Euler-Rodrigues Formula Variations, Quaternion Conjugation and Intrinsic

Total Page:16

File Type:pdf, Size:1020Kb

Euler-Rodrigues Formula Variations, Quaternion Conjugation and Intrinsic Mechanism and Machine Theory 92 (2015) 144–152 Contents lists available at ScienceDirect Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt Euler–Rodrigues formula variations, quaternion conjugation and intrinsic connections Jian S. Dai ⁎ MOE key lab for mechanism theory and equipment design, International Centre for Advanced Mechanisms and Robotics, Tianjin University, PR China Centre for Robotics Research, School of Natural Sciences and Mathematics, King's College London, University of London, United Kingdom article info abstract Article history: This paper reviews the Euler–Rodrigues formula in the axis–angle representation of rotations, Received 13 May 2014 studies its variations and derivations in different mathematical forms as vectors, quaternions Received in revised form 16 February 2015 and Lie groups and investigates their intrinsic connections. The Euler–Rodrigues formula in the Accepted 7 March 2015 Taylor series expansion is presented and its use as an exponential map of Lie algebras is discussed Available online xxxx particularly with a non-normalized vector. The connection between Euler–Rodrigues parameters and the Euler–Rodrigues formula is then demonstrated through quaternion conjugation and the Keywords: equivalence between quaternion conjugation and an adjoint action of the Lie group is subsequent- – Euler Rodrigues formula ly presented. The paper provides a rich reference for the Euler–Rodrigues formula, the variations Quaternions and their connections and for their use in rigid body kinematics, dynamics and computer graphics. Exponential map Lie groups © 2015 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY license Lie algebras (http://creativecommons.org/licenses/by/4.0/). Kinematics 1. Introduction Euler–Rodrigues formula was first revealed in Euler's equations [1] published in 1775 in the way of change of direction cosines of a unit vector before and after a rotation. This was rediscovered independently by Rodrigues [2] in 1840 with Rodrigues param- eters [3] of tangent of half the rotation angle attached with coordinates of the rotation axis, known as Rodrigues vector [4–6] sometimes called the vector–parameter [7], presenting a way for geometrically constructing a rotation matrix. The vector form of this formula was revealed by Gibbs [8], Bisshopp [9], and Bottema and Roth [10] in their presentation of the Rodrigues formu- lae in planar and spatial motion. In addition to Rodrigues parameters, Euler–Rodrigues parameters were revealed in the same paper [2] as the unit quaternion. It was illustrated by Cayley [11] that the rotation about an axis by an angle could be implement- ed by a quaternion transformation [12] that was again interpreted by Cayley [13] physically using Euler–Rodrigues parameters that we now know as the quaternion conjugation [14,15], this coincides with the result of using Rodrigues parameters in the Euler–Rodrigues formula, leading to Cayley transform [16] as a mapping between skew–symmetric matrices of Lie algebra ele- ments and special orthogonal matrices of Lie group elements. The Euler–Rodrigues formula for finite rotations [17,18] raised much interest in the second half of the 20th century. In 1969, Bisshop [9] studied the formula in vector form of the rotation tensor by presenting a derivation from rotating a vector about an axis by an angle. In 1979, Bottema and Roth [10] presented Rodrigues formulae for rigid body displacements of various motions and put forward vectorial representations. In 1980, Gray [4] reviewed Rodrigues' contribution to the combination of two rotations ⁎ Tel.: +442078482321. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.mechmachtheory.2015.03.004 0094-114X/© 2015 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). J.S. Dai / Mechanism and Machine Theory 92 (2015) 144–152 145 and presented a historical account of this development. In 1989, Cheng and Gupta [19] verified the original contribution of Euler and accounted a further contribution of Rodrigues based on Euler–Rodrigues parameters. Since 1980s, Euler–Rodrigues formula has been widely used in geometric algebra [20,21], theoretical kinematics [10,22,23], and robotics [24]. In modern mathematics, Euler–Rodri- gues formula is used as an exponential map [25] that converts Lie algebra so(3) into Lie group SO(3), providing an algorithm for the exponential map without calculating the full matrix exponent [26–28] and for multi-body dynamics [29–32]. In the 21st century, Euler–Rodrigues formula continuously attracted broad interest. In 2003, Bauchau and Trainelli [33] developed an explicit expression of the rotation tensor in terms of vector parameterization based on the Euler–Rodrigues formula and in particular utilized tangent of