Abstract: An arbitrary rigid transformation in $\mathbf{SE}(3)$ can be separated into two parts, namely, a translation and a rigid rotation. This technical report reviews, under a unifying viewpoint, three common alternatives to representing the rotation part: sets of three (yaw-pitch-roll) Euler angles, orthogonal rotation matrices from $\mathbf{SO}(3)$ and quaternions. It will be described: (i) the equivalence between these representations and the formulas for transforming one to each other (in all cases considering the translational and rotational parts as a whole), (ii) how to compose poses with poses and poses with points in each representation and (iii) how the uncertainty of the poses (when modeled as Gaussian distributions) is affected by these transformations and compositions. Some brief notes are also given about the Jacobians required to implement least-squares optimization on manifolds, an very promising approach in recent engineering literature. The text reflects which MRPT C++ library functions implement each of the described algorithms. All formulas and their implementation have been thoroughly validated by means of unit testing and numerical estimation of the Jacobians

Abstract: In this paper, we investigate the Fibonacci and Lucas quaternions. We give the generating functions and Binet formulas for these quaternions. Moreover, we derive some sums formulas for them.

Abstract: In recent years, several authors have reported that spectral saliency detection methods provide state-of-the-art performance in predicting human gaze in images (see, eg, [1---3]) We systematically integrate and evaluate quaternion DCT- and FFT-based spectral saliency detection [3,4], weighted quaternion color space components [5], and the use of multiple resolutions [1] Furthermore, we propose the use of the eigenaxes and eigenangles for spectral saliency models that are based on the quaternion Fourier transform We demonstrate the outstanding performance on the Bruce-Tsotsos (Toronto), Judd (MIT), and Kootstra- Schomacker eye-tracking data sets

Abstract: In this paper, we give a beginners guide to the practicality of using dual-quaternions to represent the rotations and translations in character-based hierarchies Quaternions have proven themselves in many fields of science and computing as providing an unambiguous, un-cumbersome, computationally efficient method of representing rotational information We hope after reading this paper the reader will take a similar view on dual-quaternions We explain how dual number theory can extend quaternions to dual-quaternions and how we can use them to represent rigid transforms (ie, translations and rotations) Through a set of examples, we demonstrate exactly how dual-quaternions relate rotations and translations and compare them with traditional Euler’s angles in combination with Matrix concatenation We give a clear-cut, step-by-step introduction to dual-quaternions, which is followed by a no-nonsense how-to approach on employing them in code The reader, I believe, after reading this paper should be able to see how dual-quaternions can offer a straightforward solution of representing rigid transforms (eg, in complex character hierarchies) We show how dual-quaternions propose a novel alternative to pure Euler-Matrix methods and how a hybrid system in combination with matrices results in a faster more reliable solution We focus on demonstrating the enormous rewards of using dual-quaternions for rigid transforms and in particular their application in complex 3D character hierarchies

Abstract: This brief considers the attitude coordination control problem for spacecraft formation flying when only a subset of the group members has access to the common reference attitude. A quaternion-based distributed attitude coordination control scheme is proposed with consideration of the input saturation and with the aid of the sliding-mode observer, separation principle theorem, Chebyshev neural networks, smooth projection algorithm, and robust control technique. Using graph theory and a Lyapunov-based approach, it is shown that the distributed controller can guarantee the attitude of all spacecraft to converge to a common time-varying reference attitude when the reference attitude is available only to a portion of the group of spacecraft. Numerical simulations are presented to demonstrate the performance of the proposed distributed controller.

Abstract: We apply recent results on robust global asymptotic stabilization of the attitude of a single rigid body to the problem of globally synchronizing the attitude of a network of rigid bodies using a decentralized strategy. The proposed hybrid feedback scheme relies on the communication of a binary logic variable between each pair of neighboring rigid bodies that determines the orientation of a torque component acting to reduce their relative error. Through a hysteretic switch of this logic variable, the hybrid feedback achieves global synchronization under the assumption that the network is connected and acyclic. The hysteresis eliminates chattering while preventing the “unwinding phenomenon” apparent in some quaternion-based attitude control schemes. The results are exercised in a numerical example.

Abstract: In this paper, we investigate some properties of generalized Fibonacci quaternions and Fibonacci-Narayana quaternions.

Abstract: By using the complex representation of quaternion matrices, the Moore-Penrose generalized inverse and the Kronecker product of matrices, the expressions of the least squares �-Hermitian solution with the least norm and the expressions of the least squares �-anti-Hermitian solution with the least norm are derived for the matrix equation AXB+CXD = E over quaternions.

