Clifford Algebra Exponentials and Planar Linkage Synthesis Equations

2004 ◽  
Vol 127 (5) ◽  
pp. 931-940 ◽  
Author(s):  
Alba Perez ◽  
J. Michael McCarthy
Keyword(s):  
Clifford Algebra ◽  
Standard Form ◽  
Algebraic Manipulation ◽  
Algebraic Solution ◽  
Linkage Design ◽  
Serial Chain ◽  
The Matrix ◽  
Basic Equations ◽  
Serial Chains ◽  
Linkage Synthesis

This paper uses the exponential defined on a Clifford algebra of planar projective space to show that the “standard-form” design equations used for planar linkage synthesis are obtained directly from the relative kinematics equations of the chain. The relative kinematics equations of a serial chain appear in the matrix exponential formulation of the kinematics equations for a robot. We show that formulating these same equations using a Clifford algebra yields design equations that include the joint variables in a way that is convenient for algebraic manipulation. The result is a single formulation that yields the design equations for planar 2R dyads, 3R triads, and nR single degree-of-freedom coupled serial chains and facilitates the algebraic solution of these equations including the inverse kinematics of the chain. These results link the basic equations of planar linkage design to standard techniques in robotics.

scholarly journals Kinematic Synthesis of Spatial Serial Chains Using Clifford Algebra Exponentials

2006 ◽  
Vol 220 (7) ◽  
pp. 953-968 ◽  
Author(s):  
A Perez-Gracia ◽  
J M McCarthy
Keyword(s):  
Clifford Algebra ◽  
Robotic System ◽  
Exponential Form ◽  
Kinematic Synthesis ◽  
End Effector ◽  
Dual Quaternions ◽  
Serial Chain ◽  
Joint Parameters ◽  
Position And Orientation ◽  
Serial Chains

This article presents a formulation of the design equations for a spatial serial chain that uses the Clifford algebra exponential form of its kinematics equations. This is the even Clifford algebra C+( P3), known as dual quaternions. These equations define the position and orientation of the end effector in terms of rotations or translations about or along the joint axes of the chain. Because the coordinates of these axes appear explicitly, specifying a set of task positions these equations can be solved to determine the location of the joints. At the same time, joint parameters or certain dimensions are specified to ensure that the resulting robotic system has specific features.

scholarly journals Algebraically Solvable Problems: Describing Polynomials as Equivalent to Explicit Solutions

10.37236/734 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Uwe Schauz
Keyword(s):  
Combinatorial Nullstellensatz ◽  
Algebraic Solution ◽  
Explicit Solutions ◽  
Polynomial Map ◽  
Combinatorial Number Theory ◽  
Special Cases ◽  
The Matrix ◽  
Grid Points ◽  
Algebraic Solutions ◽  
Solvable Problems

The main result of this paper is a coefficient formula that sharpens and generalizes Alon and Tarsi's Combinatorial Nullstellensatz. On its own, it is a result about polynomials, providing some information about the polynomial map $P|_{\mathfrak{X}_1\times\cdots\times\mathfrak{X}_n}$ when only incomplete information about the polynomial $P(X_1,\dots,X_n)$ is given.In a very general working frame, the grid points $x\in \mathfrak{X}_1\times\cdots\times\mathfrak{X}_n$ which do not vanish under an algebraic solution – a certain describing polynomial $P(X_1,\dots,X_n)$ – correspond to the explicit solutions of a problem. As a consequence of the coefficient formula, we prove that the existence of an algebraic solution is equivalent to the existence of a nontrivial solution to a problem. By a problem, we mean everything that "owns" both, a set ${\cal S}$, which may be called the set of solutions; and a subset ${\cal S}_{\rm triv}\subseteq{\cal S}$, the set of trivial solutions.We give several examples of how to find algebraic solutions, and how to apply our coefficient formula. These examples are mainly from graph theory and combinatorial number theory, but we also prove several versions of Chevalley and Warning's Theorem, including a generalization of Olson's Theorem, as examples and useful corollaries.We obtain a permanent formula by applying our coefficient formula to the matrix polynomial, which is a generalization of the graph polynomial. This formula is an integrative generalization and sharpening of:1. Ryser's permanent formula.2. Alon's Permanent Lemma.3. Alon and Tarsi's Theorem about orientations and colorings of graphs.Furthermore, in combination with the Vigneron-Ellingham-Goddyn property of planar $n$-regular graphs, the formula contains as very special cases:4. Scheim's formula for the number of edge $n$-colorings of such graphs.5. Ellingham and Goddyn's partial answer to the list coloring conjecture.

