1. Temporal Plan Execution: Dynamic Scheduling and Simple Temporal Networks
Brian C. Williams
16.412J/6.834J
December 2nd, 2002 1

2. Dynamic Scheduling and Task Dispatch
Goals and Environment Constraints
Temporal
Network Solver
Projective
Task Expansion
Temporal
Planner
Temporal Plan
Dynamic Scheduling and Task Dispatch
Task
Dispatch
Goals
Modes
Model-based
Programming& Execution
Task-Decomposition
Execution
Reactive
Task Expansion
Observations
Commands

3. Dynamic Scheduling and Task Dispatch
Goals and Environment Constraints
Temporal
Network Solver
Projective
Task Expansion
Temporal
Planner
Temporal Plan
Dynamic Scheduling and Task Dispatch
Task
Dispatch
Goals
Modes
Model-based
Programming& Execution
Task-Decomposition
Execution
Reactive
Task Expansion
Observations
Commands

4. Outline Temporal Representation
Qualitative temporal relations
Metric constraints
Temporal Constraint Networks
Temporal Plans
Temporal Reasoning for Planning and Scheduling
Flexible Execution through Dynamic Scheduling

5. 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

6. Example: Deep Space One Remote Agent Experiment
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

7. 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+)
Inequalities may be expressed as binary interval relations:
X+ - Y- < [-inf, 0]

8. Metric Constraints
Going to the store takes at least 10 minutes and at most 30 minutes.
10 < [T+(store) – T-(store)] < 30
Bread should be eaten within a day of baking.
0 < [T+(baking) – T-(eating)] < 1 day
Inequalities, X+ < Y- , may be expressed as binary interval relations:
- inf < [X+ - Y-] < 0

9. 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 . . .

10. Temporal Constraint Satisfaction Problem (TCSP)
< Xi, Ti , Tij >
Xi continuous variables
{I1, ,In } interval constraints
where Ii = [ai,bi] interval
Ti = (ai £ Xi £ bi) or or (ai £ Xi £ bi)
Tij = (a1£ Xi - Xj £ b1) or ... or (an £ Xi - Xj £ bn)
[Dechter, Meiri, Pearl, aij89]

11. TCSP Are Visualized Using Directed Constraint Graphs1 3 4 2
[10,20]
[30,40]
[60,inf]
[20,30]
[40,50]
[60,70]

12. 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

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

14. Delta_V(direction=b, magnitude=200)
A Completed Plan Forms an STN
Thrust Goals
Delta_V(direction=b, magnitude=200)
Power
Attitude
Point(a)
Turn(a,b)
Point(b)
Turn(b,a)
Engine
Off
Warm Up
Thrust (b, 200)
Off

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

16. Outline Temporal Representation
Temporal Reasoning for Planning and Scheduling
TCSP Queries
Induced Constraints
Plan Consistency
Static Scheduling
Flexible Execution through Dynamic Scheduling

17. 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?

18. 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

19. 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? Scheduling
What are the earliest possible times? Scheduling
What are the latest possible times?

20. To Query an STN Map to a 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

21. Gd Induces Constraints
Path constraint: i0 =i, i1 = . . ., ik = j
Conjoined path constraints result in the shortest path as bound:
where dij is the shortest path from i to 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. Conjoined Paths are Computed using All Pairs Shortest Path (e. g
Conjoined Paths are Computed using 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

25. Testing Plan Consistency
No negative cycles: -5 > TA – TA = 0 70 1 2 4 3 20 50
-10 40
-30
-40
-60
d-graph

26. 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

27. Solution: Latest Times
S1 = (d01, , d0n) 20 40 1 3
-10
-30
-10 20 50 2 4
-40
-60 70

28. 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

29. Solution: Earliest Times
S1 = (-d10, , -dn0) 20 40 1 3
-10
-30
-10 20 50 2 4
-40
-60 70

30. Scheduling: Feasible Values
Latest Times
X1 in [10, 20]
X2 in [40, 50]
X3 in [20, 30]
X4 in [60, 70]
Earliest Times
d-graph

31. Scheduling without Search: Solution by Decomposition
Select value for 1
15 [10,20]
d-graph

32. Scheduling without Search: Solution by Decomposition
Select value for 1
15 [10,20]
d-graph

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

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

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

36. 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

37. 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 [25,30]
d-graph

38. 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 [25,30]
d-graph

39. 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)

