Generalizations of Kempe's universality theorem

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Copyright statement on t.p. reads: ©Timothy Good Abbott, 2004-2007, ©Reid W. Barton, 2004-2007.Includes bibliographical references (p. 85-86).In 1876, A. B. Kempe presented a flawed proof of what is now called Kempe's Universality Theorem: that the intersection of a closed disk with any curve in R2 defined by a polynomial equation can be drawn by a linkage. Kapovich and Millson published the first correct proof of this claim in 2002, but their argument relied on different, more complex constructions. We provide a corrected version of Kempe's proof, using a novel contraparallelogram bracing. The resulting historical proof of Kempe's Universality Theorem uses simpler gadgets than those of Kapovich and Millson. We use our two-dimensional proof of Kempe's theorem to give simple proofs of two extensions of Kempe's theorem first shown by King: a generalization to d dimensions and a characterization of the drawable subsets of Rd. Our results improve King's by proving better continuity properties for the constructions. We prove that our construction requires only O(nd) bars to draw a curve defined by a polynomial of degree n in d dimensions, improving the previously known bounds of O(n4) in two dimensions and O(n6) in three dimensions. We also prove a matching Q(nd) lower bound in the worst case. We give an algorithm for computing a configuration above a given point on a given polynomial curve, running in time polynomial in the size of the dense representation of the polynomial defining the curve. We use this algorithm to prove the coNP-hardness of testing the rigidity of a given configuration of a linkage. While this theorem has long been assumed in rigidity theory, we believe this to be the first published proof that this problem is computationally intractable. This thesis is joint work with Reid W. Barton and Erik D. Demaine.by Timothy Good Abbott.S.M

Optimization of Planar Mechanisms by using a Minimum Distance Function

The paper examines the application of a general minimum distance error function to the dimensional kinematic synthesis of bidimensional mechanisms. The minimum distance ap- proach makes it possible to solve the problem maintaining the same generality as that of the minimum deformation energy method while solving the problems that occasionally appear in the former method involving low stiffness mechanisms. It is a general method that can deal both with unprescribed and prescribed timing problems, and is applicable for path generation problems, function generation, solid guidance, and any combination of the aforementioned requirements as introduced in the usual precision point scheme. The method exhibits good convergence and computational efficiency. The minimum distance error function is solved with a sequential quadratic programming (SQP) approach. In the study, the synthesis problem is also optimized by using SQP, and the function can be easily adapted to other methods such as genetic algorithms. In the study, the minimum distance approach is initially presented. Subsequently, an efficient SQP method is developed by using analytic derivatives for solving. The next point addresses the application of the concept for the synthesis of mechanisms by using an SQP approach with approximate derivatives. This delivers a situation where the optimization is performed on an error function that itself consists of an inner optimization function. A few examples are presented and are also compared with the minimum deformation energy method. Finally, a few conclusions and future studies are discussedThe authors acknowledge direct or indirect economic support provided by the Investigation Groups recognized by the Basque Government under section IT 947-16 , and the Spanish Ministry of Economy and Competitiveness through the project DPI2016-80372-R (AEI/FEDER. UE

Virtual articulation and kinematic abstraction in robotics

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 279-292).This thesis presents the theory, implementation, novel applications, and experimental validation of a general-purpose framework for applying virtual modifications to an articulated robot, or virtual articulations. These can homogenize various aspects of a robot and its task environment into a single unified model which is both qualitatively high-level and quantitatively functional. This is the first framework designed specifically for the mixed real/virtual case. It supports arbitrary topology spatial kinematics, a broad catalog of joints, on-line structure changes, interactive kinostatic simulation, and novel kinematic abstractions, where complex subsystems are simplified with virtual replacements in both space and time. Decomposition algorithms, including a novel method of hierarchical subdivision, enable scaling to large closed-chain mechanisms with 100s of joints. Novel applications are presented in two areas of current interest: operating high- DoF kinematic manipulation and inspection tasks, and analyzing reliable kinostatic locomotion strategies based on compliance and proprioception. In both areas virtual articulations homogeneously model the robot and its task environment, and abstractions structure complex models. For high-DoF operations the operator attaches virtual joints as a novel interface metaphor to define task motion and to constrain coordinated motion (by virtually closing kinematic chains); virtual links can represent task frames or serve as intermediate connections for virtual joints. For compliant locomotion, virtual articulations model relevant compliances and uncertainties, and temporal abstractions model contact state evolution.(cont.) Results are presented for experiments with two separate robotic systems in each area. For high-DoF operations, NASA/JPL's 36 DoF ATHLETE performs previously challenging coordinated manipulation/inspection moves, and a novel large-scale (100s of joints) simulated modular robot is conveniently operated using spatial abstractions. For compliant locomotion, two experiments are analyzed that each achieve high reliability in uncertain tasks using only compliance and proprioception: a novel vertical structure climbing robot that is 99.8% reliable in over 1000 motions, and a mini-humanoid that steps up an uncertain height with 90% reliability in 80 trials. In both cases virtual articulation models capture the essence of compliant/proprioceptive strategies at a higher level than basic physics, and enable quantitative analyses of the limits of tolerable uncertainty that compare well to experiment.by Marsette Arthur Vona, III.Ph.D