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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]

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]

AbstractThe 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]

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.

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.

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]

Damping is an important factor contributing to errors in the measurement of rotational inertia using the torsion pendulum method. Identifying the system damping allows for minimizing the measurement errors of rotational inertia, and accurate continuous sampling of torsional vibration angular displacement is the key to realizing system damping identification. To address this issue, this paper proposes a novel method for measuring the rotational inertia of rigid bodies based on monocular vision and the torsion pendulum method. In this study, a mathematical model of torsional oscillation under a linear damping condition is established, and an analytical relationship between the damping coefficient, torsional period, and measured rotational inertia is obtained. A high-speed industrial camera is used to continuously photograph the markers on a torsion vibration motion test bench. After several data processing steps, including image preprocessing, edge detection, and feature extraction, with the aid of a geometric model of the imaging system, the angular displacement of each frame of the image corresponding to the torsion vibration motion is calculated. From the characteristic points on the angular displacement curve, the period and amplitude modulation parameters of the torsion vibration motion can be obtained, and finally the rotational inertia of the load can be derived. The experimental results demonstrate that the proposed method and system described in this paper can achieve accurate measurements of the rotational inertia of objects. Within the range of 0–100 × 10−3 kg·m2, the standard deviation of the measurements is better than 0.90 × 10−4 kg·m2, and the absolute value of the measurement error is less than 2.00 × 10−4 kg·m2. Compared to conventional torsion pendulum methods, the proposed method effectively identifies damping using machine vision, thereby significantly reducing measurement errors caused by damping. The system has a simple structure, low cost, and promising prospects for practical applications. [ABSTRACT FROM AUTHOR]

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The problem of attitude stabilization of a rigid body is under consideration. The issue of realizing such a control system of the PID-controller type with distributed delay (integral term) is addressed. A theorem on the asymptotic stability of the stabilized position of the body, which justifies the possibility of constructing the desired control system, has been proven. The effectiveness of the constructed control with a distributed delay is confirmed by numerical modeling. [ABSTRACT FROM AUTHOR]

This paper aims to distinguish two main types of coarse graining, and reveal the relationship between the notions of coarse graining and emergence. In physics, some forms of coarse graining seem to be indispensable to show a physical property, and the other merely changes our descriptions of the system. To clarify the notion of coarse graining, this article investigates the cases of the renormalization group method and irreversibility, both of which have been important topics in philosophy of science, and the case of the rigid body in classical mechanics, which is an elementary case including coarse graining. The case studies reveal the distinction between substantial and mere coarse-graining. This distinction clarifies the relationships between the notions of coarse graining and emergence and further provides some implications for the issues about emergence. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of a toy known as the chatter ring. Specifically, it examines the mechanism by which the small ring rotates around the large ring, the mechanism by which the force from the large ring provides torque to the small ring, and whether the motion of the small ring is the same as that of a hula hoop. The dynamics of a chatter ring has been investigated in previous work [13-14, 15]; however, a detailed analysis has not yet been performed. Thus, to understand the mechanisms described above, the equations of motion and constraint conditions are obtained, and an analysis of the motion is performed. To simplify the problem, a model consisting of a straight rod and a washer ring is analyzed under the no-slip condition. The motion of a washer has two modes: the one point of contact (1PC) mode and two points of contact (2PC) mode. The motion of the small ring of the chatter ring is similar to that of a washer in the 2PC mode, whereas the motion of a hula hoop is similar to that of a washer in the 1PC mode. The analysis indicates that the motion of a washer with two points of contact is equivalent to free fall motion. However, in practice, the velocity reaches a constant value through energy dissipation. The washer rotates around an axis that passes through the two points of contact. The components of the forces exerted by the rod at the points of contact that are normal to the plane of the washer provide rotational torque acting at the center of mass. The components of the forces parallel to the horizontal plane are centripetal forces, which induce the circular motion of the center of mass. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose a semi-smooth Newton method and a primal-dual active set strategy to solve dynamical contact problems with friction. The conditions of contact with Coulomb’s friction can be formulated in the form of a fixed point problem related to a quasi-optimization one thanks to the semi-smooth Newton method. This method is based on the use of the primal-dual active set (PDAS) strategy. The main idea here is to find the correct subset A $\mathcal{A}$ of nodes that are in contact (active) opposed to those which are not in contact (inactive). For each case, the nonlinear boundary condition is replaced by a suitable linear one. Numerical experiments on both hyper-elastic problems and rigid granular materials are presented to show the efficiency of the proposed method.

