motion planning for ltl specifications: a satisfiability ... dynamics (with input constraints): x...

Motion Planning for LTL Specifications: ASatisfiability Modulo Convex Optimization Approach

Yasser ShoukryUC Berkeley, UCLA, and UPenn

Joint work with

Pierluigi Nuzzo (UC Berkeley), Indranil Saha (IIT Kanpur),Alberto Sangiovanni-Vincentelli (UC Berkeley), Sanjit A. Seshia (UC Berkeley),

George Pappas (UPenn), and Paulo Tabuada (UCLA)

Yasser Shoukry ExCAPE PI Meeting 2016 May 9, 2016 1 / 12

Motion Planning ProblemGiven:

Robot Dynamics (with input constraints):

xt+1 = Axt + But , x0 = x , ‖ut‖∞ ≤ u ∀t ∈ N

WorkspaceW:

all obstacles are assumed to be unions of polyhedra.

Atomic propositions Π = {π1, . . . , πm}: defined over the free-workspace.

LTL Specification Φ:

for simplicity, I will focus on reach-avoid problems: ♦π1 ∧�¬π0.

Objective:

Generate the input sequence u0, u1, . . . , uL such that the trajectory of the robotsatisfies Φ.

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WorkspaceW:

all obstacles are assumed to be unions of polyhedra.Atomic propositions Π = {π1, . . . , πm}: defined over the free-workspace.



Objective:


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WorkspaceW: all obstacles are assumed to be unions of polyhedra.

Atomic propositions Π = {π1, . . . , πm}: defined over the free-workspace.



Objective:


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WorkspaceW: all obstacles are assumed to be unions of polyhedra.Atomic propositions Π = {π1, . . . , πm}: defined over the free-workspace.



Objective:


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for simplicity, I will focus on reach-avoid problems: ♦π1 ∧�¬π0.Objective:


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LTL Specification Φ:for simplicity, I will focus on reach-avoid problems: ♦π1 ∧�¬π0.

Objective:


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LTL Specification Φ:for simplicity, I will focus on reach-avoid problems: ♦π1 ∧�¬π0.

Objective:Generate the input sequence u0, u1, . . . , uL such that the trajectory of the robotsatisfies Φ.

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Solver Architecture [0/3]

Step 0: Compute coarse,obstacle-based, discretization of the freespace.

Avoid discrete state-spaceexplosion (no abstraction ofcontinuous dynamics, no grid-basedabstractions).

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Step 1: For a given horizon L, use a SATsolver to find a candidate high-level path(Bounded Model Checking) that satisfies theLTL specification.

Increase the horizon L until suchhigh-level path is found.

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SAT-solver(Bounded Model Checking)

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CON-PLAN.CHECK

CON-PLAN.CERTIFICATE

CON-PLAN

L, φLTL

ρ = π1, π17 . . . , π28

(A,B, x , u)

u0, u1, . . . , uL−1

φcert



Step 1: For a given horizon L, use a SATsolver to find a candidate high-level path(Bounded Model Checking) that satisfies theLTL specification.

Increase the horizon L until suchhigh-level path is found. ⇡4

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π1

π2

π9

π17

π3

π10

CON-PLAN.CHECK


CON-PLAN

L, φLTL

ρ = π1, π17 . . . , π28

(A,B, x , u)

u0, u1, . . . , uL−1

φcert



Step 2: Check whether the high-leveltrajectory ρ is feasible (satisfies therobot initial state, dynamics, andinput constraints).

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π1

π2

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π3

π10

CON-PLAN.CHECK


CON-PLAN

L, φLTL

ρ = π1, π17 . . . , π28

(A,B, x , u)

u0, u1, . . . , uL−1

φcert




Can be casted as anoptimization problem.

Problem (CON-PLAN.CHECK)

min 1 i = 0, . . . , L

(initial condition) x0 = x ,

(dynamics constraints) xi+1 = Axi + Bui

(input constraints) ‖ui‖ ≤ u,

(plan constraints) xi ∈ ρi


π1

π2

π9

π17

π3

π10

CON-PLAN.CHECK


CON-PLAN

L, φLTL

ρ = π1, π17 . . . , π28

(A,B, x , u)

u0, u1, . . . , uL−1

φcert




Can be casted as a convexoptimization problem (LinearProgram).

