1. Soft Walls, Cataldo 1 Restricting the Control of Hijackers: Soft Walls Presented By Adam Cataldo UC Berkeley NASA Ames Research Center 22 November 2002

2. Soft Walls, Cataldo 2 Outline Soft Walls Overview Current Research Current Results

3. Soft Walls, Cataldo 3 Basic Idea (Edward Lee) Protect restricted airspaces using on-board avionics Non-networked solution (control on aircraft) Non-hackable solution (software protected through encryption)

4. Soft Walls, Cataldo 4 Design Objectives Keep the pilot in mind!

5. Soft Walls, Cataldo 5 Design Objectives Apply zero control when pilot is far from a no-fly zone Bias pilot’s input with a control input –No attenuation of pilot control –No instantaneous change in control inputs Always allow the pilot to turn away from no- fly zone –Restrict controls from saturating

6. Soft Walls, Cataldo 6 Unsaturated Control (Xiaojun Liu) No-fly zone Even under the maximum control bias, the pilot can make a sharper turn away from the no-fly zone

7. Soft Walls, Cataldo 7 Outline Soft Walls Overview Current Research –Reachability Approach –Simulation Interface –Crazyboard Current Results

8. Soft Walls, Cataldo 8 Reachable Sets (Mitchell, Tomlin, Sastry) Suppose we have the dynamics: where x is the state, u is the control input, and d is the disturbance Let X be the set of all possible states Let G be an unsafe region of states, called the target set, where

9. Soft Walls, Cataldo 9 Reachable Sets (Mitchell, Tomlin, Sastry) The backwards reachable set is the set of states for which safety cannot be guaranteed for all possible disturbances Target Set (unsafe states) Reachable set Safe States

10. Soft Walls, Cataldo 10 Reachable Sets (Mitchell, Tomlin, Sastry) We denote the reachable set G The reachable set is, more formally, the set of states for which there exists a disturbance d(t) such that all for any control inputs u(t) the state will reach the target set For any state outside the reachable set, we can find a control input that can guarantee the state is kept outside the reachable set

11. Soft Walls, Cataldo 11 Reachable Sets and Soft Walls (Cataldo, Mitchell) No-fly zone defines the target set Calculate the reachable set Model the pilot as the “disturbance” Design a controller that prevents the aircraft from entering the reachable set while satisfying the design objectives. Target Set x, y, h, , , ,...

12. Soft Walls, Cataldo 12 Two Dynamics Models 1.Simple constant-speed car model: –Makes tests/computations faster –Easier to visualize  V pilot inputcontrol input

13. Soft Walls, Cataldo 13 Two Dynamics Models 2.More realistic model (Menon, Sweriduk, Sridhar) includes: –Thrust T –Drag D –Mass m –Flight Path Angle  –Bank Angle  –Fuel Flow Rate Q –Lift L –Load Factor n –Height h

14. Soft Walls, Cataldo 14 Two Dynamics Models 2.More realistic model rudder and ailerons elevator throttle pilot input control input

15. Soft Walls, Cataldo 15 Simulation Interface (Chen, Ahmadi) Soft Walls interface for Microsoft Flight Simulator Real-time controller created in Ptolemy II

16. Soft Walls, Cataldo 16 Crazyboard (Neuendorffer, Cataldo) Hovercraft with controlled by two fans Created at Cal Tech by Richard Murray’s group Test bed for Soft Walls algorithm

17. Soft Walls, Cataldo 17 Outline Soft Walls Overview Current Research –Reachability Approach –Simulation Interface –Crazyboard Current Results –Controller for Simplified Dynamics Model

18. Soft Walls, Cataldo 18 Hybrid Controller (Cataldo) Given the simple dynamics model: We constructed a controller that will prevent the aircraft from entering any no-fly zone, assuming the aircraft is initially far from the no-fly zone

19. Soft Walls, Cataldo 19 Hybrid Controller Definitions: d1 d2 aircraft position (x,y) right center (x right,y right ) left center (x left,y left ) current heading minimum turning radius

20. Soft Walls, Cataldo 20 Hybrid Controller Definitions: –d left = distance of left-center point from no-fly zone –d right = distance of right-center point from no-fly zone –B = control bias that forces the aircraft to turn left at the maximum turning rate –N = no-fly zone, where N is an open subset of

21. Soft Walls, Cataldo 21 Hybrid Controller Discrete State Transitions: q0 (no bias) q2 (leftward bias) q1 (rightward bias) d left <= d2 d left > d2 d right >d2 d left > d2 d right >d2 d left <= d2 d left > d2 d right <= d2 d left >= d2 d right < d2

22. Soft Walls, Cataldo 22 Hybrid Controller Continuous Control-Input Calculation:

23. Soft Walls, Cataldo 23 Hybrid Controller Theorem: Given N, if d left (t 0 ) > d2, and d right (t 0 ) > d2, then using this hybrid controller, (x,y)  N  t > t 0 That is, this hybrid controller guarantees the aircraft never enters the no-fly zone

24. Soft Walls, Cataldo 24 Proof If d left (t) > d1 and d right (t) > d1, then (x(t),y(t))  N. If d left (t) < d2 or d right (t) < d2, there will be a non-zero bias, rightward or leftward If the bias is leftward and d left (t) = d1, then the aircraft is turning maximally left, and d left (  ) = d1   > t Similarly, if the bias is rightward and d right (t) = d1, then d right (  ) = d1   > t

25. Soft Walls, Cataldo 25 Proof Because the aircraft’s position is continuous, d left (t)  d1 and d right (t)  d1  t > t 0. That is if d left (t) < d1, then for some  < t, d left (  ) = d1.

26. Soft Walls, Cataldo 26 Controller Impracticality Because this controller may capture the aircraft (if d left (t) = d1 or d right (t) = d1), this algorithm is impractical. However, it may work without capturing the aircraft, as long as the control bias never saturates.

27. Soft Walls, Cataldo 27 Acknowledgements Edward Lee Xiaojun Liu Shankar Sastry Ian Mitchell Ashwin Ganesan Steve Neuendorffer Zhongning Chen Iman Ahmadi Claire Tomlin David Lee Paul Yang