half the angle of rotations [10,34].In2004and2006,Dai[3,35] reviewed Euler–Rodrigues parameters in the context of theoretical development of rigid body displacement historically. In 2007, Mebius [36] presented a way of obtaining the Euler–Rodrigues formula by substituting Euler–Rodrigues parameters in a 4 × 4 rotation matrix based on a quaternion represen- tation. In 2008, Senan and O'Reilly [37] illustrated rotation tensors with a direct product of quaternions and examined the parameter constraint in Euler–Rodrigues parameters. In the same year, Norris [38] applied the Euler–Rodrigues formula to developing rotation of tensors in elasticity by projecting it onto the hexagonal symmetry defined by axes of rotations with Carton decomposition of rotation tensors [39]. In 2010, Müller [40] used a Cayley transformation to obtain a modified vector parameterization that represents an extension of the Rodrigues parameters, which reduces the computational complexity while increasing accuracy. In 2012, Kovács [41] gave a new derivation of the Euler–Rodrigues formula based on a matrix transformation of three continuous rotations. In the same year, Pujol [15,42] investigated the relation between the composition of rotations and the product of quaternions [43] and related the work to Cayley's early contribution [11,44] through Euler–Rodrigues parameters. Following various studies, the use of the Euler–Rodrigues formula and of the Euler–Rodrigues-parameters formulated unit-quaternion has been extended to a broad range of research topics including vector parameterization of rotations [33,40,45], rational motions [46–49], motion generation [50–52] and planning [53],kinematicmapping[54,55],orientation[56] and attitude estimation [57–59],mechanics[60],constraint analysis [61,62], reconfiguration [63,64], mechanism analysis [65–67] and synthesis [68,69], sensing [70] and computer graphics [71–73] and vision [74]. Though various studies were made, reconciliation of different versions of the Euler–Rodrigues formula and their derivations were vaguely known. This paper is to examine all Euler–Rodrigues formula variations, present their derivations and discuss their intrinsic connections to provide readers with a complete picture of variations and connections, leading to understanding of the Euler–Rodrigues formula in its variations and uses as an exponential map and a quaternion operator. 2. Geometrical interpretation of the Euler–Rodrigues formula Rigid body rotation can be presented in the form of Rodrigues parameters [34,75] that integrate direction cosines of a rotation axis with tangent of half the rotation angle as three quantities in the form of b ¼ tan 1 θs ; b ¼ tan 1 θs ; b ¼ tan 1 θs ; ð1Þ x 2 x y 2 y z 2 z Fig. 1. Projected rhombus and the half-angle of rotation. 146 J.S. Dai / Mechanism and Machine Theory 92 (2015) 144–152 T where b =(bx, by, bz) is referred to as Rodrigues vector [4,6], the three quantities are known as Rodrigues parameters [10,14,19], where the axis of rotation is in the form of ÀÁ ; ; T s ¼ sx sy sz ð2Þ which is a unit vector. The half-angle is an essential feature [3,10] of parameterization of rotations and of the measure of pure rotation in the most elegant representation of rotations in kinematics. This can be seen in Fig. 1. In the figure, vector v1 with any magnitude rotates by angle θ about a unit axis s which is in line with axis z to form vector v2. Projection of vector v1 and that of its rotated vector v2 on xy-plane are presented as v1p and v2p with rotation angle θ. In this projection plane, a rhombus is formed by producing line P1Q parallel to rotated vector projection v2p and line P2Q parallel to original vector projection v1p. Drawing diagonals P1P2 and OQ that are perpendicular to each other, intersection point Pm can be obtained. Tangent of half the rotation angle was then given as that by Rodrigues [2] and demonstrated again by Bottema and Roth [10] in the form of θ P P tan ¼ 1 m : ð3Þ 2 OPm This can further be replaced by diagonals of the rhombus in vector form of v2p − v1p and v2p + v1p as θ v −v 2p 1p tan ¼ : ð4Þ 2 v þ v 2p 1p The use of the half angle in the study of motions led to the Euler–Rodrigues formula and to the discovery of Euler–Rodrigues parameters. A geometrical interpretation of the Euler–Rodrigues formula is illustrated in Fig. 1, leading to rotated vector v2 as v ¼ v þ v ð Þ 2 2p z 5 which is a combination of its projection on xy-plane and that on z-axis. Further from Fig. 1,vectorv2 as a result of rotation from v1 can be given as v ¼ v θ þ w θ þ v ð Þ 2 1p cos sin z 6 where v ¼ v −v ¼ v −ðÞs Á v s; ð Þ 1p 1 z 1 1 7 and w ¼ s  v : ð Þ 1 8 The above equation gives a skew-symmetric matrix As belowhavingtheeffectoftaking the vector cross product of s with a vector, 2 3 − 0 sz sy A ¼ ½¼s 4 − 5 ð Þ s sz 0 sx 9 − sy sx 0 Here, matrix As gives Lie algebra so(3) of SO(3) in the form of a 3 × 3 skew–symmetric matrix [34].