Abstract: Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the ‘three-fold way’. It is perhaps easiest to see it in the study of irreducible unitary representations of groups on complex Hilbert spaces. These representations come in three kinds: those that are not isomorphic to their own dual (the truly ‘complex’ representations), those that are self-dual thanks to a symmetric bilinear pairing (which are ‘real’, in that they are the complexifications of representations on real Hilbert spaces), and those that are self-dual thanks to an antisymmetric bilinear pairing (which are ‘quaternionic’, in that they are the underlying complex representations of representations on quaternionic Hilbert spaces). This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics. More generally, Hilbert spaces of any one of the three kinds—real, complex and quaternionic—can be seen as Hilbert spaces of the other kinds, equipped with extra structure.

Abstract: This paper proposes a singularity-free beam element with Euler–Bernoulli assumption, i.e., the cross section remains rigid and perpendicular to the tangent of the centerline during deformation. Each node of this two-nodal beam element has eight nodal coordinates, including three global positions and one normal strain to describe the rigid translation and flexible deformation of the centerline, respectively, four Euler parameters or quaternion to represent the attitude of cross section. Adopting quaternion instead of Eulerian angles as nodal variables avoids the traditionally encountered singularity problem. The rigid cross section assumption is automatically satisfied. To guarantee the perpendicularity of cross section to the deformed neutral axes, the position and orientation coordinates are coupled interpolated by a special method developed here. The proposed beam element allows arbitrary spatial rigid motion, and large bending, extension, and torsion deformation. The resulting governing equations include normalization constraint equations for each quaternion of the beam nodes, and can be directly solved by the available differential algebraic equation (DAE) solvers. Finally, several numerical examples are presented to verify the large deformation, natural frequencies and dynamic behavior of the proposed beam element.

Abstract: A new quaternion-based method for the SINS/SAR (strap-down inertial navigation system/synthetic aperture radar) integrated navigation system is presented. This method overcomes the shortcomings due to the linear SINS error model used in the currently existing SINS/SAR integrated navigation systems. A quaternion-based matrix is derived for describing the attitude of SINS. Quaternion-based nonlinear error and observation models are established for the SINS/SAR integrated navigation system. An adaptive unscented particle filtering (UPF) algorithm is developed based on the quaternion-based nonlinear models for optimal data fusion in the SINS/SAR integrated navigation system. Experimental results demonstrate that the proposed quaternion based method can effectively reduce the navigation error and improve the navigational positioning precision.

Abstract: From the point of view of classical mechanics, deriving the equations of motion for systems of coupled rigid bodies is regarded as a straightforward procedure: once a suitable set of generalized coordinates and reference frames have been chosen, what remains is to either apply Lagrange’s equations or Newton and Euler’s equations to obtain the differential equations of motion. As the complexity of multibody system increases, the need for more elegant formulation of the equation of motion becomes an issue of paramount importance. Our primary focus is on the kinematic analysis of rigid bodies and serial manipulators (robotic systems) using simultaneously, both homogeneous transformations (4x4) matrices and Dual Quaternions, for the sake of results comparisons (cost,complexity,storage capacity etc.) . This paper has been done mainly for educational and peadagogical purposes, hoping that the scientific community will finally adopt and use Dual Quaternions at least when dealing with multibody systems and specially robotics. DOI: http://dx.doi.org/10.11591/ijra.v1i1.275

Abstract: In this paper, we present a new factorized quaternion approach for determining the arm limbs' orientation using triaxial accelerometers with consideration of anatomical and sensor constraints. Typical use of the quaternion method determines the angle and axis of rotation represented by a single angle-axis quaternion. Different from the conventional approach, we propose using the factorized quaternion approach for the determination of arm motions. This approach allows the implementation of anatomical arm constraints which match the range of motion of the human arm and also reduces the ambiguity in solutions. In addition, the singularities arising from the use of triaxial accelerometers can be detected and resolved for a transient state. Measurement of the upper arm motion is demonstrated along a vertical plane and extended along a tilted plane for the forearm. Experiments have been conducted using a wireless sensor network equipped with triaxial accelerometers attached to the arm. The results have been benchmarked with a commercial inertial measurement unit to validate the feasibility and advantages of this new approach. Comparable performance in terms of accuracy has been obtained at a much reduced cost and power consumption.

