Quaternion Components Research Articles

Improving quaternion neural networks with quaternionic activation functions

In this paper, we propose novel quaternion activation functions where we modify either the quaternion magnitude or the phase, as an alternative to the commonly used split activation functions. We define criteria that are relevant for quaternion activation functions, and subsequently we propose our novel activation functions based on this analysis. Instead of applying a known activation function like the ReLU or Tanh on the quaternion elements separately, these activation functions consider the quaternion properties and respect the quaternion space H. In particular, all quaternion components are utilized to calculate all output components, carrying out the benefit of the Hamilton product in e.g. the quaternion convolution to the activation functions. The proposed activation functions can be incorporated in arbitrary quaternion valued neural networks trained with gradient descent techniques. We further discuss the derivatives of the proposed activation functions where we observe beneficial properties for the activation functions affecting the phase. Specifically, they prove to be sensitive on basically the whole input range, thus improved gradient flow can be expected. We provide an elaborate experimental evaluation of our proposed quaternion activation functions including comparison with the split ReLU and split Tanh on two image classification tasks using the CIFAR-10 and SVHN dataset. There, especially the quaternion activation functions affecting the phase consistently prove to provide better performance.

On the Quaternion Transformation and Field Equations in Curved Space-Time

In this paper, we use four-dimensional quaternionic algebra to describe space-time geodesics in curvature form. The transformation relations of a quaternionic variables are established with the help of basis-transformations of quaternion algebra. We deduce the quaternionic covariant derivative that explains how the quaternion components vary with scalar and vector fields. The quaternionic metric tensor and the geodesic equation are also discussed to describe the quaternionic line element in curved space-time. We examine a quaternionic metric tensor equation for the Riemannian Christoffel curvature tensor. We present the quaternionic Einstein’s field-like equation, which indicates that quaternionic matter and geometry are equivalent. Relevance of the work:In recent decades, hypercomplex algebra, viz., quaternion and octonions, has been widely used to explain various branches of physics. In this way, we have investigated quaternionic transformations and field equations in curved space-time. The present novel work will help to explain the characteristics of the curved space-time universe in terms of quaternion algebra. It can also be used to describe quaternionic gravitational waves, the black hole formulation, and so on.

A redundant fused MIMU attitude system algorithm based on two-stage data fusion of MEMS gyro clusters array

A redundant fused microelectromechanical inertial measurement unit (MIMU) attitude system composed of a two-stage data fusion is proposed. In the first stage, multiple gyroscopes are arranged on the axes of the MIMU, and a KF is presented to fuse measurements of the gyro arrays. In the second stage, an integrated KF is designed to estimate attitude quaternion, in which the Gauss-Newton algorithm is used to estimate quaternion to construct KF measurements. Lastly, a fused MIMU with three eight-gyro arrays is developed. The results show that the ARW and RRW are reduced by a factor of about 4.86 and 3.8 respectively, and the quaternion error solved by the fused signals is smaller than that by the original MIMU signals. Additionally, the 1σ error of quaternion components q0, q1 and q3 are each reduced by half while q2 is reduced to 13% of the corresponding error using the original MIMU signals.

Assessment of Breathing Parameters Using an Inertial Measurement Unit (IMU)-Based System.

Breathing frequency (fB) is an important vital sign that—if appropriately monitored—may help to predict clinical adverse events. Inertial sensors open the door to the development of low-cost, wearable, and easy-to-use breathing-monitoring systems. The present paper proposes a new posture-independent processing algorithm for breath-by-breath extraction of breathing temporal parameters from chest-wall inclination change signals measured using inertial measurement units. An important step of the processing algorithm is dimension reduction (DR) that allows the extraction of a single respiratory signal starting from 4-component quaternion data. Three different DR methods are proposed and compared in terms of accuracy of breathing temporal parameter estimation, in a group of healthy subjects, considering different breathing patterns and different postures; optoelectronic plethysmography was used as reference system. In this study, we found that the method based on PCA-fusion of the four quaternion components provided the best fB estimation performance in terms of mean absolute errors (<2 breaths/min), correlation (r > 0.963) and Bland–Altman Analysis, outperforming the other two methods, based on the selection of a single quaternion component, identified on the basis of spectral analysis; particularly, in supine position, results provided by PCA-based method were even better than those obtained with the ideal quaternion component, determined a posteriori as the one providing the minimum estimation error. The proposed algorithm and system were able to successfully reconstruct the respiration-induced movement, and to accurately determine the respiratory rate in an automatic, position-independent manner.