Recommended publications
  • Continuous Cayley Transform
    Continuous Cayley Transform A. Jadczyk In complex analysis, the Cayley transform is a mapping w of the complex plane to itself, given by z − i w(z) = : z + i This is a linear fractional transformation of the type az + b z 7! cz + d given by the matrix a b 1 −i C = = : c d 1 i The determinant of this matrix is 2i; and, noticing that (1 + i)2 = 2i; we can 1 as well replace C by 1+i C in order to to have C of determinant one - thus in the group SL(2; C): The transformation itself does not change after such a replacement. The Cayley transform is singular at the point z = −i: It maps the upper complex half{plane fz : Im(z) > 0g onto the interior of the unit disk D = fw : jwj < 1g; and the lower half{plane fz : Im(z) < 0g; except of the singular point, onto the exterior of the closure of D: The real line fz : Im(z) = 0g is mapped onto the unit circle fw : jwj = 1g - the boundary of D: The singular point z = −i can be thought of as being mapped to the point at infinity of the Riemann sphere - the one{point compactification of the complex plane. By solving the equation w(z) = z we easily find that there are two fixed points (z1; z2) of the Cayley transform: p p ! 1 3 1 3 z = + + i − − ; 1 2 2 2 2 p p ! 1 3 1 3 z = − + i − + : 2 2 2 2 2 In order to interpolate between the identity mapping and the Cayley trans- form we will choose a one{parameter group of transformations in SL(2; C) shar- ing these two fixed points.
    [Show full text]
  • Characterization and Computation of Matrices of Maximal Trace Over Rotations
    Characterization and Computation of Matrices of Maximal Trace over Rotations Javier Bernal1, Jim Lawrence1,2 1National Institute of Standards and Technology, Gaithersburg, MD 20899, USA 2George Mason University, 4400 University Dr, Fairfax, VA 22030, USA javier.bernal,james.lawrence @nist.gov [email protected] { } Abstract The constrained orthogonal Procrustes problem is the least-squares problem that calls for a rotation matrix that optimally aligns two cor- responding sets of points in d dimensional Euclidean space. This − problem generalizes to the so-called Wahba’s problem which is the same problem with nonnegative weights. Given a d d matrix M, × solutions to these problems are intimately related to the problem of finding a d d rotation matrix U that maximizes the trace of UM, × i.e., that makes UM a matrix of maximal trace over rotation matrices, and it is well known this can be achieved with a method based on the computation of the singular value decomposition (SVD) of M. As the main goal of this paper, we characterize d d matrices of maximal trace × over rotation matrices in terms of their eigenvalues, and for d = 2, 3, we show how this characterization can be used to determine whether a matrix is of maximal trace over rotation matrices. Finally, although depending only slightly on the characterization, as a secondary goal of the paper, for d = 2, 3, we identify alternative ways, other than the SVD, of obtaining solutions to the aforementioned problems. MSC: 15A18, 15A42, 65H17, 65K99, 93B60 Keywords: Eigenvalues, orthogonal, Procrustes, rotation, singular value decomposition, trace, Wahba 1 Contents 1 Introduction 2 2 Reformulation of Problems as Maximizations of the Trace of Matrices Over Rotations 4 3 Characterization of Matrices of Maximal Trace Over Rota- tions 6 4 The Two-Dimensional Case: Computation without SVD 17 5 The Three-Dimensional Case: Computation without SVD 22 6 The Three-Dimensional Case Revisited 28 Summary 33 References 34 1 Introduction Suppose P = p1,...,pn and Q = q1,...,qn are each sets of n points in Rd.
