Parallelotope Bundles for Polynomial Reachability

2019 International Conference on Robotics and Automation (ICRA)

et al. 2019

Hamilton-Jacobi (HJ) reachability analysis has been developed over the past decades into a widely-applicable tool for determining goal satisfaction and safety verification in nonlinear systems. While HJ reachability can be formulated very generally, computational complexity can be a serious impediment for many systems of practical interest. Much prior work has been devoted to computing approximate solutions to large reachability problems, yet many of these methods may only apply to very restrictive problem classes, do not generate controllers, and/or can be extremely conservative. In this paper, we present a new method for approximating the optimal controller of the HJ reachability problem for control-affine systems. While also a specific problem class, many dynamical systems of interest are, or can be well approximated, by controlaffine models. We explicitly avoid storing a representation of the reachability value function, and instead learn a controller as a sequence of simple binary classifiers. We compare our approach to existing grid-based methodologies in HJ reachability and demonstrate its utility on several examples, including a physical quadrotor navigation task.

Section: B Dynamic Programming With Binary Classifiersmentioning

confidence: 99%

A Classification-based Approach for Approximate Reachability

Rubies-Royo

Fridovich-Keil

2019 International Conference on Robotics and Automation (ICRA)

et al. 2019

“…For example, the methods presented in [10]- [14] have had success in analyzing relatively high-dimensional affine systems using sets of pre-specified shapes, such as polytopes or hyperplanes. Other potentially less scalable methods are able to handle systems with the more complex dynamics [11], [15]- [18]. Computational scalability varies among these different methods, with the most scalable methods requiring that the system dynamics do not involve control and disturbance variables.…”

Section: Introductionmentioning

confidence: 99%

Decomposition of Reachable Sets and Tubes for a Class of Nonlinear Systems

Chen

IEEE Trans. Automat. Contr.

Vashishtha

et al. 2018

144

133

Reachability analysis provides formal guarantees for performance and safety properties of nonlinear control systems. Here, one aims to compute the backward reachable set (BRS) or tube (BRT) -the set of states from which the system can be driven into a target set at a particular time or within a time interval, respectively. The computational complexity of current approaches scales exponentially, making application to high-dimensional systems intractable. We propose a technique that decomposes the dynamics of a general class of nonlinear systems into subsystems which may be coupled through common states, controls, and disturbances. Despite this coupling, BRSs and BRTs can be computed efficiently using our technique without incurring additional approximation errors and without the need for linearizing dynamics or approximating sets as polytopes. Computations of BRSs and BRTs now become orders of magnitude faster, and for the first time BRSs and BRTs for many high-dimensional nonlinear control systems can be computed using the Hamilton-Jacobi (HJ) formulation. In situations involving bounded adversarial disturbances, our proposed method can obtain slightly conservative results. We demonstrate our theory by numerically computing BRSs and BRTs using the HJ formulation for several systems, including the 6D Acrobatic Quadrotor and the 10D Near-Hover Quadrotor.

“…There are some methods for speeding up this computation using decomposition [8], and there are other efficient approaches that require simplified problem formulations and/or dynamics [9][10][11][12][13][14][15]. The methods in [9,[16][17][18][19], can handle more complex dynamics, but may be less scalable or unable to represent complex sets. Efficient reachability analysis remains challenging for general system dynamics and problem setups.…”

Section: Introductionmentioning

confidence: 99%

Reachability-Based Safety Guarantees using Efficient Initializations

2019 IEEE 58th Conference on Decision and Control (CDC)

Bansal

Ghosh

et al. 2019

Hamilton-Jacobi-Isaacs (HJI) reachability analysis is a powerful tool for analyzing the safety of autonomous systems. This analysis is computationally intensive and typically performed offline. Online, however, the autonomous system may experience changes in system dynamics, external disturbances, and/or the surrounding environment, requiring updated safety guarantees. Rather than restarting the safety analysis, we propose a method of "warm-start" reachability, which uses a user-defined initialization (typically the previously computed solution). By starting with an HJI function that is closer to the solution than the standard initialization, convergence may take fewer iterations.In this paper we prove that warm-starting will result in guaranteed conservative solutions by over-approximating the states that must be avoided to maintain safety. We additionally prove that for many common problem formulations, warmstarting will result in exact solutions. We demonstrate our method on several illustrative examples with a double integrator, and also on a more practical example with a 10D quadcopter model that experiences changes in mass and disturbances and must update its safety guarantees accordingly. We compare our approach to standard reachability and a recently proposed "discounted" reachability method, and find for our examples that warm-starting is 1.6 times faster than standard and 6.2 times faster than (untuned) discounted reachability.