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Computing Reachable Sets via Toolbox of Level Set Methods Mo Chen Slides adapted from Michael Vitus and Jerry Ding

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Toolbox of Level Set Methods Ian Mitchell –Professor at the University of British Columbia 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

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Backwards Reachability [Mitchell, 2005]

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Problem Formulation

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Backwards Reachable Set

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Toolbox Formulation

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Example: Double Integrator

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Start from Examples\Reachability\air3D.m

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

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Example: Double Integrator

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Hamiltonian and Partial Functions

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Hamiltonian and Partial Function

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Example: Double Integrator Set up partials function

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Results

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Additional Comments Hamiltonian overestimated reachable set underestimated Partials function (Pages 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

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Useful Dynamical Form for Partial Function

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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 of toolbox manual

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