Quaternion Conjugate Research Articles

Robot-world and hand–eye calibration based on quaternion: A new method and an extension of classic methods, with their comparisons

As the two main rigid body transformations in robotic vision systems, the problems of robot-world calibration and hand–eye calibration were raised. Classic quaternion as an advantageous representation of rotation exerts its advantage for the separable methods of the hand–eye calibration problems. While the impact of rotation propagation errors on the translation estimate in separable methods was often ignored. In this paper, after a summarization of quaternion-based methods in SO(3) and SE(3), a dual equations model using adjoint transformation was derived which converts the transformation of rigid body motion into a line transformation. Then, a closed-form rotation solution based on quaternion conjugate was proposed, and two translation closed-form solutions with different propagation error paths were given respectively. Based on these, the impact of rotation propagation errors of the four methods on translation estimate was analyzed, and an assessment of when each method works most and least applicable was given. In addition, simulations and experiments were presented to assess all methods and verify the accuracy and efficacy of the proposed method against uncalibrated commercial robot.

Multidimensional signal processing using quaternion complex Hadamard transform in sequency domain

This Letter presents the quaternion conjugate symmetric-sequency ordered complex Hadamard transform (QCS-SCHT). The purpose of this research is to present the idea of the quaternion in sequency domain that is based on four complexified terms including real and imaginary values are discussed and developed with different types of quaternions planes for QCS-SCHT. QCS-SCHT is discussed for colour image processing.

Hyperquaternions: A New Tool for Physics

A hyperquaternion formulation of Clifford algebras in n dimensions is presented. The hyperquaternion algebra is defined as a tensor product of quaternion algebras $$\mathbb {H}$$ (or a subalgebra thereof). An advantage of this formulation is that the hyperquaternion product is defined independently of the choice of the generators. The paper gives an explicit expression of the generators and develops a generalized multivector calculus. Due to the isomorphism $$\mathbb {H}\otimes \mathbb {H}\simeq m(4, \mathbb {R})$$ , hyperquaternions yield all real, complex and quaternion square matrices. A hyperconjugation is introduced which generalizes the concepts of transposition, adjunction and transpose quaternion conjugate. As applications, simple expressions of the unitary and unitary symplectic groups are obtained. Finally, the hyperquaternions are compared, in the context of physical applications, to another algebraic structure based on octonions which has been proposed recently.

A PERSPECTIVE ON THE MAGIC SQUARE AND THE "SPECIAL UNITARY" REALIZATION OF REAL SIMPLE LIE ALGEBRAS

This paper contains the last part of the minicourse "Spaces: A Perspective View" delivered at the IFWGP2012. The series of three lectures was intended to bring the listeners from the more naive and elementary idea of space as "our physical Space" (which after all was the dominant one up to the 1820s) through the generalization of the idea of space which took place in the last third of the 19th century. That was a consequence of first the discovery and acceptance of non-Euclidean geometry and second, of the views afforded by the works of Riemann and Klein and continued since then by many others, outstandingly Lie and Cartan. Here we deal with the part of the minicourse which centers on the classification questions associated to the simple real Lie groups. We review the original introduction of the Magic Square "á la Freudenthal", putting the emphasis in the role played in this construction by the four normed division algebras ℝ, ℂ, ℍ, 𝕆. We then explore the possibility of understanding some simple real Lie algebras as "special unitary" over some algebras 𝕂 or tensor products 𝕂1 ⊗ 𝕂2, and we argue that the proper setting for this construction is not to confine only to normed division algebras, but to allow the split versions ℂ′, ℍ′, 𝕆′ of complex, quaternions and octonions as well. This way we get a "Grand Magic Square" and we fill in all details required to cover all real forms of simple real Lie algebras within this scheme. The paper ends with the complete lists of all realizations of simple real Lie algebras as "special unitary" (or only unitary when n = 2) over some tensor product of two *-algebras 𝕂1, 𝕂2, which in all cases are obtained from ℝ, ℂ, ℂ′, ℍ, ℍ′, 𝕆, 𝕆′ as sets, endowing them with a *-conjugation which usually but not always is the natural complex, quaternionic or octonionic conjugation.

