Chapter 1 Rigid Body Kinematics
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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. -
Understanding Euler Angles 1
Welcome Guest View Cart (0 items) Login Home Products Support News About Us Understanding Euler Angles 1. Introduction Attitude and Heading Sensors from CH Robotics can provide orientation information using both Euler Angles and Quaternions. Compared to quaternions, Euler Angles are simple and intuitive and they lend themselves well to simple analysis and control. On the other hand, Euler Angles are limited by a phenomenon called "Gimbal Lock," which we will investigate in more detail later. In applications where the sensor will never operate near pitch angles of +/‐ 90 degrees, Euler Angles are a good choice. Sensors from CH Robotics that can provide Euler Angle outputs include the GP9 GPS‐Aided AHRS, and the UM7 Orientation Sensor. Figure 1 ‐ The Inertial Frame Euler angles provide a way to represent the 3D orientation of an object using a combination of three rotations about different axes. For convenience, we use multiple coordinate frames to describe the orientation of the sensor, including the "inertial frame," the "vehicle‐1 frame," the "vehicle‐2 frame," and the "body frame." The inertial frame axes are Earth‐fixed, and the body frame axes are aligned with the sensor. The vehicle‐1 and vehicle‐2 are intermediary frames used for convenience when illustrating the sequence of operations that take us from the inertial frame to the body frame of the sensor. It may seem unnecessarily complicated to use four different coordinate frames to describe the orientation of the sensor, but the motivation for doing so will become clear as we proceed. For clarity, this application note assumes that the sensor is mounted to an aircraft. -
Swimming in Spacetime: Motion by Cyclic Changes in Body Shape
RESEARCH ARTICLE tation of the two disks can be accomplished without external torques, for example, by fixing Swimming in Spacetime: Motion the distance between the centers by a rigid circu- lar arc and then contracting a tension wire sym- by Cyclic Changes in Body Shape metrically attached to the outer edge of the two caps. Nevertheless, their contributions to the zˆ Jack Wisdom component of the angular momentum are parallel and add to give a nonzero angular momentum. Cyclic changes in the shape of a quasi-rigid body on a curved manifold can lead Other components of the angular momentum are to net translation and/or rotation of the body. The amount of translation zero. The total angular momentum of the system depends on the intrinsic curvature of the manifold. Presuming spacetime is a is zero, so the angular momentum due to twisting curved manifold as portrayed by general relativity, translation in space can be must be balanced by the angular momentum of accomplished simply by cyclic changes in the shape of a body, without any the motion of the system around the sphere. external forces. A net rotation of the system can be ac- complished by taking the internal configura- The motion of a swimmer at low Reynolds of cyclic changes in their shape. Then, presum- tion of the system through a cycle. A cycle number is determined by the geometry of the ing spacetime is a curved manifold as portrayed may be accomplished by increasing by ⌬ sequence of shapes that the swimmer assumes by general relativity, I show that net translations while holding fixed, then increasing by (1). -
Quantum Information
Quantum Information J. A. Jones Michaelmas Term 2010 Contents 1 Dirac Notation 3 1.1 Hilbert Space . 3 1.2 Dirac notation . 4 1.3 Operators . 5 1.4 Vectors and matrices . 6 1.5 Eigenvalues and eigenvectors . 8 1.6 Hermitian operators . 9 1.7 Commutators . 10 1.8 Unitary operators . 11 1.9 Operator exponentials . 11 1.10 Physical systems . 12 1.11 Time-dependent Hamiltonians . 13 1.12 Global phases . 13 2 Quantum bits and quantum gates 15 2.1 The Bloch sphere . 16 2.2 Density matrices . 16 2.3 Propagators and Pauli matrices . 18 2.4 Quantum logic gates . 18 2.5 Gate notation . 21 2.6 Quantum networks . 21 2.7 Initialization and measurement . 23 2.8 Experimental methods . 24 3 An atom in a laser field 25 3.1 Time-dependent systems . 25 3.2 Sudden jumps . 26 3.3 Oscillating fields . 27 3.4 Time-dependent perturbation theory . 29 3.5 Rabi flopping and Fermi's Golden Rule . 30 3.6 Raman transitions . 32 3.7 Rabi flopping as a quantum gate . 32 3.8 Ramsey fringes . 33 3.9 Measurement and initialisation . 34 1 CONTENTS 2 4 Spins in magnetic fields 35 4.1 The nuclear spin Hamiltonian . 35 4.2 The rotating frame . 36 4.3 On-resonance excitation . 38 4.4 Excitation phases . 38 4.5 Off-resonance excitation . 39 4.6 Practicalities . 40 4.7 The vector model . 40 4.8 Spin echoes . 41 4.9 Measurement and initialisation . 42 5 Photon techniques 43 5.1 Spatial encoding . -
