Parametric Statistical model checking of UAV flight plan 1 Ran Bao 12 - PowerPoint PPT Presentation

Parametric Statistical model checking of UAV flight plan 1 Ran Bao 12 e 2 ıt Delahaye 2 Christian Attiogb´ Benoˆ 1 PIXIEL GROUP, Nantes, France 2 Universit´ e de Nantes - LS2N, UMR 6004 - Nantes, France SynCoP 2019 1 Thanks to Didier Lime and Paulin Fournier Benoˆ ıt Delahaye (Univ Nantes/LS2N) Parametric SMC of UAV flight plan 2019-04-06 1 / 29

Introduction Motivation UAVs flying above a crowd (Entertainment) Benoˆ ıt Delahaye (Univ Nantes/LS2N) Parametric SMC of UAV flight plan 2019-04-06 2 / 29

Introduction Motivation UAVs flying above a crowd (Entertainment) ⇒ How to ensure that the flight is safe? Benoˆ ıt Delahaye (Univ Nantes/LS2N) Parametric SMC of UAV flight plan 2019-04-06 2 / 29

Introduction Contribution We propose a model of the UAV system ◮ In the context of a flight plan ◮ Parametric: takes into account ◮ Sensor precision and failure ◮ Wind force ◮ Allows to predict the trajectory We propose and use parametric statistical model checking techniques ◮ Computes an approximation of the probability of satisfying a property ◮ as a parametric function ◮ polynomial ◮ with parametric confidence intervals Benoˆ ıt Delahaye (Univ Nantes/LS2N) Parametric SMC of UAV flight plan 2019-04-06 3 / 29

Introduction Outline Introduction Parametric Markov Chains and Properties Background - Properties Parametric Markov Chains Monte Carlo and pMCs UAV flight model Safety zones Drone components Formal model Experimental results Prototype implementation Drone model Results Discussion Benoˆ ıt Delahaye (Univ Nantes/LS2N) Parametric SMC of UAV flight plan 2019-04-06 4 / 29

Parametric Markov Chains and Properties Outline Introduction Parametric Markov Chains and Properties Background - Properties Parametric Markov Chains Monte Carlo and pMCs UAV flight model Safety zones Drone components Formal model Experimental results Prototype implementation Drone model Results Discussion Benoˆ ıt Delahaye (Univ Nantes/LS2N) Parametric SMC of UAV flight plan 2019-04-06 5 / 29

Parametric Markov Chains and Properties Background - Properties Markov Chains Definition (Markov chain) A Markov chain (MC, for short) is a tuple M = ( S , s 0 , P ) where S is a denumerable set of states, s 0 ∈ S is the initial state and P : S × S → [0 , 1] is the transition probability function. ◮ Finite run: ρ = s 0 s 1 . . . s n s.t. P ( s i , s i +1 ) > 0 ◮ Γ( l ): set of all runs of length l in M ◮ Probability of a finite run: P M ( ρ ) = Π n i =1 P ( s i − 1 , s i ) Benoˆ ıt Delahaye (Univ Nantes/LS2N) Parametric SMC of UAV flight plan 2019-04-06 6 / 29

Parametric Markov Chains and Properties Background - Properties Properties In the context of SMC, we only consider properties on bounded runs. Let r : Γ( l ) → R be a reward function. Reachability P M ( ♦ ≤ l s ). ρ | = ♦ ≤ l s , if there exists i ≤ l such that s i = s . Safety P M ( � = l E ). ρ | = � = l E , if for all i ≤ l , s i ∈ E . M ( r ) = � Expected reward E l M ( r ). E l ρ ∈ Γ( l ) P M ( ρ ) r ( ρ ) is the expected value of r on the runs of length l . Remark For any property ϕ ⊆ Γ( l ), P M ( ϕ ) = E l M ( ✶ ϕ ) ⇒ we focus on properties of the form E l M ( r ). Benoˆ ıt Delahaye (Univ Nantes/LS2N) Parametric SMC of UAV flight plan 2019-04-06 7 / 29

Parametric Markov Chains and Properties Parametric Markov Chains Outline Introduction Parametric Markov Chains and Properties Background - Properties Parametric Markov Chains Monte Carlo and pMCs UAV flight model Safety zones Drone components Formal model Experimental results Prototype implementation Drone model Results Discussion Benoˆ ıt Delahaye (Univ Nantes/LS2N) Parametric SMC of UAV flight plan 2019-04-06 8 / 29

