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AbstractThis work presents a procedure that allows us to formulate logically and orderly all the equations of position, velocity, and acceleration that must be satisfied by any rigid body moving arbitrarily in three-dimensional physical space. Using Gaussian elimination, the complex system of linear equations can be changed to a more straightforward equivalent system of linearly independent equations, eliminating redundant equations and making the solution directly obtainable without much effort. This formulation allows us to systematically obtain the kinematic parameters of position, velocity, and acceleration associated with a rigid body in arbitrary motion, including the angular velocity vector and the angular acceleration vector. The particularity of the development allows us to find a unique condition that makes the position, velocity, and acceleration equations compatible. This condition gives rise to a novel and original symbolic equation representing a cylindrical surface whose points represent the locus of the angular velocity vector of the rigid body in motion. Two fully detailed case studies illustrate the applicability of the discovery. This research may help estimate a rigid body’s attitude and use accelerometers to control machines, mechanisms, and robots moving in space. [ABSTRACT FROM AUTHOR]

AbstractWe present a comprehensive methodology for calculating angular kinematic quantities up to the fourth order using automatic differentiation with dual numbers. Two approaches are employed: the vector method, which uses position vectors and derivatives of three non-collinear points, and the body-fixed frame (BFF) method, which differentiates an attached basis to the rigid body. The BFF method avoids indeterminacy and divergence issues. Numerical examples, drawn from robotics and classical mechanics, validate the methodology and demonstrate its robustness. Our approach effectively addresses challenging inverse problems of acceleration, jerk, and jounce/snap. [ABSTRACT FROM AUTHOR]

Purpose: In this study, we investigate the minimum time (MT) for the spatial motion of a free rigid body (RB) under the influence of viscous friction and gyrostatic moments (GMs). Assuming that the body’s center of mass coinciding with the original point of two Cartesian systems of coordinates. Results: The optimal control (OC) law for the slow movements, as a main goal, is established, and the corresponding time and phase pathways are assessed. Conclusion: The resulting innovative outcomes are achieved and displayed for two new scenarios in various graphs to show the gyrostatic moment’s positive impacts. When comparing the obtained results with the previous one when the gyrostatic moment isn’t present, amazing consistency is observed, and the slight deviations between them are discussed. As a result, the acquired results generalize those that were discovered in earlier investigations. Applications: The significance of this study arises from its practical implications in everyday existence, that confine themself to applying gyroscopic theory to preserve the stability and balance of vehicles in the design of which gyroscopes are used, in addition to determining the path of aircraft and marine vehicles. [ABSTRACT FROM AUTHOR]

The method of reverberation-ray matrix (MRRM) is modified for the dynamic response analysis of beam structures with rigid components. The wave expressions for the beam structures are derived based on the Timoshenko beam theory, and the MRRM formulations are obtained through the compatibility conditions of displacements and joint coupling equations by considering the effects of rigid components. An application of this method is considered in the context of a phononic crystal with spring-mass-like resonators. The proposed resonant unit consists of a large mass-like central component connected to thin beams that serve as springs. The large central component can be approximated as a rigid body. Using the MRRM and the finite element method (FEM), the band diagrams and the transmission spectrums are calculated. The innovative cell offers the advantage of producing lower and wider bandgaps when compared to traditional lattice structures. The accuracy of the MRRM is verified with the FEM. The current research provides a vibration analysis approach for beam structures with rigid components and the analysis of phononic crystals serves as a reference for structural design. [ABSTRACT FROM AUTHOR]

ABSTRACT In this paper, we study the leader‐following attitude consensus problem for multiple uncertain rigid body systems by a sampled‐data adaptive distributed observer. Unlike the existing sampled‐data distributed observer, which can only asymptotically estimate the state of the leader, the sampled‐data adaptive distributed observer can estimate both the state and the system matrix of the leader exponentially. We synthesize a distributed control law utilizing sampled‐data communications to solve the leader‐following attitude consensus problem for multiple uncertain rigid body systems based on the sampled‐data adaptive distributed observer. Moreover, we present a sufficient condition for guaranteeing the convergence of the estimated unknown parameters to their true values. Compared with a distributed control law that uses continuous‐time communications, the distributed control law utilizing sampled‐data communications consumes fewer communication resources and is more robust to communication failures. The effectiveness of our approach is verified by a numerical example. [ABSTRACT FROM AUTHOR]

