Abstract: Attitude determination systems that use inexpensive sensors and are based on computationally efficient and robust algorithms are indispensable for real-time vehicle navigation, guidance and control applications. This paper describes an attitude determination system that is based on two vector measurements of non-zero, non-colinear vectors. The algorithm is based on a quaternion formulation of Wahba's (1966) problem, whereby the error quaternion (q/sub e/) becomes the observed state and can be cast into a standard linear measurement equation. Using the Earth's magnetic field and gravity as the two measured quantities, a low-cost attitude determination system is proposed. An iterated least-squares solution to the attitude determination problem is tested on simulated static cases, and shown to be globally convergent. A time-varying Kalman filter implementation of the same formulation is tested on simulated data and experimental data from a maneuvering aircraft. The time-varying Kalman filter implementation of this algorithm is exercised on simulated and real data collected from an inexpensive triad of accelerometers and magnetometers. The accelerometers in conjunction with the derivative of GPS velocity provided a measure of the gravitation field vector and the magnetometers measured the Earth's magnetic field vector. Tracking errors on experimental data are shown to be less than 1 degree mean and standard deviation of approximately 11 degrees in yaw, and 3 degrees in pitch and roll. Best case performance of the system during maneuvering is shown to improve standard deviations to approximately 3 degrees in yaw, and 1.5 degrees in pitch and roll.

Abstract: ENGINEERING NOTES are short manuscripts describing new developments or important results of a preliminary nature. These Notes cannot exceed 6 manuscript pages and 3 Žgures; a page of text may be substituted for a Žgure and vice versa. After informal review by the editors, they may be published within a few months of the date of receipt. Style requirements are the same as for regular contributions (see inside back cover).

Abstract: The quaternion estimation (QUEST) batch attitude-determination algorithm has been extended to work in a general Kalman-e lter framework. This has been done to allow the inclusion of a complicated dynamics model and to allow the estimation of additional quantities beyond the attitude quaternion. The QUEST algorithm, which workswith vectorattitude observations, servesasa starting pointbecause itisable to work with a poor (orno)e rst guess of the attitude. This paper’ s extended version of QUEST uses square-root information e ltering techniques and linearization of the dynamics to propagate the state and its covariance. The measurement update problem is solved by a technique that is an extension of the original QUEST algorithm’ s eigenvalue/eigenvector solution. The paperdemonstrates the new algorithm’ sperformance on an attitude determination problem that usesstar-tracker and rate-gyro measurements. The new algorithm is able to converge from initial attitude errors of 180 deg and initial rate-gyro bias errors as large as 2400 deg/h.

Abstract: An optimal control approach using variable-structure (sliding-mode) tracking for large angle spacecraft maneuvers is presented. The approach expands upon a previously derived regulation result using a quaternion parameterization for the kinematic equations of motion. This parameterization is used since it is free of singularities. The main contribution of this paper is the utilization of a simple term in the control law that produces a maneuver to the reference attitude trajectory in the shortest distance. Also, a multiplicative error quaternion between the desired and actual attitude is used to derive the control law. Sliding-mode switching surfaces are derived using an optimal-control analysis. Control laws are given using either external torque commands or reaction wheel commands. Global asymptotic stability is shown for both cases using a Lyapunov analysis. Simulation results are shown which use the new control strategy to stabilize the motion of the Microwave Anisotropy Probe spacecraft.

Abstract: As is well-known, the real quaternion division algebra ℍ is algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division octonion algebra Open image in new window can not be algebraically isomorphic to any matrix algebras over the real number field ℝ, because Open image in new window is a non-associative algebra over ℝ. However since Open image in new window is an extension of ℍ by the Cayley-Dickson process and is also finite-dimensional, some pseudo real matrix representations of octonions can still be introduced through real matrix representations of quaternions. In this paper we give a complete investigation to real matrix representations of octonions, and consider their various applications to octonions as well as matrices of octonions.

