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Computing Reachable Sets via Toolbox of Level Set Methods

Computing Reachable Sets via Toolbox of Level Set Methods. Michael Vitus (michael.vitus@gmail.com) Jerry Ding (jding@eecs.berkeley.edu) 4/16/2012. Toolbox of Level Set Methods. Ian Mitchell Professor at the University of British Columbia http://www.cs.ubc.ca/~mitchell/ Toolbox Matlab

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Computing Reachable Sets via Toolbox of Level Set Methods

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  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 yr Blunderer Evader • Kinematic Model • Position and heading angle • Inputs: turning rates • Target set • Protected zone

  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

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