Multibody Dynamics Formulation Research Articles

Review and comparison of empirical friction coefficient formulation for multibody dynamics of lubricated slotted joints

Review and comparison of empirical friction coefficient formulation for multibody dynamics of lubricated slotted joints

A Sensorized 3D-Printed Knee Test Rig for Preliminary Experimental Validation of Patellar Tracking and Contact Simulation.

Experimental validation of computational simulations is important because it provides empirical evidence to verify the accuracy and reliability of the simulated results. This validation ensures that the simulation accurately represents real-world phenomena, increasing confidence in the model's predictive capabilities and its applicability to practical scenarios. The use of musculoskeletal models in orthopedic surgery allows for objective prediction of postoperative function and optimization of results for each patient. To ensure that simulations are trustworthy and can be used for predictive purposes, comparing simulation results with experimental data is crucial. Although progress has been made in obtaining 3D bone geometry and estimating contact forces, validation of these predictions has been limited due to the lack of direct in vivo measurements and the economic and ethical constraints associated with available alternatives. In this study, an existing commercial surgical training station was transformed into a sensorized test bench to replicate a knee subject to a total knee replacement. The original knee inserts of the training station were replaced with personalized 3D-printed bones incorporating their corresponding implants, and multiple sensors with their respective supports were added. The recorded movement of the patella was used in combination with the forces recorded by the pressure sensor and the load cells, to validate the results obtained from the simulation, which was performed by means of a multibody dynamics formulation implemented in a custom-developed library. The utilization of 3D-printed models and sensors facilitated cost-effective and replicable experimental validation of computational simulations, thereby advancing orthopedic surgery while circumventing ethical concerns.

Use of patellofemoral digital twins for patellar tracking and treatment prediction: comparison of 3D models and contact detection algorithms.

Introduction: Poor patellar tracking can result in painful contact pressures, patella subluxation, or dislocation. The use of musculoskeletal models and simulations in orthopedic surgeries allows for objective predictions of post-treatment function, empowering clinicians to explore diverse treatment options for patients. Although a promising approach for managing knee surgeries, the high computational cost of the Finite Element Method hampers its clinical usability. In anticipation of minimal elastic deformations in the involved bodies, the exploration of the Multibody Dynamics approach emerged as a viable solution, providing a computationally efficient methodology to address clinical concerns related to the knee joint. Methods: This work, with a focus on high-performance computing, achieved the simulation of the patellofemoral joint through rigid-body multibody dynamics formulations. A comparison was made between two collision detection algorithms employed in the simulation of contact between the patellar and femoral implants: a generic mesh-to-mesh collision detection algorithm, which identifies potential collisions between bodies by checking for proximity or overlap between their discretized mesh surface elements, and an analytical contact algorithm, which uses a mathematical model to provide closed-form solutions for specific contact problems, but cannot handle arbitrary geometries. In addition, different digital twins (3D model geometries) of the femoral implant were compared. Results: Computational efficiency was considered, and histories of position, orientation, and contact force of the patella during the motion were compared with experimental measurements obtained from a sensorized 3D-printed test bench under pathological and treatment scenarios. The best results were achieved through a purely analytical contact detection algorithm, allowing for clinical usability and optimization of clinical outcomes.

A peridynamics approach to flexible multibody dynamics for fracture analysis of mechanical systems

The classical theory of continuum mechanics is formulated using partial differential equations (PDEs) that fail to describe structural discontinuities, such as cracks. This limitation motivated the development of peridynamics, reformulating the classical PDEs into integral-differential equations. In this theory, each material point interacts with its neighbours inside a characteristic length-scale through bond-interaction forces. However, while peridynamics can simulate complex multi-physics phenomena, its integration in the study of mechanical systems is still limited. This work presents a methodology that incorporates a peridynamics formulation into a planar multibody dynamics (MBD) formulation to allow the integration of flexible structures described by peridynamics into mechanical systems. A flexible body is described by a collection of point masses, in analogy with the meshless collocation scheme commonly used for peridynamics discretisations. Each point mass interacts with other point masses through nonlinear forces governed by a bond-based peridynamics (BBPD) formulation. The virtual bodies methodology enables the definition of kinematic joints connecting the flexible body with the neighbouring bodies. The implementation of the methodology proposed is illustrated using various mechanisms with different levels of complexity. Notched plates subjected to different loading conditions are compared with the results presented in the literature of the peridynamics field. The deformations of a flexible slider-crank mechanism compare well with the results obtained using a classical flexible MBD formulation. Additionally, three scenarios involving a rotating pendulum illustrate how the methodology proposed allows simulating impact scenarios. The results demonstrate how this methodology is capable to successfully simulate highly nonlinear phenomena, including crack propagation, in a multibody framework.