Abstract: An optimal control approach using variable-structure (sliding-mode) tracking for large angle spacecraft maneuvers is presented. The approach expands upon a previously derived regulation result using a quaternion parameterization for the kinematic equations of motion. This parameterization is used since it is free of singularities. The main contribution of this paper is the utilization of a simple term in the control law that produces a maneuver to the reference attitude trajectory in the shortest distance. Also, a multiplicative error quaternion between the desired and actual attitude is used to derive the control law. Sliding-mode switching surfaces are derived using an optimal-control analysis. Control laws are given using either external torque commands or reaction wheel commands. Global asymptotic stability is shown for both cases using a Lyapunov analysis. Simulation results are shown which use the new control strategy to stabilize the motion of the Microwave Anisotropy Probe spacecraft.

Abstract: We derive a Weierstrass-type formula for conformal Lagrangian immersions in Euclidean 4-space, and show that the data satisfies an equation similar to Dirac equation with complex potential. Alternatively this representation has a simple formulation using quaternions. We apply it to the Hamiltonian stationary case and construct all possible tori, thus obtaining a first approach to a moduli space in terms of a simple algebraic-geometric problem on the plane. We also classify Hamiltonian stationary Klein bottles and show they self-intersect.

Abstract: As is well-known, the real quaternion division algebra $ {\cal H}$ is algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division octonion algebra ${\cal O}$ can not be algebraically isomorphic to any matrix algebras over the real number field ${\cal R}$, because ${\cal O}$ is a non-associative algebra over ${\cal R}$. However since ${\cal O}$ is an extension of ${\cal H}$ by the Cayley-Dickson process and is also finite-dimensional, some pseudo real matrix representations of octonions can still be introduced through real matrix representations of quaternions. In this paper we give a complete investigation to real matrix representations of octonions, and consider their various applications to octonions as well as matrices of octonions.

Abstract: The Clifford algebra for the group of rigid body motions is described. Linear elements, that is points, lines and planes are identified as homogeneous elements in the algebra. In each case the action of the group of rigid motions on the linear elements is found. The relationships between these linear elements are found in terms of operations in the algebra. That is, incidence relations, the conditions for a point to lie on a line for example are found. Distance relations, like the distance between a point and a plane are found. Also the meet and join of linear elements, for example, the line determined by two planes and the plane defined by a line and a point, are found. Finally three examples of the use of the algebra are given: a computer graphics problem on the visibility of the apparent crossing of a pair of lines, an assembly problem concerning a double peg-in-hole assembly, and a problem from computer vision on finding epipolar lines in a stereo vision system.

Abstract: Motivated by a quaternionic formulation of quantum mechanics, we discuss quaternionic and complex linear differential equations. We touch only a few aspects of the mathematical theory, namely the resolution of the second order differential equations with constant coefficients. We overcome the problems coming out from the loss of the fundamental theorem of the algebra for quaternions and propose a practical method to solve quaternionic and complex linear second order differential equations with constant coefficients. The resolution of the complex linear Schrodinger equation, in presence of quaternionic potentials, represents an interesting application of the mathematical material discussed in this paper.

Abstract: This paper presents the quaternion color difference edge detector, a new approach to detection of edges in color images. Based on a new type of convolution, the color difference subspace and the proposed edge detector are expressed analytically by using the algebra of the quaternion. The proposed color image edge detector generates edges only where sharp changes of color occur in the original image. The experimental results have verified the improved performance of the new edge detector compared to some well known methods.

Abstract: Nous calculons, en fonction des parametres arithmetiques decrits par Borel, les sous-groupes finis d'un groupe de Klein arithmetique maximal. Ceci est notamment appliquable a l'etude des 3-varietes arithmetiques hyperboliques.