    [Show full text]
  • Rotation Matrix - Wikipedia, the Free Encyclopedia Page 1 of 22
    Rotation matrix - Wikipedia, the free encyclopedia Page 1 of 22 Rotation matrix From Wikipedia, the free encyclopedia In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy -Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv (see below for details). In two and three dimensions, rotation matrices are among the simplest algebraic descriptions of rotations, and are used extensively for computations in geometry, physics, and computer graphics. Though most applications involve rotations in two or three dimensions, rotation matrices can be defined for n-dimensional space. Rotation matrices are always square, with real entries. Algebraically, a rotation matrix in n-dimensions is a n × n special orthogonal matrix, i.e. an orthogonal matrix whose determinant is 1: . The set of all rotation matrices forms a group, known as the rotation group or the special orthogonal group. It is a subset of the orthogonal group, which includes reflections and consists of all orthogonal matrices with determinant 1 or -1, and of the special linear group, which includes all volume-preserving transformations and consists of matrices with determinant 1. Contents 1 Rotations in two dimensions 1.1 Non-standard orientation
    [Show full text]
  • Generalization of Rotational Mechanics and Application to Aerospace Systems
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Texas A&amp;M Repository GENERALIZATION OF ROTATIONAL MECHANICS AND APPLICATION TO AEROSPACE SYSTEMS A Dissertation by ANDREW JAMES SINCLAIR Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY May 2005 Major Subject: Aerospace Engineering GENERALIZATION OF ROTATIONAL MECHANICS AND APPLICATION TO AEROSPACE SYSTEMS A Dissertation by ANDREW JAMES SINCLAIR Submitted to Texas A&M University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Approved as to style and content by: John L. Junkins John E. Hurtado (Co-Chair of Committee) (Co-Chair of Committee) Srinivas R. Vadali John H. Painter (Member) (Member) Helen L. Reed (Head of Department) May 2005 Major Subject: Aerospace Engineering iii ABSTRACT Generalization of Rotational Mechanics and Application to Aerospace Systems. (May 2005) Andrew James Sinclair, B.S., University of Florida; M.S., University of Florida Co–Chairs of Advisory Committee: Dr. John L. Junkins Dr. John E. Hurtado This dissertation addresses the generalization of rigid-body attitude kinematics, dynamics, and control to higher dimensions. A new result is developed that demon- strates the kinematic relationship between the angular velocity in N-dimensions and the derivative of the principal-rotation parameters. A new minimum-parameter de- scription of N-dimensional orientation is directly related to the principal-rotation parameters. The mapping of arbitrary dynamical systems into N-dimensional rotations and the merits of new quasi velocities associated with the rotational motion are studied.
    [Show full text]
  • Medical Issue2012 4
    Journal of Science and Technology Vol. 13 June /2012 ISSN -1605 – 427X Natural and Medical Sciences (NMS No. 1) www.sustech.edu On the Development and Recent Work of Cayley Transforms [I] Adam Abdalla Abbaker Hassan Nile Valley University, Faculty of Education- Department of Mathematics ABSTRACT: This paper deals with the recent work and developments of Cayely transforms which had been rapidly developed. ا : : ﺘﺘﻌﻠﻕ ﻫﺫﻩ ﺍﻝﻭﺭﻗﺔ ﺍﻝﻌﻠﻤﻴﺔ ﺒﺘﻘﺩﻴﻡ ﺍﻝﻌﻤل ﺍﻝﺤﺩﻴﺙ ﻭ ﺍﻝﺘﻁﻭﺭﺍﺕ ﺍﻝﺴﺭﻴﻌﺔ ﻝﺘﺤﻭﻴﻼﺕ ﻜﻤ ﺎﻴﻠﻲ . INTRODUCTION In this study we shall give the basic is slightly simplified variant of that of important transforms and historical Langer (1, 2) . background of the subject, then introduce -1 the accretive operator of Cayley transform Cayley Transforms V = ( A – iI)(A+iI) and a maximal accretive and dissipative First we introduce the transformation B = operator, and relate this with Cayley -1 ( I + T*T) and Transform which yield the homography in function calculus. After this we will study C = T ( I + T*T) -1 in order to study the the fractional power and maximal Cayely transforms. accretive operators; indeed all this had been developed by Sz.Nagy and If the linear transform T is bounded , it is C. Foias (1) . The method of extending and clear that the transformation B appearing accretive (or dissipative) operator to above is also bounded , symmetric , and a maximal one via Cayley transforms such that 0 ≤ B ≤ I ; (3) . modeled on Von Neumann’s theory on symmetric operators, is due to Philips (1) C = TB is then bounded too. If T is a fractional powers of operators A in Hilbert linear transformation with dense domain, space , or even in Banachh space, such as we know that T*, and consequently also T*T, exist, but we know nothing of their – A is infinitesimal generator of a conts (4) one parameter sem–group of contractions, domains of definition .
