System Of Rigid Bodies Research Articles

Basic Principles of Deformed Objects with Methods of Analytical Mechanics

Analytical mechanics is the most fundamental discipline in this field. The basic principles of analytical mechanics should also be applicable to deformed objects. However, the virtual displacement principle proposed by analytical mechanics is only applicable to particle systems and rigid body systems, and not to general deformed objects. In this study, the basic principle, which includes the virtual displacement principle and d’Alembert–Lagrange principle (also called the virtual displacement principle of dynamics), of general deformed objects (such as, elastic, plastic, elasto-plastic, and flexible objects) is derived using analytical mechanics. First of all, according to the method of analytical mechanics, the external force, internal force, constraint reaction force and elastic recovery force of the deformed object system under the equilibrium state are analyzed, and the concepts of virtual displacement, ideal constraint and virtual work are introduced, and the virtual displacement principle (also called virtual work principle) of deformed objects is proposed; secondly, vector form, coordinate component form and generalized coordinate form of generalized virtual displacement principle of deformed object are given; thirdly, Introduce inertial force and use analytical mechanics to derive the d’Alembert–Lagrange principle of dynamic systems; fourthly, as application of the principle, the virtual displacement principle of deformed objects in plane polar coordinate system, space cylindrical coordinate system and spherical coordinate system are given; fifthly, the constitutive relationship between the gravitational strain of elastic–plastic materials was introduced, and an example of the application of the d'Alembert–Lagrange principle in elastic–plastic objects was given; finally, a brief conclusion is drawn. This study unifies the virtual displacement principle of elastic objects, plastic, elastoplastics, deformed object systems and rigid object systems using the basic analytical mechanics method. This is a basic principle for dealing with the static problems of deformed objects. This work also lays the foundation for further study of the dynamics of deformed object systems.

Towards Design Optimization of Compliant Mechanisms: A Hybrid Pseudo-Rigid-Body Model-Finite Element Method Approach and an Accurate Empirical Compliance Equation for Circular Flexure Hinges.

Innovative designs such as morphing wings and terrain adaptive landing systems are examples of biomimicry and innovations inspired by nature, which are actively being investigated by aerospace designers. Morphing wing designs based on Variable Geometry Truss Manipulators (VGTMs) and articulated helicopter robotic landing gear (RLG) have drawn a great deal of attention from industry. Compliant mechanisms have become increasingly popular due to their advantages over conventional rigid-body systems, and the research team led by the second author at Toronto Metropolitan University (TMU) has set their long-term goal to be exploiting these systems in the above aerospace applications. To gain a deeper insight into the design and optimization of compliant mechanisms and their potential application as alternatives to VGTM and RLG systems, this study conducted a thorough analysis of the design of flexible hinges, and single-, four-, and multi-bar configurations as a part of more complex, flexible mechanisms. The investigation highlighted the flexibility and compliance of mechanisms incorporating circular flexure hinges (CFHs), showcasing their capacity to withstand forces and moments. Despite a discrepancy between the results obtained from previously published Pseudo-Rigid-Body Model (PRBM) equations and FEM-based analyses, the mechanisms exhibited predictable linear behavior and acceptable fatigue testing results, affirming their suitability for diverse applications. While including additional linkages perpendicular to the applied force direction in a compliant mechanism with N vertical linkages led to improved factors of safety, the associated increase in system weight necessitates careful consideration. It is shown herein that, in this case, adding one vertical bar increased the safety factor by 100N percent. The present study also addressed solutions for the precise modeling of CFHs through the derivation of an empirical polynomial torsional stiffness/compliance equation related to geometric dimensions and material properties. The effectiveness of the presented empirical polynomial compliance equation was validated against FEA results, revealing a generally accurate prediction with an average error of 1.74%. It is expected that the present investigation will open new avenues to higher precision in the design of CFHs, ensuring reliability and efficiency in various practical applications, and enhancing the optimization design of compliant mechanisms comprised of such hinges. A specific focus was put on ABS plastic and aluminum alloy 7075, as they are the materials of choice for non-load-bearing and load-bearing structural components, respectively.

