DOCSLIB.ORG

Screw Theory Cedric Fischer and Michael Mattmann

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Screw Theory Cedric Fischer and Michael Mattmann Screw Theory Cedric Fischer and Michael Mattmann Institute of Robotics and Intelligent Systems Department of Mechanical and Process Engineering (DMAVT) ETH Zurich Screw Theory Every rigid body motion can be expressed by a rotation about an axis combined with a translation parallel to that axis. (= screw motion) Method is based on rigid body motion instead of location! Time dependent now! Different ways to describe it: - screws (geometrical description: screw parameter) - twists (mathematical description: abstract) - product of exponentials (mathematical description: homogeneous) 2 Screw Theory: Geometrical Description l M h Screw parameters . Pitch h . Ratio of translational motion to rotational motion . Axis l . Axis of rotation, line through a point . Direction of translation . Magnitude M . Amount of displacement . Net rotation and/or translation 3 Screw Theory: Mathematical Description é a b - a b ù é 0 -a a ù ê 2 3 3 2 ú ê 3 2 ú Skew-symmetric matrix a ´ b = abˆ = ê a3b1 - a1b3 ú = ê a3 0 -a1 úb ê ú ê ú a b - a b -a a 0 ëê 1 2 2 1 ûú ëê 2 1 ûú . Twist Coordinates . Twist . 6x1 vector . 4x4 matrix General description with Derivation: page 39 – A Mathematical Introduction to Robotic Manipulation 1994 4 Screw Theory: Mathematical Description é a b - a b ù é 0 -a a ù ê 2 3 3 2 ú ê 3 2 ú Skew-symmetric matrix a ´ b = abˆ = ê a3b1 - a1b3 ú = ê a3 0 -a1 úb ê ú ê ú a b - a b -a a 0 ëê 1 2 2 1 ûú ëê 2 1 ûú . Twist Coordinates . Twist . 6x1 vector . 4x4 matrix é -w ´ q ù é wˆ -w ´ q ù x = ê ú xˆ = ê ú ê w ú ê 0 0 ú Revolute joint ë û ë û é v ù é 0 v ù x = ê ú xˆ = ê ú ë 0 û ë 0 0 û Prismatic joint Derivation: page 39 – A Mathematical Introduction to Robotic Manipulation 1994 5 Rodrigues’ Formula . All rotation matrices can be written as a matrix exponential of a skew-symmetric matrix! wqˆ 2 . Rodrigues’ Formula: e = I +wˆ sinq +wˆ (1- cosq) Proof on extra slide! . Homogeneous é wqˆ wqˆ ù ˆ e I - e w ´ v + hqw Transformation exq = ê ( )( ) ú ê ú . 4x4 matrix ë 0 1 û Revolute joint Prismatic joint é wqˆ wqˆ ù é ù xqˆ e I - e (w ´ v ) xqˆ I qv e = ê ( ) ú e = ê ú ê ú 0 1 ë 0 1 û ë û 6 Screw Theory: Mathematical Description If a coordinate frame B is attached to a rigid body undergoing a screw motion, the instantaneous configuration of the coordinate frame B, relative to a fixed frame A, is given by g(0): - all joint angles defined as being zero - Describes transformation from the base frame to tool frame This transformation can be interpreted as follows: - multiplication by gab(0) maps the coordinates of a point relative to the B frame into A’s coordinates - the exponential map transforms the point to its final location (still in A coordinates). 7 Assignment 3 Two systems: System 1: . a) g(0) . Rotation and translation between CS . b) Pitch, axis and magnitude System 2: . c) Twist and twist coordinates . d) Find total homogeneous transformation g12 . e) Matlab: use twist, twistexp . plot trajectories of P1,P2,P3 ( g(θ)*Pi ) 9 Kinematics Toolbox • Figure, chart, video… Point Rotation axis skew Skew-symm. matrix skewlog q w w^ skewcoords Rotation matrix skewexp R createtwist Magnitude twistmagnitude M Twist Coord. twist Twist Axis twistaxis [6x1] [4x4] twistlog l x twistcoords x^ Pitch h twistpitch Hom. Transf. [4x4] twistexp g 10.
