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In a recent article in this journal, Nikolić et al. presented a mathematical model for computing the coordinates in the trimetric projection of a 3D object, in terms of the projections of unit areas on the three dihedral planes. We revisit this model and analytically formalize its geometric principles, noting that such projections amount to the direction cosines of the unit normal to the viewplane. Thus, we reinterpret their proposal as providing three direction numbers that define a scaled version of this unit normal. The model also derives formulas relating the trimetric parameters (i.e., trimetric angles and foreshortening ratios). We observe that these relationships, found in classical literature, admit more compact expressions through simpler derivations. Also, we compile and reexamine various methods for selecting the trimetric projection, making them more accessible. In particular, the turn and tilt rotations of interactive user interfaces provide an intuitive way to choose the direction cosines. Ultimately, any method defines the mathematical model through a transformation matrix that maps 3D world coordinates to viewplane coordinates in the viewing pipeline.

The Sagdeev pseudopotential (SP) method is used to study ion acoustic solitary waves (IASWs) in a warm, magnetized plasma with relativistic electrons. Employing the pseudopotential approach allows for the investigation of solitary wave (SW) structures across arbitrary amplitudes. The study highlights the simultaneous occurrence of compressive N > 1 subsonic M < 1 solitons, as well as rarefactive N < 1 subsonic and supersonic M > 1 solitons, under specific parametric conditions. Notably, it is seen that as the direction cosine of wave propagation k z increases, both the amplitude of SWs and the depth of the potential well decrease. The reduction in amplitude indicates a closer alignment between the magnetic field lines and the direction of wave propagation. The coexistence of compressive subsonic, rarefactive subsonic, and supersonic solitons in this plasma model is a rich and complex phenomenon that has both fundamental and practical implications in plasma physics. It reflects the intricate interplay of nonlinear effects, particle dynamics, and wave propagation in plasmas, with potential applications in both laboratory and astrophysical contexts.

The extent and forms of seismic damage vary with the tunnel depth range, according to which seismic measures should be designed. A seismic buffer of expanded polystyrene geofoam that is widely applied in retaining walls may be wrapped around the tunnel lining to improve the seismic performance. In this study, seismic buffer thicknesses are recommended based on pairs of displacement patterns (DPs) and stress patterns (SPs) in the linings. They were determined using three-dimensional displacement direction cosines and a stress-space-transformed Mohr–Coulomb criterion. Using raw data from numerical models, four displacement forms (DFs) related to practical seismic damage in four lining domains are presented, with a group of DFs denoting a DP and a group of failure statuses denoting an SP. The results demonstrate that without a buffer, at most depths, the four domains are in a composite DF denoting a perilous DP I and are in failure SP I. As the buffer thickens, the linings are gradually mitigated to lower-risk DP and SP II as well as DP and SP III, depending on whether the buffer thickness is appropriate for a certain tunnel depth. Based on the DP and SP pairs, appropriate buffer thicknesses are recommended for each tunnel depth range.

Numerical analysis based on micromechanics is an effective method to study the mechanical properties of three-dimensional (3D) braided composites, in which the establishment of micromechanics model is the basis of mechanical analysis. The direction cosine of fiber yarns and the characterization of geometric parameters of microstructure are two key issues in the modeling of microstructure. Focusing on these two key issues, the three-cell model of 3D four-directional braided composites are studied to comprehensively analyze various spatial directions of fiber yarns in the 45° division and horizontal division of unit cell, so as to obtain the detailed fiber yarn direction cosine. Aiming at the cross sections of three types of fiber yarns, the complex relationship between the braiding process parameters and the geometric parameters of the unit cell is analyzed and deduced. The unit cell structure is quantitatively characterized by using the braiding process parameters as the initial parameters, and the calculation methods of the volume fraction of fiber yarns and yarn filling coefficient are obtained. Finally, the predicted elastic constants by stiffness-volume averaging method are compared with experimental results, demonstrating the analysis results of fiber yarns directions. This paper can provide a theoretical reference for the microstructure modeling of 3D multi-directional braided composites, and has certain practical value in engineering.

Teaching kinematic rotations is a daunting task for even some of the most advanced mathematical minds. However, changing the paradigm can highly simplify envisioning and explaining the three-dimensional rotations. This paradigm change allows a high school student with an understanding of geometry to develop the matrix and explain the rotations at a collegiate level. The proposed method includes the assumption of a point (P) within the initial three-dimensional frame with axes (x^i, y^i, z^i). The method then utilizes a two-dimensional rotation view (2DRV) to measure how the coordinates of point P translate after a rotation around the initial axis. The equations are used in matrix notation to develop a rotation matrix for follow-on direction cosine matrixes. The method removes the requirement to use Euler’s formula, ultimately, providing a high school student with an elementary and repeatable process to compose and explain kinematic rotations, which are critical to attitude direction control systems commonly found in vehicles.