Semiglobal Exponential Stability for Dual-Quaternion-Based Rigid-Body Tracking Control

Semi-Global Exponential Stability (SGES) is proven for the combined attitude and position rigid-body motion tracking problem, presenting a significant upgrade compared to previously proposed controllers which can only guarantee asymptotic stability. Dual quaternions are used to jointly represent the rotational and translational tracking error dynamics of the rigid body. A novel nonlinear feedback tracking controller is proposed, and a Lyapunov-based analysis is provided to prove SGES of the closed-loop dynamics. The analysis does not place any restrictions on the reference trajectory or the feedback gains. This stronger SGES result aids in further analyzing the robustness of the rigid-body system by establishing input-to-state stability (ISS) in the presence of time-varying additive and bounded external disturbances. Motivated by the fact that, in many aerospace applications, stringent adherence to safety constraints such as approach path and input constraints is critical for overall mission success, a framework for safe control of spacecraft is presented that combines the proposed feedback controller with control barrier functions. Numerical simulations are provided to verify the SGES and ISS results and also showcase the efficacy of the proposed nonlinear feedback controller in several nontrivial scenarios, including trajectory tracking, transposition and docking, flip maneuvers, collision avoidance, and rendezvous.

ANALISIS MEKANIKA LAGRANGE PADA BENDA TEGAR DENGAN PENDEKATAN DAN PENERAPANNYA PADA SISTEM FISIKA

The Lagrange method is a method used to solve complex mechanical problems by utilizing kinetic and potential energy. This research examines the application of the Lagrange method to rigid body systems, where the position and orientation of the body are determined by general coordinates. This article uses a literature study approach from various relevant sources. The research results show that the Lagrange method is effective in solving mechanical problems in rigid body systems that cannot be solved using Newton's method.

Estimation of minimum foot clearance using a single foot-mounted inertial sensor and personalized foot geometry scan

The real-world measurement of minimum foot clearance (mFC) during the swing phase of gait is critical in efforts to understand and reduce the risk of trip-and-fall incidents in populations with gait impairments. Past research has focused on measuring clearance of a single point on a person’s foot, typically the toe—however, this may overestimate mFC and may even be the wrong region of the foot in cases of gait impairments or interventions. In this work, we present a novel method to reconstruct the swing-phase trajectory of an arbitrary number of points on a person’s shoe and estimate the instantaneous height and location of whole-foot mFC. This is achieved using a single foot-mounted inertial sensor and personalized shoe geometry scan, assuming a rigid-body IMU-shoe system. This combination allows collection and analysis using out-of-lab tests, potentially including clinical environments. Validation of single marker location using the proposed method vs. motion capture showed height errors with bias less than 0.05 mm, and 95% confidence interval of − 8.18 to + 8.09 mm. The method is demonstrated in an example data set comparing different interventions for foot drop, and it shows clear differences among no intervention, functional electrical stimulation, and ankle–foot orthosis conditions. This method offers researchers and clinicians a rich understanding of a person’s gait by providing objective 3D foot kinematics and allowing a unique opportunity to view the regions of the foot where minimum clearance occurs. This information can contribute to a more informed recommendation of specific interventions or assistive technology than is currently possible in standard clinical practice.

Antiwindup Compensation for Unstable Rigid-Body Systems with Quantized and Saturated Inputs

It is well known that actuator saturation can cause destabilization and degradation in performance; similar problems are encountered when actuation is quantized. This study proposes the design of an antiwindup compensator for systems with actuators that are limited to a finite number of quantization levels. This combination of discrete-level actuation and saturation poses a unique antiwindup problem that has not yet been solved. To surmount this combined issue, an antiwindup compensator is proposed, which provides ultimate boundedness of the system state within a prescribed region and guarantees that the state does not stray outside a larger compact set. The use of shifted ramp functions enables a less conservative bound on the control-signal error, which yields significantly lower L2 gain bounds compared to a standard sector-bound antiwindup design approach. A numerical simulation example illustrates the effectiveness on a rigid-body system, which inspired this study.

Output feedback control for hypersonic flight vehicles via distributed high-gain observer

During the hypersonic flight process of rigid hypersonic flight vehicles (HFVs), the flight-path angle and angle of attack are usually small and sensitive to flight condition variation, the accurate measurements are difficult, subsequently the formulation of states observers is an urgent issue. A distributed high-gain observer is proposed to reconstruct the admissible system state dynamics based on limited measured output. First, a distributed observer is proposed for the cascade strict-feedback system to ensure the global convergence of the state estimation errors between the estimated state in each local observer and system states, for the prescribed system initial values. Second, the distributed high-gain observer-based nonlinear control is investigated on the longitudinal dynamics of HFVs with partial measurable states. The rigid body system dynamic is divided into the altitude subsystem and the velocity subsystem. In the altitude subsystem, the proposed distributed high-gain observer is formulated to exact estimate the flight-path angle and angle of attack, and then the back-stepping design is proposed for constructing the controllers. The nonlinear dynamic inversion controller is introduced to accomplish the velocity tracking. Finally, simulation examples are served to verify that the proposed output feedback control possesses the superior properties of high tracking performance, fast convergence of state estimation error and easy-implementation.