Load more

Recommended publications
  • Screw and Lie Group Theory in Multibody Dynamics Recursive Algorithms and Equations of Motion of Tree-Topology Systems
    Multibody Syst Dyn DOI 10.1007/s11044-017-9583-6 Screw and Lie group theory in multibody dynamics Recursive algorithms and equations of motion of tree-topology systems Andreas Müller1 Received: 12 November 2016 / Accepted: 16 June 2017 © The Author(s) 2017. This article is published with open access at Springerlink.com Abstract Screw and Lie group theory allows for user-friendly modeling of multibody sys- tems (MBS), and at the same they give rise to computationally efficient recursive algo- rithms. The inherent frame invariance of such formulations allows to use arbitrary reference frames within the kinematics modeling (rather than obeying modeling conventions such as the Denavit–Hartenberg convention) and to avoid introduction of joint frames. The com- putational efficiency is owed to a representation of twists, accelerations, and wrenches that minimizes the computational effort. This can be directly carried over to dynamics formula- tions. In this paper, recursive O(n) Newton–Euler algorithms are derived for the four most frequently used representations of twists, and their specific features are discussed. These for- mulations are related to the corresponding algorithms that were presented in the literature. Two forms of MBS motion equations are derived in closed form using the Lie group formu- lation: the so-called Euler–Jourdain or “projection” equations, of which Kane’s equations are a special case, and the Lagrange equations. The recursive kinematics formulations are readily extended to higher orders in order to compute derivatives of the motions equations. To this end, recursive formulations for the acceleration and jerk are derived. It is briefly discussed how this can be employed for derivation of the linearized motion equations and their time derivatives.
    [Show full text]
  • An Historical Review of the Theoretical Development of Rigid Body Displacements from Rodrigues Parameters to the finite Twist
    Mechanism and Machine Theory Mechanism and Machine Theory 41 (2006) 41–52 www.elsevier.com/locate/mechmt An historical review of the theoretical development of rigid body displacements from Rodrigues parameters to the finite twist Jian S. Dai * Department of Mechanical Engineering, School of Physical Sciences and Engineering, King’s College London, University of London, Strand, London WC2R2LS, UK Received 5 November 2004; received in revised form 30 March 2005; accepted 28 April 2005 Available online 1 July 2005 Abstract The development of the finite twist or the finite screw displacement has attracted much attention in the field of theoretical kinematics and the proposed q-pitch with the tangent of half the rotation angle has dem- onstrated an elegant use in the study of rigid body displacements. This development can be dated back to RodriguesÕ formulae derived in 1840 with Rodrigues parameters resulting from the tangent of half the rota- tion angle being integrated with the components of the rotation axis. This paper traces the work back to the time when Rodrigues parameters were discovered and follows the theoretical development of rigid body displacements from the early 19th century to the late 20th century. The paper reviews the work from Chasles motion to CayleyÕs formula and then to HamiltonÕs quaternions and Rodrigues parameterization and relates the work to Clifford biquaternions and to StudyÕs dual angle proposed in the late 19th century. The review of the work from these mathematicians concentrates on the description and the representation of the displacement and transformation of a rigid body, and on the mathematical formulation and its progress.
    [Show full text]
  • STORM: Screw Theory Toolbox for Robot Manipulator and Mechanisms
    2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) October 25-29, 2020, Las Vegas, NV, USA (Virtual) STORM: Screw Theory Toolbox For Robot Manipulator and Mechanisms Keerthi Sagar*, Vishal Ramadoss*, Dimiter Zlatanov and Matteo Zoppi Abstract— Screw theory is a powerful mathematical tool in Cartesian three-space. It is crucial for designers to under- for the kinematic analysis of mechanisms and has become stand the geometric pattern of the underlying screw system a cornerstone of modern kinematics. Although screw theory defined in the mechanism or a robot and to have a visual has rooted itself as a core concept, there is a lack of generic software tools for visualization of the geometric pattern of the reference of them. Existing frameworks [14], [15], [16] target screw elements. This paper presents STORM, an educational the design of kinematic mechanisms from computational and research oriented framework for analysis and visualization point of view with position, velocity kinematics and iden- of reciprocal screw systems for a class of robot manipulator tification of different assembly configurations. A practical and mechanisms. This platform has been developed as a way to bridge the gap between theory and practice of application of screw theory in the constraint and motion analysis for robot mechanisms. STORM utilizes an abstracted software architecture that enables the user to study different structures of robot manipulators. The example case studies demonstrate the potential to perform analysis on mechanisms, visualize the screw entities and conveniently add new models and analyses. I. INTRODUCTION Machine theory, design and kinematic analysis of robots, rigid body dynamics and geometric mechanics– all have a common denominator, namely screw theory which lays the mathematical foundation in a geometric perspective.
