Probabilistic occupancy via forward stochastic reachability

High-level description

For reliable autonomy under uncertainty, we must have efficient ways to estimate event likelihood given a stochastic system model.

Existing methods

1. Use sampling-based methods like Monte-Carlo simulation. Hoever, this approach only have asymptotic convergence guarantees.
2. Compute the probability density via convolutions when the dynamics is linear. However, this iterative method requires gridding for numerical evalution of the convolution integral. Further, its accuracy degrades over long time horizons.
3. Use concentration inequalities like moment-based bounds to obtain crude estimates of the event probability

Proposed solution

Use Fourier transformation to obtain a grid-free, recursion-free, and sampling-free method to compute the probability density function. The event likelihood, which we call the probablistic occupancy function, can be computed via numerical integration. The $\alpha$-super level sets of the probabilitistic occupancy functions, which we call the $\alpha$-probability occupied sets, are precisely the set of states for which the event likelihood is above a pre-specified threshold.

See my paper on probabilistic occupancy via forward stochastic reachability for more details.

Applications

1. Stochastic motion planning: Navigate a robot in an environment with obtacles that have geometric shapes and uncontrolled stochastic dynamics
2. Stochastic target capture: Capture a stochastically moving target using a controller robot