К РАСЧЕТУ ПЛИТНЫХ АЭРОДРОМНЫХ ПОКРЫТИЙ ПРИ ИХ ВЗАИМОДЕЙСТВИИ С ИНЕРЦИОННОЙ НАГРУЗКОЙ ПРИ ДВИЖЕНИИ В РЕЖИМЕ ТОРМОЖЕНИЯ.

The method of calculating plates under the action of inertial loads during their movement with variable velocity is considered. The problem of the interaction of the airfield surface with the aircraft load is solved, at the moment of its movement, in braking mode, when landing on the surface in the form of an extended rectangular freely operable plate lying on an elastic base. When taking into account the nonlinear contacts of the chassis wheels with the plate, piecewise linear characteristics of contact stiffness are used. The vertical dynamics of the spatial model of the aircraft during the run is modeled by a system of rigid bodies with elastic-viscous bonds. To construct the methodology, the methods proposed earlier by the author of the article are used: a step-by-step method for solving problems of the relentless dynamics of structures and a method for calculating structures for the action of inertial and mobile force loads (the method of nodal accelerations).

Reconfigurable Microlens Array Enables Tunable Imaging Based on Shape Memory Polymers.

Microlens arrays (MLAs) with a tunable imaging ability are core components of advanced micro-optical systems. Nevertheless, tunable MLAs generally suffer from high power consumption, an undeformable rigid body, large and complex systems, or limited focal length tunability. The combination of reconfigurable smart materials with MLAs may lead to distinct advantages including programmable deformation, remote manipulation, and multimodal tunability. However, unlike photopolymers that permit flexible structuring, the fabrication of tunable MLAs and compound eyes (CEs) based on transparent smart materials is still rare. In this work, we report reconfigurable MLAs that enable tunable imaging based on shape memory polymers (SMPs). The smart MLAs with closely packed 200 × 200 microlenses (40.0 μm in size) are fabricated via a combined technology that involves wet etching-assisted femtosecond laser direct writing of MLA templates on quartz, soft lithography for MLA duplication using SMPs, and the mechanical heat setting for programmable reconfiguration. By stretching or squeezing the shape memory MLAs at the transition temperature (80 °C), the size, profiles, and spatial distributions of the microlenses can be programmed. When the MLA is stretched from 0 to 120% (area ratio), the focal length is increased from 116 to 283 μm. As a proof of concept, reconfigurable MLAs and a 3D CE with a tunable field of view (FOV, 160-0°) have been demonstrated in which the thermally triggered shape memory deformation has been employed for tunable imaging. The reconfigurable MLAs and CEs with a tunable focal length and adjustable FOV may hold great promise for developing smart micro-optical systems.

Power XShanker Distribution: Properties, Estimation, and Applications

A two-parameter extension of the XShanker distribution called power XShanker is developed and studied in this article. The properties which include moment, quantile function, moment generating function, order statistics, stress-strength reliability function, and Rény entropy were studied. The classical and Bayesian estimation procedures were studied with simulations. The usefulness of the proposed model was demonstrated using lifetime data sets.

Variational Estimation for Mechanical Systems on Lie Groups based on Geometric Mechanics

Geometric mechanics analyzes mechanical systems in the framework of variational mechanics, while accounting for the geometry of the configuration space. From the late 1970s, developments in this area produced several schemes for geometric control of mechanical systems in continuous time and discrete time. In the mid to late 2000s, geometric mechanics was first applied to state estimation of mechanical systems, particularly systems evolving on Lie groups as configuration manifolds, like rigid body systems. Much of the existing work on geometric mechanics-based estimation has been in continuous time, using deterministic, semi-stochastic and stochastic approaches. While the body of existing literature on discrete-time estimation schemes on Lie groups is not as extensive, the literature on this topic is contemporaneous with continuous-time schemes. This work describes some recent and ongoing research on geometric mechanics-based estimation schemes in continuous and discrete time from the mid-2010s, which were developed using the Lagrange-d’Alembert principle applied to rigid body systems. This approach gives (deterministic) observer designs with strong stability and robustness properties. This work concludes with potential extensions of this approach to mechanical systems with principal fiber bundles as configuration manifolds.

