Risk-bounded Programming on Continuous State...Risk-bounded Programming on Continuous State Prof....
Transcript of Risk-bounded Programming on Continuous State...Risk-bounded Programming on Continuous State Prof....
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Risk-bounded Programming on Continuous State
Prof. Brian Williams
March 30th, 2016
Cognitive Robotics (16.412J / 6.834J)
photo courtesy MIT News
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Today: Risk-bounded Motion Planning
• M. Ono and B. C. Williams, "Iterative Risk Allocation: A New Approach to Robust Model Predictive Control with a Joint Chance Constraint,” IEEE Conference on Decision and Control, Cancun, Mexico, December 2008.
• M. Ono, B. Williams and L. Blackmore, “Probabilistic Planning for Continuous Dynamic Systems under Bounded Risk,” Journal of Artificial Intelligence Research, v. 46, 2013.
After Advanced Lectures: Risk-bounded Scheduling
• C. Fang, P. Yu, and B. C. Williams, “Chance-constrained Probabilistic Simple Temporal Problems,” AAAI, Montreal, CN, 2014.
• A. Wang and B. C. Williams, “Chance-constrained Scheduling via Conflict-directed Risk Allocation,“ AAAI, Austin, TX, January, 2015.
After Advanced Lectures: Risk-bounded Probabilistic Activity Planning
• Santana, P., Thiébaux, S., Williams, B.C., "RAO*: an Algorithm for Chance-Constrained POMDP's,” AAAI, Phoenix, AZ, February 2016.
Homework:
• Changing Pset order; different from syllabus.
• IRA pset soon (next 1-2 weeks)
• Let Steve know what times work for advanced lecture dry runs
Assignments
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• Maximizing utility under bounded risk makes sense.
• Risk allocation can help us solve.
Key takeaways
October 29th, 2015
Risk Bounded Goal-directed Motion Planning 3
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• Review
• Risk-aware Trajectory Planning
• Iterative Risk Allocation (IRA)
• Generalizing to Risk-aware Systems
• Convex Risk Allocation (CRA)
Outline
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Depth Navigation for Bathymetric Mapping – Jan. 23rd, 2008
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Dynamic Execution of State Plans
Command script
00:00 Go to x1,y100:20 Go to x2,y200:40 Go to x3,y3…
04:10 Go to xn,yn
Plant
Commands
Leaute & Williams, AAAI 05© MBARI. All rights reserved. This content is excluded fromour Creative Commons license. For more information,see https://ocw.mit.edu/help/faq-fair-use/.
16.412J / 6.834J – L15 Risk-bounded Programs on Continuous States 63/30/16
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of State Plans
Sulu
Model-based Executive
Observations Commands
“Explore mapping region for at least 100s, then explore bloom region for at least 50s, then return to pickup region.
Avoid obstacles at all times”State Plan
Plant
Leaute & Williams, AAAI 05
Optimal
Robust
Dynamics
+Constraints
16.412J / 6.834J – L15 Risk-bounded Programs on Continuous States 73/30/16
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Execution Dynamic
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Sulu: Dynamic Execution of State Plans
Remain in [safe region]
Remain in
[bloom region]
e1e5
Remain in
[mapping region]e2 e3 e4End in
[pickup region]
[50,70] [40,50]
[0,300]
Obstacle 1
Obstacle 2
Mapping
Region
Bloom
RegionPickup
Region
“Explore bloom region for between 50 and 70seconds. Afterwards, explore mapping regionfor between 40s and 50s. End in the pickup region. Avoid obstacles at all times. Complete the mission within 300s”
Issue: Activities couple through time and state constraints.
Approach: Frame as Model-Predictive Controlusing Mixed Logic or Integer / Linear Programming.
[Leaute & Williams, AAAI 05]
A state plan is a model-based program that is unconditional, timed, and hybrid and provides flexibility in state and time.
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Frame Planning as a Mathematical Program
)1,1,0(
),1,0( )1,1,0(
..
),(min
maxmax
goal
start0
1
, ::
Nk
NkNk
ts
J
k
N
kkk
NNNN
uuuxxxx
gHxBuAxx
uuxx
k
11ux 11Cost function
Dynamics
Spatial constraints
Initial position and velocity
Goal position and velocity
Thrust limits
TkykxkT
kkkkk FFyxyx ,, , ux
)( Nf x
Cost-to-go function
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Encode “Remain In” Constraints, . . .