This study focuses on determining the required minimum-time (MT) for the spatial motion of a free rigid body (RB) experiencing gyrostatic moments (GMs) and viscous friction. The study assumes that the body's center of mass coincides with the original point of two Cartesian systems of coordinates. An optimal control law for slow motion is established, and the corresponding time and phase pathways are analyzed. The innovative results are presented for two new cases through various graphs highlighting the positive effects of the GMs. A comparison is achieved between the obtained results and previous outcomes that did not consider gyrostatic moments, showing remarkable consistency with slight deviations that are discussed. The practical applications of this study, which limits itself to using gyroscopic theory to maintain the stability and balance of vehicles in which gyroscopes are used, as well as figuring out the trajectory of aircraft and marine vehicles, are what make it noteworthy. [ABSTRACT FROM AUTHOR]

With the rapid development of remote sensing imaging and deep learning technology, the fine-grained recognition of rigid targets has gradually emerged. Rigid targets in remote sensing scenes usually retain relatively stable scale information and apparent structure, providing an adequate basis for their discrimination. However, existing methods need to more reasonably utilize the scale information and apparent structure, which results in scale neglect and insufficient discriminative feature extraction (DFE). In response to the aforementioned challenges, we propose our SD-Net, a training framework for fine-grained recognition tasks of rigid objects in remote sensing scenes. It consists of a fused label learning process based on probability distribution function (PDF) and a DFE branch. The PDF counts the objects' scale information by category, builds a probability model based on the sample distribution, and finally converts it into a soft label form to guide model learning. DFE extracts discriminative features along the channel and spatial dimensions of the feature map based on feature deep mining and wide-ranging. Finally, we propose the FAIR1M-OR dataset, containing 37 fine-grained categories and about 600 000 instances, to verify the method's effectiveness. The experimental results show that introducing only a small number of parameters during training, SD-Net, improves the performance of the models based on the ResNet and ViT by about 4.6 points. The code and dataset will be open source in the future.

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As commonly known, standard time integration of the kinematic equations of rigid bodies modeled with three rotation parameters is infeasible due to singular points. Common workarounds are reparameterization strategies or Euler parameters. Both approaches typically vary in accuracy depending on the choice of rotation parameters. To efficiently compute different kinds of multibody systems, one aims at simulation results and performance that are independent of the type of rotation parameters. As a clear advantage, Lie group integration methods are rotation parameter independent. However, few studies have addressed whether Lie group integration methods are more accurate and efficient compared to conventional formulations based on Euler parameters or Euler angles. In this paper, we close this gap using the R 3 × S O (3) Lie group formulation and several typical rigid multibody systems. It is shown that explicit Lie group integration methods outperform the conventional formulations in terms of accuracy. However, it turns out that the conventional Euler parameter-based formulation is the most accurate one in the case of implicit integration, while the Lie group integration method is computationally the more efficient one. It also turns out that Lie group integration methods can be implemented at almost no extra cost in an existing multibody simulation code if the Lie group method used to describe the configuration of a body is chosen accordingly. [ABSTRACT FROM AUTHOR]

