Computing Reachable Sets via Toolbox of Level Set Methods

Computing Reachable Sets via Toolbox of Level Set Methods

Mo [email protected]

Slides adapted from Michael Vitus and Jerry Ding

Toolbox of Level Set Methods• Ian Mitchell

– Professor at the University of British Columbia• http://www.cs.ubc.ca/~mitchell/

• MatLab Toolbox – http://www.cs.ubc.ca/~mitchell/ToolboxLS/inde

x.html– Computes the backwards reachable set

starting from some final target set– Fixed spacing Cartesian grid– Up to 4 or 5 dimensions

Backwards Reachability

[Mitchell, 2005]

Problem Formulation

• Dynamics: – System input: – Disturbance:

• Target set: – Final conditions– Level set representation:

– Eg. 2D unit disk centered at the origin:

Backwards Reachable Set• Solution to a Hamilton-Jacobi PDE:

where:

– For theory, see lecture 8 notes

• Terminal value HJ PDE– Converted to an initial value PDE by multiplying the

Hamiltonian by

• Provide 3 items1. Final target set

2. Hamiltonian function (multiplied by -1)

3. An upper bound on the partials functions

Toolbox Formulation

Example: Double Integrator• Dynamics

• Target set– Outside of the box

• Coding:– Mostly setting up the environment in the toolbox

t = 0

-2 0 2-1

0

1t = 1

-2 0 2-1

0

1

t = 2

-2 0 2-1

0

1t = 3

-2 0 2-1

0

1

Example: Double Integrator• Start from Examples\Reachability\air3D.m

Example: Grid and Target Set• Set-up grid and target set

– g.bdry: @addGhostExtrapolate usually– g.bdry: @addGhostPeriodic for periodic

dimensions

Example: Double Integrator– Target set: define own function OR use pre-

existing functions such as shapeCylinder, shapeRectangleByCorners

– See Section 3.3 of toolbox manual

Hamiltonian and Partial Functions

• Hamiltonian:

Hamiltonian and Partial Function

• Partials:

Hamiltonian and Partial Function• Set-up Hamiltonian

– is written as deriv{i}– is written as grid.xs{i}

Example: Double Integrator• Set up partials function

Results

t = 0

-2 0 2-1

0

1t = 1

-2 0 2-1

0

1

t = 2

-2 0 2-1

0

1t = 3

-2 0 2-1

0

1

Additional Comments• Hamiltonian overestimated reachable set

underestimated• Partials function (Pages 50-51 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

Useful Dynamical Form for Partial Function

• Pages 50-51 of Toolbox manual• Nonlinear system, linear input

• Input constraints are hyperrectangles• Analytical optimal inputs:

• Partials upper bound:

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 2.6.1 of

toolbox manual