Remain in [ R ] eEeS
ReTteT kEkS
Nk
k
x)()(0
•Thomas Léauté, "Coordinating Agile Systems through the Model-based Execution of Temporal Plans, " S. M. Thesis,
Massachusetts Institute of Technology, August 2005.
•Thomas Léauté, Brian Williams, “Coordinating Agile Systems Through the Model-based Execution of Temporal Plans,"
Proceedings of the Twentieth National Conference on Artificial Intelligence (AAAI-05), Pittsburgh, PA, July 2005, pp. 114-120.
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• Review
• Risk-aware Trajectory Planning
• Iterative Risk Allocation (IRA)
• Generalizing to Risk-aware Systems
• Convex Risk Allocation (CRA)
Outline
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16.412J / 6.834J – L15 Risk-bounded Programs on Continuous States 123/30/16
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Issue: Frequent Mission Aborts
Minimum Altitude
Planned trajectory
Actual trajectory
Attitude is less than the minimum altitude
Mission abort
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Chanced Constrained, Robust Path Planning
– “Plan optimal path to goal such that p(failure) ≤ Δ.”
p(failure) ≤ Δ
Expected path
October 29th, 2015
Risk Bounded Goal-directed Motion Planning 14
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Risk – Performance Tradeoff• Maximum probability of failure is used to
trade performance against risk-aversion.
October 29th, 2015
Risk Bounded Goal-directed Motion Planning 15
2 4 6 8 10 12 14 160
2
4
6
8
10
12
14
x(meters)
y(m
eter
s)
No uncertaintyΔ = 0.1Δ = 0.001Δ = 0.0001
10−5
10−4
10−3
10−2
10−1
60
65
70
75
80
85
90
95
100
105
110
Fue
l Use
Maximum Probability of Collision (Δ)
Method: Uniform Risk Allocation[Blackmore, PhD]
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Goal-directed, Risk-bounded Planning
Remain in [safe region]
Explore
[bloom region]
e1e5
Explore
[mapping region]e2 e3 e4End in
[pickup region]
[50,70] [40,50]
[0,300]
2. p( End in [pickup region] fails OR Remain in [safe region] fails ) < .1%.
1. p( Remain in [bloom region] fails OR Remain in [mapping region] fails ) < 5%.
Constraints on risk of failure (Chance Constraints):
1. Science Activities
2. Safety Activities
Instance of Chance-constrained Programming.
“Stay over science region with 95% success, avoid collision and achieve pickup with 99.9% success.”
Operator: Specifies acceptable risk.Executive: Decides how to use risk effectively.
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• p-Sulu: Probabilistic Sulu (plan executive)
Control Sequence
Output
Input and Output
p
p-Sulu
Input
Goal
(cc-QSP)
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Example Execution
Remain in [safe region]
Remain in
[bloom region]
e1e5
Remain in
[mapping region]e2 e3 e4End in
[pickup region]
[50,70] [40,50]
[0,300]
T(e1)=0
T(e2)=70
T(e3)=110 T(e4)=150 T(e5)=230
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Problems
Waypoint
Goal
Start
t = 1
t = 5
Convex chance-constrained traj opt
Fixed schedule
Non-convex, chance-constrained traj opt
C
Waypoint
Goal
Start
Obstacle
t = 1
t = 5
Fixed schedule
Flexible schedule (QSP)
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Deterministic Finite-Horizon Optimal Control
ttt
T
tBuAxx
1
1
0
s.t.
itt
iTt
N
i
T
tgxh
11State constraints(Convex)
Discrete-time linear dynamics
Cost function (e.g. fuel consumption)
)(min :1:1
Tu
uJT
T U
Ni
N
i
Ni
N
i
CCCC
CCCC
211
211Notations:
Logicalconjunctions
Logicaldisjunctions
Convex function
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),0(~ tt Nw
Chance-Constrained FH Optimal Control
),(~ 0,00 xxNx
itt
iTt
N
i
T
tgxh
11State constraints
Stochastic dynamics
Gaussian distribution
s.t.
tttt
T
twBuAxx
1
1
0
State estimation error
Exogenous disturbance
)(min :1:1
Tu
uJT
T U
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),0(~ tt Nw
itt
iTt
N
i
T
tgxh
11
Chance-Constrained FH Optimal Control
),(~ 0,00 xxNx
Stochastic dynamics
State constraints
s.t.
tttt
T
twBuAxx
1
1
0
)(min :1:1
Tu
uJT
T U
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),0(~ tt Nw
1Pr
11
itt
iTt
N
i
T
tgxh
Chance-Constrained FH Optimal Control
),(~ 0,00 xxNx
Chance constraint
Stochastic dynamics
Risk bound(Upper bound of the probability of failure)
Assumption: Δ < 0.5
s.t.
tttt
T
twBuAxx
1
1
0
)(min :1:1
Tu
uJT
T U
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Example: Connected Sustainable Homejoint w F. Casalegno & B. Mitchell, MIT Mobile Experience Lab
• Goal: Optimally control HVAC, window opacity, washer and dryer, e-car.