Problem (CON-PLAN.CHECK)

min 1 i = 0, . . . , L

(initial condition) x0 = x ,

(dynamics constraints) xi+1 = Axi + Bui

(input constraints) ‖ui‖ ≤ u,

(plan constraints) xi ∈ ρi


π1

π2

π9

π17

π3

π10

CON-PLAN.CHECK


CON-PLAN

L, φLTL

ρ = π1, π17 . . . , π28

(A,B, x , u)

u0, u1, . . . , uL−1

φcert



Step 3: Generate counter example.

φtriv-ce :=L∨

i=0

¬ρi ,

Repeat.

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π1

π2

π9

π17

π3

π10

CON-PLAN.CHECK


CON-PLAN

L, φLTL

ρ = π1, π17 . . . , π28

(A,B, x , u)

u0, u1, . . . , uL−1

φcert




φtriv-ce :=L∨

i=0

¬ρi ,

Repeat.⇡4

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π1

π2

π9

π17

π3

π10

CON-PLAN.CHECK


CON-PLAN

L, φLTL

ρ = π1, π17 . . . , π28

(A,B, x , u)

u0, u1, . . . , uL−1

φcert




φtriv-ce :=L∨

i=0

¬ρi ,

Repeat.⇡3

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π1

π2

π9

π17

π3

π10

CON-PLAN.CHECK


CON-PLAN

L, φLTL

ρ = π1, π17 . . . , π28

(A,B, x , u)

u0, u1, . . . , uL−1

φcert


Succinct Counter-example Generation

Detecting the earliest possible occurrence of an infeasible transition between tworegions rules out a broader class of assignments to Boolean variables.

Example 1: φcounter-example := ¬ρ0 ∨ ¬ρ1 ∨ ¬ρ2 ∨ ¬ρ3

Example 2: φcounter-example := ¬ρ0 ∨ ¬ρ1

ρ0

ρ1 = 0

ρ2 = 0

ρ3 = 0(0, 0, 0)

ρ3 = 1(0, 0, 1)

ρ2 = 1

ρ3 = 0(0, 1, 0)

ρ3 = 1(0, 1, 1)

ρ1 = 1

ρ2 = 0

ρ3 = 0(1, 0, 0)

ρ3 = 1(1, 0, 1)

ρ2 = 1

ρ3 = 0(1, 1, 0)

ρ3 = 1(1, 1, 1);


Succinct Counter-example Generation

Detecting the earliest possible occurrence of an infeasible transition between tworegions rules out a broader class of assignments to Boolean variables.

Example 1: φcounter-example := ¬ρ0 ∨ ¬ρ1 ∨ ¬ρ2 ∨ ¬ρ3

Example 2: φcounter-example := ¬ρ0 ∨ ¬ρ1

ρ0

ρ1 = 0

ρ2 = 0

ρ3 = 0(0, 0, 0)

ρ3 = 1(0, 0, 1)

ρ2 = 1

ρ3 = 0(0, 1, 0)

ρ3 = 1(0, 1, 1)

ρ1 = 1

ρ2 = 0

ρ3 = 0(1, 0, 0)

ρ3 = 1(1, 0, 1)

ρ2 = 1

ρ3 = 0(1, 1, 0)

ρ3 = 1(1, 1, 1)


Succinct Counterexample Generation

Searching for succinct counterexample can be performed by checking thefeasibility of prefixes of ρ.

Leads to solving multiple optimization problems.

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Succinct Counterexample Generation

Key insight: we can check feasibility of high-level plans and generate theshortest counter example by solving a single linear program.

Satisfiability modulo convex-optimization approach.


