Quaternion Model Research Articles

Analytic LQR Design for Spacecraft Control System Based on Quaternion Model

In this paper, linear-quadratic regulator (LQR) method is proposed for the design of nonlinear spacecraft control systems. The proposed design is based on the linearized spacecraft model that involves only three components of the quaternion. By using a simple and special structure of the linearized reduced quaternion model, an analytical solution for the controller. It is shown that the analytical solution of the state feedback matrix is an explicit function of the cost matrices Q and R. The analytic solution makes it convenient in design practice to tune the feedback matrix and the cost matrices Q and R to balance the requirements between response performance and fuel consumption. It is shown that the designed controller globally stabilizes the nonlinear spacecraft system, whereas it locally optimizes the spacecraft performance. A design example is provided to show the effectiveness of the design method.

QUATERNION BASED COLOR IMAGE QUALITY ASSESSMENT INDEX

A quaternion model for describing color image is proposed in order to evaluate its quality. Local variance distribution of luminance layer is calculated. Color information is taken into account by using quaternion matrix. The description method is a combination of luminance layer and color information. The angle between the singular value feature vectors of the quaternion matrices corresponding to the reference image and the distorted image is used to measure the structural similarity of the two color images. When the reference image and distorted images are of unequal size it can also assess their quality. Results from experiments show that the proposed method is better consistent with the human visual characteristics than MSE, PSNR and MSSIM. The resized distorted images can also be assessed rationally by this method.

Solutions of Dynamic Equations in Quaternion Model of Relativity

Exact and approximate solutions of direct and inverse problems for particle’s dynamic equations deduced in the framework of quaternion model of relativity are suggested. Point particles in the approach are regarded as bodies of reference represented by quaternion triads subject to transformations preserving form invariance of quaternion multiplication. Values of forces causing non-inertial motion of frames are estimated from viewpoints of immobile observer and a moving frame. Dynamical characteristics of several types of non-inertial motions (uniformly rectilinearly accelerated frame, relativistic oscillator) are analyzed. A simple variant of relativistic twobody problem is considered in detail, and motion integrals for a particle in a field of central force are found under natural conditions imposed upon the force’s potential. A relativistic Kepler-type problem is examined, and deceleration of the space probe Pioneer-10 is estimated.

Quaternion based model for momentum biased nadir pointing spacecraft

In this paper, we propose a quaternion based spacecraft model that describes the nadir pointing spacecraft with momentum wheel under gravity gradient torque disturbance. This state space model uses only three components of the quaternion. From this nonlinear model, we derive a linearized state space model for the spacecraft system. We show that unlike all existing quaternion models, this linearized state space model is fully controllable. Therefore, all modern control system design methods can be directly applied to the attitude control system design for momentum biased nadir pointing spacecraft.

Quaternion Model of Relativity: Solutions for Non-Inertial Motions and New Effects

Quaternion Model of Relativity: Solutions for Non-Inertial Motions and New Effects

Analysis of optimal motion control for a material point in a central field with application of quaternions

This work is devoted to a survey and generalization of results obtained in the theory of optimal motion control for a material point in the central Newtonian gravitational field using the Pontryagin’s maximum principle and quaternion models of orbital motion. This theory is very important in space flight mechanics, being the background of the solution of optimal control problems of the motion of the center of mass of a space vehicle. In the first part of this work, a survey of quaternion models of the motion of a material point in a central Newtonian gravitational field is given, their advantages and disadvantages are analyzed. The formulation of the optimal control problem of the motion of a material point in the central Newtonian gravitational field and its correlation with the optimal control problem of the motion of the center of mass of a space vehicle is considered. The main problems arising in the solution of optimal control problems of the motion of a material point using the maximum principle, including instability in the Lyapunov’s sense of solutions to adjoint equations, are studied. It is shown that efficiency of analytical investigation and numerical solution of the corresponding boundary-value problems can be increased by the application of quaternion models of orbital motion.

Quaternion-MUSIC for vector-sensor array processing

This paper considers the problem of direction of arrival (DOA) and polarization parameters estimation in the case of multiple polarized sources impinging on a vector-sensor array. The quaternion model is used, and a data covariance model is proposed using quaternion formalism. A comparison between long vector orthogonality and quaternion vector orthogonality is also performed, and its implications for signal subspace estimation are discussed. Consequently, a MUSIC-like algorithm is presented, allowing estimation of wave's DOAs and polarization parameters. The algorithm is tested in numerical simulations, and performance analysis is conducted. When compared with other MUSIC-like algorithms for vector-sensor array, the newly proposed algorithm results in a reduction by half of memory requirements for representation of data covariance model and reduces the computational effort, for equivalent performance. This paper also illustrates a compact and elegant way of dealing with multicomponent complex-valued data.