Dual Quaternion Involutions and Anti-Involutions

An involution or anti-involution is a self-inverse linear mapping. In this paper, we present involutions and anti-involutions of dual quaternions. In order to do this, quaternion conjugate, dual conjugate and total conjugate are defined for a dual quaternion and these conjugates are used in some transformations in order to check whether involution or anti-involution axioms are being satisfied or not by these transformations. Finally, geometric interpretations of real quaternion and dual quaternion involutions and anti-involutions are given.

Algorithms for quaternion polynomial root-finding

In 1941 Niven pioneered root-finding for a quaternion polynomial P(x), proving the fundamental theorem of algebra (FTA) and proposing an algorithm, practical if the norm and trace of a solution are known. We present novel results on theory, algorithms and applications of quaternion root-finding. Firstly, we give a new proof of the FTA resulting in explicit formulas for both exact and approximate quaternion roots of P(x) in terms of exact and approximate complex roots of the real polynomial F(x)=P(x)P¯(x), where P¯(x) is the conjugate polynomial. In particular, if |F(c)|≤ϵ, then for a computable quaternion conjugate q of c, |P(q)|≤ϵ. Consequences of these include relevance of root-finding methods for complex polynomials, computation of bounds on zeros, and algebraic solution of special quaternion equations. Secondly, working directly in the quaternion space, we develop Newton and Halley methods and analyze their local behavior. Surprisingly, even for a quadratic quaternion polynomial Newton’s method may not converge locally. Finally, we derive an analogue of the Bernoulli method in the quaternion space for computing the dominant root in certain cases. This requires the development of an independent theory for the solution of quaternion homogeneous linear recurrence relations. These results also lay a foundation for quaternion polynomiography.

Involutions of Complexified Quaternions and Split Quaternions

An involution or anti-involution is a self-inverse linear mapping. Involutions and anti-involutions of real quaternions were studied by Ell and Sangwine [15]. In this paper we present involutions and antiinvolutions of biquaternions (complexified quaternions) and split quaternions. In addition, while only quaternion conjugate can be defined for a real quaternion and split quaternion, also complex conjugate can be defined for a biquaternion. Therefore, complex conjugate of a biquaternion is used in some transformations beside quaternion conjugate in order to check whether involution or anti-involution axioms are being satisfied or not by these transformations. Finally, geometric interpretations of real quaternion, biquaternion and split quaternion involutions and anti-involutions are given.

On the transposition anti-involution in real Clifford algebras III: the automorphism group of the transposition scalar product on spinor spaces

In Abłamowicz and Fauser [R. Abłamowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras I: The transposition map, Linear Multilinear Alg. (to appear)] a signature ϵ = (p, q)-dependent transposition anti-involution of real Clifford algebras Cℓ p,q for non-degenerate quadratic forms was introduced. In Abłamowicz and Fauser [R. Abłamowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras II: Stabilizer groups of primitive idempotents, Linear Multilinear Alg. (to appear)] we showed that, depending on the value of (p − q) mod 8, the map gives rise to transposition, complex Hermitian or quaternionic Hermitian conjugation of representation matrices in spinor representation. The resulting scalar product is in general different from the two known standard scalar products [R. Abłamowicz and B. Fauser, Clifford and Grassmann Hopf algebras via the BIGEBRA package for Maple, Comput. Phys. Commun. 170 (2005), pp. 115–130]. We provide a full signature (p, q)-dependent classification of the invariance groups of this product for p + q ≤ 9. The map is identified as the ‘star’ map known [D.S. Passman, The Algebraic Structure of Group Rings, Robert E. Krieger Publishing Company, Malabar, Florida, 1985] from the theory of (twisted) group algebras where the Clifford algebra Cℓ p,q is seen as a twisted group ring n = p + q. We discuss important subgroups of a stabilizer group G p,q (f) of a primitive idempotent f and we relate their transversals to spinor bases in spinor spaces realized as minimal left ideals Cℓ p,q f.