• Objective: Minimize energy cost.
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Qualitative State Plan (QSP)
Maintain room
temperature
Wake
up
Wake
up
Maintain room
temperature
Go to
work
Home
from
work
Go to
sleep
Maintain comfortable
sleeping temperature
[24 hours]
“Maintain room temperature after waking up until I go to work. No temperature constraints while I’m at work, but when I get home, maintain room temperatureuntil I go to sleep. Maintain a comfortable sleeping temperature while I sleep.”
……
[1-3 hour] [6-8 hours] [7-8 hours][7-9 hours]
Sulu [Leaute & Williams, AAAI05]
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Maintain room
temperature
Wake
up
Wake
up
Maintain room
temperature
Go to
work
Home
from
work
Go to
sleep
Maintain comfortable
sleeping temperature
[24 hours]
(p)Sulu Results
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• Review
• Risk-aware Trajectory Planning
• Iterative Risk Allocation (IRA)
• Generalizing to Risk-aware Systems
• Convex Risk Allocation (CRA)
Outline
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Example: Race Car Path Planning
Start
Goal
Walls
Planned Path
Actual PathSafety Margin
Safety Margin
Idea:
• Create safety margin that satisfies the risk bound from start to the goal.
• Reduce to simpler, deterministic optimization problem.
Problem
Find the fastest path to the goal, while limiting the probability of crashthroughout the race to 0.1%
Risk bound
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Executive creates safety margins that satisfy risk bounds while maximizing expected utility
Start
Goal
Safety margin
Walls
(a) Uniform width safety margin
(b) results in better path → takes risk when most beneficial
Start
Goal
Walls
Safety margin
(b) Uneven width safety margin
[Ono & Williams, AAAI 08]Approach: Algorithmic Risk Allocation
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q
Key Idea - Risk Allocation
CornerNarrow safety margin= higher risk
StraightawayWide safety margin= lower risk
• Taking risk at the cornerresults in a shorter path, than taking the same risk at the straightaway.
• Sensitivity of path length to risk is higher at the corner.
• Risk Allocation
– Optimize allocation of risk to time steps and constraints.
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Iterative Risk Allocation (IRA) Algorithm
Iteration
•Descent algorithm
•Starts from a suboptimal risk allocation•Improves it by iteration
)()()( 2*
1*
0* δδδ JJJ
(Refer to paper for proof)
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Iterative Risk Allocation AlgorithmAlgorithm IRA
1 Initialize with arbitrary risk allocation
2 Loop3 Compute the best
available path given the current risk allocation
4 Decrease the risk where the constraint is inactive
5 Increase the risk where the constraint is active
6 End loopStart
Goal
Safety margin
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Iterative Risk Allocation AlgorithmAlgorithm IRA
1 Initialize with arbitrary risk allocation
2 Loop3 Compute the best
available path given the current risk allocation
4 Decrease the risk where the constraint is inactive
5 Increase the risk where the constraint is active
6 End loopStart
Goal
Safety margin
No gap = Constraint is active
Gap = constraint is inactive
Best available path given the safety margin
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Iterative Risk Allocation AlgorithmAlgorithm IRA
1 Initialize with arbitrary risk allocation
2 Loop3 Compute the best
available path given the current risk allocation
4 Decrease the risk where the constraint is inactive
5 Increase the risk where the constraint is active
6 End loopStart
Goal
Safety margin
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Iterative Risk Allocation AlgorithmAlgorithm IRA
1 Initialize with arbitrary risk allocation
2 Loop3 Compute the best
available path given the current risk allocation
4 Decrease the risk where the constraint is inactive
5 Increase the risk where the constraint is active
6 End loopStart
Goal
Safety margin
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Iterative Risk Allocation AlgorithmAlgorithm IRA
1 Initialize with arbitrary risk allocation
2 Loop3 Compute the best
available path given the current risk allocation
4 Decrease the risk where the constraint is inactive
5 Increase the risk where the constraint is active
6 End loopStart
Goal
Safety margin
ActiveInactive
Inactive
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Iterative Risk Allocation AlgorithmAlgorithm IRA
1 Initialize with arbitrary risk allocation