Abstract: A state estimation scheme that does not depend on the statistical distribution of bounded measurement noise is presented. This scheme is used to provide state estimates for feedback in an attitude tracking control scheme that exhibits almost global asymptotically stable tracking of a desired attitude trajectory with perfect state measurements. The control and estimation schemes use the global, unique representation of rigid body attitude provided by rotation matrices. Attitude and angular velocity state estimate updates are obtained from discrete multi-rate measurements using a deterministic filtering scheme. Propagation of discrete state estimates is carried out with a Lie group variational integrator, which preserves the orthogonality of rotation matrices during numerical propagation without reprojection. This integrator is also used to numerically simulate the feedback system. The performance of this attitude tracking control scheme is then compared with that of a recently reported quaternion observer-based continuous feedback attitude tracking scheme. This quaternion-based attitude tracking scheme is shown to exhibit unstable, unwinding behavior. Numerical

Abstract: Defining the generalized charge, potential, current and generalized fields as complex quantities where real and imaginary parts represent gravitation and electromagnetism respectively, corresponding field equation, equation of motion and other quantum equations are derived in manifestly covariant manner. It has been shown that the field equations are invariant under Lorentz as well as duality transformations. It has been shown that the quaternionic formulation presented here remains invariant under quaternion transformations.

Abstract: This paper develops a unified methodology for obtaining both the general equations of motion describing the rotational dynamics of a rigid body using quaternions as well as its control. This is achieved in a simple systematic manner using the so-called fundamental equation of constrained motion that permits both the dynamics and the control to be placed within a common framework. It is shown that a first application of this equation yields, in closed form, the equations of rotational dynamics, whereas a second application of the self-same equation yields two new methods for explicitly determining, in closed form, the nonlinear control torque needed to change the orientation of a rigid body. The stability of the controllers developed is analysed, and numerical examples showing the ease and efficacy of the unified methodology are provided.

Abstract: A series representation for continuous-time quaternion random signals is given. The series expansion is based on augmented statistics and provides uncorrelated scalar real-valued random variables. The proposed technique implies a dimension reduction of the four-dimensional original problem to a one-dimensional problem. As a particular case, the quaternion Karhunen-Loeve expansion is obtained. Finally, two illustrative applications to the quaternion widely linear detection and estimation problems are presented.

Abstract: In this paper, we examine a unit quaternion based method to design the optimal paths with maximum sun exposure for unmanned aerial vehicles (UAVs) equipped with photovoltaic cells on their wings. The mission of traveling between two specified boundary points with fixed flying time and constant speed is considered. Since the solar power is the sole source of energy for these UAVs during the flight, we consider the problem of maximizing the incoming solar radiation throughout their trajectory. As the attitude of the UAV directly determines solar intensity normal to the vertical surface of the wing, we use a unit quaternion based method to control the attitude maneuver during the flight interval. Subsequently, the aircraft kinematics are expressed as quadratic functions in terms of unit quaternions which can be solved by a branch and bound approach. Simulation results in two and three dimensions are presented.

Abstract: Let f : T ! S be a finite flat morphism of degree 2 of regular integral schemes of dimension � 2 (with 2 invertible), having regular branch divisor DS. We establish a bijection between Azumaya quaternion algebras on T and quadric surface bundles with simple degeneration along D. This is a manifestation of the exceptional isomorphism 2 A1 = D2 degenerating to the exceptional isomorphism A1 = B1. In one direction, the even Clifford algebra yields the map. In the other direction, we show that the classical algebra norm functor can be uniquely extended over the discriminant divisor. Along the way, we study the orthogonal group schemes, which are smooth yet nonreduc- tive, of quadratic forms with simple degeneration. Finally, we provide two applications: constructing counter-examples to the local-global principle for isotropy, with respect to discrete valuations, of quadratic forms over surfaces; and a new proof of the global Torelli theorem for very general cubic fourfolds containing a plane.

Abstract: Starting from known results, due to Y. Tian in [Ti; 00], referring to the real matrix representations of the real quaternions, in this paper we will investigate the left and right real matrix representations for the complex quaternions and we give some examples in the special case of the complex Fibonacci quaternions.

Abstract: Several attitude error representations are developed for describing the tracking orientation error kinematics. Compact forms of attitude error equation are derived for each case. The attitude error is initially de ned as the quaternion (rotation) error between the current and the estimated orientation. The nonlinear kinematic models are valid for arbitrarily large relative rotations and rotation rates. These modes have been developed for supporting the development of nonlinear spacecraft maneuver formulations. All of the kinematic formulations assume that a reference state has been de ned. These results are expected to be broadly useful for generalizing extended Kalman ltering formulations. The bene ts of paper are discussed.

Abstract: Bearings only target tracking is concerned with estimating the trajectory of an object from noise-corrupted bearing (phase) measurements. Traditionally this problem has been formulated as real valued for the Cartesian coordinate system or modified polar coordinate system. In this study, the authors introduce the bearings only tracking problem for the complex and quaternion domains to take advantage of the natural representation offered by these domains, for multivariate real signals, as well as the greater insights provided into the dynamics of tracking. Moreover, the authors introduce the augmented complex and quaternion extended Kalman filters for the modelling of second-order non-circular complex and quaternion valued signals, for which a widely linear model is shown to be more suitable than a strictly linear model.