Landslides and landslide-generated tsunami waves arbitrarily happen and cause tremendous hazards to human lives worldwide. Among the recent landslide tsunami events, the rigid submarine landslide sources are becoming more and more concerned due to their mysteries. Dynamical properties of rigid submarine landslides (e.g., velocity) play an important role in the generated tsunami waves. In this study, an empirical velocity profile of rigid submarine landslide-generated waves on an inclined plane is proposed. This new velocity profile is suggested from the velocity profile in a physical experiment of the horizontal rigid submarine landslidegenerated waves. A new FLOW-3D HYDRO package is employed to conduct the rigid submarine landslidegenerated waves experiment. We first verified the chosen numerical model to the experimental data of rigid submarine landslides moving horizontally. Following the rigid velocity profile on the horizontal plane, we propose a new method to apply it on the inclined plane. The error norm technique is used to find out the empirical velocity profile among 10 critical velocity profiles applied. The novel empirical velocity profile of the critical case of m = 3 as well as critical velocity value of Vc = 0.82 m/s is used to generate tsunami waves, which are in good agreement with the experimental data and better than those of other similar three-dimensional numerical models. [ABSTRACT FROM AUTHOR]

Pose estimation of rigid objects is a practical challenge in optical metrology and computer vision. This paper presents a novel stochastic-geometrical modeling framework for object pose estimation based on observing multiple feature points. This framework utilizes mixture models for feature point densities in object space and for interpreting real measurements. Advantages are the avoidance to resolve individual feature correspondences and to incorporate correct stochastic dependencies in multi-view applications. First, the general modeling framework is presented, second, a general algorithm for pose estimation is derived, and third, two example models (camera and lateration setup) are presented. Numerical experiments show the effectiveness of this modeling and general algorithm by presenting four simulation scenarios for three observation systems, including the dependence on measurement resolution, object deformations and measurement noise. Probabilistic modeling utilizing mixture models shows the potential for accurate and robust pose estimations while avoiding the correspondence problem. [ABSTRACT FROM AUTHOR]

This paper presents a study of wave propagation through an infinite periodic structure that consists of elastic Timoshenko beams interconnected with rigid bodies. This is a generalized approach in which the beams are not coaxial and the centre of mass of each rigid body is placed away from the intersection of their neutral axes. An analytical approach is used by applying the transfer matrix method (TMM), along with the Floquet–Bloch theorem for elastic wave propagation. Subsequent parametric analysis is performed with visualization of resulting band diagrams of a representative structure. These results are verified through comparison with solutions obtained using the finite-element method (FEM). In this manner, a comprehensive dynamical analysis of tailored metastructures is provided. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 2)'. [ABSTRACT FROM AUTHOR]

The problem of a planar rigid rocking block subject to dynamic loading is of significant interest to the earthquake engineering community due to its wide application. Despite its simple configuration, rocking itself is a highly nonlinear phenomenon which exhibits non-smooth dynamic behavior. The non-smoothness in dynamics comes from the sudden impact of the rocking body with ground when the sign of rotation angle reverses. Therefore the impact model used in the simulation for rocking problem has a profound influence on the dynamic behavior. Most of existing impact models assume that the impulse during impact is concentrated at the pivot point of the block, ignoring the potential distribution of impulses. This study proposes a distributed impact model for dynamic analysis of planar rigid rocking body. The distributed impulses are achieved through the integration of normal forces induced by penetration over a dummy time period during impact. The distributed impact model is subsequently used to evaluate the post-impact quantities of interest. Numerical simulations suggest Housner's impact model is a special case of the proposed model. The developed impact model is also compared with published experimental result, showing good accuracy. Finally, evaluation of overturning spectra under analytical pulse excitations suggests that using the distributed impact model can give substantially different results compared to the classic concentrated impact model for relatively stocky blocks. Similar observations can also be made when evaluating rocking responses under seismic excitations. [ABSTRACT FROM AUTHOR]

Teaching damped simple harmonic motion (shm) is a pedagogical challenge that requires additional strategies for the teaching and learning process of the associated physical concepts and mathematical structures, thus representing a greater effort in a non-face-to-face mode. A pedagogical guide for remote work was implemented, based on the homemade torsion pendulum model, supported by computational, digital, and ict tools for the study of damped shm. The experiment was conducted with three cohorts of second-year Physics students. It was demonstrated that the torsion pendulum is a valid model for learning concepts associated with damped harmonic motion and for developing competencies in experimental work. [ABSTRACT FROM AUTHOR]

