup() Stack (, ) (on ) Unstack (, ) Putdown () Pickup () Stack (, ) (on ) Pickup () Stack (,) Goal Is it Optimal? an we do it with less actions? 38

39 The Sussmann nomaly: Non Linear (Optimal) Solution (on ) Unstack (, ) Putdown () (on ) Pickup () Stack (, ) (on ) Pickup() Stack(, ) Goal 39

40 Linear Planning and the Goal Stack dvantages Reduced search space, since goals are solved one at a time, and not all possible goal orderings are considered dvantageous if goals are (mainly) independent Linear planning is sound Disadvantages Linear planning may produce suboptimal solutions (based on the number of operators in the plan) Planner's efficiency is sensitive to goal orderings ontrol knowledge for the right ordering Random restarts Iterative deepening What about completeness? 40

41 One Way Rocket (Veloso 89) 41

42 Space Non Linear Planning Extend linear planning: From stack to set of goals Include in the search space all possible interleaving of goals -space nonlinear planning is complete 42

43 Prodigy4.0 (Veloso et al. 90) 1. Terminate if the goal statement is satisfied in the current state. Initially the set of applicable relevant operators is empty. 2. ompute the SET of pending goals G, and the SET of applicable relevant operators. goal is pending if it is a precondition, not satisfied in the current state, of a relevant operator already in the plan. relevant operator is applicable when all its preconditions are satisfied in the state. 3. hoose a pending goal G in G or choose a relevant applicable operator in. 4. If the pending goal G has been chosen, then Expand goal G, i.e., get the set O of relevant instantiated operators that could achieve G, hoose an operator O from O, as a relevant operator for goal G. Go to step If a relevant operator has been selected as directly applicable, then pply, Go to step 1. 43

44 Prodigy4.0 Search Representation Head plan gap Tail plan pplying and Operator (moving it to the head) dding and operator to the tail plan 44

45 fter all, it is all about graph exploration Multiple solutions are possible. No need to explore the whole graph, but you should be able to do it! 45

46 Planning issues representation The frame problem The choice of predicates (e.g., On-table (x), On (x, table), On-table-, On-table-, ) ction representation Many alternative definitions Reduce to needed definition onditional effects Uncertainty Quantification Functions Generation planning algorithm(s) 46

47 Wrap-up slide on Planning and Plan Generation What should remain from this lecture? Planning: selecting one sequence of actions (operators) that transform (apply to) an initial state to a final state where the goal statement is true. Means-ends analysis: identify and reduce, as soon as possible, differences between state and goals. Linear planning: backward chaining with means-ends analysis using a stack of goals, potentially efficient, possibly unoptimal, incomplete; GPS Nonlinear planning with means-ends analysis: backward chaining using a set of goals; reason about when to reduce the differences; Prodigy4.0. References S. Russell, P. Norvig. «rtificial Intelligence: Modern pproach». hapter 11: Planning, pages Pearson,