New kinematic parameters of the finite rotation of a rigid body

A new family of kinematic parameters for the orientation of a rigid body (global and local) is presented and described. All the kinematic parameters are obtained by mapping the variables onto a corresponding orientated subspace (hyperplane). In particular, a method of stereographically projecting a point belonging to a five-dimensional sphere S⁵ ⊂ R⁶ onto an orientated hyperplane R⁵ is demonstrated in the case of the classical direction cosines of the angles specifying the orientation of two systems of coordinates. A family of global kinematic parameters is described, obtained by mapping the Hopf five-dimensional kinematic parameters defined in the space R⁵ onto a four-dimensional orientated subspace R⁴. A correspondence between the five-dimensional and four-dimensional kinematic parameters defined in the corresponding spaces is established on the basis of a theorem on the homeomorphism of two topological spaces (a four-dimensional sphere S⁴ ⊂ R⁵ with one deleted point and an orientated hyperplane in R⁴). It is also shown to which global four-dimensional orientation parameters–quaternions defined in the space R⁴ the classical local parameters, that is, the three-dimensional Rodrigues and Gibbs finite rotation vectors, correspond. The kinematic differential rotational equations corresponding to the five-dimensional and four-dimensional orientation parameters are obtained by the projection method. All the rigid body kinematic orientation parameters enable one, using the kinematic equations corresponding to them, to solve the classical Darboux problem, that is, to determine the actual angular position of a body in a three-dimensional space using the known (measured) angular velocity of rotation of the object and its specified initial position.     

Stable finite-element calculation of incompressible flows using the rotation form of convection

Conforming finite-element approximations are considered forthe incompressible Navier–Stokes equations with nonlinearterms written in the convection or rotation forms. Implicittime integration results in nice stability properties of auxiliaryproblems which can be solved by efficient numerical algorithms.The original nonlinear system admits relatively simple stabilizationstrategies. The paper presents in a unified form the convergenceanalysis, including the design of stabilization parameters,for linearized equations in both convection and rotation forms.Moreover, it is shown that a Galerkin discretization of thepressure-regularized Oseen problem with skew-symmetric termsin rotation form possesses better stability properties and,being much easier to solve, can be used as a predictor in implicitcalculations.     

A unified analysis of linear quaternion dynamic equations on time scales

Over the last years, considerable attention has been paid to the role of the quaternion differential equations (QDEs) which extend the ordinary differential equations. The theory of QDEs was recently well established and has wide applications in physics and life science. This paper establishes a systematic frame work for the theory of linear quaternion dynamic equations on time scales (QDETS), which can be applied to wave phenomena modeling, fluid dynamics and filter design. The algebraic structure of the solutions to the QDETS is actually a left- or right- module, not a linear vector space. On the non-commutativity of the quaternion algebra, many concepts and properties of the classical dynamic equations on time scales (DETS) can not be applied. They should be redefined accordingly. Using $q$-determinant, a novel definition of Wronskian is introduced under the framework of quaternions which is different from the standard one in DETS. Liouville formula for QDETS is also analyzed. Upon these, the solutions to the linear QDETS are established. The Putzer''s algorithms to evaluate the fundamental solution matrix for homogeneous QDETS are presented. Furthermore, the variation of constants formula to solve the nonhomogeneous QDETs is given. Some concrete examples are provided to illustrate the feasibility of the proposed algorithms.     

SM-stability of operator-difference schemes

The spectral mimetic (SM) properties of operator-difference schemes for solving the Cauchy problem for first-order evolutionary equations concern the time evolution of individual harmonics of the solution. Keeping track of the spectral characteristics makes it possible to select more appropriate approximations with respect to time. Among two-level implicit schemes of improved accuracy based on Padé approximations, SM-stability holds for schemes based on polynomial approximations if the operator in an evolutionary equation is self-adjoint and for symmetric schemes if the operator is skew-symmetric. In this paper, additive schemes (also called splitting schemes) are constructed for evolutionary equations with general operators. These schemes are based on the extraction of the self-adjoint and skew-symmetric components of the corresponding operator.     

Gradient stability of numerical algorithms in local nonequilibrium problems of critical dynamics

The critical dynamics of a spatially inhomogeneous system are analyzed with allowance for local nonequilibrium, which leads to a singular perturbation in the equations due to the appearance of a second time derivative. An extension is derived for the Eyre theorem, which holds for classical critical dynamics described by first-order equations in time and based on the local equilibrium hypothesis. It is shown that gradient-stable numerical algorithms can also be constructed for second-order equations in time by applying the decomposition of the free energy into expansive and contractive parts, which was suggested by Eyre for classical equations. These gradient-stable algorithms yield a monotonically nondecreasing free energy in simulations with an arbitrary time step. It is shown that the gradient stability conditions for the modified and classical equations of critical dynamics coincide in the case of a certain time approximation of the inertial dynamics relations introduced for describing local nonequilibrium. Model problems illustrating the extended Eyre theorem for critical dynamics problems are considered.     