    [Show full text]
  • Chapter 1 Introduction and Overview
    Chapter 1 Introduction and Overview ‘ 1.1 Purpose of this book This book is devoted to the kinematic and mechanical issues that arise when robotic mecha- nisms make multiple contacts with physical objects. Such situations occur in robotic grasp- ing, workpiece fixturing, and the quasi-static locomotion of multilegged robots. Figures 1.1(a,b) show a many jointed multi-fingered robotic hand. Such a hand can both grasp and manipulate a wide variety of objects. A grasp is used to affix the object securely within the hand, which itself is typically attached to a robotic arm. Complex robotic hands can imple- ment a wide variety of grasps, from precision grasps, where only only the finger tips touch the grasped object (Figure 1.1(a)), to power grasps, where the finger mechanisms may make multiple contacts with the grasped object (Figure 1.1(b)) in order to provide a highly secure grasp. Once grasped, the object can then be securely transported by the robotic system as part of a more complex manipulation operation. In the context of grasping, the joints in the finger mechanisms serve two purposes. First, the torques generated at each actuated finger joint allow the contact forces between the finger surface and the grasped object to be actively varied in order to maintain a secure grasp. Second, they allow the fingertips to be repositioned so that the hand can grasp a very wide variety of objects with a range of grasping postures. Beyond grasping, these fingers’ degrees of freedom potentially allow the hand to manipulate, or reorient, the grasped object within the hand.
    [Show full text]
  • Exponential and Cayley Maps for Dual Quaternions
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by LSBU Research Open Exponential and Cayley maps for Dual Quaternions J.M. Selig Abstract. In this work various maps between the space of twists and the space of finite screws are studied. Dual quaternions can be used to represent rigid-body motions, both finite screw motions and infinitesimal motions, called twists. The finite screws are elements of the group of rigid-body motions while the twists are elements of the Lie algebra of this group. The group of rigid-body displacements are represented by dual quaternions satisfying a simple relation in the algebra. The space of group elements can be though of as a six-dimensional quadric in seven-dimensional projective space, this quadric is known as the Study quadric. The twists are represented by pure dual quaternions which satisfy a degree 4 polynomial relation. This means that analytic maps between the Lie algebra and its Lie group can be written as a cubic polynomials. In order to find these polynomials a system of mutually annihilating idempotents and nilpotents is introduced. This system also helps find relations for the inverse maps. The geometry of these maps is also briefly studied. In particular, the image of a line of twists through the origin (a screw) is found. These turn out to be various rational curves in the Study quadric, a conic, twisted cubic and rational quartic for the maps under consideration. Mathematics Subject Classification (2000). Primary 11E88; Secondary 22E99. Keywords. Dual quaternions, exponential map, Cayley map.
    [Show full text]
  • A New Formulation Method for Solving Kinematic Problems of Multiarm Robot Systems Using Quaternion Algebra in the Screw Theory Framework
    Turk J Elec Eng & Comp Sci, Vol.20, No.4, 2012, c TUB¨ ITAK˙ doi:10.3906/elk-1011-971 A new formulation method for solving kinematic problems of multiarm robot systems using quaternion algebra in the screw theory framework Emre SARIYILDIZ∗, Hakan TEMELTAS¸ Department of Control Engineering, Istanbul˙ Technical University, 34469 Istanbul-TURKEY˙ e-mails: [email protected], [email protected] Received: 28.11.2010 Abstract We present a new formulation method to solve the kinematic problem of multiarm robot systems. Our major aims were to formulize the kinematic problem in a compact closed form and avoid singularity problems in the inverse kinematic solution. The new formulation method is based on screw theory and quaternion algebra. Screw theory is an effective way to establish a global description of a rigid body and avoids singularities due to the use of the local coordinates. The dual quaternion, the most compact and efficient dual operator to express screw displacement, was used as a screw motion operator to obtain the formulation in a compact closed form. Inverse kinematic solutions were obtained using Paden-Kahan subproblems. This new formulation method was implemented into the cooperative working of 2 St¨aubli RX160 industrial robot-arm manipulators. Simulation and experimental results were derived. Key Words: Cooperative working of multiarm robot systems, dual quaternion, industrial robot application, screw theory, singularity-free inverse kinematic 1. Introduction Multiarm robot configurations offer the potential to overcome many difficulties by increased manipulation ability and versatility [1,2]. For instance, single-arm industrial robots cannot perform their roles in the many fields in which operators do the job with their 2 arms [3].