Алгоритм начальной инициализации кватерниона пространственной ориентации в параметрах Родрига-Гамильтона

Introduction. The correction time reduction for the spatial orientation evaluation of a solid under the orientation system initiation is considered. Solid spatial orientation is measured by the integrated values from three orthogonal angular speed sensors. Difference between the actual spatial orientation and the orientation estimated by sensors is adjusted due to the data obtained from other sensors, such as accelerometers and magnetometers. Major existing algorithms deduct data acquired from accelerometers and magnetometers from the evaluation of angular rate through the correction factor thus correcting the spatial orientation assessment error. The higher the solid inclination angle in relation to horizon when the orientation system is switched on, the greater the spatial orientation error is. The algorithm presented herein corrects the spatial orientation evaluation in quaternion components without angular rate sensors which allows minimizing the spatial orientation assessment error within shorter time in comparison with the existing algorithms. Materials and Methods. To implement the correction algorithm, the MPU6050 sensor is used. It is made with microelectromechanical technology, and its body includes three orthogonally located angular velocity sensors and three orthogonally located accelerometers. The output data from MPU6050 sensor is processed by dsPIC33EP256MU806 microchip. The spatial orientation is calculated by the Rodrigues-Hamilton parameters in the quaternion components. The result is input to the Matlab software package which executes the program of visualizing the dependencies on the time of four quaternion components graphically. Research Results . In existing algorithms that use the Rodrigues-Hamilton parameters, at the initial initialization of the orientation system, it is suggested to increase the correction factor, or to use the trigonometric formulas to find the Euler angles and translate them into the RodriguesHamilton parameters. In the first case, the initial initialization time remains sufficiently long, in the second case, due to the use of Euler angles, a phenomenon of “gimbal lock” can be observed. The proposed algorithm performs the initial initialization in a time equivalent to the initialization time in the Euler angles parameters, but it applies only the RodriguesHamilton parameters. Discussion and Conclusions. Using the proposed algorithm will allow a minimum of 5-fold reduction in the initial initialization time of the spatial orientation quaternion. In consequence, the total time required for activating the system will be also reduced due to the fact that the initial initialization is necessary every time the orientation system is switched on. For the correct determination of the spatial orientation according to the proposed algorithm, the necessary condition is the absence of any acceleration on the body other than the gravitational acceleration because the initialization occurs only upon the accelerometer readings.

Low read‐only memory distributed arithmetic implementation of quaternion multiplier using split matrix approach

In most algorithms that use quaternion numbers, the key operation is a quaternion multiplication, of which the efficiency and accuracy obviously determine the same properties of the whole computational scheme of a filter or transform. A digit (L-bit)-serial quaternion multiplier based on the distributed arithmetic (DA) using the splitting of the multiplication matrix is presented. The circuit provides the facility to compute several products of quaternion components concurrently as well as to reduce the memory capacity by half in comparison with the known DA-based multiplier, and it is well suited for field programmable gate array (FPGA)-based fixed-point implementations of the algorithms. Apart from a theoretical development, the experimental design results which are obtained using a Xilinx Virtex 6 FPGA are reported.

