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Computing Reachable Sets via Toolbox of Level Set Methods
Michael Vitus
Jerry Ding ([email protected])
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– 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
v
rx
ry
va
b
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