Predicting dynamic responses of continuous deformable bodies:A graph-based learning approach

Predicting dynamic responses of continuous deformable bodies (CDBs) is essential for various research fields, including civil engineering and computer graphics. Machine learning models, unlike traditional physically-based models, learn from data without predefined physical properties and have significantly advanced the simulation of rigid body systems and fluids. Graph neural network (GNN)-based simulators excel in this area due to their ability to naturally represent physical bodies and interactions using graphs composed of nodes and edges. However, predicting the dynamic responses of CDBs remains challenging because interactions in a CDB are complex, heterogeneous, and dynamic. The complexity of interactions arises from the spatial variability of internal stress, making it challenging to encapsulate such multifaceted interactions of one edge into one edge attribute vector. Additionally, those interactions are heterogeneous and dynamic due to the material and geometric nonlinearity of CDBs. CDBs with elastic, plastic, and elastoplastic materials follow different deformation rules during vibrations. These rules also vary under small and large deformations for a single material type. To address these challenges, we introduce a new GNN-based simulator called physics-informed edge recurrent simulator (Piers) for learning CDB dynamics. We first formulate the CDB simulation as a sequence-to-sequence input–output relationship modeling problem and incorporate a recurrent neural network (RNN) to learn edge updates within short timesteps, during which stiffness changes are negligible. To accurately capture complex interactions, we initialize the RNN module’s hidden states with prior physical knowledge and equip Piers with a physics-informed loss function by assigning the physical properties of interactions as the target edge output. Extensive experimental results demonstrate that Piers can simulate the dynamics of elastic, plastic, and elastoplastic CDBs with smaller response prediction errors than alternative baselines. Piers reduces prediction errors by 78% to 99% across four typical CDB physical systems.

Synchronization and decoherence in a self-excited inertia-wheel multiple rigid-body dynamical system.

We investigate the synchronization and decoherence of a self-excited inertia wheel multiple rigid-body dynamical system. We employ an Euler-Lagrange formulation to derive a nondimensional state space that governs the dynamics of a coupled pendula array where each element incorporates an inertia wheel. The dynamical system exhibits multiple equilibria, periodic limit-cycle oscillations, quasiperiodic, and chaotic oscillations and rotations. We make use of a combined approach including a singular perturbation multiple time scale and numerical bifurcation methodologies to determine the existence of synchronized and decoherent solutions in both weakly and strongly nonlinear regimes, respectively. The analysis reveals that synchronous oscillations are in-phase, whereas quasiperiodic oscillations are anti-phase. Furthermore, the non-stationary rotations are found to exhibit combinations of oscillations and rotations of the individual elements that are asynchronous. A Kuramoto order parameter analysis of representative solutions in various bifurcation regimes reveals the existence of chimera-like solutions where two elements are synchronized, whereas the third is desynchronized. Moreover, synchronous solutions were found to coexist with stable chimera solutions with a constant phase difference between the oscillators.

Event-Triggered Distributed Observer for a Rigid Body Leader System Over Acyclic Switching Networks and its Application

The distributed observer for a rigid body leader system plays a key role in solving the leader-following consensus problem of multiple rigid body systems. Recently, the authors of this paper established an event-triggered distributed observer over jointly connected networks, which consumes less communication and actuator energy compared with the analog distributed observer. However, a drawback of this event-triggered distributed observer is that the upper bounds of two key design parameters were only shown to exist without giving an explicit estimate of the upper bounds. In this paper, by assuming that the communication network is acyclic, we further show that these two design parameters can take any positive value by choosing other parameters appropriately. Furthermore, we will apply the event-triggered distributed observer to solve the leader-following consensus problem of multiple uncertain rigid body systems.

A Method for Measuring the Absolute Position and Attitude Parameters of a Moving Rigid Body Using a Monocular Camera

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.

Optimal Control Formulation for Manual Wheelchair Locomotion Simulations: Influence of Anteroposterior Stability.