40. In General Solving TCSPs is NP Hard
Forward(I1, . . ., Ii)
1. if i = m then
2. M M union Solve-STP(I1, . . ., Im), and
3. Go-Back(I1, . . ., Im);
4. Ci empty;
5. for every Ij in Di+1 do
6. if Consistent-STP (I1, , Ii , Ij), then
7. for i, j := 1 to n do
dij min{dij, dik + dkj};

41. Outline Temporal Representation
Temporal Reasoning for Planning and Scheduling
Flexible Execution through Dynamic Scheduling

42. Dynamic Scheduling and Task Dispatch
Goals and Environment Constraints
Temporal
Network Solver
Projective
Task Expansion
Temporal
Planner
Temporal Plan
Dynamic Scheduling and Task Dispatch
Task
Dispatch
Goals
Modes
Model-based
Programming& Execution
Task-Decomposition
Execution
Reactive
Task Expansion
Observations
Commands

43. Executing Flexible Temporal Plans [Muscettola, Morris, Pell et al.]
Handling delays and fluctuations in task duration:
Least commitment temporal plans leave room to adapt
EXECUTIVE
CONTROLLED SYSTEM
flexible execution adapts through dynamic scheduling
Assigns time to event when executed.

44. Issues in Flexible Execution
How do we minimize execution latency?
How do we schedule at execution time?

45. Time Propagation Can Be Costly
EXECUTIVE
CONTROLLED SYSTEM

46. Time Propagation Can Be Costly
EXECUTIVE
CONTROLLED SYSTEM

47. Time Propagation Can Be Costly
EXECUTIVE
CONTROLLED SYSTEM

48. Time Propagation Can Be Costly
EXECUTIVE
CONTROLLED SYSTEM

49. Time Propagation Can Be Costly
EXECUTIVE
CONTROLLED SYSTEM

50. Time Propagation Can Be Costly
EXECUTIVE
CONTROLLED SYSTEM

51. Issues in Flexible Execution
How do we minimize execution latency?
Propagate through a small set of neighboring constraints.
How do we schedule at execution time?

52. Compile to Efficient Network
EXECUTIVE
CONTROLLED SYSTEM

53. Compile to Efficient Network
EXECUTIVE
CONTROLLED SYSTEM

54. Compile to Efficient Network
EXECUTIVE
CONTROLLED SYSTEM

55. Issues in Flexible Execution
How do we minimize execution latency?
Propagate through a small set of neighboring constraints.
How do we schedule at execution time?

56. Issues in Flexible Execution
How do we minimize execution latency?
Propagate through a small set of neighboring constraints.
How do we schedule at execution time?
Through decomposition?

57. Dynamic Scheduling by Decomposition10 2 -2 -1 1
Equivalent Distance Graph Representation A C B D
[0,10]
[-2,2]
[-1,1]
Simple Example

58. Dynamic Scheduling by Decomposition
Compute APSP graph
Decomposition enables assignment without search A C B D 10 2 -2 -1 1 11
Equivalent All Pairs Shortest Paths (APSP) Representation A C B D 10 2 -2 -1 1
Equivalent Distance Graph Representation

59. Assignment by Decomposition11 10
<0,10> -1 B 1 A D
<2,11>
t = 0 -1 1 -2 C 2 10
<0,10> -2

60. Assignment by Decomposition11
t = 3 10 -1 B 1
<0,0> A D
<2,11> -1 1 -2 C 2 10
<0,10> -2

61. Assignment by Decomposition
Solution:
Assignments must monotonically increase in value.
Execute first, all APSP neighbors with negative delays.
But C now has to be executed at t =2, which is already in the past! 11
t = 3 10 -1 B 1
<0,0> A D
<4,4> -1 1 -2 C 2 10
<2,2> -2

62. Dispatching Execution Controller
Execute an event when enabled and active
Enabled - APSP Predecessors are completed
Predecessor – a destination of a negative edge that starts at event.
Active - Current time within bound of task.

