Formal Verification of Curved Flight Collision Avoidance Maneuvers - PowerPoint PPT Presentation

Formal Verification of Curved Flight Collision Avoidance Maneuvers A Case Study Andr´ e Platzer Edmund M. Clarke Carnegie Mellon University, Computer Science Department, Pittsburgh, PA Formal Methods, FM, Eindhoven, November 2009 0.5 0.4 0.3 0.2 1.0 0.1 0.8 0.6 0.4 0.2 Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 1 / 17

Outline Motivation 1 Differential Dynamic Logic for Hybrid Systems 2 Compositional Verification Logic Differential Invariants Curved Flight Air Traffic Collision Avoidance Maneuver 3 Compositional Verification Plan Verifying Roundabout Flight Safe Flyable Entry Separation Safe Exit Separation Successful Negotiation & Synchronization Flyable Tangential Roundabout Maneuver 4 Experimental Results 4 Conclusions & Future Work 5 Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 1 / 17

Air Traffic Control: Straight Lines & Instant Turns Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 2 / 17

Air Traffic Control: Straight Lines & Instant Turns Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 2 / 17

Air Traffic Control: Straight Lines & Instant Turns Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 2 / 17

Air Traffic Control: Hybrid Systems & Curves Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 3 / 17

Air Traffic Control: Hybrid Systems & Curves Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 3 / 17

Air Traffic Control: Hybrid Systems & Curves Hybrid Systems continuous evolution along differential equations + discrete change Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 3 / 17

Air Traffic Control: Hybrid Systems & Curves ς y 2 ̺ ω e x 2 d x 1 y 1   x ′ 1 = − v 1 + v 2 cos ϑ + ω x 2 x ′   2 = v 2 sin ϑ − ω x 1   ϑ ′ = ̺ − ω Hybrid Systems continuous evolution along differential equations + discrete change Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 3 / 17

Air Traffic Control: Hybrid Systems & Curves ς y 2 ̺ ω e x 2 d x 1 y 1   x ′ 1 = − v 1 + v 2 cos ϑ + ω x 2 x ′   2 = v 2 sin ϑ − ω x 1   ϑ ′ = ̺ − ω Example (“Solving” differential equations) x 1 ( t ) = 1 � x 1 ω̺ cos t ω − v 2 ω cos t ω sin ϑ + v 2 ω cos t ω cos t ̺ sin ϑ − v 1 ̺ sin t ω ω̺ � 1 − sin ϑ 2 sin t ω + x 2 ω̺ sin t ω − v 2 ω cos ϑ cos t ̺ sin t ω − v 2 ω � + v 2 ω cos ϑ cos t ω sin t ̺ + v 2 ω sin ϑ sin t ω sin t ̺ . . . Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 3 / 17

Air Traffic Control: Hybrid Systems & Curves ς y 2 ̺ ω e x 2 d x 1 y 1   x ′ 1 = − v 1 + v 2 cos ϑ + ω x 2 x ′   2 = v 2 sin ϑ − ω x 1   ϑ ′ = ̺ − ω Example (“Solving” differential equations) 1 � ∀ t ≥ 0 x 1 ω̺ cos t ω − v 2 ω cos t ω sin ϑ + v 2 ω cos t ω cos t ̺ sin ϑ − v 1 ̺ sin t ω ω̺ � 1 − sin ϑ 2 sin t ω + x 2 ω̺ sin t ω − v 2 ω cos ϑ cos t ̺ sin t ω − v 2 ω � + v 2 ω cos ϑ cos t ω sin t ̺ + v 2 ω sin ϑ sin t ω sin t ̺ . . . Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 3 / 17

Air Traffic Control: Hybrid Systems & Curves Hybrid Systems continuous evolution along differential equations + discrete change Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 3 / 17

Introduce: Flyable Roundabout Maneuver Problem ⇒ Solution Unrealistic instant turns can cause problems Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 4 / 17

Introduce: Flyable Roundabout Maneuver Problem ⇒ Solution Unrealistic instant turns can cause problems ( ⇒ smooth curves) Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 4 / 17

Introduce: Flyable Roundabout Maneuver Problem ⇒ Solution Unrealistic instant turns can cause problems ( ⇒ smooth curves) Geometric intuition can be misleading Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 4 / 17