In this contribution, we apply space-time formulation on constrained rigid body dynamics. In particular, we discretize directly Hamilton's principle using appropriate space-time approximation spaces for the variational problem. Moreover, we make use of a rotationless formulation for the rigid bodies, and thus we have to define appropriate approximation spaces for the Lagrange multipliers as well. Moreover, we make use of Livens' principle, introducing independent quantities for the position, velocity, and momentum, where the latter can be considered as Lagrange multipliers, and we apply this concept to the space-time rigid body formulation. Finally, we demonstrate the convergence of the different approaches and the superiority in terms of computational effort, and thus total energy consumption of dynamical simulations. [ABSTRACT FROM AUTHOR]

We solve the boundary output-feedback tracking problem for a moving rigid body-liquid system described by the linearized Saint-Venant PDEs. The control input is an external force acting on the rigid body. The control law is a boundary state-feedback type augmented with a Luenberger boundary observer, leading to a dynamic feedback law. The main complexity lies in the fact that we can only measure the position and velocity of the rigid body. To get around this limitation, the observer estimates the liquid levels at the cavity walls. We utilize a control Lyapunov functional methodology (with two different functionals for the observer and the dynamic feedback law) and achieve a slosh-free asymptotic convergence to the prespecified equilibrium point (in the sense of an appropriate norm). Finally, through a simulation, we demonstrate the feasibility of the boundary output-feedback controller for the (linear and nonlinear) system. [ABSTRACT FROM AUTHOR]

In this paper, rigid body static optimization is investigated on the Riemannian manifold of rigid body motion groups. This manifold, which is also a matrix manifold, provides a framework to formulate translational and rotational motions of the body, while considering any coupling between those motions, and uses members of the special orthogonal group SO (3) to represent the rotation. Hence, it is called the special Euclidean group SE (3) . Formalism of rigid body motion on SE (3) does not fall victim to singularity or non-uniqueness issues associated with attitude parameterization sets. Benefiting from Riemannian matrix manifolds and their metrics, a generic framework for unconstrained static optimization and a customizable framework for constrained static optimization are proposed that build a foundation for dynamic optimization of rigid body motions on SE (3) and its tangent bundle. The study of Riemannian manifolds from the perspective of rigid body motion introduced here provides an accurate tool for optimization of rigid body motions, avoiding any biases that could otherwise occur in rotational motion representation if attitude parameterization sets were used. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of a small rigid body in a viscous incompressible fluid in dimension two and three. More precisely we investigate the trajectory of the rigid body in the limit when its mass and its size tend to zero. We show that the velocity of the center of mass of the rigid body coincides with the background fluid velocity in the limit. We are able to consider the limit when the volume of the rigid bodies converges to zero while their densities are a fixed constant. [ABSTRACT FROM AUTHOR]

Abstract This paper aims to explore the rotatory spatial motion of an asymmetric rigid body (RB) under constant body-fixed torques and a nonzero first component gyrostatic moment vector (GM). Euler's equations of motion are used to derive a set of dimensionless equations of motion, which are then proposed for the stability analysis of equilibrium points. Specifically, this study develops 3D phase space trajectories for three distinct scenarios; two of them are applied constant torques that are directed on the minor and major axes, while the third one is the action of applied constant torque on the body’s middle axis. Novel analytical and simulation results for both scenarios of constant torque applied along the minor and middle axes are provided in the context of separatrix surfaces, equilibrium manifolds, periodic or non-periodic solutions, and periodic solutions’ extreme. Concerning the scenario of a directed torque on the major axis, a numerical solution for the problem is presented in addition to a simulation of the graphed results for the angular velocities' trajectories in various regions. Moreover, the influence of GM is examined for each case and a full modeling for the body's stability has been present. The exceptional impact of these results is evident in the development and assessment of systems involving asymmetric RBs, such as satellites and spacecraft. It may serve as a motivating factor to explore different angles within the GM in similar cases, thereby influencing various industries, including engineering and astrophysics applications.

