Covariance matrix transformation method in all-earth integrated navigation considering coordinate frame conversion

In the exploration of polar region, navigation is one of the most important issues to be resolved. To avoid the limitations of single navigation coordinate frame, the navigation systems usually use different navigation coordinate frames in polar and nonpolar region, such as the north-oriented geographic frame and the grid frame. However, the error states and covariance matrix are related with the definition of navigation coordinate frame, since the coordinate frame conversion will cause the integrated navigation Kalman filter overshoot and error discontinuity. To solve this problem, the transformation relationship of error states defined in different frames is deduced, whereby the covariance matrix transformation relationship is also analyzed. On this basis, covariance transformation-based the open-loop and the closed-loop Kalman filter integrated navigation algorithms are proposed. The effectiveness of algorithms is verified by flight tests with rotational strapdown inertial navigation system (RSINS)/global navigation satellite system (GNSS) integrated navigation system.

Cartesian Meshing Spherical Earth (CMSE): A Code Package to Incorporate the Spherical Earth in SPECFEM3D Cartesian Simulations

The SPECFEM3D_Cartesian code package is widely used in simulating seismic wave propagation on local and regional scales due to its computational efficiency compared with the one-chunk version of the SPECFEM3D_Globe code. In SPECFEM3D_Cartesian, the built-in meshing tool maps a spherically curved cube to a rectangular cube using the Universal Transverse Mercator projection (UTM). Meanwhile, the geodetic east, north, and up directions are assigned as the local x–y–z directions. This causes coordinate orientation issues in simulating waveform propagation in regions larger than 6° × 6° or near the Earth’s polar regions. In this study, we introduce a new code package, named Cartesian Meshing Spherical Earth (CMSE), that can accurately mesh the 3D geometry of the Earth’s surface under the Cartesian coordinate frame, while retaining the geodetic directions. To benchmark our new package, we calculate the residual amplitude of the CMSE synthetics with respect to the reference synthetics calculated by SPECFEM3D_Globe. In the regional scale simulations with an area of 1300 km × 1300 km, we find a maximum of 5% amplitude residual for the SPECFEM3D_Cartesian synthetics using the mesh generated by the CMSE, much smaller than the maximum amplitude residual of 100% for the synthetics based on its built-in meshing tool. Therefore, our new meshing tool CMSE overcomes the limitations of the internal mesher used by SPECFEM3D_Cartesian and can be used for more accurate waveform simulations in larger regions beyond one UTM zone. Furthermore, CMSE can deal with regions at the south and north poles that cannot be handled by the UTM projection. Although other external code packages can be used to mesh the curvature of the Earth, the advantage of the CMSE code is that it is open-source, easy to use, and fully integrated with SPECFEM3D_Cartesian.

Quaternion methods and models of regular celestial mechanics and astrodynamics

This paper is a review, which focuses on our work, while including an analysis of many works of other researchers in the field of quaternionic regularization. The regular quaternion models of celestial mechanics and astrodynamics in the Kustaanheimo-Stiefel (KS) variables and Euler (Rodrigues-Hamilton) parameters are analyzed. These models are derived by the quaternion methods of mechanics and are based on the differential equations of the perturbed spatial two-body problem and the perturbed spatial central motion of a point particle. This paper also covers some applications of these models. Stiefel and Scheifele are known to have doubted that quaternions and quaternion matrices can be used efficiently to regularize the equations of celestial mechanics. However, the author of this paper and other researchers refuted this point of view and showed that the quaternion approach actually leads to efficient solutions for regularizing the equations of celestial mechanics and astrodynamics.This paper presents convenient geometric and kinematic interpretations of the KS transformation and the KS bilinear relation proposed by the present author. More general (compared with the KS equations) quaternion regular equations of the perturbed spatial two-body problem in the KS variables are presented. These equations are derived with the assumption that the KS bilinear relation was not satisfied. The main stages of the quaternion theory of regularizing the vector differential equation of the perturbed central motion of a point particle are presented, together with regular equations in the KS variables and Euler parameters, derived by the aforementioned theory. We also present the derivation of regular quaternion equations of the perturbed spatial two-body problem in the Levi-Civita variables and the Euler parameters, developed by the ideal rectangular Hansen coordinates and the orientation quaternion of the ideal coordinate frame.This paper also gives new results using quaternionic methods in the perturbed spatial restricted three-body problem.

Regular Frames for Spherically Symmetric Black Holes Revisited

We consider a space-time of a spherically symmetric black hole with one simple horizon. As a standard coordinate frame fails in its vicinity, this requires continuation across the horizon and constructing frames which are regular there. Up to now, several standard frames of such a kind are known. It was shown in the literature before, how some of them can be united in one picture as different limits of a general scheme. However, some types of frames (the Kruskal–Szekeres and Lemaître ones) and transformations to them from the original one remained completely disjoint. We show that the Kruskal–Szekeres and Lemaître frames stem from the same root. Overall, our approach in some sense completes the procedure and gives the most general scheme. We relate the parameter of transformation e0 to the specific energy of fiducial observers and show that in the limit e0→0, a homogeneous metric under the horizon can be obtained by a smooth limiting transition.

