1. Computing Reachable Sets via Toolbox of Level Set Methods Mo Chen mochen72@berkeley.edu Slides adapted from Michael Vitus and Jerry Ding

2. Toolbox of Level Set Methods Ian Mitchell –Professor at the University of British Columbia http://www.cs.ubc.ca/~mitchell/ MatLab Toolbox –http://www.cs.ubc.ca/~mitchell/ToolboxLS/ind ex.htmlhttp://www.cs.ubc.ca/~mitchell/ToolboxLS/ind ex.html –Computes the backwards reachable set starting from some final target set –Fixed spacing Cartesian grid –Up to 4 or 5 dimensions

3. Backwards Reachability [Mitchell, 2005]

4. Problem Formulation

5. Backwards Reachable Set

6. Toolbox Formulation

7. Example: Double Integrator

8. Start from Examples\Reachability\air3D.m

9. Example: Grid and Target Set Set-up grid and target set –g.bdry : @addGhostExtrapolate usually –g.bdry : @addGhostPeriodic for periodic dimensions

10. Example: Double Integrator

11. Hamiltonian and Partial Functions

12. Hamiltonian and Partial Function

14. Example: Double Integrator Set up partials function

15. Results

16. Additional Comments Hamiltonian overestimated  reachable set underestimated Partials function (Pages 50-51 of Toolbox manual) –Underestimation  numerical instability –Overestimation  rounded corners or worst case underestimation of reachable set Computation –The solver grids the state space –Tractable only up to 4-5 continuous states –Advanced: Can also define avoid sets Toolbox –Coding: ~90% is setting up the environment

17. Useful Dynamical Form for Partial Function

18. Other Tools Plotting utilities –Kernel\Helper\Visualization –visualizeLevelSet.m –spinAnimation.m Initial condition helpers –Cylinders, hyperrectangles Advice –Start with a small example –Look over air3D.m along with Section 2.6.1 of toolbox manual