    [Show full text]
  • Complex Variables Visualized
    Diplomarbeit Revised Version - June 17, 2014 Computer Algebra and Analysis: Complex Variables Visualized Ausgef¨uhrtam Research Institute for Symbolic Computation (RISC) der Johannes Kepler Universit¨atLinz unter Anleitung von Univ.-Prof. Dr. Peter Paule durch Thomas Ponweiser Stockfeld 15 4283 Bad Zell Datum Unterschrift ii Kurzfassung Ziel dieser Diplomarbeit ist die Visualisierung einiger grundlegender Ergeb- nisse aus dem Umfeld der Theorie der modularen Gruppe sowie der modularen Funktionen unter Zuhilfenahme der Computer Algebra Software Mathematica. Die Arbeit gliedert sich in drei Teile. Im ersten Kapitel werden f¨urdiese Ar- beit relevante Begriffe aus der Gruppentheorie zusammengefasst. Weiters wer- den M¨obiusTransformationen eingef¨uhrtund deren grundlegende geometrische Eigenschaften untersucht. Das zweite Kapitel ist der Untersuchung der modularen Gruppe aus al- gebraischer und geometrischer Sicht gewidmet. Der kanonische Fundamen- talbereich der modularen Gruppe sowie die daraus abgeleitete Kachelung der oberen Halbebene wird eingef¨uhrt.Weiters wird eine generelle Methode zum Auffinden von Fundamentalbereichen f¨urUntergruppen der modularen Gruppe vorgestellt, welche sich zentral auf die Konzepte der 2-dimensionalen hyper- bolischen Geometrie st¨utzt. Im dritten Kapitel geben wir einige konkrete Beispiele, wie die aufge- baute Theorie f¨urdie Visualisierung bestimmter mathematischer Sachverhalte angewendet werden kann. Neben der Visualisierung von Graphen modularer Funktionen stellt sich auch der Zusammenhang zwischen modularen Transfor- mationen und Kettenbr¨uchen als besonders sch¨onesErgebnis dar. iii iv Abstract The aim of this diploma thesis is the visualization of some fundamental results in the context of the theory of the modular group and modular functions. For this purpose the computer algebra software Mathematica is utilized. The thesis is structured in three parts.
    [Show full text]
  • Metaplectic Group, Symplectic Cayley Transform, and Fractional Fourier Transforms
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector J. Math. Anal. Appl. 416 (2014) 947–968 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Metaplectic group, symplectic Cayley transform, and fractional Fourier transforms MauriceA.deGossona,∗,FranzLuefb a University of Vienna, Faculty of Mathematics (NuHAG), 1090 Vienna, Austria b Department of Mathematics, Norwegian University of Science and Technology, 7491 Trondheim, Norway article info abstract Article history: We begin with a survey of the standard theory of the metaplectic group with some Received 15 February 2013 emphasis on the associated notion of Maslov index. We thereafter introduce the Available online 14 March 2014 Cayley transform for symplectic matrices, which allows us to study in detail the Submitted by P.G. Lemarie-Rieusset spreading functions of metaplectic operators, and to prove that they are basically Keywords: quadratic chirps. As a non-trivial application we give new formulae for the fractional Metaplectic and symplectic group Fourier transform in arbitrary dimension. We also study the regularity of the Fractional Fourier transform solutions to the Schrödinger equation in the Feichtinger algebra. Weyl operator © 2014 The Authors. Published by Elsevier Inc. This is an open access article Feichtinger algebra under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Contents 1. Introduction.....................................................................