Introduce: Flyable Roundabout Maneuver Problem ⇒ Solution Unrealistic instant turns can cause problems ( ⇒ smooth curves) Geometric intuition can be misleading ( ⇒ hybrid system model) Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 4 / 17

Introduce: Flyable Roundabout Maneuver Problem ⇒ Solution Unrealistic instant turns can cause problems ( ⇒ smooth curves) Geometric intuition can be misleading ( ⇒ hybrid system model) ⇒ Introduce smoothly curved flyable maneuver as hybrid system model Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 4 / 17

Introduce: Flyable Roundabout Maneuver S ∧ far ∧ compat r a f ∧ S agree free exit S ∧ far c entry entry r ω < 0 ω > 0 exit S ∧ T x r r circ y circ h S ∧ T Problem ⇒ Solution Unrealistic instant turns can cause problems ( ⇒ smooth curves) Geometric intuition can be misleading ( ⇒ hybrid system model) ⇒ Introduce smoothly curved flyable maneuver as hybrid system model Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 4 / 17

Introduce: Flyable Roundabout Maneuver S ∧ far ∧ compat r a f ∧ S agree free exit S ∧ far c entry entry r ω < 0 ω > 0 exit S ∧ T x r r circ y circ h S ∧ T Problem ⇒ Solution Unrealistic instant turns can cause problems ( ⇒ smooth curves) Geometric intuition can be misleading ( ⇒ hybrid system model) ⇒ Introduce smoothly curved flyable maneuver as hybrid system model Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 4 / 17

Introduce: Flyable Roundabout Maneuver S ∧ far ∧ compat r a f ∧ S agree free exit S ∧ far c entry entry r ω < 0 ω > 0 exit S ∧ T x r r circ y circ h S ∧ T Problem ⇒ Solution Unrealistic instant turns can cause problems ( ⇒ smooth curves) Geometric intuition can be misleading ( ⇒ hybrid system model) ⇒ Introduce smoothly curved flyable maneuver as hybrid system model Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 4 / 17

Introduce: Flyable Roundabout Maneuver S ∧ far ∧ compat r a f ∧ S agree free exit S ∧ far c entry entry r ω < 0 ω > 0 exit S ∧ T x r r circ y circ h S ∧ T Problem ⇒ Solution Unrealistic instant turns can cause problems ( ⇒ smooth curves) Geometric intuition can be misleading ( ⇒ hybrid system model) ⇒ Introduce smoothly curved flyable maneuver as hybrid system model Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 4 / 17

Introduce: Flyable Roundabout Maneuver S ∧ far ∧ compat r a f ∧ S agree free t S ∧ far i x c e entry entry r ω < 0 ω > 0 exit S ∧ T x r r circ y circ h S ∧ T Problem ⇒ Solution Unrealistic instant turns can cause problems ( ⇒ smooth curves) Geometric intuition can be misleading ( ⇒ hybrid system model) ⇒ Introduce smoothly curved flyable maneuver as hybrid system model Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 4 / 17

Introduce: Flyable Roundabout Maneuver S ∧ far ∧ compat r a f ∧ S agree free t S ∧ far i x c e entry entry r ω < 0 ω > 0 exit S ∧ T x r r circ y circ h S ∧ T Problem ⇒ Solution Unrealistic instant turns can cause problems ( ⇒ smooth curves) Geometric intuition can be misleading ( ⇒ hybrid system model) ⇒ Introduce smoothly curved flyable maneuver as hybrid system model Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 4 / 17

Introduce: Flyable Roundabout Maneuver S ∧ far ∧ compat r a f ∧ S agree free exit S ∧ far c entry entry r ω < 0 ω > 0 exit S ∧ T x r r circ y circ h S ∧ T Problem ⇒ Solution Unrealistic instant turns can cause problems ( ⇒ smooth curves) Geometric intuition can be misleading ( ⇒ hybrid system model) ⇒ Introduce smoothly curved flyable maneuver as hybrid system model Verification for: nonlinear curve dynamics + mode switching? Andr´ e Platzer, Edmund M. Clarke (CMU) Formal Verification of Curved Flight Collision Avoidance FM’09 4 / 17