Extended Kalman filter based statistical orbit determination for geostationary and geosynchronous satellite orbits in BeiDou constellation

BeiDou Navigation Satellite System (BDS) is composed of satellites in geostationary Earth orbit (GEO), medium Earth orbit (MEO) and inclined geosynchronous orbit (IGSO). However, the orbit determination of geostationary Earth orbits and of geosynchronous orbits (GSO) with small inclination angle and small eccentricity is a challenging task that is addressed in this paper using Extended Kalman Filter (EKF). The satellite positions were predicted in Earth-centred inertial (ECI) reference frame when propagated through Keplerian model and perturbation force model for different values of right ascension of ascending node (RAAN). Root mean square (RMS) errors of 9.61 cm, 6.73 cm and 11.46 cm were observed in ECI X, Y and Z satellite position coordinates of GSO respectively, whereas, the RMS errors for GEO satellite were 8.89 cm, 7.92 cm, and 0.93 cm respectively in ECI X, Y and Z coordinates; for perturbation force model with maximum value of RAAN when compared with dynamic orbit determination model. Kolmogorov-Smirnov test for EKF reported a p-value > 0.05, indicating a good fit of perturbation force model for orbit propagation. Orbit determination using EKF with perturbation force model were compared with that using EKF with Kepler's model. Wilcoxon Rank Sum test was used to compare the residuals from EKF algorithm through Kepler's model and perturbation force model. EKF with Perturbation force model showed improvement in predicting the satellite positions as compared to Kepler's model. EKF with Perturbation force model was further applied to International GNSS Service (IGS) station data and kilometre level accuracy was achieved. RMS errors of 0.75 km, 2.53 km and 1.91 km were observed in ECI X, Y and Z satellite position coordinates of GSO, respectively, whereas, the RMS errors for GEO satellite were 3.89 km, 4.20 km and 6.66 km respectively in ECI X, Y and Z coordinates for perturbation force model.

The history force on a small particle in a linearly stratified fluid

The hydrodynamic force experienced by a small spherical particle undergoing an arbitrary time-dependent motion in a weakly density-stratified fluid is investigated theoretically. The study is carried out under the Oberbeck–Boussinesq approximation and in the limit of small Reynolds and small Péclet numbers. The force acting on the particle is obtained by using matched-asymptotic expansions. In this approach, the small parameter is given by $a/\ell $, where $a$ is the particle radius and $\ell $ is the stratification length, as defined by Ardekani & Stocker (Phys. Rev. Lett., vol. 105, 2010, article 084502), which depends on the Brunt–Väisälä frequency, on the fluid kinematic viscosity and on the thermal or the concentration diffusivity (depending on the case considered). The matching procedure used here, which is based on series expansions of generalized functions, slightly differs from that generally used in similar problems. In addition to the classical Stokes drag, it is found that the particle experiences a memory force given by two convolution products, one of which involves, as usual, the particle acceleration and the other one, the particle velocity. Owing to the stratification, the transient behaviour of this memory force, in response to an abrupt motion, consists of an initial fast decrease followed by a damped oscillation with an angular frequency corresponding to the Brunt–Väisälä frequency. The perturbation force eventually tends to a constant which provides us with correction terms that should be added to the Stokes drag to accurately predict the settling time of a particle in a diffusive stratified fluid.

Human standing: does the control strategy preprogram a rigid knee?

Human standing requires control of multisegmental configuration. Does the postural system normally allow flexible adjustment of configuration, or does it minimize degrees of freedom at the ankle, knee, and hip joints? Gentle, external, unpredictable, sagittal, mechanical perturbations (randomized force, 1–10 N; duration, 0.2–2 s; and leg) were applied to either knee of 24 healthy participants who stood symmetrically for 200 s. The translation of knee perturbation force to ankle, knee, and hip joint rotations in the perturbed and unperturbed legs was studied. We assessed whether consequent joint rotations indicated a stiff configuration-conserving or viscous energy-absorbing relationship to the knee perturbation. Two distinctive response patterns were observed. Twenty-two participants showed limited knee flexion and high ankle stiffness, whereas two participants showed substantial knee flexion, low ankle stiffness, measurable internal rotation of the unperturbed hip (0.4 ± 0.3 vs. 3.0 ± 1°, 5.7 ± 17 vs. 0.5 ± 0.3 N/°, 1.1 ± 0.4°, respectively; mean ± SD), and a viscous relationship between perturbation force and subsequent ankle flexion, knee flexion, and perturbed and unperturbed hip internal rotation. The size of knee-flexion response to knee perturbations was uncorrelated with the extent of unperturbed standing sway. Normal standing conceals a large interindividual range in leg control strategies, indicating adaptive potential to progress with development and skill acquisition and decline with age, disease, injury, and fear. Commonly, leg configuration was maintained stiffly. Less commonly, a bilateral, low-stiffness, energy-absorbing strategy utilizing the available degrees of freedom was shown. We propose that identification of individual coordination strategy has diagnostic and prognostic potential in relation to perceptual-posture-movement-fall interactions.

Interfacial instability due to MHD mode coupling in aluminium reduction cells

This paper analyses instabilities on the cryolite/aluminium interface in an aluminium reduction cell. The simplified cell model is a finite rectangular tank containing the two fluid layers, and carrying a uniform normal current. The magnetic field is assumed to be a linear function of position. Several previous studies have considered waves consisting of a single Fourier component but here we consider perturbations which are a general combination of the normal gravity-wave modes. We derive a system of coupled ordinary differential equations for the time-development of the mode amplitudes, and show that instability can occur via mode interactions, the electromagnetic perturbation force due to one mode feeding energy into the other. Growth rates are determined by computing the eigenvalues of an interaction matrix, and an approximate method using only the three leading diagonals is developed. If two modes have similar frequencies they may resonate and become unstable at a very low threshold current. We consider the influence of various cell parameters and draw some general conclusions about cell design.

An asymptotic theory of incompressible turbulent boundarylayer flow over a small hump

A rational asymptotic theory describing the perturbed flow in a turbulent boundary layer encountering a small two-dimensional hump is presented. The theory is valid in the limit of very high Reynolds number in the case of an aerodynamically smooth surface, or in the limit of small drag coefficient in the case of a rough surface. The method of matched asymptotic expansions is used to obtain a multiple-structured flow, along the general lines of earlier laminar studies. The leading-order velocity perturbations are shown to be precisely the inviscid, irrotational, potential flow solutions over most of the domain. The Reynolds stresses are found to vary across a thin layer adjacent to the surface, and display a singular behaviour near the surface which needs to be resolved by an even thinner wall layer. The Reynolds stress perturbations are calculated by means of a second-order closure model, which is shown to be the minimum level of sophistication capable of describing these variations. The perturbation force on the hump is also calculated, and its order of magnitude is shown to depend on the level of turbulence closure; a cruder turbulence model gives rise to spuriously large forces.