Abraham P. Vinod

Forward Stochastic Reachability Analysis for Uncontrolled Linear Systems using Fourier Transforms

We propose a scalable method for forward stochastic reachability analysis for uncontrolled linear systems with affine disturbance. Our method uses Fourier transforms to efficiently compute the forward stochastic reach probability measure (density) and the forward stochastic reach set. This method is applicable to systems with bounded or unbounded disturbance sets. We also examine the convexity properties of the forward stochastic reach set and its probability density. Motivated by the problem of a robot attempting to capture a stochastically moving, non-adversarial target, we demonstrate our method on two simple examples. Where traditional approaches provide approximations, our method provides exact analytical expressions for the densities and probability of capture.

Computation of forward stochastic reach sets: Application to stochastic, dynamic obstacle avoidance

We propose a method to efficiently compute the forward stochastic reach (FSR) set and its probability measure. We consider nonlinear systems with an affine disturbance input, that is stochastic and bounded. This model includes uncontrolled systems and systems with an a priori known controller, and often arises in problems in obstacle avoidance in mobile robotics. When used as a constraint in finite horizon controller synthesis, the FSR set and its probability measure facilitate probabilistic collision avoidance. This is in contrast to the traditional game-theoretic approaches which presume the obstacles are adversaries, generating hard constraints that cannot be violated. We tailor our approach to accommodate the geometry of the rigid body obstacles, and show convexity is assured when the rigid body shape of each obstacle is convex. We extend existing methods for multi-obstacle avoidance through mixed integer programming (with linear robot and obstacle dynamics) to accommodate chance constraints derived using the FSR analysis. We use our method to synthesize a receding horizon controller that drives a robot to a desired goal while avoiding several rigid-body obstacle with stochastic dynamics. Our approach can provide solutions when approaches that presume a worst-case action from the obstacle fail.

Dynamic risk tolerance: Motion planning by balancing short-term and long-term stochastic dynamic predictions

Identifying collision-free paths over long time windows in environments with stochastically moving obstacles is difficult, in part because long-term predictions of obstacle positions typically have low fidelity, and the region of possible obstacle occupancy is typically large. As a result, planning methods that are restricted to identifying paths with a low probability of collision may not be able to find a valid path. However, allowing paths with a higher probability of collision may limit detection of imminent collisions. In this paper, we present Dynamic Risk Tolerance (DRT), a framework that dynamically evaluates risk tolerance, a function which is formulated as a time-varying upper bound on the acceptable likelihood of collision for a given path. DRT is implemented with forward stochastic reachable sets to predict the exact distribution of obstacles in a scalable manner over an arbitrarily long time window. In effect, DRT identifies actions that balance risks posed by both near and far obstacles. We empirically compare DRT to other state of the art methods that are capable of generating real-time solutions in highly crowded environments, and demonstrate the success rates for DRT that is 46% higher than the best performing comparison method, in the most difficult problem tested.

A Deterministic Attitude Estimation Using a Single Vector Information and Rate Gyros

This paper proposes a deterministic estimator for the estimation of the attitude of a rigid body. A deterministic estimator uses a minimal set of information and does not try to minimize a cost function or lit the measurements into a stochastic process. The proposed estimator obtains the attitude estimation utilizing only the properties of the rotational group SO(3). The information set required by the proposed estimator is a single vector information and rate gyro readings. For systems in which one of the rotational freedom is constrained, the proposed estimator provides an accurate estimate of the reduced attitude. The performance of the algorithm is verilied on different experimental testbeds.

Scalable Underapproximation for the Stochastic Reach-Avoid Problem for High-Dimensional LTI Systems Using Fourier Transforms

We present a scalable underapproximation of the terminal hitting time stochastic reach-avoid probability at a given initial condition, for verification of high-dimensional stochastic LTI systems. While several approximation techniques have been proposed to alleviate the curse of dimensionality associated with dynamic programming, these techniques cannot handle larger, more realistic systems. We present a scalable method that uses Fourier transforms to compute an underapproximation of the reach-avoid probability for systems with disturbances with arbitrary probability densities. We characterize sufficient conditions for Borel-measurability of the value function. We exploit fixed control sequences parameterized by the initial condition (an open-loop control policy) to generate the underapproximation. For Gaussian disturbances, the underapproximation can be obtained using existing efficient algorithms by solving a convex optimization problem. Our approach produces non-trivial lower bounds and is demonstrated on a 40-D chain of integrators.

User-interface design for MIMO LTI human-automation systems through sensor placement

We propose a method for user-interface design for a MIMO LTI human-automation system by solving a corresponding sensor placement problem. We derive suitable metrics for optimization by transforming human factors guidelines for human-automation interaction into objective functions and constraints. We consider operation under 1) both nominal and off-nominal conditions, which have vastly different user requirements, and 2) noisy and noise-free sensors. The resulting optimization problems are combinatorial. We analyze optimal solutions, and provide algorithms to construct sub-optimal solutions when optimal solutions are not readily available. We apply this method to the problem of designing a user-interface for the remote operation of a fleet of UAVs.

Scalable Underapproximative Verification of Stochastic LTI Systems using Convexity and Compactness

We present a scalable algorithm to construct a polytopic underapproximation of the terminal hitting time stochastic reach-avoid set, for the verification of high-dimensional stochastic LTI systems with arbitrary stochastic disturbance. We prove the existence of a polytopic underapproximation by characterizing the sufficient conditions under which the stochastic reach-avoid set and the proposed open-loop underapproximation are compact and convex. We construct the polytopic underapproximation by formulating and solving a series of convex optimization problems. These set-theoretic properties also characterize circumstances under which the stochastic reach-avoid problem admits a bang-bang optimal Markov policy. We demonstrate the scalability of our algorithm on a 40D chain of integrators, the highest dimensional example demonstrated to date for stochastic reach-avoid problems, and compare its performance with existing approaches on a spacecraft rendezvous and docking problem.

Validation of cognitive models for collaborative hybrid systems with discrete human input

We present a method to validate a cognitive model, based on the cognitive architecture ACT-R, in dynamic human-automation systems with discrete human input. We are inspired by the general problem of K-choice games as a proxy for many decision making applications in dynamical systems. We model the human as a Markovian controller based on gathered experimental data, that is, a non-deterministic control input with known likelihoods of control actions associated with certain configurations of the state-space. We use reachability analysis to predict the outcome of the resulting discrete-time stochastic hybrid system, in which the outcome is defined as a function of the system trajectory. We suggest that the resulting expected outcomes can be used to validate the cognitive model against actual human subject data. We apply our method to a two-choice game in which the human is tasked with maximizing net coverage of a robotic swarm that can operate under rendezvous or deployment dynamics. We validate the corresponding ACTR cognitive model generated with the data from eight human subjects. The novelty of this work is (1) a method to compute expected outcome in a hybrid dynamical system with a Markov chain model of the humans discrete choice, and (2) application of this method to validation of cognitive models with a database of actual human subject data.

Viable set approximation for linear-Gaussian systems with unknown, bounded variance

Computation of stochastic reachable and viable sets enables assurances of safety and feasibility through the synthesis of optimal control policies. These control policies are typically generated under the assumption of accurate characterization of additive noise processes. We consider the case in which independent noise processes are not fully characterized. Specifically, we consider linear time-invariant dynamics with additive noise, with known mean and bounded (but unknown) variance. We propose a method to compute a conservative underapproximation to the stochastic viable set for problems with convex viable and target sets. We underapproximate probability values through linear transformation based on bounds on the unknown variance. We demonstrate this method (via dynamic programming) on a simple example.