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Planning with Continuous Linear Change: COLIN
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Planning: How it all Started
(: action pickup_from_block (?b1 ?b2 – block)
:precondition (and (on ?b1 ?b2)
(arm_free))
:effect (and (holding ?b1)
(clear ?b2))
(not (arm_free))
(not (on ?b1 ?b2)))
Initial State: (on C A) (ontable B) (ontable A)
(arm_free) (clear B) (clear C)
Goal: (on B C) (on A B)
(: action put_down_on_block (?b1 ?b2 – block)
:precondition (and (holding ?b1)
(clear ?b2))
:effect (and (holding ?b1)
(clear ?b2))
(not (arm_free))
(not (on ?b1 ?b2)))
…And where it is Going
What Matters?
Time: Deadlines, Windows of Opportunity;
Numeric Quantities: Discrete and Continuous, Optimisation;
The environment: Uncertainty, Predictable Exogenous Change.
The Reality
Solar generation
State of Charge
Minimum state of charge at nightfall
Classical Planning as Forward Search
Classical Planning Gives Us:
Propositional Relaxation Heuristics: RPG, Causal Graph;
Search Guidance: Helpful Actions/Preferred Operators.
Search Techniques, Enforced Hill-Climbing, Multi Open List Search, Memoisation;
Continuous Linear Change
Numeric quantities so far we have seen change instantaneously:
e.g. (at end (decrease (battery) 1))
(at end (decrease (battery) (+ (*0.5 (temperature)) (*3 (?duration))))
V’ = W . V + C
In reality numeric values often change continuously, rather than discretely.
While the bus is running dbattery/dt = -40
(increase (battery) (*#t -40))
Of course change is not always linear, but that’s another lecture.
Today we will deal with linear change only.
battery’ =
temperature
V’ = W . V + C
Continuous Linear Change
Numeric quantities so far we have seen change instantaneously:
e.g. (at end (decrease (battery) 1))
(at end (decrease (battery) (+ (*0.5 (temperature)) (*3 (?duration))))
V’ = W . V + C
In reality numeric values often change continuously, rather than discretely.
While the bus is running dbattery/dt = -40
(increase (battery) (*#t -40))
Of course change is not always linear, but that’s another lecture.
Today we will deal with linear change only.
Continuous Linear Change: D1
Route1 D1 B1
duration = 2
Done Route1
Battery >= 10
Battery >= 30
dBattery/dt -=40
(increase (battery) (*#t -40))
“Temporal Planning in Domains with Linear Processes.” A. J. Coles, A. I. Coles, M. Fox, and D. Long. IJCAI (2009).
“COLIN: Planning with Continuous Linear Numeric Change.” A. J. Coles, A. I. Coles, M. Fox and D. Long. JAIR (44) (2013)
On Route B1
Run Engine B1
Battery <= 100
dBattery/dt +=60
dCost/dt +=1
duration >= 0.25
EngineOff B1
¬EngineOff B1
EngineOff B1
Cost += 10
BR1⊣ >= 30
BE1 ⊢ >= 10
B’R1⊢ >= 10
B’E1⊢ >= 10
BE1⊣ >= 10
B’E1⊣ >= 10
BR1⊣ >= 10
B’E1⊢ <= 100
BE1⊣ <= 100
Continuous Linear Change: 1 D1 B1
duration = 2
Battery >= 10
Battery >= 30
dBattery/dt -=40
Run Engine B1
Battery <= 100
dBattery/dt +=60
dCost/dt +=1
duration >= 0.25
Constraints:
R1⊣ – R⊢ = 2
E1 ⊢ >= R1⊢ + ε
E1⊣ – E1⊢ >= 0.25
R1 ⊣ >= E1 ⊣ + ε
Cost += 10
BE1⊢ = BR1 ⊢ + (E1⊢ – R1⊢)*-40
BE1 ⊢ >= 10
CE1⊢ = C’R1⊢
B’E1⊢ = BE1⊢
B’E1⊢ >= 10, <= 100
C’E1⊢ = CE1 ⊢ + 10
BE1⊣ = BE1⊢ + (E1⊣ - E1⊢)*20
BE1⊣ >= 10, <=100
CE1⊣ = C’R1⊢ + (E1⊣ - E1⊢) * 1
B’E1⊣ = BE1 ⊣
B’E1⊣ >= 10
C’E1⊣ = CE1 ⊣
BR1⊣ = BE1⊣ + (R1⊣ – E1⊣)*-40
BR1⊣ >= 10, >= 30
CR1⊣ = C’E1 ⊣
B’R1⊣ = BE1 ⊣,
B’R1⊣ >= 10, <= 100
C’R1⊣ = CE1 ⊣
"Temporal Planning in Domains with Linear Processes." A. J. Coles, A. I. Coles, M. Fox, and D. Long. IJCAI (2009).
"COLIN: Planning with Continuous Linear Numeric Change." A. J. Coles, A. I. Coles, M. Fox and D. Long. JAIR (44) (2013)
BR1⊢ = 50
B’R1⊢=BR1⊢
BE1⊢=B’R1⊢+(E1⊢-R1⊢)*-40
B’E1⊢=BE1⊢
Numeric Constraints:
B’E1⊣= BE1⊣
BR1⊣=B’E1⊣+(R1⊣-E1⊣)*-40
BE1⊣=B’E1⊢+(E1⊣-E1⊢)*20
B’E1⊣ = BE1 ⊣
Continuous Linear Change: 1 D1 B1
duration = 2
Battery >= 10
Battery >= 30
dBattery/dt -=40
Run Engine B1
Battery <= 100
dBattery/dt +=60
dCost/dt +=1
duration >= 0.25
Constraints:
R1⊣ – R⊢ = 2
E1 ⊢ >= W⊢ + ε
E1⊣ – E1⊢ >= 0.25
R1 ⊢ >= E1⊢ + ε
Cost += 10
BE1⊢ = BR1 ⊢ + (E1⊢ – R1⊢)*-40
BE1 ⊢ >= 10
CE1⊢ = C’R1⊢
B’E1⊢ = BE1⊢
B’E1⊢ >= 10, <= 100
C’E1⊢ = CE1 ⊢ + 10
BE1⊣ = BE1⊢ + (E1⊣ - E1⊢)*20
BE1⊣ >= 10, <=100
CE1⊣ = C’R1⊢ + (E1⊣ - E1⊢) * 1
B’E1⊣ = BE1 ⊣
B’E1⊣ >= 10
C’E1⊣ = CE1 ⊣
BR1⊣ = BE1⊣ + (R1⊣ – E1⊣)*-40
BR1⊣ >= 10, >= 30
CR1⊣ = C’E1 ⊣
B’R1⊣ = BE1 ⊣,
B’R1⊣ >= 10, <= 100
C’R1⊣ = CE1 ⊣
"Temporal Planning in Domains with Linear Processes." A. J. Coles, A. I. Coles, M. Fox, and D. Long. IJCAI (2009).
"COLIN: Planning with Continuous Linear Numeric Change." A. J. Coles, A. I. Coles, M. Fox and D. Long. JAIR (44) (2013)
C’R1⊢=CR1⊢
CE1⊢=C’R1⊢
C’E1⊢=CE1⊢+10
Numeric Constraints:
C’E1⊣= CE1⊣
CR1⊣=C’E1⊣
CE1⊣=C’E1⊢+(E1⊣-E1⊢)*1
C’E1⊣ = CE1 ⊣
Linear Programming
General form:
Maximise: z = wz .v
Subject to:
w1 .v {≤, ≥, =} c1
w2 .v {≤, ≥, =} c2
wn .v {≤, ≥, =} cn
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