2 Loop3 Compute the best
available path given the current risk allocation
4 Decrease the risk where the constraint is inactive
5 Increase the risk where the constraint is active
6 End loopStart
Goal
Safety margin
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Iterative Risk Allocation AlgorithmAlgorithm IRA
1 Initialize with arbitrary risk allocation
2 Loop3 Compute the best
available path given the current risk allocation
4 Decrease the risk where the constraint is inactive
5 Increase the risk where the constraint is active
6 End loopStart
Goal
Safety margin
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Iterative Risk Allocation AlgorithmAlgorithm IRA
1 Initialize with arbitrary risk allocation
2 Loop3 Compute the best
available path given the current risk allocation
4 Decrease the risk where the constraint is inactive
5 Increase the risk where the constraint is active
6 End loopStart
Goal
Safety margin
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Monterey Bay Mapping Example
Sea floor level
Risk allocation ( )%5Ono & Williams, AAAI 08
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• Review
• Risk-aware Trajectory Planning (IRA)
• Iterative Risk Allocation
• Generalizing to Risk-aware Systems
• Convex Risk Allocation (CRA)
Outline
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Risk-Sensitive Architectures for Exploration• In collaboration with JPL, WHOI and Caltech.
• Initial year study, funded by Keck Institute for Spaces Sciences.
• 2 year follow on for demonstration.
NereId Under IceVenus Sage © Woods Hole Oceanographic Institution. All rights reserved. Thiscontent is excluded from our Creative Commons license. For moreinformation, see https://ocw.mit.edu/help/faq-fair-use/.
This image is in the public domain.
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Risk-aware Planning & Execution
UserKirk
Pike
Sulu
Goals &
models
in RMPL
Control
Commands
Enterprise
Coordinates and monitors task risk
Plans paths within risk bounds
Sketches mission and assigns tasks within risk-bounds
Burton
Plans actions within risk bounds.
Bones
Diagnoses incipient failures
Uhura
Collaboratively adapts goal to reduce risk
Start
GoalWalls
Safety margin
11:00 a.m. 4:00 p.m. 10:00 p.m.
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Falkor Cruise – March-April, 2015
Falkor - Schmidt Ocean Institute
Slocum Glider
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© Schmidt Ocean Institute. All rights reserved. This contentis excluded from our Creative Commons license. For moreinformation, see https://ocw.mit.edu/help/faq-fair-use/.
© Schmidt Ocean Institute. All rights reserved. This contentis excluded from our Creative Commons license. For moreinformation, see https://ocw.mit.edu/help/faq-fair-use/.
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Glider Primer
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User Kirk
Pike
Sulu-lite
Goals &
models
in RMPL
Control
Commands
Enterprise
Coordinates and monitors tasks
Plans paths
Sketches mission and assigns tasks
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Activity Planning in Scott Reef
region1
no-fly zone
region2region3
region4
no-fly zone
ship1
Iver1
glider
Iver2
10:00-11:30am
10:30-13:00
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START POINT
END GOAL
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April/May 2016 Deployment:Slocum Glider at Santa Barbara• Mission goal:
– Use miniaturized mass spectrometer to find and characterize oil seeps off the coast of Santa Barbara.
• Primary research goal:– Adaptive science using planning and rule-based algorithms.
• Secondary research goal:– Risk-aware path planning.
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Advisory System for WHOI Cruise AT26-06: San Francisco, California to Los Angeles
• Find and sample methane seeps near the coast.
• Locate and sample a 60 year-old DDT dumping site.
• Recover and replace incubators on the seafloor.
Courtesy WHOI
Yu, Fang, Williams, Camilli, Fall 13
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• [Day 1] Jason failed after 30 min into its first dive, entered an uncontrollable spin and broke its optic fiber tether.
• [Day 1] The new camera installed on Sentry did not work well in low light situations. It had been replaced during its second dive.
Everything Can Go Wrong
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User
PTS
pUhura
pKirk
pSulu
Goals in
Natural
Language
Control
Commands
pEnterprise
Collaboratively diagnoses and resolves goal failures
Plans mission and contingencies.