A benchmark problem with singularities for multibody system dynamics formulations with constraints

A benchmark problem with singularities for multibody system dynamics formulations with constraints

A unified framework for corotational flexible multibody system dynamics formulations.

Corotational formulations play an important role for flexible multibody dynamics systems, because they reflect the nature of many technical systems undergoing arbitrarily large rigid body motions but small deformations within each body. This paper defines flexible multibody dynamics and corotational formulations in this context. Furthermore, the «ingredients» and workflow of a flexible multibody dynamics simulation are briefly addressed for the reader less familiar with the topic. This part also points to major review papers and textbooks in the field, and embeds the unified formulations in the literature. The paper's main part presents state-of-the-art corotational flexible multibody dynamics formulations in a systematic and unified way. In this formulation part, the standard integral-based floating frame of reference formulation with modal reduction and with the conventionally employed lumped mass approximation is presented, and its drawbacks highlighted. Then, the so-called nodal-based, i.e., space-wise discretized, equations of motion are presented for several up-to-date nodal-displacement-based formulations within a unified framework. This approach clearly shows the equivalence of the presented formulations, and highlights the fact that the formulations differ only in the choice of degrees of freedom. Moreover, this contribution also intends to reduce the information and complexity within the scientific literature, since this unified framework allows the derivation of these formulations with significantly less effort..

Hamiltonian formulation with reduced variables for flexible multibody systems under linear constraints: Theory and experiment

Mechanical structures, such as robot arms in space stations and rockets that are used to support futuristic activities and carry loads in modern society, tend to be flexible owing to their reduced weight. Flexible multibody dynamics is an effective method for analyzing the dynamic motion of flexible structures. Although the Hamiltonian formulation based on the canonical theory is commonly used in multibody dynamics, certain drawbacks exist, such as complexity and computational cost. In this study, a novel flexible multibody dynamics formulation based on canonical theory is proposed with reduced variables to improve computational efficiency. This method formulates the equation of motion using only independent generalized coordinates with the assumption of linear constraint equations and the introduction of the first derivative of the constraint condition. The proposed method is combined with absolute nodal coordinate formulation and the motions of flexible structures are simulated to verify its accuracy and effectiveness. In comparison with the conventional methods, the proposed method requires less calculation time while maintaining accuracy. In addition, a wind tunnel experiment was conducted to validate the proposed method. The concurrence between the simulation and experimental results further verifies the accuracy of the proposed method.

Real-Time Modeling of Vehicle’s Longitudinal-Vertical Dynamics in ADAS Applications

The selection of an appropriate method for modeling vehicle dynamics heavily depends on the application. Due to the absence of human intervention, the demand for an accurate and real-time model of vehicle dynamics for intelligent control increases for autonomous vehicles. This paper develops a multibody vehicle model for longitudinal-vertical dynamics applicable to advanced driver assistance (ADAS) applications. The dynamic properties of the chassis, suspension, and tires are considered and modeled, which results in accurate vehicle dynamics and states. Unlike the vehicle dynamics models built into commercial software packages, such as ADAMS and CarSim, the proposed nonlinear dynamics model poses the equations of motion using a subset of relative coordinates. Therefore, the real-time simulation is conducted to improve riding performance and transportation safety. First, a vehicle system is modeled using a semi-recursive multibody dynamics formulation, and the vehicle kinematics and dynamics are accurately calculated using the system tree-topology. Second, a fork-arm removal technique based on the rod-removal technique is proposed to reduce the number of bodies, relative coordinates, and equations constrained by loop-closure. This increase the computational efficiency even further. Third, the dynamic simulations of the vehicle are performed on bumpy and sloping roads. The accuracy and efficiency of the numerical results are compared to the reference data. The comparative results demonstrate that the proposed vehicle model is effective. This efficient model can be utilized for the intelligent control of vehicle ADAS applications, such as forward collision avoidance, adaptive cruise control, and platooning.