In this paper we analyze the performance of different combinations of: (1) parameterization of the rotational degrees of freedom (DOF) of multibody systems, and (2) choice of the integration scheme, in the context of direct optimal control discretized according to the direct multiple-shooting method. The considered representations include quaternions and Direction Cosine Matrices, both having the peculiarity of being non-singular and requiring more than three parameters to describe an element of the Special Orthogonal group SO ( 3 ) . These representations yield invariants in the dynamics of the system, i.e., algebraic conditions which have to be satisfied in order for the model to be representative of physical reality. The investigated integration schemes include the classical explicit Runge–Kutta method, its stabilized version based on Baumgarte’s technique, which tends to reduce the drift from the underlying manifold, and a structure-preserving alternative, namely the Runge–Kutta Munthe-Kaas method, which preserves the invariants by construction. The performances of the combined choice of representation and integrator are assessed by solving thousands of planning tasks for a nonholonomic, underactuated cart-pendulum system, where the pendulum can experience arbitrarily large 3D rotations. The aspects analyzed include success rate, average number of iterations and CPU time to convergence, and quality of the solution. The results reveal how structure-preserving integrators are the only choice for lower accuracies, whereas higher-order, non-stabilized standard integrators seem to be the computationally most competitive solution when higher levels of accuracy are pursued. Overall, the quaternion-based representation is the most efficient in terms of both iterations and CPU time to convergence, albeit at the cost of lower success rates and increased probability of being trapped by higher local minima.

The prominence of Indoor Navigation Systems (INS) has been on an upward trajectory in recent years. While the Global Positioning System (GPS) commonly utilizes radio waves from artificial satellites for positioning information, its precision is compromised indoors due to potential radio wave obstruction by buildings. In contrast, pedometers, a critical component of INS, can provide invaluable insights into health, exercise, and user itineraries by detecting the number of steps and pinpointing optimal indoor positions. This paper proposes the development of a high-accuracy pedometer system. The proposed tracking system capitalizes on data harvested from accelerometers, sensors integrated into mobile devices, to furnish indoor tracking predicated on a straightforward pedometer approach. Online measurements and tests were conducted in residential settings, and the recorded tests were subsequently simulated offline via MATLAB. The performance of the system was evaluated in a real-world indoor residential scenario using an iPhone6 mobile device, with the discussion encompassing potential usability aspects of the approach. The devised system mitigated the drift of sensor readings by amalgamating the data from the gyroscope and accelerometer. The experimental results revealed a percentage error of 4.33% for the proposed method, translating to an error of 0.65 meters from an average walking distance of approximately 3 meters, out of 15 meters. Future research endeavors will concentrate on enhancing the accuracy of the approach by implementing data filtering and interference reduction techniques.

This paper mainly studies how to use the stereo vision system that combines the monocular vision with parallel path search to locate the target. When the unmanned aerial vehicle (UAV) searches in the mission area according to the parallel path, the SSD image detection algorithm based on deep learning is adopted to detect and identify the target in the area. The image coordinate information is inversely calculated by using the pixel coordinate information fed back by machine vision. The auxiliary coordinate system is established according to the relationship of angle position between the track line and the basic coordinate system in the parallel path. Combining the position relation and the attitude direction information of UAV, the target position conversion relation between the imaging coordinate system and the auxiliary coordinate system is solved by using the direction cosine matrix. Combined with the coordinate information of UAV, the coordinate position of the target point in the basic coordinate system is finally solved through three coordinate conversion operations. In order to avoid the single calculating error of the target coordinates, the weighted average operation is carried out. On the basis of not changing the search trip of the parallel path, the target location function is preliminarily realized through the reverse solution and the weighted average operation of the target coordinates.

Positioning and navigation data for unmanned surface vehicles (USVs) are extracted using the Global Positioning System (GPS) and the Inertial Navigation System (INS) integrated with an inertial measurement unit (IMU). The integration of quaternion with direction cosine matrix (DCM) with the aim of obtaining high accuracy with complete system independence has been effectively used to supply position and attitude information for autonomous navigation of marine crafts. A DCM integrated with a quaternion provided an advanced technique for precise USV attitude estimation and position determination using low-cost sensors. This paper presents the implementation of an INS developed by the integration of DCM and quaternion.

Modern kinematics derives directly from developments in the 1700s, and in their current instantiation, have been adopted as standard realizations…or templates that seem unquestionable. For example, so-called aerospace sequences of rotations are ubiquitously accepted as the norm for aerospace applications, owing from a recent heritage in the space age of the late twentieth century. With the waning of the space-age as a driver for technology development, the information age has risen with the advent of digital computers, and this begs for re-evaluation of assumptions made in the former era. The new context of the digital computer defines the use of the term “information age” in the manuscript title and further highlights the novelty and originality of the research. The effects of selecting different Direction Cosine Matrices (DCM)-to-Euler Angle rotations on accuracy, step size, and computational time in modern digital computers will be simulated and analyzed. The experimental setup will include all twelve DCM rotations and also includes critical analysis of necessary computational step size. The results show that the rotations are classified into symmetric and non-symmetric rotations and that no one DCM rotation outperforms the others in all metrics used, yielding the potential for trade space analysis to select the best DCM for a specific instance. Novel illustrations include the fact that one of the ubiquitous sequences (the “313 sequence”) has degraded relative accuracy measured by mean and standard deviations of errors, but may be calculated faster than the other ubiquitous sequence (the “321 sequence”), while a lesser known “231 sequence” has comparable accuracy and calculation-time. Evaluation of the 231 sequence also illustrates the originality of the research. These novelties are applied to spacecraft attitude control in this manuscript, but equally apply to robotics, aircraft, and surface and subsurface vehicles.