Perturbed motions of a rigid body, close to the Lagrange case, under the action of restoring and perturbation torques of forces are investigated in the paper. The following problem is formulated: investigating solutions behavior of system of equations of motion for nonzero values of small parameter on a sufficiently long time interval. To analyze a nonlinear system of equations of motion, the averaging method is used. The problem can be decomposed into slowly and quickly changing variables. Conditions for the possibility of averaging the equations of motion with respect to the phase of nutation angle are presented and averaging procedure for slow variables of a perturbed motion of a rigid body in the first approximation is described. As an example of the developed procedure, we investigate a perturbed motion, close to Lagrange case, taking into account constant dissipative and small torque, and dissipative torques depending on slow time. A new class of rotational motions of a dynamically symmetric rigid body about a fixed point has been investigated with restoring and perturbation torques of forces being taken into account.

In physical therapy, exercises improve range of motion, muscle strength, and flexibility, where motion-tracking devices record motion data during exercises to improve treatment outcomes. Cameras and inertial measurement units (IMUs) are the basis of these devices. However, issues such as occlusion, privacy, and illumination can restrict vision-based systems. In these circumstances, IMUs may be employed to focus on a patient's progress quantitatively during their rehabilitation. In this study, a 3D rigid body that can substitute a human arm was developed, and a two-stage algorithm was designed, implemented, and validated to estimate the elbow joint angle of that rigid body using three IMUs and incorporating the Madgwick filter to fuse multiple sensor data. Two electro-goniometers (EGs) were linked to the rigid body to verify the accuracy of the joint angle measuring algorithm. Additionally, the algorithm's stability was confirmed even in the presence of external acceleration. Multiple trials using the proposed algorithm estimated the elbow joint angle of the rigid body with a maximum RMSE of 0.46°. Using the IMU manufacturer's (WitMotion) algorithm (Kalman filter), the maximum RMSE was 1.97°. For the fourth trial, joint angles were also calculated with external acceleration, and the RMSE was 0.996°. In all cases, the joint angles were within therapeutic limits. [ABSTRACT FROM AUTHOR]

In this paper we study the interaction of a small rigid body in a viscous compressible fluid. The system occupies a bounded three dimensional domain. The object it allowed to freely move and its dynamics follows the Newton's laws. We show that as the size of the object converges to zero the system fluid plus rigid body converges to the compressible Navier-Stokes system under some mild lower bound on the mass and the inertia momentum. It is a first result of homogenization in the case of fluid-structure interaction in the compressible situation. As a corollary we slightly improved the result on the influence of a vanishing obstacle in a compressible fluid for γ ≥ 6. [ABSTRACT FROM AUTHOR]

Rigid body with rotors is a widespread mechanical system modeled after the direct product S O (3) × 1 × 1 × 1 , which under mild assumptions is the symmetry group of the system. In this paper, the authors present and compare different Lagrangian reduction procedures: Euler–Poincaré reduction by the whole group and reduction by stages in different orders or using different connections. The exposition keeps track of the equivalence of equations as well as corresponding conservation laws. [ABSTRACT FROM AUTHOR]

We study degenerate singularities of the well-known multiparametric family of integrable Zhukovsky cases of rigid-body dynamics, i.e., Euler tops with added constant gyrostatic moment. For an axisymmetric rigid body and systems close to it, it is proved that degenerate local and semilocal singularities are parabolic and cuspidal singularities, respectively, for all values of the set of system parameters, excluding some hypersurfaces. It was established that these singularities belonging to the preimage of the cusp of the bifurcation curve satisfy the parabolicity criterion of A.V. Bolsinov, L. Guglielmi, and E.A. Kudryavtseva. Therefore, they are structurally stable for small perturbations of the system in the class of integrable systems, in particular, for a small change in the principal moments of inertia, the components of the gyrostatic moment, and the values of the area integral. [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. [ABSTRACT FROM AUTHOR]

The motion of a homogeneous ellipsoid on a fixed horizontal plane in a uniform gravity field is considered. The plane is considered to be perfectly smooth and the semiaxes of the ellipsoid are different. There is a position of stable equilibrium, when the ellipsoid rests on the plane with the lowest point of its surface. The nonlinear oscillations of the ellipsoid in the vicinity of this equilibrium are studied. Analysis is performed using the methods of the classical perturbation theory, the Kolmogorov–Arnold–Moser (KAM) theory, and computer algebra algorithms. The normal form of the Hamiltonian function of the perturbed motion is obtained including the terms of the sixth degree relative to deviations from the equilibrium position. An approximate analytical representation of the Kolmogorov set of conditionally periodic oscillations and an estimate of the measure of this set are presented. The problem of the existence and orbital stability of periodic motions occurring from the stable equilibrium in the resonant and nonresonant cases is studied. [ABSTRACT FROM AUTHOR]