Navigates within risk bounds
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• Review
• Risk-aware Trajectory Planning
• Iterative Risk Allocation (IRA)
• Generalizing to Risk-aware Systems
• Convex Risk Allocation (CRA)– Intuitions
– Math (optional)
Outline
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Problems
Waypoint
Goal
Start
t = 1
t = 5
Convex chance-constrained traj opt
Fixed schedule
Non-convex, chance-constrained traj opt
C
Waypoint
Goal
Start
Obstacle
t = 1
t = 5
Fixed schedule
Flexible schedule (QSP)
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Risk-Allocation Overview: One Variable
Idea 1: We easily solve a chance constrained problem with one linear constraint C and one normally distributed random variable x, by reformulating C to a deterministic constraint C’ on .
< 1%
C(x)Chance constraint:
Risk < 1%x m
x
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C '(x) C(x)m
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Risk-Allocation Overview: Many Variables
Idea 2: Generalize to a single constraint over an
N-dimensional random variable,
by projecting its distribution onto the axis perpendicular to the constraint boundary.
99.9%
99%
90%
80%
Chance constraint:
Risk < 1%
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Risk-Allocation Overview: Many Constraints
99.9%
99%
90%
80%
Chance constraint:
Risk < 1%
Idea 3: Generalize to a joint chance-constraint over multiple constraints C1, C2,
by distributing risk.
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Risk-Allocation Overview: Many Constraints
99.9%
99%
90%
80%
Chance constraint:
Risk < 1%
Find a solution such that:
1. Each constraint Ci takes less than δi risk, and
2. Σi δi ≤ 1%
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Risk-Allocation Overview: Conservatism
99.9%
99%
90%
80%
Chance constraint:
Risk < 1%
Conservatism
Significantly less conservative than the elliptic approximation, especially in a high-dimensional problem.
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• Review
• Risk-aware Trajectory Planning
• Iterative Risk Allocation (IRA)
• Generalizing to Risk-aware Systems
• Convex Risk Allocation (CRA)– Intuitions
– Math (optional)
Outline
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),0(~ tt Nw
1Pr
11
itt
iTt
N
i
T
tgxh
Reformulation: Now Lets Do the Math!
),(~ 0,00 xxNx
Chance constraint
Stochastic dynamics
Risk bound(Upper bound of the probability of failure)
Assumption: Δ < 0.5
s.t.
tttt
T
twBuAxx
1
1
0
)(min :1:1
Tu
uJT
T U
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Conversion of Joint Chance Constraint
1Pr
00
itt
iTt
N
i
T
tgxhJoint chance
constraint
Intractable- Requires computation of an integral over a multivariate Gaussian.
1
A set of individual chance constraints.- involves univariate Gaussian distribution.
2
A set of deterministic state constraint.
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Decomposition of Joint Chance Constraint
1Pr
00
itt
iTt
N
i
T
tgxhJoint chance
constraint
Using Boole’s inequality (union bound)
BABA PrPrPr
1
Where A and B denote constraint failures
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1Pr
00
itt
iTt
N
i
T
tgxhJoint chance
constraint
is implied by:
0
1Pr
it11
,
11
N
i
T
t
it
it
it
itt
iTt
N
i
T
tgxh
Individual chanceconstraints
Upper bound of the probability of violating ithconstraint at time t
Upper bound of the probability of violating anyconstraints over the planning horizon
Risk allocation:
NT 2
111 ,δ
1Decomposition of Joint Chance Constraint
Variable
Constant
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)(min:1
UJT
Tu U
),0(~ tt Nw
),(~ 0,00 xxNx
it
it
it
itt
iTtit
gxh
,
1Pr
Individual chanceconstraints
1Pr
00
itt
iTt
N
i
T
tgxhJoint chance
constraint
δmin
Risk allocationoptimization
1Decomposition of Joint Chance Constraint
s.t.
tttt
T
twBuAxx
1
1
0
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itit
itt
iTt mgxh Deterministic constraint
it
itt
iTt gxh 1PrChance constraint
Conversion to Deterministic Constraint2
tiTt xh
itg
Mean state (deterministic variable)
Prob. Distribution of t
iTt xh
it
it
itm
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itit
itt
iTt mgxh Deterministic constraint
Safety Margin
it
itt
iTt gxh 1PrChance constraint
1
00,, )(
t
nx
Tnw
ntx AA
),(~ ,txtt xNx
12erf2 1, i
tittx
iTt
it
it hhm
where
(Inverse of cdf of Gaussian)
[Charnes et. al. 1959]
t = 1
t = 2
t = 3t = 4t = 5
Safety Margin
Mean stateNominal path
11 m
22 m
Conversion to Deterministic Constraint2
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)(min :1:1
Tu
uJT
T U
),0(~ tt Nw
),(~ 0,00 xxNx
it
it
it
itt
iTt
I
i
T
tgxh
,
111Pr
Individual chanceconstraints
δmin
s.t.