Many widely different problems have a common mathematical structure wherein limited knowledge leads to ambiguity that can be captured conveniently using a concept of invisibility that requires the introduction of negative values for quantities that are inherently positive. Here I analyze three examples taken from perception theory, rigid body mechanics, and quantum measurement. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of attitude trajectory tracking of rigid bodies subject to external disturbances. The attitude control problem has been studied in the last decades and novel solutions have been proposed. Nevertheless, most of the proposed controllers only achieve local or almost global asymptotic stability without considering external disturbances. Contrary to other works, we focus on the problem of achieving almost global exponential stability of the attitude tracking errors in the presence of external disturbances using continuous control laws. We propose an attitude trajectory tracking controller based on the exponential coordinates of rotation in combination with a disturbance observer that estimates the exogenous signals. To solve this problem, we adopt a hierarchical approach, where the proposed control law is divided into a kinematic controller (outer control loop) and a velocity tracking controller (inner control loop). To design the disturbance observer, it is assumed that exogenous signals can be generated by a linear exosystem, i.e. the magnitude and phase of the disturbance are unknown. The almost global exponential stability of the closed-loop dynamics' equilibrium point was proved by a strict Lyapunov function. The performance of the proposed approach is assessed by numerical simulations and experimental tests on a low-cost quadrotor. [ABSTRACT FROM AUTHOR]

This paper aims to explore the rotatory spatial motion of an asymmetric rigid body (RB) under constant body-fixed torques and a nonzero first component gyrostatic moment vector (GM). Euler's equations of motion are used to derive a set of dimensionless equations of motion, which are then proposed for the stability analysis of equilibrium points. Specifically, this study develops 3D phase space trajectories for three distinct scenarios; two of them are applied constant torques that are directed on the minor and major axes, while the third one is the action of applied constant torque on the body's middle axis. Novel analytical and simulation results for both scenarios of constant torque applied along the minor and middle axes are provided in the context of separatrix surfaces, equilibrium manifolds, periodic or non-periodic solutions, and periodic solutions' extreme. Concerning the scenario of a directed torque on the major axis, a numerical solution for the problem is presented in addition to a simulation of the graphed results for the angular velocities' trajectories in various regions. Moreover, the influence of GM is examined for each case and a full modeling for the body's stability has been present. The exceptional impact of these results is evident in the development and assessment of systems involving asymmetric RBs, such as satellites and spacecraft. It may serve as a motivating factor to explore different angles within the GM in similar cases, thereby influencing various industries, including engineering and astrophysics applications. [ABSTRACT FROM AUTHOR]

This work focuses on the dynamical rotatory motion of an asymmetric rigid body (RB) subjected to a constant body-fixed torque and a vector of a gyrostatic moment (GM). The motion is considered in the absence of the first two components of the GM. The novelty of the current work is the conversion of the stability analysis of this body from a two dimensional phase plane to three ones phase space. Specifically, two scenarios of stationary torques are considered, the first one is directed on the minor or major axis, while the latter is presumed to act on the middle axis. Therefore, three dimensional phase space routes are generated. In both scenarios some novel analytical and simulation outcomes are provided regarding equilibrium manifolds, periodic solutions or non-periodic ones, separator surfaces, as well as the extreme periodic solutions. The significance of this work is due to its great applications, especially those that use the gyro's theory. [ABSTRACT FROM AUTHOR]

The problem of tracking control is addressed for rigid bodies with interior shallow-water sloshing. The liquid motion is modeled by the Saint-Venant equations, coupled with the ODE of the rigid body, leading to a global system with an ODE-hyperbolic PDE cascade structure. The paper aims to design an innovative boundary feedback framework for a pre-specified position to deal with rigid body tracking errors. Using only one control force applied to the rigid body, the formulated strategy efficiently stabilizes both the finite- and infinite-dimensional states. The main complexity lies in the fact that no sensor can be implemented in the liquid domain. Indeed, the proposed stabilizing feedback law simply requires measurements of (i) the rigid body position error and velocity and (ii) the liquid pressure at the cavity walls (liquid boundary). The asymptotic stability of the closed-loop system is analyzed using the Lyapunov direct method and LaSalle’s invariance principle without any discretization, reduction, and linearization. Additional controller features are highlighted by simulation results, including its benefits in contrast to the corresponding PD controller and its robustness to time delay and system uncertainty.