Manual wheelchair (MWC) locomotion exposes the user's upper-body to large and repetitive loads, which can lead to upper limbs pain and injuries. A thinner understanding of the influence of MWC settings on propulsion biomechanics could allow for a better adaptation of MWC configuration to the user, thus limiting the risk of developing such injuries. Advantageously compared to experimental studies, simulation methods allow numerous configurations to be tested. Recent studies have developed predictive locomotion simulation using optimal control methods. However, those models do not consider MWC anteroposterior stability, potentially resulting in unreasonable propulsion strategies. To this extent, this study aimed at confirming if constraining MWC anteroposterior stability in the optimal control formulation could lead to a different simulated movement. For this purpose, a four-link rigid-body system was used in a forward dynamics optimization paired with an anteroposterior stability constraint to predict MWC locomotion dynamics of the upper limbs during both startup and steady-state propulsion. Simulation results indicated the occurrence of MWC tipping when stability was not constrained, and that the constrained optimal control algorithm predicted different propulsion strategies. Hence, further proceedings of MWC locomotion simulation and optimal control investigations should take the anteroposterior stability into account to achieve more realistic simulations. Additionally, the implementation of the anteroposterior stability constrains unexpectedly resulted in a reduction of the computational time.

Effects of Crosswind on Pantograph–Catenary Wear Using Nonlinear Multibody System Dynamic Algorithms

In this study, a multibody system (MBS) computational framework is developed to determine the exact location of the contact point and wear prediction resulting from the pantograph–catenary interaction. The railroad vehicle models in the MBS computational framework comprise rigid-body railroad vehicles, rigid-body pantograph systems, and flexible catenary systems. To avoid incremental rotation, the nonlinear finite element absolute nodal coordinate formulation is used to model a flexible catenary system in the MBS computational framework. To avoid co-simulation processes, the rigid-body railroad vehicle and the pantograph and flexible catenary systems were integrated into the MBS algorithms. The pantograph–catenary interaction is modeled using an elastic contact formulation developed to include the effect of pantograph–catenary separation and sliding contact. The proposed MBS approach evaluates the location of the contact point, contact force, and normal wear rate (NWR) from the mechanical and electrical contributions. This investigation considers the vibration caused by a crosswind scenario and determines the numerical result in the case of a steady crosswind scenario. The steady crosswind scenario contains the advantage of pantograph–catenary aerodynamic design, and the vibration of the catenary system remains significant after the excitation of a steady crosswind. In the case of a steady crosswind, the higher value of the steady crosswind effect significantly increases the mean contact force and the NWR from the mechanical contribution. After crosswind load disturbances, the mean contact force decreases, but the standard deviation of the contact force increases. Therefore, the NWR from the electrical contribution increases significantly. However, the total NWR increases with the crosswind velocity.

Synchronization of multiple rigid body systems: A survey.

The multi-agent system has been a hot topic in the past few decades owing to its lower cost, higher robustness, and higher flexibility. As a particular multi-agent system, the multiple rigid body system received a growing interest for its wide applications in transportation, aerospace, and ocean exploration. Due to the non-Euclidean configuration space of attitudes and the inherent nonlinearity of the dynamics of rigid body systems, synchronization of multiple rigid body systems is quite challenging. This paper aims to present an overview of the recent progress in synchronization of multiple rigid body systems from the view of two fundamental problems. The first problem focuses on attitude synchronization, while the second one focuses on cooperative motion control in that rotation and translation dynamics are coupled. Finally, a summary and future directions are given in the conclusion.

Mechanical Characterization of Supernumerary Robotic Tails for Human Balance Augmentation

Abstract Humans are intrinsically unstable in quiet stance from a rigid body system viewpoint; however, they maintain balance, thanks to neuro-muscular sensory control properties. With increasing levels of balance related incidents in industrial and ageing populations globally each year, the development of assistive mechanisms to augment human balance is paramount. This work investigates the mechanical characteristics of kinematically dissimilar one and two degrees-of-freedom (DoF) supernumerary robotic tails for balance augmentation. Through dynamic simulations and manipulability assessments, the importance of variable coupling inertia in creating a sufficient reaction torque is highlighted. It is shown that two-DoF tails with solely revolute joints are best suited to address the balance augmentation issue. Within the two-DoF options, the characteristics of open versus closed loop tails are investigated, with the ultimate design selection requiring trade-offs between environmental workspace, biomechanical factors, and manufacturing ease to be made.

Adaptive leader-following attitude consensus of multiple rigid body systems with resilience to communication link faults

Adaptive leader-following attitude consensus of multiple rigid body systems with resilience to communication link faults