Nanocrystalline Nanowires: I. Structure
Nanocrystalline Nanowires: I. Structure Philip B. Allen Department of Physics and Astronomy, State University of New York, Stony Brook, NY 11794-3800∗ and Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, NY 11973-5000 (Dated: September 7, 2006) Geometric constructions of possible atomic arrangements are suggested for inorganic nanowires. These are fragments of bulk crystals, and can be called \nanocrystalline" nanowires (NCNW). To minimize surface polarity, nearly one-dimensional formula units, oriented along the growth axis, generate NCNW's by translation and rotation. PACS numbers: 61.46.-w, 68.65.La, 68.70.+w Introduction. Periodic one-dimensional (1D) motifs twinned16; \core-shell" (COHN)4,17; and longitudinally are common in nature. The study of single-walled car- heterogeneous (LOHN) NCNW's. bon nanotubes (SWNT)1,2 is maturing rapidly, but other Design principle. The remainder of this note is about 1D systems are at an earlier stage. Inorganic nanowires NCNW's. It offers a possible design principle, which as- are the topic of the present two papers. There is much sists visualization of candidate atomic structures, and optimism in this field3{6. Imaging gives insight into provides a template for arrangements that can be tested self-assembled nanostructures, and incentives to improve theoretically, for example by density functional theory growth protocols. Electron microscopy, diffraction, and (DFT). The basic idea is (1) choose a maximally linear, optical spectroscopy on individual wires7 provide detailed charge-neutral, and (if possible) dipole-free atomic clus- information. Device applications are expected. These ter containing a single formula unit, and (2) using if possi- systems offer great opportunities for atomistic modelling. -
Foundations of Newtonian Dynamics: an Axiomatic Approach For
Foundations of Newtonian Dynamics: 1 An Axiomatic Approach for the Thinking Student C. J. Papachristou 2 Department of Physical Sciences, Hellenic Naval Academy, Piraeus 18539, Greece Abstract. Despite its apparent simplicity, Newtonian mechanics contains conceptual subtleties that may cause some confusion to the deep-thinking student. These subtle- ties concern fundamental issues such as, e.g., the number of independent laws needed to formulate the theory, or, the distinction between genuine physical laws and deriva- tive theorems. This article attempts to clarify these issues for the benefit of the stu- dent by revisiting the foundations of Newtonian dynamics and by proposing a rigor- ous axiomatic approach to the subject. This theoretical scheme is built upon two fun- damental postulates, namely, conservation of momentum and superposition property for interactions. Newton’s laws, as well as all familiar theorems of mechanics, are shown to follow from these basic principles. 1. Introduction Teaching introductory mechanics can be a major challenge, especially in a class of students that are not willing to take anything for granted! The problem is that, even some of the most prestigious textbooks on the subject may leave the student with some degree of confusion, which manifests itself in questions like the following: • Is the law of inertia (Newton’s first law) a law of motion (of free bodies) or is it a statement of existence (of inertial reference frames)? • Are the first two of Newton’s laws independent of each other? It appears that -
The Rolling Ball Problem on the Sphere
S˜ao Paulo Journal of Mathematical Sciences 6, 2 (2012), 145–154 The rolling ball problem on the sphere Laura M. O. Biscolla Universidade Paulista Rua Dr. Bacelar, 1212, CEP 04026–002 S˜aoPaulo, Brasil Universidade S˜aoJudas Tadeu Rua Taquari, 546, CEP 03166–000, S˜aoPaulo, Brasil E-mail address: [email protected] Jaume Llibre Departament de Matem`atiques, Universitat Aut`onomade Barcelona 08193 Bellaterra, Barcelona, Catalonia, Spain E-mail address: [email protected] Waldyr M. Oliva CAMGSD, LARSYS, Instituto Superior T´ecnico, UTL Av. Rovisco Pais, 1049–0011, Lisbon, Portugal Departamento de Matem´atica Aplicada Instituto de Matem´atica e Estat´ıstica, USP Rua do Mat˜ao,1010–CEP 05508–900, S˜aoPaulo, Brasil E-mail address: [email protected] Dedicated to Lu´ıs