tttt
T
twBuAxx
1
1
0
Conversion to Deterministic Constraint2
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)(min :1:1
Tu
uJT
T U
it
it
it
it
itt
iTt
I
i
T
tmgxh
,
11
δmin
s.t.
ttt
T
tBuxAx
1
1
0
Conversion to Deterministic Constraint2
Convex if δ < 0.5
1.00.50
it
m ConvexConvex optimization
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• Maximizing utility under bounded risk makes sense.
• Risk allocation can help us solve.
Key takeaways
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Summary: Risk Allocation
Iteration
1. IRA: reallocates risk manually.2. CRA,NRA: standard solver reallocates risk.
)()()( 2*
1*
0* δδδ JJJ
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Approach: Programming Cognitive Systems
1. An embedded programming language elevated to the goal-level through partial specification and operations on state (RMPL).
2. A language executive that achieves robustness by reasoning over constraint-based models and by bounding risk (Enterprise).
3. Interfaces that support natural human interaction fluidly and at the cognitive level (Uhura).
This image is in the public domain.
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Risk-aware Planning & Execution
User
Kirk
Pike
Sulu
Goals &
models
in RMPL
Control
Commands
Enterprise
Coordinates and monitors task risk
Plans paths within risk bounds
Sketches mission and assigns tasks within risk-bounds
Burton
Plans actions within risk bounds.
Bones
Diagnoses incipient failures
Uhura
Collaboratively adapts goal to reduce risk
Start
Goal
Walls
Safety margin
11:00 a.m. 4:00 p.m. 10:00 p.m.
© source unknown. All rights reserved. This content isexcluded from our Creative Commons license. For moreinformation, see https://ocw.mit.edu/help/faq-fair-use/.
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APPENDIX: RISK-BOUNDED PLANNING FOR NON-CONVEX PROBLEMS
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Next …
Waypoint
Goal
Start
t = 1
t = 5
Convex chance-constrained trajectory optimization
Fixed schedule
Non-convex, chance-constrained traj opt
C
Waypoint
Goal
Start
Obstacle
t = 1
t = 5
Fixed schedule
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Non-Convex Problem Formulation
Non-convex state constraint
C
Waypoint
Goal
Start
Obstacle
t = 1
t = 5
Fixed schedule
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Problem Formulation: Non-Convex Chance Constraint
1Pr ..
)(min
11
:1
ntt
nTt
N
n
T
t
ti
u
gxhts
uJT
T U
CC over convex state constraints
1
A set of individualchance constraints
2
A set of deterministicstate constraints
A joint chance constraints
NT
nt
nt
nt
nt
ntt
nTt
N
n
T
t
ti
u
mgxhts
uJT
T
,
1,1
11
..
)(min:1
U
Convex Deterministic Program
Risk allocation
78Risk Bounded Goal-directed Motion Planning
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Problem Formulation: Non-Convex Chance Constraint
1Pr ..
)(min
11
:1
ntt
nTt
N
n
T
t
ti
u
gxhts
uJT
T U
CC over convex state constraints
NT
nt
nt
nt
knt
kntt
kTnt
K
k
N
n
T
t
ti
u
mgxhts
uJT
T
,
1,1
,,,
111
..
)(min:1
U
Convex Disjunctive Deterministic Program
NT
nt
nt
nt
nt
ntt
nTt
N
n
T
t
ti
u
mgxhts
uJT
T
,
1,1
11
..
)(min:1
U
Convex Deterministic Program
1Pr ..
)(min
,,
111
:1
kntt
kTnt
K
k
N
n
T
t
ti
u
gxhts
uJT
T U
CC over non-convex state constraints
Risk allocation Risk allocationRisk selection
79
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Solving Disjunctive Programusing Branch and Bound
NT
nt
nt
nt
knt
kntt
kTnt
K
k
N
n
T
t
ti
u
mgxhts
uJT
T
,
1,1
,,,
111
..
