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
– 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
b
• Kinematic Model
• Target set
– Protected zone
y
xr
– Position and heading angle
– Inputs: turning rates
v
r
yr
Blunderer
v
r
a
Evader
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