    [Show full text]
  • Superconnection Character Forms and the Cayley Transform
    0040 9283 RX so3 00 + 00 Pergamon Press plc SUPERCONNECTION CHARACTER FORMS AND THE CAYLEY TRANSFORM DANIEL QUILLEN (Received June 1987) $0. INTRODUCTION LET E = E+ @E- be a super vector bundle equipped with inner product, and let D be a connection on E compatible with the grading and inner product. Let x= where T:E’-+E- is a vector bundle map. In [l l] we introduced the superconnection character form (1) O(E, D,X)=tr,exp u(X’+[D,X]~+D*) which generalizes the well-known definition of the Chern character in de Rham co- homology. This construction was motivated by the problem of proving a local version of the Atiyah-Singer index theorem for families of Dirac operators. The formula (1) is used,in an infinite dimensional context in Bismut’s solution of this problem [3]. From the cohomological viewpoint the differential form (1) is not very interesting. Its cohomology class remains unchanged as X is contracted to zero by tX, so the class doesn’t depend on X. In the first part of this paper we remedy this defect by showing that (1) has a meaning when X is allowed to become infinite in a certain sense. We show that this formula can be written in terms of the (ungraded) subbundle of E given by the graph of T. Moreover the resulting expression makes sense for an arbitrary subbundle W. Thus superconnection character forms can be defined for any correspondence between E+ and E-. We also show that the corresponding cohomology class is the Chern character of the difference W-E-.
    [Show full text]
  • Cayley Transformation and Numerical Stability of Calibration Equation
    Int J Comput Vis (2009) 82: 156–184 DOI 10.1007/s11263-008-0193-x Cayley Transformation and Numerical Stability of Calibration Equation F.C. Wu · Z.H. Wang · Z.Y. Hu Received: 25 May 2008 / Accepted: 22 October 2008 / Published online: 5 December 2008 © US Government 2008 Abstract The application of Cayley transformation to en- it is proved that the standard calibration equation, the Cay- hance the numerical stability of camera calibration is inves- ley calibration family and the S-Cayley calibration family tigated. First, a new calibration equation, called the standard are all some special cases of this generic calibration family. calibration equation, is introduced using the Cayley transfor- mation and its analytical solution is obtained. The standard Keywords Camera calibration · The absolute conic · calibration equation is equivalent to the classical calibration Calibration equation · Cayley transformation · Numerical equation, but it exhibits remarkable better numerical stabil- stability ity. Second, a one-parameter calibration family, called the Cayley calibration family which is equivalent to the stan- dard calibration equation, is obtained using also the Cayley 1 Introduction transformation and it is found that this family is composed Camera calibration is a necessary step to recover 3D met- of those infinite homographies whose rotation has the same ric information from 2D images. Many methods and tech- axis with the rotation between the two given views. The con- niques for camera calibration have been proposed in the last dition number of equations in the Cayley calibration family decades. The existing methods can be roughly classified into varies with the parameter value, and an algorithm to deter- two categories: methods based on a reference calibration mine the best parameter value is provided.
