This thesis about immobilization of two- and three-dimensional objects. Immobilization is crucial to fixturing in manufacturing processes and to robot hand grasping. There are many types of immobility depending on the finger types and the analyses, such as form closure, force closure and second-order immobility. Among these, form closure is the basis of immobility. Form closure was formulated by Reuleaux in 1876. He defined a rigid body to be in form closure if a set of fingers along its boundary constrained all finite and infinitesimal motions of the body. Chater 1 and 2 introduce the background and the concepts of immobility of various types. Chater 3 to Chapter 8 are about efficiently identifying all these immobilizing grasps. For efficient synthesis of all form-closure grasps of planar objects, we use the geometric formulation of form closure in three-dimensional wrench space. Our problem in the formulation in wrench space states as follows: Given a set of shapes in three-dimensional space, report all sets of shapes, such that each set has four points whose convex hull contains the origin inside. We transform this geometric problem into intersection search problems on planes, which can be handled with techniques in Computational Geometry. In Chapter 3 and 4, we propose efficient algorithms to enumerate all combinations of concave vertices and edges or arcs of a polygon and a semi-algebraic set that allow at least one form-closure grasp with at most four frictionless point fingers. In Chapter 5, we efficiently identify all immobilizing grasps (called force-closure grasps) of a polygon and a semi-algebraic set with two or three frictional point fingers. In Chapter 6, we propose efficient output-sensitive algorithms to enumerate all immobilizing grasps called second-order immobility grasps, with two and three frictionless point fingers for a polygon. Chapter 7 is about grasps that are tolerant to misplacements of fingers. We first divide the edges into small pieces of length varepsilon. Then we compute all sets of concave vertices and edges of a polygon, such that any placement of three or four frictionless point fingers inside these edges will put the polygon in form closure. We call such a combination of concave vertices and edges an independent form-closure grasp region. Chapter 8 is on efficiently computing all combinations of faces, concave edges and concave vertices, such that four to seven frictionless point fingers on each of these combinations allow at least one form-closure grasp. In Chapter 9, we study the problem of immobilizing a serial chain of hinged polygons in a given placement with frictionless point fingers. We define new notions of immobility and robust immobility, which are comparable to second-order immobility and to form closure for a single rigid object. For each case, we show how many fingers can immobilize or robustly immobilize a given serial chain of hinged polygons, by constructing a grasp
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