)(min:1
U
Convex Disjunctive Programming
nt
kt
ktt
kTttk mgxhCKNT ,1,1,1 ;2,1,2
nt
knt
kntt
kTnt
K
k
N
n
T
tmgxh ,,,
111
)()( 22211211 CCCC
Example:
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Stochastic DLP Branch and Bound
)()( 22211211 CCCC
)( 222111 CCC )( 222112 CCC
2111 CC 2211 CC 2112 CC 2212 CC
Convex Optimization Problems
Repeat until no clauses left:1. Select clause.2. Split on disjuncts.
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)()( 22211211 CCCC
Stochastic DLP Branch and Bound
)( 222111 CCC )( 222112 CCC
2111 CC 2211 CC 2112 CC 2212 CC
Convex Optimization Problems
Waypoint
Goal
Start
t = 1
t = 5
Convex, chance-constrained
Fixed scheduleRepeat until no clauses left:
1. Select clause.2. Split on disjuncts.
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Bound Sub-ProblemsThrough Convex Relaxation
)()( 22211211 CCCC
)( 222111 CCC )( 222112 CCC
2111 CC 2211 CC 2112 CC 2212 CC
• Bound: Remove all disjunctive clauses [Li & Williams 2005].
• Issue: Computing bound is slow!!
• Cause: Sub-problems include non-linear constraints.
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B&B Subproblem (non-linear)
),0(~ tt Nw
s.t. ttttTtwBuAxx
110
),(~ 0,00 xxNx
)(min :1:1
TuuJ
T
0 ,1
)()()(
1
t
T
tt
tti
tti
ttTti
t
T
tmgxh
Nonlinear
1.00.50
it
m
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B&B Subproblem Relaxation (Linear)
),0(~ tt Nw
s.t. ttttTtwBuAxx
110
),(~ 0,00 xxNx
)(min :1:1
TuuJ
T
0 ,1
)()()(
1
t
T
tt
tti
tti
ttTti
t
T
tmgxh
Nonlinear
t
Fixed RiskRelaxation
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B&B Subproblem Relaxation (Linear)
),0(~ tt Nw
s.t. ttttTtwBuAxx
110
),(~ 0,00 xxNx
)(min :1:1
TuuJ
T
)()()(
1
tit
titt
Ttit
T
tmgxh
Constant
t
Fixed RiskRelaxation
•All constraints are linear (FRR is typically LP or QP).
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FRR Intuition
• Results in an infeasible solution to the original problem.
• Gives lower bound on the cost of the original problem.
Start
Goal
Safety margin
Start
Goal
FRR Safety margin
FRROriginal problemSets safety margin for all constraints to max risk Δ.
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Problems
Waypoint
Goal
Start
t = 1
t = 5
Convex traj opt
Fixed schedule
Non-convex, traj opt
C
Waypoint
Goal
Start
Obstacle
t = 1
t = 5
Fixed schedule
C
Waypoint
Goal
Start
Obstacle
[1 3]
[2 4]
[0 5]
Simple temporalconstraints
Goal-directed qualitative state plan traj opt 88
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APPENDIX: OVERVIEW AND ALTERNATIVE APPROACHES
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What are the uncertainties and risks?
Airport
UAV
NFZ
Scenic region
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Robust Model Predictive Control
• Receding horizon (MPC) planners react to uncertainty after something goes wrong.
• Can we take precautionary actions before something goes wrong?
•Ali A. Jalali and Vahid Nadimi, “A Survey on Robust Model Predictive Control from 1999-2006.”
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Robust versus Chance ConstrainedR
obust
Pre
dic
tive C
ontr
ol
x
y
Assume bounds on
uncertainty.
Ch
an
ce-c
onstr
ain
ed
Pre
dic
tive
Co
ntr
ol Wind speed
Wind speed
99%
90%80% x
y
Assume probability
distribution that
characterizes uncertainty.
No Fly Zone
No Fly Zone
• Predicted position has bounded uncertainty.
• Problem: Find control sequence that satisfies
constraints for all realizations of uncertainty.
t =1 t =2
99.9%99%
90%80%
99.9%
99%
90%
80%
t =1 t =2
• Predicted position has probabilistic uncertainty.
• Problem: Find control sequence that satisfies
constraints within a probability bound
(Chance Constraint). 92October 29th, 2015
- 99
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Incorporating Uncertainty
• Deterministic discrete-time LTI model.