Regularity theorems for Maxwell's equations

Methods using the theory of distributions and Hilbert space operators have been very powerful in the past to achieve uniqueness and existence results for Maxwell's equations. In this paper conditions are given when such abstract “Hilbert space”-solutions represent differentiable “regular” functions which satisfy Maxwell's equations, boundary conditions, and transmission conditions in the classical sense.     

Numerical analysis of differential operators on raw point clouds

3D acquisition devices acquire object surfaces with growing accuracy by obtaining 3D point samples of the surface. This sampling depends on the geometry of the device and of the scanned object and is therefore very irregular. Many numerical schemes have been proposed for applying PDEs to regularly meshed 3D data. Nevertheless, for high precision applications it remains necessary to compute differential operators on raw point clouds prior to any meshing. Indeed differential operators such as the mean curvature or the principal curvatures provide crucial information for the orientation and meshing process itself. This paper reviews a half dozen local schemes which have been proposed to compute discrete curvature-like shape indicators on raw point clouds. All of them will be analyzed mathematically in a unified framework by computing their asymptotic form when the size of the neighborhood tends to zero. They are given in terms of the principal curvatures or of higher order intrinsic differential operators which, in return, characterize the discrete operators. All considered local schemes are of two kinds: either they perform a polynomial local regression, or they compute directly local moments. But the polynomial regression of order 1 is demonstrated to play a special role, because its iterations yield a scale space. This analysis is completed with numerical experiments comparing the accuracies of these schemes. We demonstrate that this accuracy is enhanced for all operators by applying previously the scale space.     

An Experimental Study of Inertial Waves in a Closed Cone

An experimental study of inertial waves in a closed cone is presented in which the inertial waves are excited by a slight periodic oscillation superimposed on the cone's basic rotation rate. The dynamic pressure field is measured along the cone's rotation axis; no standing modal structure is observed, confirming Greenspan's (1969) argument that the closed cone appears open to inertial oscillations and the inertial wave spectrum is continuous. Similar pressure measurements made in the frustum of a right circular cone show that removal of the singular apex of the cone leads to standing wave modes.     

Geometry of a five link mechanism with two degrees of freedom

The paper is devoted to the solution of straight and inverse geometrical tasks of five link mechanism with two degrees of freedom. The solution of the mentioned problem is very important in order to determine kinematic parameters of actuators. The problem can be divided into two parts. The first part is considered when we are given the coordinates of the output link of the mechanism and the necessity arises of determining the angles of rotation of actuators. On the other hand, it is very important to determine the position of the output link when the angles of rotation of the actuators are known. Here we consider that the mechanism is composed only of five classes of rotating kinematic pairs and the actuators are situated at the junctions of frames and links of the examined mechanism. The solution of the said problem is based on utilization of homogenous coordinates. On the basis of the obtained equations of motion, one can calculate the trajectories of motion of the output link as well angles of rotation of the actuators by taking into consideration preliminary given kinematic parameters of the mechanism. Here we also obtain equations for calculating values of speed and acceleration of the links of the mechanism. The calculations differ from known methods in simplicity and high performance, which would be useful for programming actuators mounted in the joints of the linkage.     

The problem of the time-optimal control of spacecraft reorientation

The use of Pontryagin's maximum principle to solve spacecraft motion control problems is demonstrated. The problem of the optimal control of the spatial reorientation of a spacecraft (as a rigid body) from an arbitrary initial angular position to an assigned final angular position in the minimum rotation time is investigated in detail. The case in which velocity parameters of the motion are constrained is considered. An analytical solution of the problem is obtained in closed form using the method of quaternions, and mathematical expressions for synthesizing the optimal control programme are given. The kinematic problem of spacecraft reorientation is solved completely. A design scheme for solving the maximum principle boundary-value problem for arbitrary turning conditions and inertial characteristics of the spacecraft is given. A solution of the problem of the optimal control of spatial reorientation for a dynamically symmetrical spacecraft is presented in analytical form (to expressions in elementary functions). The results of mathematical modelling of the motion of a spacecraft under optimal control, which confirm the practical feasibility of the control algorithm developed, are given. Estimates have shown that the turn time of modern spacecraft with a constrained magnitude of the angular momentum can be reduced by 15–25% compared with conventional reorientation methods. The greatest effect is achieved for turns through large angles (90° or more) when the final rotation vector is equidistant from the longitudinal axis and the transverse plane of the spacecraft.     