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Convex Optimization

min u0,...,uL∈Rm

v0,...,vL∈Rm

su0 ,...,s

uL∈R

sv0 ,...,s

vL∈R

x1,...,xL+1∈Rn

∑Li=0 su

i + svi

subject tox0 = x ,hX→W(xi ) ∈ ρW

i , i = 1, . . . , L + 1xi+1 = Axi + Bui + B′vi i = 0, . . . , L‖ui‖ ≤ u + su

i , i = 0, . . . , L‖vi‖ ≤ sv

i , i = 0, . . . , L0 ≤ su

i , 0 ≤ svi i = 0, . . . , L

sε

(∑i−1k=0 su

k + svk

)≤ su

i + svi i = 1, . . . , L

L, φLTL

ρ = π1, π17 . . . , π28

(A,B, x , u)

u0, u1, . . . , uL−1

φcert


Case Study 1: Dubin’s Vehicle

Robot dynamics (Dubin car): x = v cos θ y = v sin θ θ = ω

Dynamics can be transformed into a linear chain of integrators usingdynamic feedback linearization.

Workspace: 30m× 30m maze-like workspace.

We increase number of passages from 1 to 4.

We compare execution-time against (1) standard RRT and (2) LTL OPT tool(mixed-integer linear program).

0 10 20 300

10

20

30

x [m]

y[m

]

0 10 20 300

10

20

30

x [m]

y[m

]


Case Study 1: Dubin’s Vehicle

Robot dynamics (Dubin car): x = v cos θ y = v sin θ θ = ω

Dynamics can be transformed into a linear chain of integrators usingdynamic feedback linearization.

Workspace: 30m× 30m maze-like workspace.

We increase number of passages from 1 to 4.

We compare execution-time against (1) standard RRT and (2) LTL OPT tool(mixed-integer linear program).

Number SMT-Based Motion Planner [s] RRT LTLof Discrete DIS-PLAN CON-PLAN [s] OPT

passages abstraction [s]1 1.9975 0.1360 0.2542 10.4381 > 72002 7.1461 1.1290 0.9294 122.3017 time out3 19.3267 3.6495 1.0053 423.6957 time out4 43.0985 4.0913 1.9204 1002.4193 time out


Case Study 2: Scalability Results

Maze-like workspace with increasing number of passages.

We increase the number of states n (randomly generate the matrices A and B).

Take average across 10 runs.

5 10 15 20 25 30 35 40 45 50

100

101

102

Number of states n

Exe

cutio

ntim

e(s

)

1-passage maze 2-passage maze 3-passage maze 4-passage maze


Final Remarks

What is still missing ?

We need to compare with other methods like RRT∗ and PRM.We need to compare against existing benchmarks.

SAT + Convex optimization ?= tools:

Probabilistic CTL (PCTL) verification of Markov Decision Processes.Security: secure state estimation.Sensor Networks: localization.Motion planning (this work).We are currently developing a comprehensive theory of SatisfiabilityModulo Convex Optimization.


Final Remarks


We need to compare with other methods like RRT∗ and PRM.

We need to compare against existing benchmarks.




Final Remarks






Final Remarks




Probabilistic CTL (PCTL) verification of Markov Decision Processes.

Security: secure state estimation.Sensor Networks: localization.Motion planning (this work).We are currently developing a comprehensive theory of SatisfiabilityModulo Convex Optimization.


Final Remarks




Probabilistic CTL (PCTL) verification of Markov Decision Processes.Security: secure state estimation.

Sensor Networks: localization.Motion planning (this work).We are currently developing a comprehensive theory of SatisfiabilityModulo Convex Optimization.


Final Remarks




Probabilistic CTL (PCTL) verification of Markov Decision Processes.Security: secure state estimation.Sensor Networks: localization.

Motion planning (this work).We are currently developing a comprehensive theory of SatisfiabilityModulo Convex Optimization.


Final Remarks




Probabilistic CTL (PCTL) verification of Markov Decision Processes.Security: secure state estimation.Sensor Networks: localization.Motion planning (this work).

We are currently developing a comprehensive theory of SatisfiabilityModulo Convex Optimization.


Final Remarks






motion planning for ltl specifications: a satisfiability ... dynamics (with input constraints): x...

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