1. Computing Reachable Sets via Toolbox of Level Set Methods Michael Vitus (michael.vitus@gmail.com) Jerry Ding (jding@eecs.berkeley.edu) 4/16/2012

2. Toolbox of Level Set Methods Ian Mitchell –Professor at the University of British Columbia http://www.cs.ubc.ca/~mitchell/ Toolbox –Matlab –Computes the backwards reachable set –Fixed spacing Cartesian grid –Arbitrary dimension (computationally limited)

3. Problem Formulation Dynamics: –System input: –Disturbance input: Target set: –Unsafe final conditions

4. Backwards Reachable Set Solution to a Hamilton-Jacobi PDE: where: Terminal value HJ PDE –Converted to an initial value PDE by multiplying the H(x,p) by -1

5. Toolbox Formulation No automated method Provide 3 items –Hamiltonian function (multiplied by -1) –An upper bound on the partials function –Final target set

6. General Comments Hamiltonian overestimated  reachable set underestimated Partials function –Most difficult –Underestimation  numerical instability –Overestimation  rounded corners or worst case underestimation of reachable set Computation –The solver grids the state space –Tractable only up to 6 continuous states Toolbox –Coding: ~90% is setting up the environment

7. Useful Dynamical Form Nonlinear system, linear input Input constraints are hyperrectangles Analytical optimal inputs: Partials upper bound:

8. Example: Two Identical Vehicles Kinematic Model –Position and heading angle –Inputs: turning rates Target set –Protected zone Blunderer Evader rr

9. Example Optimal Hamiltonian: Partials:

10. Results

11. Toolbox Plotting utilities –Kernel\Helper\Visualization –visualizeLevelSet.m –spinAnimation.m Initial condition helpers –Cylinders, hyperrectangles Advice: Start small… Walk through example