    [Show full text]
  • Internship Report
    Internship Report Assessment of influence of play in joints on the end effector accuracy in a novel 3DOF (1T-2R) parallel manipulator Vinayak J. Kalas (MSc – Mechanical Engineering, University of Twente) Advisors: ir. A.G.L. Hoevenaars (Delft Haptics Lab, Delft University of Technology) prof. dr. ir J.L. Herder (University of Twente) Acknowledgement I would like to express my sincere gratitude to my supervisor from TU Delft, Teun Hoevenaars, for his constant guidance that kept me on track with the defined objectives and helped me in having clarity of thought at different stages of my research. I would like to extend my appreciation to prof. Just Herder for his guidance and support. Finally, I would like to thank all the researchers at the Delft Haptics lab for interesting discussions and valuable inputs at various stages of the research. Preface This internship assignment was carried out at TU Delft and was a compulsory module as a part of my Master studies in Mechanical Engineering at the University of Twente. This internship was carried out for a period of 14 weeks, from 16th of November 2015 to 19th of February 2016, and during this period I was exposed to different research projects being carried out at the Delft Haptics lab, apart from in-depth knowledge of the project I was working on. The main objectives of this internship assignment were to get acquainted with Parallel Kinematics, which included deep understanding of Screw theory and its application to parallel manipulator analysis, and to get acquainted with various methods to assess the effects of play at the joints of parallel mechanisms on the end effector accuracy.
    [Show full text]
  • Time Derivatives of Screws with Applications to Dynamics and Stiffness Harvey Lipkin
    Time derivatives of screws with applications to dynamics and stiffness Harvey Lipkin To cite this version: Harvey Lipkin. Time derivatives of screws with applications to dynamics and stiffness. Mechanism and Machine Theory, Elsevier, 2005, 40 (3), 10.1016/j.mechmachtheory.2003.07.002. hal-01371131 HAL Id: hal-01371131 https://hal.archives-ouvertes.fr/hal-01371131 Submitted on 24 Sep 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License body systems, and a symposium in his honor well illustrates the breadth and variety of modern screw theory applications [14]. Comprehensive treatments in kinematics and mechanism theory are presented by Bottema and Roth [2], Hunt [11], and Phillips [18]. Although it would appear that the time derivative of screw quantity should be well under- stood, that is not the case. There is active discussion in the literature of: the general topic by [21,25,29];applications to acceleration analysis by [22,8], with historical background in [20]; applications to stiffness by [9,5,10,26,24];and applications to dynamics by [15,7,13,4,17,27].
    [Show full text]
  • Dual Quaternions in Spatial Kinematics in an Algebraic Sense
    Turk J Math 32 (2008) , 373 – 391. c TUB¨ ITAK˙ Dual Quaternions in Spatial Kinematics in an Algebraic Sense Bedia Akyar Abstract This paper presents the finite spatial displacements and spatial screw motions by using dual quaternions and Hamilton operators. The representations are considered as 4 × 4 matrices and the relative motion for three dual spheres is considered in terms of Hamilton operators for a dual quaternion. The relation between Hamilton operators and the transformation matrix has been given in a different way. By considering operations on screw motions, representation of spatial displacements is also given. Key Words: Dual quaternions, Hamilton operators, Lie algebras. 1. Introduction The matrix representation of spatial displacements of rigid bodies has an important role in kinematics and the mathematical description of displacements. Veldkamp and Yang-Freudenstein investigated the use of dual numbers, dual numbers matrix and dual quaternions in instantaneous spatial kinematics in [9] and [10], respectively. In [10], an application of dual quaternion algebra to the analysis of spatial mechanisms was given. A comparison of representations of general spatial motion was given by Rooney in [8]. Hiller- Woernle worked on a unified representation of spatial displacements. In their paper [7], the representation is based on the screw displacement pair, i.e., the dual number extension of the rotational displacement pair, and consists of the dual unit vector of the screw axis AMS Mathematics Subject Classification: 53A17, 53A25, 70B10. 373 AKYAR and the associate dual angle of the amplitude. Chevallier gave a unified algebraic approach to mathematical methods in kinematics in [4]. This approach requires screw theory, dual numbers and Lie groups.
