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

Optimization-based Framework for Stability and Robustness of Bipedal Walking Robots

As robots become more sophisticated and move out of the laboratory, they need to be able to reliably traverse difficult and rugged environments. Legged robots -- as inspired by nature -- are most suitable for navigating through terrain too rough or irregular for wheels. However, control design and stability analysis is inherently difficult since their dynamics are highly nonlinear, hybrid (mixing continuous dynamics with discrete impact events), and the target motion is a limit cycle (or more complex trajectory), rather than an equilibrium. For such walkers, stability and robustness analysis of even stable walking on flat ground is difficult. This thesis proposes new theoretical methods to analyse the stability and robustness of periodic walking motions. The methods are implemented as a series of pointwise linear matrix inequalities (LMI), enabling the use of convex optimization tools such as sum-of-squares programming in verifying the stability and robustness of the walker. To ensure computational tractability of the resulting optimization program, construction of a novel reduced coordinate system is proposed and implemented. To validate theoretic and algorithmic developments in this thesis, a custom-built “Compass gait” walking robot is used to demonstrate the efficacy of the proposed methods. The hardware setup, system identification and walking controller are discussed. Using the proposed analysis tools, the stability property of the hardware walker was successfully verified, which corroborated with the computational results

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