Characterization of Workspaces of Parallel Manipulators

1992 ◽  
Vol 114 (3) ◽  
pp. 368-375 ◽  
Author(s):  
V. Kumar
Keyword(s):  
Parallel Manipulators ◽  
Geometric Optimization ◽  
Reachable Workspace ◽  
Serial Chain ◽  
Kinematic Performance ◽  
Serial Chains ◽  
Manipulator Control ◽  
General Method

The workspaces and kinematic characterization of serial chain manipulator geometries and the geometric optimization have been studied extensively. Much less is known about workspaces for manipulation systems which possess several serial chains arranged in parallel. In this paper, two well known workspaces, the reachable workspace and the dexterous workspace, are investigated for parallel manipulators. A general method for obtaining these workspaces is presented. The existence of numerous special configurations in the workspace present problems in manipulator control. Therefore the controllably dexterous workspace is proposed as a useful measure of kinematic performance. The methodology of delineating the workspaces and its limitations are illustrated with examples.

scholarly journals A Design System for Eight-Bar Linkages as Constrained 4R Serial Chains

2015 ◽  
Vol 8 (1) ◽  
Author(s):  
Kaustubh H. Sonawale ◽  
J. Michael McCarthy
Keyword(s):  
Random Search ◽  
Design System ◽  
End Effector ◽  
Serial Chain ◽  
Straight Line ◽  
Tolerance Zones ◽  
Serial Chains

This paper presents a design system for planar eight-bar linkages that adds three RR constraints to a user-specified 4R serial chain. R denotes a revolute, or hinged, joint. There are 100 ways in which these constraints can be added to yield as many as 3951 different linkages. An analysis routine based on the Dixon determinant evaluates the performance of each linkage candidate and determines the feasible designs that reach the task positions in a single assembly. A random search within the user-specified tolerance zones around the task specifications is iterated in order to increase the number of linkage candidates and feasible designs. The methodology is demonstrated with the design of rectilinear eight-bar linkages that guide an end-effector through five parallel positions along a straight line.

scholarly journals Determination of Settings of a Tilted Head Cutter for Generation of Hypoid and Spiral Bevel Gears

1988 ◽  
Vol 110 (4) ◽  
pp. 495-500 ◽  
Author(s):  
F. L. Litvin ◽  
Y. Zhang ◽  
M. Lundy ◽  
C. Heine
Keyword(s):  
Spiral Bevel Gears ◽  
Computation Procedure ◽  
Bevel Gears ◽  
Pressure Angle ◽  
The Matrix ◽  
Spiral Bevel ◽  
Basic Equations

Kinematics of mechanisms of hypoid and spiral bevel cutting machines is considered. These mechanisms are designated to install the position and tilt of the head cutter. The tilt of the head cutter with standard blades provides the required pressure angle. The authors have developed the matrix presentation of kinematics of these mechanisms and basic equations for the required settings. An example is presented based on the developed computation procedure.

scholarly journals Matrix 3-Lie superalgebras and BRST supersymmetry

2017 ◽  
Vol 14 (11) ◽  
pp. 1750160 ◽  
Author(s):  
Viktor Abramov
Keyword(s):  
Clifford Algebra ◽  
Lie Algebra ◽  
Vector Fields ◽  
Jacobi Identity ◽  
Lie Superalgebras ◽  
Infinite Dimensional ◽  
Algebra Approach ◽  
The Matrix ◽  
Infinite Dimensional Lie Algebra

Given a matrix Lie algebra one can construct the 3-Lie algebra by means of the trace of a matrix. In the present paper, we show that this approach can be extended to the infinite-dimensional Lie algebra of vector fields on a manifold if instead of the trace of a matrix we consider a differential 1-form which satisfies certain conditions. Then we show that the same approach can be extended to matrix Lie superalgebras [Formula: see text] if instead of the trace of a matrix we make use of the supertrace of a matrix. It is proved that a graded triple commutator of matrices constructed with the help of the graded commutator and the supertrace satisfies a graded ternary Filippov–Jacobi identity. In two particular cases of [Formula: see text] and [Formula: see text], we show that the Pauli and Dirac matrices generate the matrix 3-Lie superalgebras, and we find the non-trivial graded triple commutators of these algebras. We propose a Clifford algebra approach to 3-Lie superalgebras induced by Lie superalgebras. We also discuss an application of matrix 3-Lie superalgebras in BRST-formalism.