A recursive formulation for open-loop gyroelastic multibody dynamics

A recursive formulation for open-loop gyroelastic multibody dynamics

Rigid-flexible coupling dynamic modeling and analysis of dumbbell-shaped spacecraft

In this paper, the dynamic modeling of a new configuration spacecraft is investigated. The significance of dumbbell-shaped spacecraft to deep space exploration and the configuration of dumbbell-shaped spacecraft are introduced firstly. Then the dynamic equations of the single flexible body and the kinematic recursive relation between adjacent bodies are deduced. The rigid-flexible coupling dynamic model of the system is established by Jourdain's velocity variation principle and the forward recursive construction formulation for flexible multibody dynamics. The validity of the dynamic model is verified by comparing results with ADAMS. Moreover, the simplification of the dynamic model is investigated to meet the requirements of controller design and the feasibility of conventional simplification method is analyzed. The simulation results show that the conventional simplified model of spacecraft cannot accurately describe the dynamic behaviors of dumbbell-shaped spacecraft. Lastly, a modified attitude dynamic model is proposed and an attitude controller is designed based on this model. The numerical simulations show that the modified model can precisely reflect the dynamic characteristics of dumbbell-shaped spacecraft.

State Estimation in a Hydraulically Actuated Log Crane Using Unscented Kalman Filter

Multibody system dynamics approaches together with state estimation methods can reduce the need for a large number of sensors, especially in the digital twin of working mobile machinery. To demonstrate this, a hydraulically actuated machine was modeled using the double-step semi-recursive multibody formulation and lumped fluid theory in terms of system independent states. Next, because of the high non-linearity of the modeled system and with respect to the reported performance degradation of the Extended Kalman Filters (EKF), which are mostly related to the linearization procedure of this filter, the Unscented Kalman Filter (UKF) was implemented to achieve high accuracy and performance. The methodology of the proposed approaches was applied to a mobile log crane model PATU 655. The implementation of the proposed estimation algorithms is demonstrated with three different multibody based simulation models: the synthetic real system producing the artificial measurements, the simulation model, and the estimation model. Encoders and pressure sensors, installed on the synthetic real system, provided synthetic sensor measurement data. To mimic real-world conditions, the estimation and simulation models used in the processing of the state estimation algorithm were assumed to have errors in the initial conditions and force model. The proposed UKF was applied to the estimation model with the synthetic sensor measurement data. The minimum percent normalized root mean square errors in the associated measured and unmeasured states of case example were 0.11% and 1.86%, respectively. The UKF using the multibody system dynamics formulations is able to estimate the case example states despite 15% and 60% errors in mass and inertial properties of bodies and Payload, respectively, confirming the accuracy and performance of the algorithm

Dynamic modeling, analysis, and comparative study of a quadruped with bio-inspired robotic tails

Looking to nature, animals frequently utilize tails to work alongside or in place of their legs to maneuver, stabilize, and/or propel to achieve highly agile motions. Although the single-link robotic tail shows its dynamical superiority and practical effectiveness in mobile platform maneuvering, most tails observed in nature have multi-link structures. Therefore, to investigate this novel tail structure, bio-inspired and biomimetic multi-link robotic tails were proposed and implemented. However, due to the lack of a whole-body dynamic model, previous research focused on investigating the tail subsystem independently without considering the mobile platform’s motions, which introduces deficiencies on both analysis and control. To bridge this theoretical gap, this paper presents a unified dynamics model that incorporates both the quadruped and the tail subsystems as a complete coupled dynamic system. Classical multibody dynamics formulation based on the principle of virtual work is utilized to derive the dynamic model. Based on the new whole-body dynamic model, three typical tail structures, including a single-link pendulum tail, a multi-link rigid tail, and a multi-link flexible tail are evaluated. The results indicate that by using a center of mass-based benchmark, the multi-link tail structure is dynamically equivalent to the single-link tail structure for bending motion. However, for rolling motions, the multi-link structure illustrates noticeable dynamical benefits compared to a single-link structure due to its higher inertia. In addition, a multi-link flexible structure shows significant oscillations and uncontrollable dynamic behaviors due to its under-actuation feature, which may limit its usage for highly dynamic applications.