The problem of tracking control is addressed for rigid bodies with interior shallow-water sloshing. The liquid motion is modeled by the Saint-Venant equations, coupled with the ODE of the rigid body, leading to a global system with an ODE-hyperbolic PDE cascade structure. The paper aims to design an innovative boundary feedback framework for a pre-specified position to deal with rigid body tracking errors. Using only one control force applied to the rigid body, the formulated strategy efficiently stabilizes both the finite- and infinite-dimensional states. The main complexity lies in the fact that no sensor can be implemented in the liquid domain. Indeed, the proposed stabilizing feedback law simply requires measurements of (i) the rigid body position error and velocity and (ii) the liquid pressure at the cavity walls (liquid boundary). The asymptotic stability of the closed-loop system is analyzed using the Lyapunov direct method and LaSalle’s invariance principle without any discretization, reduction, and linearization. Additional controller features are highlighted by simulation results, including its benefits in contrast to the corresponding PD controller and its robustness to time delay and system uncertainty.

The present paper is devoted to the development of algorithms for dynamic grid adaptation to the geometry of an immersed moving rigid body and tracking its position on the grid. Hierarchical octree-based mesh adaptation is dynamically performed for a Cartesian base grid so that 2:1 grid cell balancing is strictly maintained in calculations and the body surface is always located in a cloud of grid cells of the lowest adaptation level. The latter means that all cut cells and all their neighbors at each time step belong to the lowest adaptation level. The details of corresponding implementations on multi-core CPUs using OpenMP and GPUs using CUDA, performance results are also provided. According to performed estimation, proposed algorithms brings only 3 overhead to CFD solver runtime. To demonstrate the ability of our implementation to process complex geometry, a DLR F6 aircraft model is used as rigid body in the tests conducted. [ABSTRACT FROM AUTHOR]

This paper presents a novel and systematic approach for obtaining the angular acceleration vector of a moving rigid body. The novelty of the proposed method lies in the particular form of writing the pose of the moving rigid body, as well as in the procedure to compute its time derivatives. The derivation process goes directly to the very foundations of rotational motion and exploits the phenomenological connection between orientation, angular velocity, angular acceleration, and spatial motion of a rigid body. Hence, as a remarkable result, a symbolic expression for the angular acceleration vector arises naturally without the need to solve the inverse acceleration problem. The novel and general expression of the angular acceleration vector involves relationships between the position, velocity, and acceleration vectors of three non-collinear points of the body, which can be easily understood and physically interpreted without particular knowledge of specialized techniques or advanced mathematical tools. Due to its vector nature, the expression for the angular acceleration vector proposed in this paper is relatively simple, as well as, it is very robust against computational singularities. Two fully detailed case studies demonstrate the robustness of the proposed angular acceleration vector compared with other expressions appearing in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study the large time behavior of the solutions of the Stokes fluid-solid system. We compute the asymptotic expansion of the solution for L q integrable initial data U 0. We show that the asymptotic profile of the solution is a linear combination of the Stokes fundamental solution for the data with extra condition | x | U 0 ∈ L 1. We also show that L 1 integrability of the solution is strongly related to the net force exerted by the fluid on the boundary of the solid. [ABSTRACT FROM AUTHOR]