Parametric Markov Chains and Properties Parametric Markov Chains Parametric Markov Chains (pMCs) Definition (Parametric Markov chain) A Parametric Markov chain is a tuple M = ( S , s 0 , P , X ) such that S is a finite set of states, s 0 ∈ S is the initial state, X is a finite set of parameters, and P : S × S → Poly ( X ) is a parametric transition probability function. If v ∈ R X is a valuation of the parameters, ◮ P v : transition probabilities under v : P v ( s , s ′ ) = P ( s , s ′ )( v ) ◮ v is valid if ( S , s 0 , P v ) is a MC ◮ M v = ( S , s 0 , P v ) ◮ Runs and probabilities are similar to MC, but parametric Benoˆ ıt Delahaye (Univ Nantes/LS2N) Parametric SMC of UAV flight plan 2019-04-06 9 / 29

Parametric Markov Chains and Properties Parametric Markov Chains Example 1 q 1 p r 1 2 0 . 5 p 0 1 0 . 5 3 4 q r pMC M 1 Benoˆ ıt Delahaye (Univ Nantes/LS2N) Parametric SMC of UAV flight plan 2019-04-06 10 / 29

Parametric Markov Chains and Properties Parametric Markov Chains Example 1 q 1 0 . 5 1 p r 0 . 5 1 2 1 2 0 . 5 0 . 5 p 0 . 5 0 0 1 1 0 . 5 0 . 5 3 4 3 4 q 0 . 5 r MC M v pMC M 1 1 for parameter valuation v such that v ( p ) = v ( q ) = 0 . 5 and v ( r ) = 0 Benoˆ ıt Delahaye (Univ Nantes/LS2N) Parametric SMC of UAV flight plan 2019-04-06 10 / 29

Monte Carlo and pMCs Outline Introduction Parametric Markov Chains and Properties Background - Properties Parametric Markov Chains Monte Carlo and pMCs UAV flight model Safety zones Drone components Formal model Experimental results Prototype implementation Drone model Results Discussion Benoˆ ıt Delahaye (Univ Nantes/LS2N) Parametric SMC of UAV flight plan 2019-04-06 11 / 29

Monte Carlo and pMCs Monte Carlo for MCs 0 . 5 1 0 . 5 1 2 0 . 5 0 . 5 0 1 0 . 5 3 4 0 . 5 Benoˆ ıt Delahaye (Univ Nantes/LS2N) Parametric SMC of UAV flight plan 2019-04-06 12 / 29

Monte Carlo and pMCs Monte Carlo for MCs 0 . 5 1 0 . 5 1 2 0 . 5 0 . 5 0 1 0 . 5 3 4 0 . 5 ◮ Run n simulations ρ i of length l Benoˆ ıt Delahaye (Univ Nantes/LS2N) Parametric SMC of UAV flight plan 2019-04-06 12 / 29

Monte Carlo and pMCs Monte Carlo for MCs 0 . 5 1 ρ 1 = 0 · 1 · 1 · 1 · 1 · 1 0 . 5 ρ 2 = 0 · 1 · 0 · 3 · 4 · 4 1 2 ρ 3 = 0 · 3 · 2 · 2 · 2 · 2 0 . 5 ρ 4 = 0 · 1 · 0 · 1 · 0 · 3 0 . 5 0 ρ 5 = 0 · 3 · 4 · 4 · 4 · 1 1 ρ 6 = 0 · 3 · 2 · 2 · 2 · 2 0 . 5 ρ 7 = 0 · 1 · 0 · 3 · 2 · 2 3 4 0 . 5 ρ 8 = 0 · 1 · 0 · 3 · 4 · 4 ◮ Run n simulations ρ i of length l Benoˆ ıt Delahaye (Univ Nantes/LS2N) Parametric SMC of UAV flight plan 2019-04-06 12 / 29

Monte Carlo and pMCs Monte Carlo for MCs 0 . 5 1 ρ 1 = 0 · 1 · 1 · 1 · 1 · 1 0 . 5 ρ 2 = 0 · 1 · 0 · 3 · 4 · 4 1 2 ρ 3 = 0 · 3 · 2 · 2 · 2 · 2 0 . 5 ρ 4 = 0 · 1 · 0 · 1 · 0 · 3 0 . 5 0 ρ 5 = 0 · 3 · 4 · 4 · 4 · 1 1 ρ 6 = 0 · 3 · 2 · 2 · 2 · 2 0 . 5 ρ 7 = 0 · 1 · 0 · 3 · 2 · 2 3 4 0 . 5 ρ 8 = 0 · 1 · 0 · 3 · 4 · 4 ◮ Run n simulations ρ i of length l ◮ r ( ρ i ) = 1 if ρ i reaches 4 Benoˆ ıt Delahaye (Univ Nantes/LS2N) Parametric SMC of UAV flight plan 2019-04-06 12 / 29