Parallel manipulator dynamics embedded in singularity free domains of functionality

Ordinary differential equations of motion for a broad spectrum of parallel manipulators are derived and implemented in manipulator input coordinates for system simulation and controller design. Equations of motion are systematically embedded in singularity free domains of manipulator configuration space, assuring controllability and existence of a unique solution of the equations of motion. The kinematic foundation for the formulation is a mechanics-based model of the underlying mechanism with forward and inverse kinematic mappings that are evaluated on singularity free domains of manipulator configuration space. The kinetic foundation is d’Alembert’s variational formulation of multibody dynamics that, with forward kinematic mappings, yields explicit ordinary differential equations of motion. No ad-hoc derivations of Lagrange’s equations, Newton–Euler equations, or Lagrange multiplier-based differential-algebraic equations are required. Differential equations of motion for a planar front-end loader and a spatial Stewart platform manipulator are implemented to illustrate the formulation.

An efficient formulation for flexible multibody dynamics using a condensation of deformation coordinates

This paper introduces a more efficient formulation for flexible multibody dynamics based on the floating frame of reference formulation (FFRF). A two-step condensation method is proposed. The first condensation involves coordinate reduction using a modal transformation, from nodal coordinates to modal coordinates, in the context of model order reduction (MOR). The second condensation involves re-fabrication of the equations of motion using a dynamic condensation of the deformation modal coordinates obtained from the first condensation. In the first condensation, constraint modes are corrected by condensation of the truncated modes that represent the effects of residual flexibility. As a result, the first condensation offers a more accurate deformation field in the reduced order model, which consequently leads to a more accurate flexible multibody simulation. In the second condensation, the equations of motion of a flexible multibody are once more reduced to the equations of motion in terms of absolute coordinates for the body reference frames. In this way, the proposed method deals with smaller sized equations of motion than in the conventional FFRF, where they are represented in terms of both absolute coordinates and deformation modal coordinates. The effect of the first condensation was verified using a benchmark example of a slider–crank mechanism with a flexible connecting rod. The efficiency of the proposed condensation method was also investigated using a computational complexity analysis with an $n$ -flexible body pendulum.

Direct sensitivity analysis of multibody systems with holonomic and nonholonomic constraints via an index-3 augmented Lagrangian formulation with projections

Optimizing the dynamic response of mechanical systems is often a necessary step during the early stages of product development cycle. This is a complex problem that requires to carry out the sensitivity analysis of the system dynamics equations if gradient-based optimization tools are used. These dynamics equations are often expressed as a highly nonlinear system of ordinary differential equations or differential-algebraic equations, if a dependent set of generalized coordinates with its corresponding kinematic constraints is used to describe the motion. Two main techniques are currently available to perform the sensitivity analysis of a multibody system, namely the direct differentiation and the adjoint variable methods. In this paper, we derive the equations that correspond to the direct sensitivity analysis of the index-3 augmented Lagrangian formulation with velocity and acceleration projections. Mechanical systems with both holonomic and nonholonomic constraints are considered. The evaluation of the system sensitivities requires the solution of a tangent linear model that corresponds to the Newton–Raphson iterative solution of the dynamics at configuration level, plus two additional nonlinear systems of equations for the velocity and acceleration projections. The method was validated in the sensitivity analysis of a set of examples, including a five-bar linkage with spring elements, which had been used in the literature as benchmark problem for similar multibody dynamics formulations, a point-mass system subjected to nonholonomic constraints, and a full-scale vehicle model.