Fourier Methods for Kinematic Synthesis of Coupled Serial Chain Mechanisms

2005 ◽  
Vol 127 (2) ◽  
pp. 232-241 ◽  
Author(s):  
Xichun Nie ◽  
Venkat Krovi
Keyword(s):  
Low Cost ◽  
Synthesis Method ◽  
Path Following ◽  
Degree Of Freedom ◽  
Dimensional Synthesis ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Serial Chain ◽  
End Effectors ◽  
Serial Chains

Single degree-of-freedom coupled serial chain (SDCSC) mechanisms are a class of mechanisms that can be realized by coupling successive joint rotations of a serial chain linkage, by way of gears or cable-pulley drives. Such mechanisms combine the benefits of single degree-of-freedom design and control with the anthropomorphic workspace of serial chains. Our interest is in creating articulated manipulation-assistive aids based on the SDCSC configuration to work passively in cooperation with the human operator or to serve as a low-cost automation solution. However, as single-degree-of-freedom systems, such SDCSC-configuration manipulators need to be designed specific to a given task. In this paper, we investigate the development of a synthesis scheme, leveraging tools from Fourier analysis and optimization, to permit the end-effectors of such manipulators to closely approximate desired closed planar paths. In particular, we note that the forward kinematics equations take the form of a finite trigonometric series in terms of the input crank rotations. The proposed Fourier-based synthesis method exploits this special structure to achieve the combined number and dimensional synthesis of SDCSC-configuration manipulators for closed-loop planar path-following tasks. Representative examples illustrate the application of this method for tracing candidate square and rectangular paths. Emphasis is also placed on conversion of computational results into physically realizable mechanism designs.

scholarly journals Design of a Linkage System to Write in Cursive

2017 ◽  
Vol 17 (3) ◽  
Author(s):  
Yang Liu ◽  
J. Michael McCarthy
Keyword(s):  
Mechanical System ◽  
Complex Plane ◽  
Design Methodology ◽  
Plane Curves ◽  
Chinese Characters ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Cursive Handwriting ◽  
Serial Chain ◽  
Serial Chains

This paper presents a design methodology for a system of linkages that can trace planar Bezier curves that represent cursive handwriting of the alphabet and Chinese characters. This paper shows that the standard degree n Bezier curve can be reparameterized so that it takes the form of a trigonometric curve that can be drawn by a one degree-of-freedom coupled serial chain consisting of 2n links. A series of cubic Bezier curves that define a handwritten name yields a series of six-link coupled serial chains that trace these curves. We then show how to simplify this system using cubic trigonometric Bezier curves to obtain a series of four-link serial chains that approximate the system of Bezier curves. The result is a methodology for the design of a mechanical system that draws complex plane curves such as the cursive alphabet and Chinese characters.

Unitary Transformations of the Dirac Hamiltonian

2007 ◽  
Author(s):  
Kenneth G. Dyall ◽  
Knut Faegri
Keyword(s):  
Unitary Transformation ◽  
Free Particle ◽  
Algebraic Manipulation ◽  
Negative Energy ◽  
Positive Energy ◽  
Small Component ◽  
The Matrix ◽  
Component Wave ◽  
Energy States ◽  
Dirac Hamiltonian

The separation of the spin-dependent terms in the Dirac Hamiltonian enables us to make an approximation in which the spin-free terms are included in the orbital optimization and the spin-dependent terms may be treated later as a perturbation. In this process, the parameter space required to treat the large and small components has not changed. Even with the extraction of (σ ·p) from the small component, we still have to calculate integrals involving pfL, which essentially regenerates the original small component space and so the integral work has not really changed. What has been achieved is the ability to use the machinery of spin algebra from nonrelativistic theory, but we are left with a large and a small component. The obvious next step is to separate the large and small components, or the positiveand negative-energy states. The small component can be eliminated from the Dirac equation by algebraic manipulation, but this leaves the energy in the denominator. It would be preferable to obtain an energy-independent Hamiltonian that acted only on positive-energy states and that could therefore be represented as two-component spinors. If, following this separation, it were possible to separate out the spin-free and spin-dependent terms, we would have a spin-free Hamiltonian that would operate on a one-component wave function, and we would then be able to use all the machinery of nonrelativistic quantum chemistry but with modified one- and two-electron integrals. The matrix form of the Dirac Hamiltonian suggests that we should seek a unitary transformation that will make it diagonal with respect to the large- and small-component spinor spaces. Such a transformation is called a Foldy–Wouthuysen transformation (Foldy and Wouthuysen 1950). Although in their original paper only the free-particle transformation was derived, together with an iterative decoupling procedure that will be described later in this chapter, the term Foldy–Wouthuysen transformation has come to mean any unitary transformation that decouples the large and small components, either exactly or approximately, and we will use it in this sense.

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