Monte Carlo and pMCs Monte Carlo for MCs 0 . 5 1 ρ 1 = 0 · 1 · 1 · 1 · 1 · 1 r ( ρ 1 ) = 0 0 . 5 ρ 2 = 0 · 1 · 0 · 3 · 4 · 4 r ( ρ 2 ) = 1 1 2 ρ 3 = 0 · 3 · 2 · 2 · 2 · 2 r ( ρ 3 ) = 0 0 . 5 ρ 4 = 0 · 1 · 0 · 1 · 0 · 3 r ( ρ 4 ) = 0 0 . 5 0 ρ 5 = 0 · 3 · 4 · 4 · 4 · 1 r ( ρ 5 ) = 1 1 ρ 6 = 0 · 3 · 2 · 2 · 2 · 2 r ( ρ 6 ) = 0 0 . 5 ρ 7 = 0 · 1 · 0 · 3 · 2 · 2 r ( ρ 7 ) = 0 3 4 0 . 5 ρ 8 = 0 · 1 · 0 · 3 · 4 · 4 r ( ρ 8 ) = 1 ◮ Run n simulations ρ i of length l ◮ r ( ρ i ) = 1 if ρ i reaches 4 Benoˆ ıt Delahaye (Univ Nantes/LS2N) Parametric SMC of UAV flight plan 2019-04-06 12 / 29

Monte Carlo and pMCs Monte Carlo for MCs 0 . 5 1 ρ 1 = 0 · 1 · 1 · 1 · 1 · 1 r ( ρ 1 ) = 0 0 . 5 ρ 2 = 0 · 1 · 0 · 3 · 4 · 4 r ( ρ 2 ) = 1 1 2 ρ 3 = 0 · 3 · 2 · 2 · 2 · 2 r ( ρ 3 ) = 0 0 . 5 ρ 4 = 0 · 1 · 0 · 1 · 0 · 3 r ( ρ 4 ) = 0 0 . 5 0 ρ 5 = 0 · 3 · 4 · 4 · 4 · 1 r ( ρ 5 ) = 1 1 ρ 6 = 0 · 3 · 2 · 2 · 2 · 2 r ( ρ 6 ) = 0 0 . 5 ρ 7 = 0 · 1 · 0 · 3 · 2 · 2 r ( ρ 7 ) = 0 3 4 0 . 5 ρ 8 = 0 · 1 · 0 · 3 · 4 · 4 r ( ρ 8 ) = 1 ◮ Run n simulations ρ i of length l ◮ r ( ρ i ) = 1 if ρ i reaches 4 � r ( ρ i ) ◮ E l ⇒ Here, E 5 M ( r ) ∼ M ( r ) ∼ 0 . 375 (exact: 0.3125) n Benoˆ ıt Delahaye (Univ Nantes/LS2N) Parametric SMC of UAV flight plan 2019-04-06 12 / 29

Monte Carlo and pMCs Intuition for pMCs q 1 p r 1 2 0 . 5 p 0 1 0 . 5 3 4 q r ◮ How to run simulations? Benoˆ ıt Delahaye (Univ Nantes/LS2N) Parametric SMC of UAV flight plan 2019-04-06 13 / 29

Monte Carlo and pMCs Intuition for pMCs 0.33 — q 1 0.33 — p 0.33 — r 1 2 0 . 5 0.33 — p 0 1 0 . 5 3 4 0.33 — q 0.33 — r ◮ How to run simulations? Use a normalization function f (uniform?) → M f Benoˆ ıt Delahaye (Univ Nantes/LS2N) Parametric SMC of UAV flight plan 2019-04-06 13 / 29

Monte Carlo and pMCs Intuition for pMCs 0.33 — q 1 ρ 1 = 0 · 1 · 1 · 2 · 2 · 2 0.33 — p 0.33 — r ρ 2 = 0 · 1 · 0 · 3 · 3 · 4 1 2 ρ 3 = 0 · 3 · 2 · 2 · 2 · 2 0 . 5 ρ 4 = 0 · 1 · 0 · 1 · 1 · 0 0.33 — p 0 ρ 5 = 0 · 3 · 4 · 4 · 4 · 4 1 ρ 6 = 0 · 3 · 3 · 3 · 4 · 4 0 . 5 ρ 7 = 0 · 1 · 0 · 3 · 2 · 2 3 4 0.33 — q ρ 8 = 0 · 1 · 2 · 2 · 2 · 2 0.33 — r ◮ How to run simulations? Use a normalization function f (uniform?) → M f Benoˆ ıt Delahaye (Univ Nantes/LS2N) Parametric SMC of UAV flight plan 2019-04-06 13 / 29