Optimal approximation of linear systems by artificial immune response

This paper puts forward a novel artificial immune response algorithm for optimal approximation of linear systems. A quaternion model of artificial immune response is proposed for engineering computing. The model abstracts four elements, namely, antigen, antibody, reaction rules among antibodies, and driving algorithm describing how the rules are applied to antibodies, to simulate the process of immune response. Some reaction rules including clonal selection rules, immunological memory rules and immune regulation rules are introduced. Using the theorem of Markov chain, it is proofed that the new model is convergent. The experimental study on the optimal approximation of a stable linear system and an unstable one show that the approximate models searched by the new model have better performance indices than those obtained by some existing algorithms including the differential evolution algorithm and the multi-agent genetic algorithm.

INS algorithm using quaternion model for low cost IMU

This paper presents a generic inertial navigation system (INS) error propagation model that does not rely on small misalignment angles assumption. The modelling uses quaternions in the computer frame approach. Based on this model, an INS algorithm is developed for low cost inertial measurement unit (IMU) to solve the initial attitudes uncertainty using in-motion alignment. The distribution approximation filter (DAF) is used to implement the non-linear data fusion algorithm.

Multivector Dirac Equation and ℤ 2-Gradings of Clifford Algebras

We generalize certain aspects of Hestenes's approach to Dirac theory to obtain multivector Dirac equations associated to a large class of representations of the gamma matrices. This is done by replacing the usual even/odd decomposition of the space-time algebra with more general \(\mathbb{Z}\)2-gradings. Some examples are given and the chiral case, which is not addressed by the usual approach, is considered in detail. A Lagrangian formulation is briefly discussed. A relationship between this work and certain quaternionic models of the (usual) quantum mechanics is obtained. Finally, we discuss under what conditions the Hestenes's form can be recovered and we suggest a geometrical interpretation for the corresponding situation.

Experimental status of quaternionic quantum mechanics

Analysis of the logical foundations of quantum mechanics indicates the possibility of constructing a theory using quaternionic Hilbert spaces. Whether this mathematical structure reflects reality is a matter for experiment to decide. We review the only direct search for quaternionic quantum mechanics yet carried out and outline a recent proposal by the present authors to look for quaternionic effects in correlated multiparticle systems. We set out how such experiments might distinguish between the several quaternionic models proposed in the literature.

Testing models of the oculomotor velocity-to-position transformation

1. Do neural computations in premotor circuits mirror the physical properties of the systems they control? In 1987, Tweed and Vilis showed that oculomotor theories where a neural integrator converts eye angular velocity commands into position commands cannot be correct, because angular position is not the integral of angular velocity. Recently Schnabolk and Raphan proposed that an angular velocity integrator is nevertheless used to generate tonic commands in the oculomotor system. Here we test the Schnabolk-Raphan (S-R) model against Tweed and Vilis's quaternion (Q) model of the velocity to position transformation. 2. The S-R model predicts large (up to 7 degrees) transient (approximately 700 ms) deviations ("blips") in torsional eye position during attempted horizontal and vertical saccades. The Q model predicts no blips. Search coil recordings of saccades by 7 normal human subjects showed no large blips. 3. For approximately 200 saccades by each subject, we plotted the area under the torsional blip versus the product of saccade eccentricity and magnitude. According to the S-R model, this graph should form a straight line with slope 1.00. According to the Q model, the slope should be zero. Measured slopes averaged 0.016 (range -0.073 to 0.061) for saccade targets at 20 degrees eccentricity and 0.040 (range 0.004-0.076) for targets at 40 degrees. 4. No parameter change can significantly improve the S-R model, but lowering one parameter eradicates the tiny inaccuracy in the Q model. We show that the fundamental reason for the S-R model's failure is its use of a commutative controller to steer a noncommutative plant.

Modeling three-dimensional velocity-to-position transformation in oculomotor control

1. A considerable amount of attention has been devoted to understanding the velocity-position transformation that takes place in the control of eye movements in three dimensions. Much of the work has focused on the idea that rotations in three dimensions do not commute and that a "multiplicative quaternion model" of velocity-position integration is necessary to explain eye movements in three dimensions. Our study has indicated that this approach is not consistent with the physiology of the types of signals necessary to rotate the eyes. 2. We developed a three-dimensional dynamical system model for movement of the eye within its surrounding orbital tissue. The main point of the model is that the eye muscles generate torque to rotate the eye. When the eye reaches an orientation such that the restoring torque of the orbital tissue counterbalances the torque applied by the muscles, a unique equilibrium point is reached. The trajectory of the eye to reach equilibrium may follow any path, depending on the starting eye orientation and eye velocity. However, according to Euler's theorem, the equilibrium reached is equivalent to a rotation about a fixed axis through some angle from a primary orientation. This represents the shortest path that the eye could take from the primary orientation in reaching equilibrium. Consequently, it is also the shortest path for returning the eye to the primary orientation. Thus the restoring torque developed by the tissue surrounding the eye was approximated as proportional to the product of this angle and a unit vector along this axis. The relationship between orientation and restoring torque gives a unique torque-orientation relationship. 3. Once the appropriate torque-orientation relationship for eye rotation is established the velocity-position integrator can be modeled as a dynamical system that is a direct extension of the one-dimensional velocity-position integrator. The linear combination of the integrator state and a direct pathway signal is converted to a torque signal that activates the muscles to rotate the eyes. Therefore the output of the integrator is related to a torque signal that positions the eyes. It is not an eye orientation signal. The applied torque signal drives the eye to an equilibrium orientation such that the restoring torque equals the applied torque but in the opposite direction. The eye orientation reached at equilibrium is determined by the unique torque-orientation relation. Because torque signals are vectors, they commute.(ABSTRACT TRUNCATED AT 400 WORDS)