Relative Pose Determination of Uncooperative Spacecraft Based on Circle Feature

This paper investigates the problem of spacecraft relative navigation with respect to an unknown target during the close-proximity operations in the on-orbit service system. The serving spacecraft is equipped with a Time-of-Flight (ToF) camera for object recognition and feature detection. A fast and robust relative navigation strategy for acquisition is presented without any extra information about the target by using the natural circle features. The architecture of the proposed relative navigation strategy consists of three ingredients. First, a point cloud segmentation method based on the auxiliary gray image is developed for fast extraction of the circle feature point cloud of the target. Secondly, a new parameter fitting method of circle features is proposed including circle feature calculation by two different geometric models and results’ fusion. Finally, a specific definition of the coordinate frame system is introduced to solve the relative pose with respect to the uncooperative target. In order to validate the efficiency of the segmentation, an experimental test is conducted based on real-time image data acquired by the ToF camera. The total time consumption is saved by 94%. In addition, numerical simulations are carried out to evaluate the proposed navigation algorithm. It shows good robustness under the different levels of noises.

Analyses of wave-phase variation of Reynolds shear stress underneath surface wave using streamline coordinates

The dependence of Reynolds shear stress on wave phase is investigated for initially isotropic turbulence distorted by a progressive surface wave through direct numerical simulations. A wave-following streamline coordinate frame is used to analyse the turbulence dynamics such that the information of the varying direction of wave orbital motions is embedded into the coordinate system, which helps capture the effect of flow curvature on the turbulence and quantify momentum exchange between the near-surface and deep regions in the wavy domain accurately. It is found that the Reynolds shear stress is enhanced under the backward slope of the wave and can be scaled by the wave steepness and the streamwise velocity fluctuations. Analyses of the budget of Reynolds shear stress indicate that such a variation with the wave phase is caused by the variation in the production of the Reynolds shear stress and the effect of pressure fluctuations. Further investigation shows that the production of the Reynolds shear stress is closely associated with the wave surface curvature. A model that includes a correction term for the curvature effect for the pressure–strain correlation term is examined and is found to agree reasonably well with the simulation result. The correction term is found to make an appreciable contribution to the model, further supporting our finding that the wave curvature plays an important role in the turbulence dynamics near the surface.

The determination of point groups from imprecise molecular geometries

We present a new approach for the assignment of a point group to a molecule when the structure conforms only approximately to the symmetry. It proceeds by choosing a coordinate frame that minimises a measure of symmetry breaking that is computed efficiently as a simple function of the molecular coordinates and point group specification.

A Deep Learning Approach for Foot Trajectory Estimation in Gait Analysis Using Inertial Sensors

Gait performance is an important marker of motor and cognitive decline in older adults. An instrumented gait analysis resorting to inertial sensors allows the complete evaluation of spatiotemporal gait parameters, offering an alternative to laboratory-based assessments. To estimate gait parameters, foot trajectories are typically obtained by integrating acceleration two times. However, to deal with cumulative integration errors, additional error handling strategies are required. In this study, we propose an alternative approach based on a deep recurrent neural network to estimate heel and toe trajectories. We propose a coordinate frame transformation for stride trajectories that eliminates the dependency from previous strides and external inputs. Predicted trajectories are used to estimate an extensive set of spatiotemporal gait parameters. We evaluate the results in a dataset comprising foot-worn inertial sensor data acquired from a group of young adults, using an optical motion capture system as a reference. Heel and toe trajectories are predicted with low errors, in line with reference trajectories. A good agreement is also achieved between the reference and estimated gait parameters, in particular when turning strides are excluded from the analysis. The performance of the method is shown to be robust to imperfect sensor-foot alignment conditions.

An improved minimal error model for the robotic kinematic calibration based on the POE formula

Summary The conventional product of exponentials $\left(\rm POE\right)$ -based methods dissatisfy the parametric minimality for the kinematic calibration of serial robots due to overlooking the magnitude and pitch constraints. Thus, the minimal kinematic model is presented to solve this problem, which can be developed further. This paper puts forward an improved algorithm for the minimal parameter calibration. An actual kinematic model with the minimal parameters $\left(\rm MP\right)$ is constructed according to the geometric properties of actual joint twists in the auxiliary frames established on the basis of the nominal joint axes. Then, the initial pose error is defined in the tool coordinate frame, which is expressed as the exponential map of the twist, and all twist descriptions are unified, so as to give a unified kinematic model in mathematics. By differentiating the kinematic model, a minimal error model is derived in explicit form. Subsequently, we propose a novel parameter identification method, which identifies the orientation error and position error parameters separately by the iterative least-squares method and updates the MP uniformly. Finally, the simulations and experiments on the different serial robots are conducted to verify the correctness and effectiveness of the proposed algorithm. The simulation results show our calibration algorithm outperforms the existing ones in the accuracy aspect, and the experiment result shows that the absolute pose accuracy of the UR5 industrial robot is upgraded about 9 times under a statistics sense after the calibration.