Quaternion non-linear filter for estimation of rotating body attitude

A problem of optimal estimation of a rotating vehicle's attitude with both differential equations of motion and observations of stars by onboard equipment is considered. The problem is formulated like a determinate non-linear Kalman filter problem in quaternion terms. An exact analytical solution of the problem is also found in quaternion terms. Various types of solutions in accordance with a mutual arrangement of observed stars are presented. An example of only one observed star is demonstrated to support the method. The method is also suited for geodetic applications, e.g. geodetic positioning and navigation systems.     

A note on sturmian comparison theorems for linear partial differential equations of second order

We show that by a modification of Sturm's classical method it is possible to obtain results for special operators of mixed type as well as for nonhomogeneous equations     

Motion correction of a statistically uncertain system under communication constraints

The problem of motion correction of a controlled system is considered. It is required to minimize a terminal functional under statistical uncertainty of the disturbance and incomplete information about the state of the system. Reachable sets of filtering and prediction equations are used; these sets are uniquely determined for a given time moment by the realizations of the observed signal and the chosen control. A modification of the correction problem under communication constraints is considered, in which the bounded capacity of the digital data transmission channel is taken into account. It is assumed that the object is equipped with a computing facility that can remember the measured information, process it with a high level of accuracy, transmit it, and receive encoded signals via communication channels. Signals in the form of words of bounded length consisting of integers come to the control and information processing center (CIPC) at discrete time moments. For simplicity, the communication channel is assumed to be noise- and delay-free. The encoding device in the communication channel is used for transmitting information about the measured parameters of the object to the CIPC and the control action from the CIPC to the object. At the CIPC, the information about the parameters is decoded and used for calculating the correction moments and optimal control. Relations between the reconstruction accuracy of the measured parameters and the optimal value of the functional are obtained as well as between the length of the transmitted word and the transmission frequency. Several results are exemplified.     

On the Comparisons of Unit Dual Quaternion and Homogeneous Transformation Matrix

This paper reveals the differences and similarities between two popular unified representations, i.e. the UDQ (unit dual quaternion) and the HTM (homogeneous transformation matrix), for transformation in the solution to the kinematic problem, in order to provide a clear, concise and self-contained introduction into dual quaternions and to further present a cohesive view for the UDQ and HTM representations as used in robotics. Specifically, after investigating some fundamental algebraic properties of the UDQ, it is revealed that the kinematical equations represented by the UDQ and the HTM are accordant, and afterwards the direct relationship of UDQ-based error kinematical models in spatialframe and in body-frame are further discussed, with conclusion that either error kinematic model can be chosen for designing kinematical control laws. Finally, the comparative study on the proportional control algorithms based on the logarithmical mapping of the HTM and the UDQ shows that the UDQ-based control law is indeed higher in computational efficiency.     

自由漂浮空间非合作目标的运动预测

空间非合作目标的运动预测是航天器在轨服务中的一个重要问题.在获得非合作目标的运动预测结果后,追踪星即可规划运动轨迹以接近目标并对其进行捕获.该文提出了一种自由漂浮空间非合作目标的运动预测方法.该方法的核心思想是首先辨识出目标的姿态动力学参数和目标的质心运动学参数,然后利用参数辨识结果和目标的动力学方程实现对目标的运动预测.在姿态动力学参数的辨识过程中,首先对目标的惯性参数进行初步辨识,然后采用自适应无迹Kalman滤波器对姿态动力学参数进行粗略辨识,最后通过最优化方法进一步提高姿态动力学参数的辨识精度.该文通过数值仿真验证了所提运动预测方法的有效性.仿真结果表明,无论目标是做单轴旋转还是翻滚运动,所提运动预测方法都能够实现对目标的长时间高精度的运动预测.     

Coupled spring equations

Coupled spring equations for modelling the motion of two springs with weights attached, hung in series from the ceiling are described. For the linear model using Hooke's Law, the motion of each weight is described by a fourth-order linear differential equation. A nonlinear model is also described and damping and external forcing are considered. The model has many features that permit the meaningful introduction of many concepts including: accuracy of numerical algorithms, dependence on parameters and initial conditions, phase and synchronization, periodicity, beats, linear and nonlinear resonance, limit cycles, harmonic and subharmonic solutions. These solutions produce a wide variety of interesting motions and the model is suitable for study as a computer laboratory project in a beginning course on differential equations or as an individual or a small-group undergraduate research project.     

On effects of discretization and the selection of constitutive equations on stability of continua

In classical continuum mechanics the set of basic equations consists of the equation of motion, the kinematic equation and the constitutive equations. The study concentrates on constitutive modeling and the effects of discretization on stability problems. The method of investigation is analytic, the spectra of linear mapping operators of continuous and discrete dynamical systems are studied. As results cases are found, when a hidden incursive nature of a material model leads to unstable behavior of the continuum. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

ITERATIVE ALGORITHMS FOR DATA ASSIMILATION PROBLEMS

InroductlonThe Investigation ofglobal changes has Increased the Interest to