This paper is devoted to dealing with the problem of global attitude synchronization for quaternion-based multiple rigid bodies, regardless of the general directed topologies of networks and arbitrary initial orientations of rigid bodies. A novel canonical quaternion is constructed to represent all physical attitudes of rigid bodies such that the pseudo-synchronization of their quaternion representations (namely, the quaternions' vector parts of all rigid bodies reach agreement on some identical value, whereas their scalar parts do not) can be precluded. Moreover, to reduce unnecessary communication requirements of rigid bodies, a hybrid triggering mechanism involving both the time regulation and neighbors' non-real-time information is proposed, with which a distributed protocol is developed by leveraging the constructed canonical quaternion. It is shown that the presented protocol for rigid bodies over directed networks can simultaneously realize the global attitude synchronization and naturally exclude the Zeno behavior. In addition, these observations are also validated via the application of our hybrid triggering protocol to networked spacecraft. • Develop a hybrid triggering mechanism and a quaternion-based distributed protocol. • Achieve global attitude synchronization regardless of arbitrary initial attitudes. • Address the two-fold covering map problem in the synchronization process. • Exclude Zeno behavior without requiring additional assumptions about attitudes. [ABSTRACT FROM AUTHOR]

To verify the dynamic model of a paddy-field laser grader and solve the problem of the unstable high-speed operation of the grader, a method for measuring the absolute pose parameters of a moving rigid body using a monocular camera is proposed. The proposed method does not require calibration beforehand. Using more than six non-coplanar control points symmetrically arranged in the rigid-body and world coordinate systems, the matrices of rotation and translation between the camera and two coordinate systems are obtained and the absolute pose of the rigid body is measured. In this paper, formulas of the absolute pose measurement of a moving rigid body are deduced systematically and the complete implementation is presented. Position and attitude measurement experiments carried out on a three-axis precision turntable show that the average absolute error in the attitude angle of a moving rigid body measured by an uncalibrated camera at different positions changes by no more than 0.2 degrees. An analysis of the three-dimensional coordinate errors of the centroid of a moving rigid body shows little deviation in measurements taken at the three camera positions, with the maximum deviation of the average absolute error being 0.53 cm and the maximum deviation of the standard deviation being 0.66 cm. The proposed method can measure the absolute pose of a rigid body and is insensitive to the position of the camera in the measurement process. This work thus provides guidance for the repeated measurement of the absolute pose of a moving rigid body using a monocular camera. [ABSTRACT FROM AUTHOR]

In this work, we present advanced investigations and treatments for the problem of a restricted vibrating motion of a connected gyrostat with a spring. It is supposed that the gyrostat spins slowly about the minor or major principal axis of the inertia ellipsoid. The gyrostat is acted upon by a gyrostatic couple vector besides the action of Newtonian and electromagnetic fields. The approach of the large parameter is applied to obtain the periodic solutions for the governing system of equations of motion of the gyrostat. A geometric illustration using the angles of Euler is given for such motion to evaluate and analyze the gyrostatic motion at any instant. The analysis of the obtained solutions is considered in terms of numerical data throughout computer programs. Characterized parametric data are assumed through one of the numerical methods for obtaining numerical solutions that prove the validity of the analytical obtained periodic solutions. The obtained solutions, besides the phase diagrams, have been drawn to describe these solutions' periodicity and stability procedures. The novelty of this work comes from the imposition of a new initial condition that does not restrict movement around the dynamic symmetry axis. This assumption allows the use of a new technique for the solution called the large parameter. This technique gives solutions in a completely new domain that are different from the ones studied in previous works. [ABSTRACT FROM AUTHOR]

This study delves into moment of inertia in rigid body mechanics, a crucial concept in physics. While it’s traditionally taught through integration, we propose an alternative method using discrete mass segments. This approach simplifies calculations, particularly for symmetric bodies, without complex integration procedures. Our goal is to enhance students’ understanding of fundamental physics principles, making the subject more accessible. We also provide visual aids and online animations to aid comprehension.