Abstract: We present in this paper some fundamental tools for developing matrix analysis over the complex quaternion algebra As applications, we consider generalized inverses, eigenvalues and eigenvectors, similarity, determinants of complex quaternion matrices, and so on

Abstract: We use sliding mode control theory to design a 3D vision based controller that is robust to bounded parametric estimation errors. First, a model of an eye-in-hand robotic system is derived and sources of uncertainties are listed. Additionally, bounds on the different uncertainties are discussed and their influence on the overall gain of the system is derived. Due to an appropriate selection of the sliding surface, based on the quaternion representation for rotations, a switching controller is proposed. Six degrees of freedom vision based tracking experiments under weak calibration conditions emphasize the practical efficiency of the algorithm.

Abstract: This paper presents the development of the aircraft kinematic transformation equations in terms of four different attitude representations, including the wellknown Euler angles, the Euler-axis rotation parameters, the direction cosines and the Euler-Rodrigues quaternion. The emphasis of the paper is directed at the application of the quaternion formulation to aircraft flight simulation. Results are presented which reinforce the observation that the quaternion formulation, typically implemented to eliminate the singularities associated with the Euler angle formulation, is far superior to the other commonly used formulations based on computational efficiency alone. A development of the quaternion constraints necessary to independently constrain roll, pitch, yaw, bank angle, elevation angle, and/or azimuth angle is presented. For verification of simulation codes, a general closed-form solution to the quaternion formulation, for the special case of constant rotation, is also presented. Additionally, the paper provides a discussion of the numerical integration of the quaternion formulation. This discussion is especially important for simulations that may still utilize a common error reduction scheme originally developed for analog computers. The paper includes a detailed review of the literature on attitude representation dating from the early work of Euler and Hamilton to recent publications in fields such as navigation and control. Nomenclature

Abstract: The structure multivector is a new approach for analyzing the local properties of a two-dimensional signal (e.g. image). It combines the classical concepts of the structure tensor and the analytic signal in a new way. This has been made possible using a representation in the algebra of quaternions. The resulting method is linear and of low complexity. The filter-response includes local phase, local amplitude and local orientation of intrinsically one-dimensional neighborhoods in the signal. As for the structure tensor, the structure multivector field can be used to apply special filters to it for detecting features in images.

Abstract: We develop an efficient technique for computing values at s = 1 of Hecke L-functions. We apply this technique to the computation of relative class numbers of non-abelian CM-fields N which are abelian extensions of some totally real subfield L. We note that the smaller the degree of L the more efficient our technique is. In particular, our technique is very efficient whenever instead of simply choosing L = N + (the maximal totally real subfield of N) we can choose L real quadratic. We finally give examples of computations of relative class numbers of several dihedral CM-fields of large degrees and of several quaternion octic CM-fields with large discriminants.

Abstract: This paper contains a critical comparison of estimators minimizing Wahba's loss function. Some new results are presented for the QUaternion ESTimator (QUEST) and EStimators of the Optimal Quaternion (ESOQ and ESOQ2) to avoid the computational burden of sequential rotations in these algorithms. None of these methods is as robust in principle as Davenport's q method or the Singular Value Decomposition (SVD) method, which are significantly slower. Robustness is only an issue for measurements with widely differing accuracies, so the fastest estimators, the modified ESOQ and ESOQ2, are well suited to sensors that track multiple stars with comparable accuracies. More robust forms of ESOQ and ESOQ2 are developed that are intermediate in speed.

Abstract: By using tensor analysis, we find a connection between normed algebras and the parallelizability of the spheres S$^1$, S$^3$ and S$^7.$ In this process, we discovered the analogue of Hurwitz theorem for curved spaces and a geometrical unified formalism for the metric and the torsion. In order to achieve these goals we first develope a proof of Hurwitz theorem based in tensor analysis. It turns out that in contrast to the doubling procedure and Clifford algebra mechanism, our proof is entirely based in tensor algebra applied to the normed algebra condition. From the tersor analysis point of view our proof is straightforward and short. We also discuss a possible connection between our formalism and the Cayley-Dickson algebras and Hopf maps.