Lattice Boltzmann implementation of the three-dimensional Ben-Naim potential for water-like fluids

We develop a three-dimensional lattice Boltzmann (LB) model accounting for directional interactions between water-like molecules, based on the so-called Ben-Naim (BN) potential [A. Ben-Naim, Molecular Theory of Water and Aqueous Solutions: Part I: Understanding Water (World Scientific Publishing Company, 2010); "Statistical mechanics of 'waterlike' particles in two dimensions. I. Physical model and application of the Percus-Yevick equation," J. Chem. Phys. 54, 3682 (1971)]. The water-like molecules are represented by rigid tetrahedra, with two donors and two acceptors at the corners and interacting with neighboring tetrahedra, sitting on the nodes of a regular lattice. The tetrahedra are free to rotate about their centers under the drive of the torque arising from the interparticle potential. The orientations of the water molecules are evolved in time via an overdamped Langevin dynamics for the torque, which is solved by means of a quaternion technique. The resulting advection-diffusion-reaction equation for the quaternion components is solved by a LB method, acting as a dynamic minimizer for the global energy of the fluid. By adding thermal fluctuations to the torque equation, the model is shown to reproduce some microscopic features of real water, such as an average number of hydrogen bonds per molecules (HBs) between 3 and 4, in a qualitative agreement with microscopic water models. Albeit slower than a standard LB solver for ordinary fluids, the present scheme opens up potentially far-reaching scenarios for multiscale applications based on a coarse-grained representation of the water solvent.

3D Keyframe Animation Watermarking Based on Orientation Interpolator

This paper presents 3D keyframe animation watermarking using orientation interpolators. 3D keyframe animation consists of a number of transform nodes, including a geometrical node from the initial model and several interpolator nodes that represent object movement. Therefore, the proposed algorithm randomly selects transform nodes with orientation interpolator nodes, then resamples the quaternion components to maintain a uniform key time. Thereafter, watermark bits are embedded into quaternion components with large rotation angles. Experimental results verify the robustness of the proposed algorithm to geometrical and timeline attacks, along with the subjective and objective quality of its invisibility.

Determination of the Quasistatic Component of Microaccelerations at the MirStation

We describe the determination of the quasistatic component of microaccelerations, as it was done during the space experiments with the instruments DACON and ALICE-2. This component was calculated using the telemetric information related to the motion of the station with respect to its center of mass. The information consists of the values of the station angular velocity vector and the quaternion which specifies its orientation, determined at discrete instants of time. The quaternion is determined with a step of about 1 min, the angular velocity with a step of about 10 s. The information is used in the following manner. At first, the quaternion components corresponding to some time interval are smoothed by splines. Then, employing the obtained splines and kinematic equations, the angular velocity and acceleration of the station are calculated on this interval. Finally, the microacceleration is calculated as a function of time at the point of the location of the instrument. The data of measurements of the angular velocity are used for the purpose of control. As a rule, the available telemetric information allows one to find the quasi-static component for the entire time interval of carrying out the experiment. Examples of determination of this component for some experiments are presented. A comparison with the results of calculating the microacceleration quasistatic component by other methods is made.

A closed-form forward kinematics solution for the 6-6/sup p/ Stewart platform

Studies the forward kinematics of a special 6-6 Stewart platform, in which both the base and the mobile platforms are similar hexagons. A concise closed-form solution to the forward kinematics of the Stewart platform is obtained by introducing a quaternion to represent the transformation matrix. It is very interesting to find that the eight possible solutions have a very simple inner symmetry relationship. So, the solutions obtained from the approach presented in the paper are concise, represented as the squares of the four quaternion components. Besides, unnecessary complex roots are avoided automatically. Finally, only univariate quadratic equations are required to solve in this approach.

Inertial rotation sensing in three dimensions using coupled electromagnetic ring-gyroscopes

A model gyroscope consisting of three coupled, spatially orthogonal electromagnetic ring-resonators is analysed. It is shown that, through an appropriate choice of inter-ring coupling components, the gyroscope can support a three-dimensional circulating or `gyrating` wave which, when subjected to an arbitrarily changing inertial rotation rate, can be described by an evolving wave amplitude state vector or `gyror`. When the effective resonant mode order takes the value N= 1/2 , the state vector transforms as a spinor and the gyrating wave becomes an inertially stable spinning wave. Despite its low sensitivity, this spin-gyroscope device is of interest since the measurable components of its state spinor are directly related to the parameters (angle-axis variables, quaternion components or Cayley-Klein parameters) which specify the orientation of the gyroscope relative to a fixed inertial frame.

Generalized Quaternions with Quaternion Components

Generalized Quaternions with Quaternion Components