Magalh˜aes and Carlos Rocha on the occasion of their 60th birthdays Abstract. By a sequence of rolling motions without slipping or twist- ing along arcs of great circles outside the surface of a sphere of radius R, a spherical ball of unit radius has to be transferred from an initial state to an arbitrary final state taking into account the orientation of the ball. Assuming R > 1 we provide a new and shorter prove of the result of Frenkel and Garcia in [4] that with at most 4 moves we can go from a given initial state to an arbitrary final state. Important cases such as the so called elimination of the spin discrepancy are done with 3 moves only. 1991 Mathematics Subject Classification. Primary 58E25, 93B27. Key words: Control theory, rolling ball problem. -
MATH 237 Differential Equations and Computer Methods
Queen’s University Mathematics and Engineering and Mathematics and Statistics MATH 237 Differential Equations and Computer Methods Supplemental Course Notes Serdar Y¨uksel November 19, 2010 This document is a collection of supplemental lecture notes used for Math 237: Differential Equations and Computer Methods. Serdar Y¨uksel Contents 1 Introduction to Differential Equations 7 1.1 Introduction: ................................... ....... 7 1.2 Classification of Differential Equations . ............... 7 1.2.1 OrdinaryDifferentialEquations. .......... 8 1.2.2 PartialDifferentialEquations . .......... 8 1.2.3 Homogeneous Differential Equations . .......... 8 1.2.4 N-thorderDifferentialEquations . ......... 8 1.2.5 LinearDifferentialEquations . ......... 8 1.3 Solutions of Differential equations . .............. 9 1.4 DirectionFields................................. ........ 10 1.5 Fundamental Questions on First-Order Differential Equations............... 10 2 First-Order Ordinary Differential Equations 11 2.1 ExactDifferentialEquations. ........... 11 2.2 MethodofIntegratingFactors. ........... 12 2.3 SeparableDifferentialEquations . ............ 13 2.4 Differential Equations with Homogenous Coefficients . ................ 13 2.5 First-Order Linear Differential Equations . .............. 14 2.6 Applications.................................... ....... 14 3 Higher-Order Ordinary Linear Differential Equations 15 3.1 Higher-OrderDifferentialEquations . ............ 15 3.1.1 LinearIndependence . ...... 16 3.1.2 WronskianofasetofSolutions . ........ 16 3.1.3 Non-HomogeneousProblem -
The Exponential of a Matrix
5-28-2012 The Exponential of a Matrix The solution to the exponential growth equation dx kt = kx is given by x = c e . dt 0 It is natural to ask whether you can solve a constant coefficient linear system ′ ~x = A~x in a similar way. If a solution to the system is to have the same form as the growth equation solution, it should look like At ~x = e ~x0. The first thing I need to do is to make sense of the matrix exponential eAt. The Taylor series for ez is ∞ n z z e = . n! n=0 X It converges absolutely for all z. It A is an n × n matrix with real entries, define ∞ n n At t A e = . n! n=0 X The powers An make sense, since A is a square matrix. It is possible to show that this series converges for all t and every matrix A. Differentiating the series term-by-term, ∞ ∞ ∞ ∞ n−1 n n−1 n n−1 n−1 m m d At t A t A t A t A At e = n = = A = A = Ae . dt n! (n − 1)! (n − 1)! m! n=0 n=1 n=1 m=0 X X X X At ′ This shows that e solves the differential equation ~x = A~x. The initial condition vector ~x(0) = ~x0 yields the particular solution At ~x = e ~x0. This works, because e0·A = I (by setting t = 0 in the power series). Another familiar property of ordinary exponentials holds for the matrix exponential: If A and B com- mute (that is, AB = BA), then A B A B e e = e + . -
Phoronomy: Space, Construction, and Mathematizing Motion Marius Stan
To appear in Bennett McNulty (ed.), Kant’s Metaphysical Foundations of Natural Science: A Critical Guide. Cambridge University Press. Phoronomy: space, construction, and mathematizing motion Marius Stan With his chapter, Phoronomy, Kant defies even the seasoned interpreter of his philosophy of physics.1 Exegetes have given it little attention, and un- derstandably so: his aims are opaque, his turns in argument little motivated, and his context mysterious, which makes his project there look alienating. I seek to illuminate here some of the darker corners in that chapter. Specifi- cally, I aim to clarify three notions in it: his concepts of velocity, of compo- site motion, and of the construction required to compose motions. I defend three theses about Kant. 1) his choice of velocity concept is ul- timately insufficient. 