A multivectorial Dirac equation

The multivectorial generalization of the Cartan map, for 𝒞(1,3) space-time Clifford algebra and an arbitrary gauge group in an isotopic space, is applied to the standard Dirac equation to generate the multivectorial Dirac equation. Using both geometrical and physical reasoning, a particular case discussed by Reifler and Morris [J. Math. Phys. 26, 2059 (1985)] is projected from the general multivectorial Dirac equation, to discuss the properties and limitations associated to their quaternion model. The use of the general multivectorial Dirac equation, which can be defined on any space-time manifold, is also illustrated.

Geometric relations of eye position and velocity vectors during saccades

Measurements of angular position and velocity vectors of the eye in three human and three monkey subjects showed that: (1) position vectors lie roughly in a single plane, in accordance with Listing's law, between and during saccades; (2) primary position of the eye is often far from the centre of the oculomotor range. (3) saccades have nearly-fixed rotation axes, which tilt out of Listing's plane in a systematic way depending on current eye position. Findings 1 and 3 show that saccadic control signals accurately reflect the properties of three-dimensional rotations, as predicted by a new quaternion model of the saccadic system; models that approximate rotational kinematics using vectorial addition and integration do not predict these findings.

Implications of rotational kinematics for the oculomotor system in three dimensions.

1. This paper develops three-dimensional models for the vestibuloocular reflex (VOR) and the internal feedback loop of the saccadic system. The models differ qualitatively from previous, one-dimensional versions, because the commutative algebra used in previous models does not apply to the three-dimensional rotations of the eye. 2. The hypothesis that eye position signals are generated by an eye velocity integrator in the indirect path of the VOR must be rejected because in three dimensions the integral of angular velocity does not specify angular position. Computer simulations using eye velocity integrators show large, cumulative gaze errors and post-VOR drift. We describe a simple velocity to position transformation that works in three dimensions. 3. In the feedback control of saccades, eye position error is not the vector difference between actual and desired eye positions. Subtractive feedback models must continuously adjust the axis of rotation throughout a saccade, and they generate meandering, dysmetric gaze saccades. We describe a multiplicative feedback system that solves these problems and generates fixed-axis saccades that accord with Listing's law. 4. We show that Listing's law requires that most saccades have their axes out of Listing's plane. A corollary is that if three pools of short-lead burst neurons code the eye velocity command during saccades, the three pools are not yoked, but function independently during visually triggered saccades. 5. In our three-dimensional models, we represent eye position using four-component rotational operators called quaternions. This is not the only algebraic system for describing rotations, but it is the one that best fits the needs of the oculomotor system, and it yields much simpler models than do rotation matrix or other representations. 6. Quaternion models predict that eye position is represented on four channels in the oculomotor system: three for the vector components of eye position and one inversely related to gaze eccentricity and torsion. 7. Many testable predictions made by quaternion models also turn up in models based on other mathematics. These predictions are therefore more fundamental than the specific models that generate them. Among these predictions are 1) to compute eye position in the indirect path of the VOR, eye or head velocity signals are multiplied by eye position feedback and then integrated; consequently 2) eye position signals and eye or head velocity signals converge on vestibular neurons, and their interaction is multiplicative.(ABSTRACT TRUNCATED AT 400 WORDS)

Predictions of quaternionic quantum mechanics for CP nonconservation in the B and D meson systems.

We apply the quaternionic quantum mechanics model for CP nonconservation, previously developed by Adler for K mesons, to heavy-flavor systems. We show that unlike superweak models, it can accommodate anomalously large ${D}^{0}$-D\ifmmode\bar\else\textasciimacron\fi{} $^{0}$ mixing. On the other hand, in common with superweak models, the quaternionic model predicts zero partial-decay-rate asymmetries for decays of charged B and D mesons, and predicts final-state-independent partial-decay-rate asymmetries for the decays of neutral B and D mesons into CP eigenstates.

Statistical analysis based on quaternion normal random variables

The quaternion normal distribution is derived and a number of characteristics are highlighted. The maximum likelihood estimation procedure in the quaternion case is examined and the conclusion is reached that the estimation procedure is simplified if the unknown parameters of the associated real probability density function are estimated. The quaternion estimator is then obtained by regarding these estimators as the components of the quaternion estimator. By means of a example attention is given to a test criterium which can be used in the quaternion model.

The O(5) nonlinear σ-model as a SU(2) gauge theory

We show how the O(5) nonlinear σ-model, coupled with “gravity” in a generally covariant way, can be translated into the quaternionic model HP 1, invariant under a local SU (2) gauge, and we discuss the classical solutions.