    [Show full text]
  • Three Problems in Robotics
    Three Problems in Robotics J.M. Selig School of Computing, Information Systems & Mathematics South Bank University London SE1 0AA, U.K. Abstract Three rather different problems in robotics are studied using the same tech- nique from screw theory. The first problem concerns systems of springs. The potential function is differentiated in the direction of an arbitrary screw to find the equilibrium position. The second problem is almost identical in terms of the computations, the least squares solution to the problem of finding the rigid motion undergone by a body given only data about points on the body is sought. In the third problem the Jacobian of a Stewart platform is found. Again this is achieved by differentiating with respect to a screw. Further, second order properties of the first two problems are studied. The Hessian of second derivatives is computed and hence the stability properties of the equilibrium positions of the spring system are found. 1 Notation n R n-dimensional Euclidean space. 3 RP 3-dimensional projective space. SE(3) The group of proper Euclidean motions in 3 dimensions, that is rigid body motions. SO(3) The group of rotations in 3 dimensions. R A 3 × 3 rotation matrix. M A 4 × 4 matrix representing a rigid transformation, sometimes called the homoge- neous representation. a; b The position vectors of points in 3-dimensional space. a˜; b˜ Points in 3-dimensional space represented as points in 4-dimensional space, in partitioned form, a˜T = (aT ; 1) for example. A; B 3-dimensional vectors represented as 3×3 anti-symmetric matrices, so that Ax = a × x for any vector x.
    [Show full text]
  • Design and Fabrication of Complex Flexible Structures and Microarchitectures
    Design and Fabrication of Complex Flexible Structures and Microarchitectures The ability to design and fabricate complex flexible structures is becoming increasingly important for enabling next‐generation technologies (e.g., microarchitectures that possess radically superior properties compared with those of naturally occurring materials, high‐ precision multi‐degree‐of‐freedom flexure‐based motion systems, and tip‐tilt‐piston micro‐ mirror arrays that steer light with unprecedented speeds). In contrast with their rigid‐body counterparts, flexible structures achieve their motions through the deformation of their compliant elements and thus they (i) achieve high resolution and repeatability, (ii) require no lubrication, (iii) experience minimal friction and wear, (iv) are easily maintained, and (v) often cost significantly less than other competing precision bearing technologies (e.g., magnetic or air bearings). Despite their many advantages, many engineers avoid using flexible structures because design tools for creating them are limited and modeling tools for analyzing their kinematics and stiffness characteristics are complex and unapproachable due to substantial nonlinearities. The Freedom, Actuation, and Constraint Topologies (FACT) design approach has, however, been successfully applied to the design and analysis of flexible structures of any complexity. The basis for FACT is a comprehensive library of geometric shapes that represent the mathematics of screw theory, projective geometry, and constraint‐based design. These shapes enable
    [Show full text]
  • CONNECTIONS BETWEEN SCREW THEORY and CARTAN's CONNECTIONS Diego Colón ∗Laboratório De Automaç˜Ao E Controle
    Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014 CONNECTIONS BETWEEN SCREW THEORY AND CARTAN'S CONNECTIONS Diego Colon´ ∗ ∗Laborat´orio de Automa¸c~aoe Controle - LAC/PTC, Escola Polit´ecnica da USP Av. Prof. Luciano Gualberto, trav 3, 158 Cidade Universit´aria, S~ao Paulo, BRAZIL Email: [email protected] Abstract| In this paper some relations that exist between expressions in Screw Theory of Robotics and Differential Geometry, from the point of view of Cartan, are presented. The ideas were inspired in books of Differential Geometry, Lie groups and Lie algebras and its applications. The Cartan's connection is the initial concept, followed by the covariant derivative. It is presented how kinematics and dynamics are obtained, in case of pure rotations and general movement in Euclidean space. The generalization of these concepts to Riemannian manifolds are then presented, and some conclusions are drawn concerning the possibility of kinematic chain calculations in such space (propagation of velocities). Finally, the idea of using connections and covariant derivatives are extended to multi-frame cases (as is common in Robotics) in Euclidean space. Keywords| Screw Theory, Riemannian geometry, Cartan's Connection. Resumo| Este artigo apresenta algumas rela¸c~oesexistentes entre express~oese formula¸c~oes em Rob´otica (em particular na Teoria das Helic´oides) com Geometria Diferencial do ponto de vista de Cartan. A id´eiade partida foi inspirada em livros de Geometria Diferential e suas aplica¸c~oes.A conex~ao de Cartan ´eo ponto de partida para a interpreta¸c~aoposterior, assim como a derivada covariante.
    [Show full text]