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## Computing Reach Sets as long as Hybrid Systems 1 2 3 K modes 1 2 3 n iterations initial

Lawry, Terri, News Director has reference to this Academic Journal, PHwiki organized this Journal Computing Reach Sets as long as Hybrid Systems 1 2 3 K modes 1 2 3 n iterations initial reach set unsafe Reach Sets: Initialize 1 2 3 K modes 1 2 3 n iterations unsafe unsafe safe

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Reach Sets: uncontrollable predecessor 1 2 3 K modes 1 2 3 n iterations uncontrolled transition unsafe safe Reach Sets: controllable predecessor 1 2 3 K modes 1 2 3 n iterations safe controlled transition safe Reach Sets: Variational Inequality 1 2 3 K modes 1 2 3 n iterations States which reach G without hitting E first: where subject to

Reach Sets: Iterate 1 2 3 K modes 1 2 3 n iterations Numerical computation of reach sets In our problem, the evolution of is governed by: Boundary of region is defined implicitly by is the distance from to the boundary at time is negative inside region in addition to positive outside Create a level set function such that: Propagating regions with level sets: Numerical computation of reach sets Level set methods: Convergent numerical algorithms to compute viscosity solution Non-oscillatory, high accuracy spatial derivative approximation Stable, consistent numerical Hamiltonian Variation diminishing, high order, explicit time integration Example (2 player zero sum game): 5 u v d v y [http://www.cs.ubc.ca/~mitchell/ToolboxLS/index.html]

Collision Avoidance Control [Mitchell, Tomlin 01] Example: Aircraft Autol in addition to er Aircraft must stay within safe flight envelope during l in addition to ing: Bounds on velocity ( ), flight path angle ( ), height ( ) Control over engine thrust ( ), angle of attack ( ), flap settings Model flap settings as discrete modes of hybrid automata Terms in continuous dynamics may depend on flap setting inertial frame wind frame body frame [Mitchell, Bayen, Tomlin 01] L in addition to ing Example: No Mode Switches Envelopes Safe sets

L in addition to ing Example: Mode Switches Envelopes Safe sets L in addition to ing Example: Synthesizing Control For states at the boundary of the safe set, results of reach-avoid computation determine What continuous inputs (if any) maintain safety What discrete jumps (if any) are safe to per as long as m Level set values in addition to gradients provide all relevant data Application to Autol in addition to Interface Controllable flight envelopes as long as l in addition to ing in addition to Take Off / Go Around (TOGA) maneuvers may not be the same Pilots cockpit display may not contain sufficient in as long as mation to distinguish whether TOGA can be initiated revised interface existing interface controllable flare envelope controllable TOGA envelope intersection

Aircraft Simulator Tests Setup Commercial flight simulator, B767 pilot Digital video of primary flight display Maneuver Go-around at low speed, high descent rate Goal Determine whether problematic behavior predicted by our model is possible in aircraft flight simulator (movie) Aircraft Simulator Results Produced unexpected behavior Non-st in addition to ard procedure; Unable to duplicate Validated types of problems addressed by this method Example: Closely Spaced Parallel Approaches San Mateo Bridge San Francisco Airport 750 ft separation CSPA to SFO video Restrictions in Instrument Meteorological Conditions (IMC)

Example: Closely Spaced Parallel Approaches Three emergency escape maneuvers (EEMs): Evader accelerates straight ahead Evader accelerates, turns to the right 45 deg Evader turns to the right 60 deg evader Tested on the Stan as long as d DragonFly UAVs Dragonfly 3 Dragonfly 2 Ground Station [Jang, Teo, Tomlin] Flight Demo 1 – Sept 2003 EEM alert Separation distance (m) North (m) East (m) time (s) Above threshold Accelerate in addition to turn EEM Put video here Evader, DF 2 (red in addition to yellow aircraft) DF 2, the evader, is the larger blob

Flight Demo 2 Sept 2003 EEM alert Separation distance (m) North (m) East (m) time (s) Above threshold Put video here Coast in addition to turn EEM Evader, DF 2 (red in addition to yellow aircraft) DF 2, the evader, is the larger blob

## Lawry, Terri News Director

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