63. Dispatching Execution Controller
Initially:
E = Time points w/o predecessors
S = {}
Repeat:
Wait until current_time has advanced st
Some TP in E is active
All time points in E are still enabled.
Set TP’s execution time to current_time.
Add TP to S.
Propagate time of execution to TP’s APSP immediate neighbors.
Add to A, all immediate neighbors that became enabled.
TPx enabled if all negative edges starting at TPx have their destination in S.
EXECUTIVE

64. Propagation is Focused
Propagate forward along positive edges to tighten upper bounds.
forward prop along negative edges is useless.
Propagate backward along negative edges to tighten lower bounds.
Backward prop along positive edges useless.

65. Propagation Example S = {A} B A D C 11 10 -1 <0,0> -1 1 1 -1 -2-2 C 9 2 -2

66. Propagation Example S = {A} B A D C 11 <1,10> 10 -1 <0,0>-2 C 9 2 -2

67. Propagation Example S = {A} B A D C 11 <1,10> 10 -1 <0,0>-2 C 9 2
<0,9> -2

68. Propagation Example S = {A} B A D C 11 <1,10> 10 -1 <2,11>
<0,0> -1 1 A D 1 -1 -2 C 9 2
<0,9> -2

69. Propagation Example S = {A} E = { C } B A D C 11 <1,10> 10 -1
<2,11>
<0,0> -1 1 A D 1 -1 -2 C 9 2
<0,9> -2

70. Reducing Execution Latency
Filtering:
some edges are redundant
remove redundant edges
Execution time is O(n)
worst case
best case 11 10
t = 3 -1 B 1
<0,0> A D
<2,11> -1 1 -2 C 2 10
<0,10> -2

71. Edge Dominance
Eliminate edge that is redundant due to the triangle inequality AB + BC = AC C A B
150
-100
100 80
-50
-20

72. Edge Dominance
Eliminate edge that is redundant due to the triangle inequality AB + BC = AC
150 A B
-100 80
100
-50
-20 C

73. Edge Dominance
Eliminate edge that is redundant due to the triangle inequality AB + BC = AC
150 A B
-100 80
-50
-20 C

74. Edge Dominance
Eliminate edge that is redundant due to the triangle inequality AB + BC = AC
150 A B
-100 80
-50 C

75. An Example of Edge Filtering
Start off with the APSP network A C B D -1 10 9 2 -2 1 11

76. An Example of Edge Filtering
Start at A-B-C triangle 11 10 B -1 -1 1 A D 1 -1 -2 C 9 2 -2

77. An Example of Edge Filtering
Look at B-D-C triangle 11 B -1 -1 1 A D 1 -1 -2 C 9 2 -2

78. An Example of Edge Filtering
Look at B-D-C triangle 11 B -1 -1 1 A D 1 -1 -2 C 9 2

79. An Example of Edge Filtering
Look at D-A-B triangle 11 B -1 -1 1 A D 1 -1 C 9 2

80. An Example of Edge Filtering
Look at D-A-C triangle 11 B -1 1 A D 1 -1 C 9 2

81. An Example of Edge Filtering
Look at B-C-D triangle B -1 1 A D 1 -1 C 9 2

82. An Example of Edge Filtering
Look at B-C-D triangle B -1 1 A D 1 -1 C 9

83. An Example of Edge Filtering
Resulting network has less edges than the original 9 1 1 A C B D -1 -1 A C B D 10 2 -2 -1 1

84. Avoiding Intermediate Graph Explosion
Problem:
APSP consumes O(n2) space.
Solution:
Interleave process of APSP construction with edge elimination
Never have to build whole APSP graph

85. Additional Filtering Node Contraction
Collapse two events with fixed time between them A
|LA|
|AL| L B
|BL|
|LB|

86. Additional Filtering Node Contraction
Collapse two events with fixed time between them A
|LA|
|AL| L B
|BL|
|LB|

87. Additional Filtering Node Contraction
Collapse two events with fixed time between them XY B
|BL|
|LB|

88. Dynamic Scheduling and Task Dispatch
Goals and Environment Constraints
Temporal
Network Solver
Projective
Task Expansion
Temporal
Planner
Temporal Plan
Dynamic Scheduling and Task Dispatch
Task
Dispatch
Goals
Modes
Model-based
Programming& Execution
Task-Decomposition
Execution
Reactive
Task Expansion
Observations
Commands