• Additive uncertainty
• Multiplicative uncertainty
ttt BuAxx 1
ttt BuxAAx )(1
tttt wBuAxx 1
x
y
W
wt W
xy
p(x,y)
p(wt )= N(wt,P0 )
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What to Minimize? (Bounded Uncertainty)
• Minimize the worst case cost
• Minimize nominal cost
yuncertaint Bounded :
..
Cost when :)(min
Ww
gxhts
wJitt
iTtWw
0U,XU
yuncertaint Bounded :
..
)(maxmin
Ww
gxhts
Jitt
iTtWw
Ww
UX,
U
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What to Minimize? (Stochastic Uncertainty)
• Utilitarian approach
• Chance constrained optimization
)()(min UU,XU
pfJ
Probability of failurePenalty (constant)
)( ..
)(min
U
U,XU
fts
J
Probability of failure Risk boundOctober 29th, 2015
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Solution Methods forChance-Constrained Problems• Sampling based methods
– Scenario-based• Bernardini and Bemporad, 2009
– Particle control• Blackmore et al., 2010
• Non-sampling-based methods
– Elliptic approximation (direct extension of robust predictive control)
• van Hessem, 2004
– Risk allocation• Ono and Williams, 2008
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Particle Control1. Use particles to sample random variables.
Obstacle 1
Obstacle 2
Goal Region
Initial state distribution.
Particles approximating initial state distribution.
)(~)(t
it p )(~ 0,
)(0, c
ic p xx Ni 1 Ft 0
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Particle Control2. Calculate future state trajectory for each particle, leaving explicit,
dependence on control inputs u0:T-1.
Obstacle 1
Obstacle 2
Goal Region
)(,
)(0,
)(:0,
iFc
ic
iTc
x
xx
Particle 1 for u = uB
Particle 1 for u = uA
t=0
t=1
t=2
t=3
t=4
t=0
t=1
t=2
t=3
t=4
),,( )(1:0
)(0,1:0
)( it
ictt
it f xux
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Particle Control2. Calculate future state trajectory for each particle, leaving explicit,
dependence on control inputs u0:T-1.
Obstacle 1
Obstacle 2
Goal RegionParticles 1…Nfor u = uB
Particles 1…Nfor u = uA
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Particle Control3. Express chance-constraints of optimization problem approximately in
terms of particles.
Probability of failureTrue expectation approximated by the approximated byfraction of failing sample mean of costparticles.function:.
Sample meanapproximates state mean.
Obstacle 1
Obstacle 2
Goal Region
t=0t=1
t=2
t=3
t=4
E h(u0:F1,x1:F )
1N
h(u0:F1,x1:F(i) )
i1
N
LB4
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Particle Control4. Solve approximate deterministic optimization problem for u0:F-1.
Obstacle 1
Obstacle 2
Goal Region
t=0
t=1
t=2
t=3
t=4
10% of particles fail in optimal solution.
δ = 0.1
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Convergence
– As N∞, approximation becomes exact.
Obstacle 1
Obstacle 2
Goal Region
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Convergence
– As N∞, approximation becomes exact.
Obstacle 1
Obstacle 2
Goal Region
10% probability of failure.
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Solution Methods forChance-Constrained Problems• Sampling based methods
– Scenario-based• Bernardini and Bemporad, 2009
– Particle control• Blackmore et al., 2010
• Non-sampling-based methods
– Elliptic approximation (direct extension of robust predictive control)
• van Hessem, 2004
– Risk allocation• Ono and Williams, 2008
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Elliptic ApproximationNo Fly Zone
1. Derive probability distribution over future states as a function of control inputs.
Chance constraint:
Risk < 1%
Note: When planning in an N-dimensional state space over time steps, a joint distribution over an N-dimensional space must be considered.
99.9%
99%
90%
80%
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Elliptic ApproximationNo Fly Zone
1. Derive probability distribution over future states as a function of control inputs.
2. Find a 99% probability ellipse.
Chance constraint:
Risk < 1%
99.9%
99%
90%
80%
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Elliptic ApproximationNo Fly ZoneChance constraint:
Risk < 1%
1. Derive probability distribution over future states as a function of control inputs.
2. Find a 99% probability ellipse.
3. Find control sequence that makes sure the probability ellipse is within the constraint boundaries.
99.9%
99%
90%
80%
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Conservatism of Elliptic ApproximationNo Fly Zone 99.9%
99%
90%
80%
Issue: often very conservative
Real probability of failure
Probability density function
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Conservatism of Elliptic ApproximationNo Fly Zone 99.9%
99%
90%
80%
Issue: often very conservative.
Conservatism
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