77 research outputs found
A Primal-Dual Method for Optimal Control and Trajectory Generation in High-Dimensional Systems
Presented is a method for efficient computation of the Hamilton-Jacobi (HJ)
equation for time-optimal control problems using the generalized Hopf formula.
Typically, numerical methods to solve the HJ equation rely on a discrete grid
of the solution space and exhibit exponential scaling with dimension. The
generalized Hopf formula avoids the use of grids and numerical gradients by
formulating an unconstrained convex optimization problem. The solution at each
point is completely independent, and allows a massively parallel implementation
if solutions at multiple points are desired. This work presents a primal-dual
method for efficient numeric solution and presents how the resulting optimal
trajectory can be generated directly from the solution of the Hopf formula,
without further optimization. Examples presented have execution times on the
order of milliseconds and experiments show computation scales approximately
polynomial in dimension with very small high-order coefficients.Comment: Updated references and funding sources. To appear in the proceedings
of the 2018 IEEE Conference on Control Technology and Application
The Stochastic Reach-Avoid Problem and Set Characterization for Diffusions
In this article we approach a class of stochastic reachability problems with
state constraints from an optimal control perspective. Preceding approaches to
solving these reachability problems are either confined to the deterministic
setting or address almost-sure stochastic requirements. In contrast, we propose
a methodology to tackle problems with less stringent requirements than almost
sure. To this end, we first establish a connection between two distinct
stochastic reach-avoid problems and three classes of stochastic optimal control
problems involving discontinuous payoff functions. Subsequently, we focus on
solutions of one of the classes of stochastic optimal control problems---the
exit-time problem, which solves both the two reach-avoid problems mentioned
above. We then derive a weak version of a dynamic programming principle (DPP)
for the corresponding value function; in this direction our contribution
compared to the existing literature is to develop techniques that admit
discontinuous payoff functions. Moreover, based on our DPP, we provide an
alternative characterization of the value function as a solution of a partial
differential equation in the sense of discontinuous viscosity solutions, along
with boundary conditions both in Dirichlet and viscosity senses. Theoretical
justifications are also discussed to pave the way for deployment of
off-the-shelf PDE solvers for numerical computations. Finally, we validate the
performance of the proposed framework on the stochastic Zermelo navigation
problem
Online Update of Safety Assurances Using Confidence-Based Predictions
Robots such as autonomous vehicles and assistive manipulators are
increasingly operating in dynamic environments and close physical proximity to
people. In such scenarios, the robot can leverage a human motion predictor to
predict their future states and plan safe and efficient trajectories. However,
no model is ever perfect -- when the observed human behavior deviates from the
model predictions, the robot might plan unsafe maneuvers. Recent works have
explored maintaining a confidence parameter in the human model to overcome this
challenge, wherein the predicted human actions are tempered online based on the
likelihood of the observed human action under the prediction model. This has
opened up a new research challenge, i.e., \textit{how to compute the future
human states online as the confidence parameter changes?} In this work, we
propose a Hamilton-Jacobi (HJ) reachability-based approach to overcome this
challenge. Treating the confidence parameter as a virtual state in the system,
we compute a parameter-conditioned forward reachable tube (FRT) that provides
the future human states as a function of the confidence parameter. Online, as
the confidence parameter changes, we can simply query the corresponding FRT,
and use it to update the robot plan. Computing parameter-conditioned FRT
corresponds to an (offline) high-dimensional reachability problem, which we
solve by leveraging recent advances in data-driven reachability analysis.
Overall, our framework enables online maintenance and updates of safety
assurances in human-robot interaction scenarios, even when the human prediction
model is incorrect. We demonstrate our approach in several safety-critical
autonomous driving scenarios, involving a state-of-the-art deep learning-based
prediction model.Comment: 7 pages, 3 figure
How river rocks round: resolving the shape-size paradox
River-bed sediments display two universal downstream trends: fining, in which
particle size decreases; and rounding, where pebble shapes evolve toward
ellipsoids. Rounding is known to result from transport-induced abrasion;
however many researchers argue that the contribution of abrasion to downstream
fining is negligible. This presents a paradox: downstream shape change
indicates substantial abrasion, while size change apparently rules it out. Here
we use laboratory experiments and numerical modeling to show quantitatively
that pebble abrasion is a curvature-driven flow problem. As a consequence,
abrasion occurs in two well-separated phases: first, pebble edges rapidly round
without any change in axis dimensions until the shape becomes entirely convex;
and second, axis dimensions are then slowly reduced while the particle remains
convex. Explicit study of pebble shape evolution helps resolve the shape-size
paradox by reconciling discrepancies between laboratory and field studies, and
enhances our ability to decipher the transport history of a river rock.Comment: 11 pages, 5 figure
TIRA: Toolbox for Interval Reachability Analysis
This paper presents TIRA, a Matlab library gathering several methods for the
computation of interval over-approximations of the reachable sets for both
continuous- and discrete-time nonlinear systems. Unlike other existing tools,
the main strength of interval-based reachability analysis is its simplicity and
scalability, rather than the accuracy of the over-approximations. The current
implementation of TIRA contains four reachability methods covering wide classes
of nonlinear systems, handled with recent results relying on contraction/growth
bounds and monotonicity concepts. TIRA's architecture features a central
function working as a hub between the user-defined reachability problem and the
library of available reachability methods. This design choice offers increased
extensibility of the library, where users can define their own method in a
separate function and add the function call in the hub function
- …