2) he sided with the rationalist faction in the early- modern debate on directed quantities. 3) it remains an open question if his algebra of motion is a priori, though he believed it was. I begin in § 1 by explaining Kant’s notion of phoronomy and its argu- ment structure in his chapter. In § 2, I present four pictures of velocity cur- rent in Kant’s century, and I assess the one he chose. My § 3 is in three parts: a historical account of why algebra of motion became a topic of early modern debate; a synopsis of the two sides that emerged then; and a brief account of his contribution to the debate. Finally, § 4 assesses how general his account of composite motion is, and if it counts as a priori knowledge. -
Chapter 5 ANGULAR MOMENTUM and ROTATIONS
Chapter 5 ANGULAR MOMENTUM AND ROTATIONS In classical mechanics the total angular momentum L~ of an isolated system about any …xed point is conserved. The existence of a conserved vector L~ associated with such a system is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian) is invariant under rotations, i.e., if the coordinates and momenta of the entire system are rotated “rigidly” about some point, the energy of the system is unchanged and, more importantly, is the same function of the dynamical variables as it was before the rotation. Such a circumstance would not apply, e.g., to a system lying in an externally imposed gravitational …eld pointing in some speci…c direction. Thus, the invariance of an isolated system under rotations ultimately arises from the fact that, in the absence of external …elds of this sort, space is isotropic; it behaves the same way in all directions. Not surprisingly, therefore, in quantum mechanics the individual Cartesian com- ponents Li of the total angular momentum operator L~ of an isolated system are also constants of the motion. The di¤erent components of L~ are not, however, compatible quantum observables. Indeed, as we will see the operators representing the components of angular momentum along di¤erent directions do not generally commute with one an- other. Thus, the vector operator L~ is not, strictly speaking, an observable, since it does not have a complete basis of eigenstates (which would have to be simultaneous eigenstates of all of its non-commuting components). This lack of commutivity often seems, at …rst encounter, as somewhat of a nuisance but, in fact, it intimately re‡ects the underlying structure of the three dimensional space in which we are immersed, and has its source in the fact that rotations in three dimensions about di¤erent axes do not commute with one another. -
Euler Quaternions
Four different ways to represent rotation my head is spinning... The space of rotations SO( 3) = {R ∈ R3×3 | RRT = I,det(R) = +1} Special orthogonal group(3): Why det( R ) = ± 1 ? − = − Rotations preserve distance: Rp1 Rp2 p1 p2 Rotations preserve orientation: ( ) × ( ) = ( × ) Rp1 Rp2 R p1 p2 The space of rotations SO( 3) = {R ∈ R3×3 | RRT = I,det(R) = +1} Special orthogonal group(3): Why it’s a group: • Closed under multiplication: if ∈ ( ) then ∈ ( ) R1, R2 SO 3 R1R2 SO 3 • Has an identity: ∃ ∈ ( ) = I SO 3 s.t. IR1 R1 • Has a unique inverse… • Is associative… Why orthogonal: • vectors in matrix are orthogonal Why it’s special: det( R ) = + 1 , NOT det(R) = ±1 Right hand coordinate system Possible rotation representations You need at least three numbers to represent an arbitrary rotation in SO(3) (Euler theorem). Some three-number representations: • ZYZ Euler angles • ZYX Euler angles (roll, pitch, yaw) • Axis angle One four-number representation: • quaternions ZYZ Euler Angles φ = θ rzyz ψ φ − φ cos sin 0 To get from A to B: φ = φ φ Rz ( ) sin cos 0 1. Rotate φ about z axis 0 0 1 θ θ 2. Then rotate θ about y axis cos 0 sin θ = ψ Ry ( ) 0 1 0 3. Then rotate about z axis − sinθ 0 cosθ ψ − ψ cos sin 0 ψ = ψ ψ Rz ( ) sin cos 0 0 0 1 ZYZ Euler Angles φ θ ψ Remember that R z ( ) R y ( ) R z ( ) encode the desired rotation in the pre- rotation reference frame: φ = pre−rotation Rz ( ) Rpost−rotation Therefore, the sequence of rotations is concatentated as follows: (φ θ ψ ) = φ θ ψ Rzyz , , Rz ( )Ry ( )Rz ( ) φ − φ θ θ ψ − ψ cos sin 0 cos 0 sin cos sin 0 (φ θ ψ ) = φ φ ψ ψ Rzyz , , sin cos 0 0 1 0 sin cos 0 0 0 1− sinθ 0 cosθ 0 0 1 − − − cφ cθ cψ sφ sψ cφ cθ sψ sφ cψ cφ sθ (φ θ ψ ) = + − + Rzyz , , sφ cθ cψ cφ sψ sφ cθ sψ cφ cψ sφ sθ − sθ cψ sθ sψ cθ ZYX Euler Angles (roll, pitch, yaw) φ − φ cos sin 0 To get from A to B: φ = φ φ Rz ( ) sin cos 0 1.