A Contraction Theory Approach to Stochastic Incremental Stability

We investigate the incremental stability properties of It\^o stochastic dynamical systems. Specifically, we derive a stochastic version of nonlinear contraction theory that provides a bound on the mean square distance between any two trajectories of a stochastically contracting system. This bound can be expressed as a function of the noise intensity and the contraction rate of the noise-free system. We illustrate these results in the contexts of stochastic nonlinear observers design and stochastic synchronization.Comment: 23 pages, 2 figure

Analysis of Discrete and Hybrid Stochastic Systems by Nonlinear Contraction Theory

We investigate the stability properties of discrete and hybrid stochastic nonlinear dynamical systems. More precisely, we extend the stochastic contraction theorems (which were formulated for continuous systems) to the case of discrete and hybrid resetting systems. In particular, we show that the mean square distance between any two trajectories of a discrete (or hybrid resetting) contracting stochastic system is upper-bounded by a constant after exponential transients. Using these results, we study the synchronization of noisy nonlinear oscillators coupled by discrete noisy interactions.Comment: 6 pages, 1 figur

Vision based estimation, localization, and mapping for autonomous vehicles

In this dissertation, we focus on developing simultaneous localization and mapping (SLAM) algorithms with a robot-centric estimation framework primarily using monocular vision sensors. A primary contribution of this work is to use a robot-centric mapping framework concurrently with a world-centric localization method. We exploit the differential equation of motion of the normalized pixel coordinates of each point feature in the robot body frame. Another contribution of our work is to exploit a multiple-view geometry formulation with initial and current view projection of point features. We extract the features from objects surrounding the river and their reflections. The correspondences of the features are used along with the attitude and altitude information of the robot. We demonstrate that the observability of the estimation system is improved by applying our robot-centric mapping framework and multiple-view measurements. Using the robot-centric mapping framework and multiple-view measurements including reflection of features, we present a vision based localization and mapping algorithm that we developed for an unmanned aerial vehicle (UAV) flying in a riverine environment. Our algorithm estimates the 3D positions of point features along a river and the pose of the UAV. Our UAV is equipped with a lightweight monocular camera, an inertial measurement unit (IMU), a magnetometer, an altimeter, and an onboard computer. To our knowledge, we report the first result that exploits the reflections of features in a riverine environment for localization and mapping. We also present an omnidirectional vision based localization and mapping system for a lawn mowing robot. Our algorithm can detect whether the robotic mower is contained in a permitted area. Our robotic mower is modified with an omnidirectional camera, an IMU, a magnetometer, and a vehicle speed sensor. Here, we also exploit the robot-centric mapping framework. The estimator in our system generates a 3D point based map with landmarks. Concurrently, the estimator defines a boundary of the mowing area by using the estimated trajectory of the mower. The estimated boundary and the landmark map are provided for the estimation of the mowing location and for the containment detection. First, we derive a nonlinear observer with contraction analysis and pseudo-measurements of the depth of each landmark to prevent the map estimator from diverging. Of particular interest for this work is ensuring that the estimator for localization and mapping will not fail due to the nonlinearity of the system model. For batch estimation, we design a hybrid extended Kalman smoother for our localization and robot-centric mapping model. Finally, we present a single camera based SLAM algorithm using a convex optimization based nonlinear estimator. We validate the effectiveness of our algorithms through numerical simulations and outdoor experiments