The use of general tensegrity systems that incorporate rigid bodies beyond axially loaded members has garnered increasing attention in practical applications. Recent preliminary studies have been conducted on the analysis and form design of general tensegrity systems with disconnecting rigid bodies. However, existing methods cannot account for connections between different rigid bodies. In practical applications, general tensegrity systems may have interconnected rigid bodies, rendering the analysis method proposed in previous studies inapplicable. To address this issue, this work proposes a comprehensive and unified analysis method for general tensegrity systems. The proposed formulation allows for the incorporation of connections between rigid bodies and general tensegrity systems with supports into the developed framework, enabling uniform analysis. Equilibrium and compatibility equations are derived through an energy approach combined with the Lagrange multiplier method. Self-stress states and mechanism modes are then computed based on these formulations. The stiffness of the mechanism mode is analyzed and validated using both the product force method and the reduced geometric stiffness matrix method. Furthermore, a self-stress design approach based on semi-definite programming (SDP) is proposed to determine feasible member forces that can stabilize general tensegrity systems. Illustrative examples are presented to verify the effectiveness of the proposed approach. This study expands the scope of the analysis theory for tensegrity systems and provides a fundamental and unified analysis approach that can be applied to any type of tensegrity system. [ABSTRACT FROM AUTHOR]

This study develops stability conditions for general tensegrity regarding the three stability criteria (e.g., prestress stability, general stability, and super stability) based on an energy approach. The coupling relations of specific nodal degrees of freedom caused by the rigid bodies are described by a set of constraint functions and incorporated into the potential energy function by the Lagrangian multiplier method. Stiffness matrices of general tensegrity systems are derived from the augmented potential energy function. It shows that the stability of general tensegrity can be evaluated through the stiffness matrices expressed in the constrained motion space generated by the constraint functions. Elegant formulations are developed for the evaluation of the three types of stability conditions for general tensegrity systems. Moreover, it turns out that the formulations developed in this study will degenerate into those for the stability analysis of classic tensegrity if no rigid bodies exist in the system, which indicates that the proposed approach is a general framework for the stability analysis of any type of tensegrity systems. [ABSTRACT FROM AUTHOR]

We consider a mathematical model of the plane-parallel action of a medium on a rigid body whose surface contains a cone-shaped being part. A complete system of equations of motion under in the quasi-stationary case is presented. A new family of phase portraits on the phase cylinder of quasi-velocity is obtained. Also, we consider a mathematical model of the influence of the medium on an axisymmetric body whose surface contains a cone-shaped part. We examine the stability with respect to a part of variables of the key mode, namely, the spatial rectilinear translational deceleration of the body. The problem of integrability is discussed. [ABSTRACT FROM AUTHOR]

This paper proposes a novel method of transporting a safety-critical rigid payload with multiple quadrotor UAVs using a geometric control approach. A simple and tractable model of the connected system has been derived directly from the configuration space of the interconnected system utilizing the framework of Lagrangian and geometric mechanics. A tractable geometric controller has been proposed directly on the configuration space of the connected system for trajectory tracking of both the payload and the quadrotors. The controller has been derived on the configuration space of the system, and the mathematical asymptotic convergence of the errors has been provided. The proposed geometric controller does not require link information and is practically less complex. Since the controller does not require linear/angular acceleration of payload, the proposed controller can be easily implemented in a practical scenario. A realistic simulation in Gazebo and ROS is carried out to validate the tracking performance. [ABSTRACT FROM AUTHOR]

Computational fluid dynamics has been widely used in the design and analysis of various fluid systems. The proper treatment of boundary conditions is crucial for the accurate simulation of fluid flow. However, in particle methods, such as smoothed particle hydrodynamics method and moving particle semi-implicit method, the treatment of boundary conditions has been a challenging problem. In this paper, starting from the incompressibility condition, we present a new theory for the unified treatment of both rigid bodies and wall boundaries, which allows the strong coupling of rigid bodies and incompressible fluids. Because the boundary models are based on signed distance functions and do not use particles, these models can avoid several problems, such as resolution dependence of particle representations, and unavoidable unevenness of surfaces. We also provide a way to efficiently handle rigid-body boundaries and wall boundaries without particles by finding the fundamental boundary weight function that does not change with time. Several numerical examples are presented to demonstrate the capability of our models and to compare them with theoretical and experimental results.