Vision-Based Control on Lie groups with Application to Needle Steering

This thesis presents vision-based control algorithms for systems evolving on Lie groups. The thesis consists of two parts: (1) task-induced symmetry and reduction and its application to needle steering and (2) kernel-based visual servoing. The core of this thesis is motivated by image-guided control of flexible bevel-tip needles. Image guidance promises to improve targeting accuracy and broaden the scope of medical procedures performed with needles. We build upon a previously proposed nonholonomic kinematic model of flexible bevel-tip needle steering in which the needle is inserted and rotated at its base in order to steer it in six degrees of freedom. As a first step for control, we show that the needle tip can be automatically guided to a planar slice of tissue as it is inserted by a physician; our approach keeps the physician in the loop to control insertion speed. The distance of the needle tip position from the plane of interest is used to drive an observer-based feedback controller. We prove that the complete six degree-offreedom pose of the needle tip can be estimated from just the three-dimensional needle tip position measurements over time. This enables us to develop dead-beat and asymptotic observers to recover needle-tip orientation for control. ii ABSTRACT The task of driving a needle tip to a desired plane induces symmetry resulting in a reduced system which greatly simplifies controller and observer design. We propose a method to perform such reduction for generic nonholonomic kinematic systems on Lie groups with left-invariant vector fields. This technique is used to develop controllers for curve-following of a unicycle and subspace-following in needle steering. These subspace controllers for needle steering are designed to work in conjunction with subspace planners for the needle tip to reach a desired location in human tissue. We show that this taskinduced reduction lifts to mechanical systems as well. In the second part of the thesis, we present kernel-based visual servoing algorithms. In visual servoing, the goal is to control the motion of the robot/scene such that a set of image features converge to a known constellation; this requires tracking these feature points in every frame. Moving away form the traditional visual servoing approaches that have treated tracking and control as two isolated problems, kernel-based visual servoing paradigm fuses tracking and control by removing the need to explicitly track features in a scene. In this method, a weighted average of the image (or its transform) is used as the signal to the controller; the weighting function is a smooth kernel and the weighted average is called the kernel measurement. Using smooth kernel functions, we design, develop, and test controllers to navigate a robot to reach a desired goal in the three translational and roll degrees of freedom for an eye-in-hand configuration. This work provides a new framework to design vision-based controllers on natural images and their formal stability characterization using Lyapunov theory.The task of driving a needle tip to a desired plane induces symmetry resulting in a reduced system which greatly simplifies controller and observer design. We propose a method to perform such reduction for generic nonholonomic kinematic systems on Lie groups with left-invariant vector fields. This technique is used to develop controllers for curve-following of a unicycle and subspace-following in needle steering. These subspace controllers for needle steering are designed to work in conjunction with subspace planners for the needle tip to reach a desired location in human tissue. We show that this taskinduced reduction lifts to mechanical systems as well. In the second part of the thesis, we present kernel-based visual servoing algorithms. In visual servoing, the goal is to control the motion of the robot/scene such that a set of image features converge to a known constellation; this requires tracking these feature points in every frame. Moving away form the traditional visual servoing approaches that have treated tracking and control as two isolated problems, kernel-based visual servoing paradigm fuses tracking and control by removing the need to explicitly track features in a scene. In this method, a weighted average of the image (or its transform) is used as the signal to the controller; the weighting function is a smooth kernel and the weighted average is called the kernel measurement. Using smooth kernel functions, we design, develop, and test controllers to navigate a robot to reach a desired goal in the three translational and roll degrees of freedom for an eye-in-hand configuration. This work provides a new framework to design vision-based controllers on natural images and their formal stability characterization using Lyapunov theory.