    [Show full text]
  • Spectral Theory of Unbounded Self-Adjoint Operators in Hilbert Spaces
    University of Helsinki Department of Mathematics and Statistics Pro Gradu Thesis Spectral Theory of Unbounded Self-adjoint Operators in Hilbert spaces Timo Weckman 013626675 Instructor: Jari Taskinen April 30, 2020 HELSINGIN YLIOPISTO HELSINGFORS UNIVERSITET UNIVERSITY OF HELSINKI Tiedekunta/Osasto Fakultet/Sektion Faculty Laitos Institution Department Faculty of Science Department of Mathematics and Statistics Tekijä Författare Author Timo Weckman Työn nimi Arbetets titel Title Spectral Theory of Unbounded Self-adjoint Operators in Hilbert spaces Oppiaine Läroämne Subject Mathematics Työn laji Arbetets art Level Aika Datum Month and year Sivumäärä Sidoantal Number of pages Pro gradu -thesis April 2020 44 p. Tiivistelmä Referat Abstract In this work we present a derivation of the spectral theorem of unbounded spectral operators in a Hilbert space. The spectral theorem has several applications, most notably in the theory of quantum mechanics. The theorem allows a self-adjoint linear operator on a Hilbert space to be represented in terms of simpler operators, projections. The focus of this work are the self-adjoint operators in Hilbert spaces. The work begins with the int- roduction of vector and inner product spaces and the denition of the complete inner product space, the Hilbert space. Three classes of bounded linear operators relevant for this work are introduced: self-adjoint, unitary and projection operators. The unbounded self-adjoint operator and its proper- ties are also discussed. For the derivation of the spectral theorem, the basic spectral properties of operators in Hilbert space are presented. The spectral theorem is rst derived for bounded operators. With the denition of basic spectral properties and the introduction of the spectral family, the spectral theorem for bounded self-adjoint operators is presented with a proof.
    [Show full text]
  • Hybrid Attitude Control and Estimation on SO(3)
    Western University Scholarship@Western Electronic Thesis and Dissertation Repository 11-29-2017 4:00 PM Hybrid Attitude Control and Estimation On SO(3) Soulaimane Berkane The University of Western Ontario Supervisor Tayebi, Abdelhamid The University of Western Ontario Graduate Program in Electrical and Computer Engineering A thesis submitted in partial fulfillment of the equirr ements for the degree in Doctor of Philosophy © Soulaimane Berkane 2017 Follow this and additional works at: https://ir.lib.uwo.ca/etd Part of the Acoustics, Dynamics, and Controls Commons, Controls and Control Theory Commons, and the Navigation, Guidance, Control and Dynamics Commons Recommended Citation Berkane, Soulaimane, "Hybrid Attitude Control and Estimation On SO(3)" (2017). Electronic Thesis and Dissertation Repository. 5083. https://ir.lib.uwo.ca/etd/5083 This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. Abstract This thesis presents a general framework for hybrid attitude control and estimation de- sign on the Special Orthogonal group SO(3). First, the attitude stabilization problem on SO(3) is considered. It is shown that, using a min-switch hybrid control strategy de- signed from a family of potential functions on SO(3), global exponential stabilization on SO(3) can be achieved when this family of potential functions satisfies certain properties. Then, a systematic methodology to construct these potential functions is developed. The proposed hybrid control technique is applied to the attitude tracking problem for rigid body systems.
    [Show full text]
  • Math 512 / Problem Set 11 (Three Pages) • Study/Read: Inner Product
    Due: Wednesday, December 3, at 4 PM Math 512 / Problem Set 11 (three pages) • Study/read: Inner product spaces & Operators on inner product spaces - Ch. 5 & 6 from LADW by Treil. - Ch. 8 & 9 from Linear Algebra by Hoffman & Kunze. (!) Make sure that you understand perfectly the definitions, examples, theorems, etc. R 1 Legendre Polynomials Recall that (f, g) = −1 f(x)g(x)dx is an inner product on C(I, R), where I = [−1, 1]. The system of polynomials (L0(t), L1(t), ... , Ln−1(t)) resulting from the n−1 Gram-Schmidt orthonormalization of the standard basis E := (1, t, ... , t ) of Pn(R) are called the Legendre polynomials. Google the term “Legendre polynomials” and learn about their applications. 1) Answer / do the following: a) Compute L0, L1, L2, L3. 3 b) Write t − t + 1 as a linear combination of L0, L1, L2, L3. c) Show that in general, one has deg(Lk) = k. k d) What is the coefficient of t in Lk(t)? 2) Which of the following pairs of matrices are unitarily equivalent: 0 1 0 2 0 0 0 1 0 1 0 0 1 0! 0 1! a) & b) −1 0 0 & 0 −1 0 c) −1 0 0 & 0 −i 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 i [Hint: Unitarily equivalent matrices have the same characteristic polynomial (WHY), thus the same eigenvalues including multiplicities, the same trace, determinant, etc.] 3) For each of the following matrices over F = R decide whether they are unitarily diago- nalizable, and if so, find the unitary matrix P over R such that P AP −1 is diagonal.
    [Show full text]