Constrained multi-body dynamics for modular underwater robots — Theory and experiments

This paper investigates the problem of modelling a system of interconnected underwater robots with highly coupled dynamics. The objective is to develop a mathematical description of the system that captures its most significant dynamics. The proposed modelling method is based on active constraint enforcement by utilising the Udwadia-Kalaba Formulation for multi-body dynamics. The required description of a rigid constraint is defined, derived and implemented into a system of interconnected sub-models. An exhaustive experimental validation is conducted on a two-vehicle system, including towing tank tests on a BlueROV vehicle to determine the model parameters. The applicability of the modelling approach is assessed by comparing experimental data to simulations of an equivalent model synthesised using the proposed theory.

An index 0 differential-algebraic equation formulation for multibody dynamics: Nonholonomic constraints

ABSTRACTA method is presented for formulating and numerically integrating index 0 differential-algebraic equations of motion for multibody systems with holonomic and nonholonomic constraints. Tangent space coordinates are defined in configuration and velocity spaces as independent generalized coordinates that serve as state variables in the formulation. Orthogonal dependent coordinates and velocities are used to enforce position, velocity, and acceleration constraints to within specified error tolerances. Explicit and implicit numerical integration algorithms are presented and used in solution of three examples: one planar and two spatial. Numerical results verify that accurate results are obtained, satisfying all three forms of kinematic constraint to within error tolerances embedded in the formulation.

Assessment of Linearization Approaches for Multibody Dynamics Formulations

Formulating the dynamics equations of a mechanical system following a multibody dynamics approach often leads to a set of highly nonlinear differential-algebraic equations (DAEs). While this form of the equations of motion is suitable for a wide range of practical applications, in some cases it is necessary to have access to the linearized system dynamics. This is the case when stability and modal analyses are to be carried out; the definition of plant and system models for certain control algorithms and state estimators also requires a linear expression of the dynamics. A number of methods for the linearization of multibody dynamics can be found in the literature. They differ in both the approach that they follow to handle the equations of motion and the way in which they deliver their results, which in turn are determined by the selection of the generalized coordinates used to describe the mechanical system. This selection is closely related to the way in which the kinematic constraints of the system are treated. Three major approaches can be distinguished and used to categorize most of the linearization methods published so far. In this work, we demonstrate the properties of each approach in the linearization of systems in static equilibrium, illustrating them with the study of two representative examples.

An Ordinary Differential Equation Formulation for Multibody Dynamics: Nonholonomic Constraints

A method is presented for formulating and numerically integrating ordinary differential equations of motion for nonholonomically constrained multibody systems. Tangent space coordinates are defined in configuration and velocity spaces as independent generalized coordinates that serve as state variables in the formulation, yielding ordinary differential equations of motion. Orthogonal-dependent coordinates and velocities are used to enforce constraints at position, velocity, and acceleration levels. Criteria that assure accuracy of constraint satisfaction and well conditioning of the reduced mass matrix in the equations of motion are used as the basis for updating local coordinates on configuration and velocity constraint manifolds, transparent to the user and at minimal computational cost. The formulation is developed for multibody systems with nonlinear holonomic constraints and nonholonomic constraints that are linear in velocity coordinates and nonlinear in configuration coordinates. A computational algorithm for implementing the approach is presented and used in the solution of three examples: one planar and two spatial. Numerical results using a fifth-order Runge–Kutta–Fehlberg explicit integrator verify that accurate results are obtained, satisfying all the three forms of kinematic constraint, to within error tolerances that are embedded in the formulation.

An index 0 Differential-Algebraic equation formulation for multibody dynamics: Holonomic constraints

ABSTRACTThe Lagrange multiplier form of index 3 differential-algebraic equations of motion for holonomically constrained multibody systems is transformed using tangent space generalized coordinates to an index 0 form that is equivalent to an ordinary differential equation. The index 0 formulation includes embedded tolerances that assure satisfaction of position, velocity, and acceleration constraints and is solved using established explicit and implicit numerical integration methods. Numerical experiments with two spatial applications show that the formulation accurately satisfies constraints, preserves invariants due to conservation laws, and behaves as if applied to an ordinary differential equation.