1. Temporal Planning, Scheduling and Execution
Brian C. Williams
16.412J/6.834J
November 25th, 2002 1

2. Cassini Saturn Orbital Insertion
courtesy JPL

3. Programming& Execution
Histories(?)
Goals
Projective
Task Expansion
Temporal
Planner
Scheduler
Flexible Sequence (Plans)
Task
Dispatch
Plan Runner
Goals
Modes
Model-based
Programming& Execution
Task-Decomposition
Execution
Reactive
Task Expansion
Observations
Commands

4. Outline Temporal Planning Representing Time
Temporal Consistency and Scheduling
Execution with Dynamic Scheduling

5. Flexible Sequence (Plans)
Histories(?)
Goals
Projective
Task Expansion
Temporal
Planner
Scheduler
Flexible Sequence (Plans)
Task
Dispatch
Plan
Runner
Goals
Modes
Model-based
Execution
(w planner)
Task-Decomposition
Execution
Reactive
Task Expansion
Observations
Commands

6. Flexible Sequence (Plans)
Histories(?)
Goals
Projective
Task Expansion
Temporal
Planner
Scheduler
Flexible Sequence (Plans)
Task
Dispatch
Plan
Runner
Goals
Modes
Model-based
Execution
(w planner)
Task-Decomposition
Execution
Reactive
Task Expansion
Observations
Commands

7. Outline Temporal Planning Representing Time
Temporal Consistency and Scheduling
Execution with Dynamic Scheduling

8. Qualitative Temporal Constraints (Allen 83)
x before y
x meets y
x overlaps y
x during y
x starts y
x finishes y
x equals y
y after x
y met-by x
y overlapped-by x
y contains x
y started-by x
y finished-by x
y equals x X Y X Y X Y Y X X Y Y X Y X

9. Deep Space One Example: Temporal Constraints
Timer
Idle
Max_Thrust
Idle
Th_Seg
contained_by
equals
meets
Start_Up
Shut_Down
Thr_Boundary
Thrust
Standby
Th_Sega
Idle_Seg
Accum_NO_Thr
Accum_Thr
CP(Ips_Tvc)
SEP_Segment
Accum
SEP Action
Attitude
Poke

10. Qualitative Temporal Constraints maybe Expressed as Inequalities (Vilain, Kautz 86)
x before y X+ < Y-
x meets y X+ = Y-
x overlaps y (Y- < X+) & (X- < Y+)
x during y (Y- < X-) & (X+ < Y+)
x starts y (X- = Y-) & (X+ < Y+)
x finishes y (X- < Y-) & (X+ = Y+)
x equals y (X- = Y-) & (X+ = Y+)

11. Metric Time: Quantitative Temporal Constraint Networks (Dechter, Meiri, Pearl 91)
A set of time points Xi at which events occur.
Unary constraints (a0 < Xi < b0 ) or (a1 < Xi < b1 ) or . . .
Binary constraints (a0 < Xj - Xi < b0 ) or (a1 < Xj - Xi < b1 ) or . . .

12. Visualize TCSP as Directed Constraint Graph1 3 4 2
[10,20]
[30,40]
[60,inf]
[20,30]
[40,50]
[60,70]

13. Simple Temporal Networks (Dechter, Meiri, Pearl 91)
A set of time points Xi at which events occur.
Unary constraints (a0 < Xi < b0 ) or (a1 < Xi < b1 ) or . . .
Binary constraints (a0 < Xj - Xi < b0 ) or (a1 < Xj - Xi < b1 ) or . . .
Sufficient to represent:
most Allen relations
simple metric constraints
Can’t represent:
Disjoint tokens

14. Simple Temporal Network
Tij = (aij£ Xi - Xj £ bij) 1 3 4 2
[10,20]
[30,40]
[60,inf]
[20,30]
[40,50]
[60,70]

15. A Completed Plan Forms an STN








16. TCSP Queries (Dechter, Meiri, Pearl, AIJ91)
Is the TCSP consistent?
What are the feasible times for each Xi?
What are the feasible durations between each Xi and Xj?
What is a consistent set of times?
What are the earliest possible times?
What are the latest possible times?

17. TCSP Queries (Dechter, Meiri, Pearl, AIJ91)
Is the TCSP consistent? Planning
What are the feasible times for each Xi?
What are the feasible durations between each Xi and Xj?
What is a consistent set of times?
What are the earliest possible times? Execution
What are the latest possible times?

18. Outline Temporal Planning Representing Time
Temporal Consistency and Scheduling
Execution with Dynamic Scheduling

19. To Query STN Map to Distance Graph Gd = < V,Ed >
Edge encodes an upper bound on distance to target from source.
Xj - Xi £ bij
Xi - Xj £ - aij
Tij = (aij£ Xj - Xi £ bij) 1 3 4 2
[10,20]
[30,40]
[40,50]
[60,70] 70 1 3 4 2 20 50
-10 40
-30
-40
-60

20. Induced Constraints for Gd
Path constraint: i0 =i, i1 = . . ., ik = j
Intersected path constraints:
where dij is the shortest path from i to j

21. Compute Intersected Paths by All Pairs Shortest Path (e. g
Compute Intersected Paths by All Pairs Shortest Path (e.g., Floyd-Warshall’s algorithm )
1. for i := 1 to n do dii 0;
2. for i, j := 1 to n do dij aij;
3. for k := 1 to n do
4. for i, j := 1 to n do
dij min{dij, dik + dkj}; k i j

22. Shortest Paths of Gd 70 1 2 4 3 20 50
-10 40
-30
-40
-60
d-graph

23. STN Minimum Network
d-graph
STN minimum network

24. Test Consistency: No Negative Cycles70 1 2 4 3 20 50
-10 40
-30
-40
-60
d-graph

25. Latest Solution 20 40 1 2 -10 -30 -10 20 50 3 4 -40 -60 70 d-graph
Node 0 is the reference. 20 40 1 2
-10
-30
-10 20 50 3 4
-40
-60 70
d-graph

26. Earliest Solution 20 40 1 2 -10 -30 -10 20 50 3 4 -40 -60 70 d-graph
Node 0 is the reference. 20 40 1 2
-10
-30
-10 20 50 3 4
-40
-60 70
d-graph

27. Feasible Values X1 in [10, 20] X2 in [40, 50] X3 in [20, 30]
d-graph

28. Solution by Decomposition
Select value for 1
15 [10,20]
d-graph

29. Solution by Decomposition
Select value for 1 15
Select value for 2, consistent with 1
45 [40,50], 15+[30,40]
d-graph

30. Solution by Decomposition
Select value for 1 15
Select value for 2, consistent with 1 45
Select value for 3, consistent with 1 & 2
30 [20,30], 15+[10,20],45+[-20,-10]
d-graph

31. Solution by Decomposition
Select value for 1 15
Select value for 2, consistent with 1 45
Select value for 3, consistent with 1 & 2 30
d-graph
Select value for 4, consistent with 1,2 & 3
O(N2)

32. Outline Temporal Planning Representing Time
Temporal Consistency and Scheduling
Execution with Dynamic Scheduling