Computing Reachable Sets via Toolbox of Level Set Methods Michael Vitus Jerry Ding 4/16/2012

Toolbox of Level Set Methods Ian Mitchell –Professor at the University of British Columbia Toolbox –Matlab –Computes the backwards reachable set –Fixed spacing Cartesian grid –Arbitrary dimension (computationally limited)

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

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

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

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

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

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

Example Optimal Hamiltonian: Partials:

Results

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