On the transposition anti-involution in real Clifford algebras II: stabilizer groups of primitive idempotents

In the first article [R. Abłamowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras I: The transposition map, Linear Multilinear Algebra, to appear] we showed that real Clifford algebras Cℓ(V, Q) possess a unique transposition anti-involution . The map reduces to reversion (resp. conjugation) for any Euclidean (resp. anti-Euclidean) signature. When applied to a general element of the algebra, it results in transposition of associated matrix of that element in the left regular representation of the algebra. In this article we show that, depending on the value of (p − q) mod 8, where ϵ = (p, q) is the signature of Q, the anti-involution gives rise to transposition, Hermitian complex and Hermitian quaternionic conjugation of representation matrices in spinor representations. We realize spinors in minimal left ideals S = Cℓ p,q f generated by a primitive idempotent f. The map allows us to define a dual spinor space S*, and a new spinor norm on S, which is different, in general, from two spinor norms known to exist. We study a transitive action of generalized Salingaros' multiplicative vee groups G p,q on complete sets of mutually annihilating primitive idempotents. Using the normal stabilizer subgroup G p,q (f), we construct left transversals, spinor bases and maps between spinor spaces for different orthogonal idempotents f i summing up to 1. We classify the stabilizer groups according to the signature in simple and semisimple cases.

Hamiltonian square roots of skew Hamiltonian quaternionic matrices

Criteria for existence of Hamiltonian quaternionic matrices that are square roots of a given skewHamiltonian quaternionic matrix are developed. The criteria are formulated in terms of respective canonical forms of skewHamiltonian quaternionic matrices. The Hamiltonian property is understood with respect to either the quaternionic conjugation, or an involutory antiautomorphism of the quaternions which is different from the quaternionic conjugation. Many results are stated and proved in a more general framework of symmetric and skewsymmetric matrices with respect to an invertible matrix which is skewsymmetric relative to an involutory antiautomorphism.

Quaternion involutions and anti-involutions

An involution or anti-involution is a self-inverse linear mapping. In this paper we study quaternion involutions and anti-involutions. We review formal axioms for such involutions and anti-involutions. We present two mappings, one a quaternion involution and one an anti-involution, and a geometric interpretation of each as reflections. We present results on the composition of these mappings and show that the quaternion conjugate may be expressed using three mutually perpendicular anti-involutions. Finally, we show that projection of a vector or quaternion can be expressed concisely using three mutually perpendicular anti-involutions.

Canonical forms for mixed symmetric–skewsymmetric quaternion matrix pencils

Canonical forms are described for pairs of quaternionic matrices, or equivalently matrix pencils, where one matrix is symmetric and the other matrix is skewsymmetric, under strict equivalence and symmetry respecting congruence. The symmetry is understood in the sense of a fixed involutory antiautomorphism of the skew field of the real quaternions; the involutory antiautomorphism is assumed to be nonstandard, i.e., other than the quaternionic conjugation. Some applications are developed, such as canonical forms for quaternionic matrices under symmetry respecting congruence, and canonical forms for matrices that are skewsymmetric with respect to a nondegenerate symmetric or skewsymmetric quaternion valued inner product.

Charged Representations of the Infinite Fermi and Clifford Algebras

The real and quaternionic charge conjugation operators invariant under the infinite-dimensional Clifford algebra, or compatible with the Fermi algebra, are determined. There results a maze of inequivalent irreducible charged representations, all of which are non-Fock. The representation vectors and their charges admit two interpretations besides those of spinors or states of quantum fields: as wavelets on the circle, with charge conjugations acting via ordinary complex conjugation; and as infinite-dimensional numbers, with charge conjugations acting by automorphisms.

Hardy spaces on the Quaternions

In this paper, the Quaternion-valued Hardy spaces and conjugate Hardyspaces on $$\mathbb{R}^{3} $$ are characterized. In analogy with the decomposition of square-integrable function space on the real line $$\mathbb{R}$$ into the direct sum of Hardy space and conjugate Hardy space, the square-integrable Quaternion -valued function space on $$\mathbb{R}^{3} $$ is decomposed into the orthogonal sum of the Quaternion Hardy and conjugate Hardy spaces.

On the quaternionic form of linear equations for the gravitational field

In this letter we show that the linear equations for the gravitational field with Heavisidian monopoles could be expressed in quaternion form using a quaternion conjugate operator as well as quaternion differential operator.