Abstract: , u both decompositions beingconsidered with respect to the quaternion structure of the ambient manifold. Taking intoaccount the research done till now, we may conclude that quaternion CR-submanifoldsand Qii-submanifolds have very little in common, and that there is much room for newresults on their geometry.According to a result of Bejancu (see [3, Theorem 3.3]), any totally umbilical properQii-submanifold M of a quaternion Kaehlerian manifol

Abstract: This paper presents a synthesis procedure for a robot that guides an end-effector as close as possible to a user-specified trajectory. The technique maps the goal workspace from the group of spatial displacements, SE(3), to the group of 4 × 4 rotations, SO(4), in order to obtain a bi-invariant metric for the design procedure. Double quaternions are used to provide a convenient parameterization for SO(4). An example is presented that compares designs obtained using dual quaternions to those for double quaternions, for varying locations of the fixed frame, in order to demonstrate the technique.

Abstract: A maximum-likelihood-based (ML-based) filter is proposed for attitude determination via the GPS carrier phase observables. The quaternion representation is adopted here to describe the attitude. Hence, the norm constraint on the quaternion should be considered. The ML estimation with Lagrange multipliers can be used to consider simultaneously the evolution equation and the constraint, and to minimize the error covariance matrix. The attitude determination via GPS carrier phase observables is fulfilled in two steps. The first step is the GPS carrier phase ambiguity resolution. After the integer ambiguities being fixed, the ML-based filter is used to determine the optimal attitude. The advantage of adopting the quaternion as the state vector to describe the kinematic behavior is that no singular problems arise. To verify our algorithm, the simulation has been conducted. In the simulation, the white noises are added on the carrier phase observables to assess the performance of the proposed method. The body frame is formed by three non-colinear GPS antennae which are mounted on a platform with two aluminum bars representing the baseline vectors. According to the simulation, our method is sound and effective.

Abstract: This paper discusses algorithms for attitude determination using GPS differential phase measurements, assuming that the cycle integer ambiguities are known. The problem of attitude determination is posed as a parameter optimization problem where a new quaternion-based cost function is used. Since the new cost function is not a simple quadratic form and therefore Davenport's q-Method is not applicable in this case. Three algorithms for finding the optimal quaternion are derived, two of which are discrete. The third one is a continuous version of the Newton-Raphson algorithm. This continuous version is new and has a guaranteed exponential convergence to the closest local minimum located on the gradient direction in regions where the associated Hessian matrix is positive definite. The algorithms presented in this paper can handle cases of planar antenna arrays and thus cover a deficiency in earlier algorithms. The efficiency of the new algorithms is demonstrated through numerical examples.

Abstract: One of the most interesting partial differential equations in complex analysis is the Beltrami equation. We will give an overview of possible generalizations of this equation in case of quaternions together with properties of these equations.

Abstract: : This thesis develops an extended Kalman filter for real-time estimation of rigid body motion altitude. The filter represents rotations using quaternions rather than Euler angles, which eliminates the long-standing problem of singularities associated with those angles. A process model for rigid body angular motions and angular rate measurements is defined. The process model converts angular rates into quaternion rates, which are in turn integrated to obtain quaternions. The outputs of the model are values of three-dimensional angular rates, three-dimensional linear accelerations, and three-dimensional magnetic field vector. Gauss-Newton iteration is utilized to find the best quaternion that relates the measured linear accelerations and earth magnetic field in the body coordinate frame to calculated values in the earth coordinate frame. The quaternion obtained from the optimization algorithm is used as part of the observations for the Kalman filter. As a result, the measurement equations become linear. A new approach to attitude estimation is introduced in this thesis. The computational requirements related to the extended Kalman filter developed using this approach are significantly reduced, making it possible to estimate attitude in real-time. Extensive static and dynamic simulation of the filter using Matlab proved it to be robust. Test cases included the presence of large initial errors as well as high noise levels. In all cases the filter was able to converge and accurately track attitude.