Multibody Algorithms Research Articles

A Musculoskeletal Multibody Algorithm Based on a Novel Rheonomic Constraints Definition Applied to the Lower Limb.

In this paper, a multibody model was developed in the framework of biotribology of lower limb artificial joints. The presented algorithm performs the inverse dynamics of musculoskeletal systems with the aim to achieve a tool for the calculation of the joint reaction forces. The revolute joint, the cam joint, the spherical joint and the free joint were considered in the analyzed lower limb system by introducing a novel analytical formulation of the rheonomic constraint equations based on the quaternions theory. Within the kinematical analysis, the curved muscle paths were modeled by simulating their geodesic wrapping over bony surfaces while the muscle actuations were formulated through the Hill muscle model. The developed theoretical model was developed in matlab environment allowing to follow the classical musculoskeletal analysis pipeline: kinematical analysis, inverse dynamics, and static optimization, applied to the lower limb during the gait kinematics. The validation of the results was obtained by comparing the calculated hip joint reactions with the ones obtained in vivo by Bergmann and calculated by Opensim software, showing a satisfactory agreement. The proposed model and algorithm represent a fully open and controllable synovial joint tribological configuration generator tool, useful to be coupled with numerical lubrication/contact models in the framework of the in silico artificial joints tribological optimization.

Absolute nodal coordinate particle finite element to the free-surface flow problems combined with multibody algorithms

In this paper, a fully Lagrangian formulation for solving rigid–liquid coupling system with free surface is proposed, in which the spatial absolute nodal coordinates and the pressure of liquid are taken as the independent variables, combining the stabilized particle finite element method and multibody algorithms for unified modeling. The virtual contact elements generated in the remeshing process are used to identify the contact nodes on the liquid–solid interface, and the free-slip boundary constraints are imposed by the Lagrange multipliers. For the position deviation of the interface nodes on the concave-curved boundary at a large time step, a position correction strategy is given. In addition, in order to control the numerical dissipation in the simulation, the implicit generalized-α time integration strategy with second-order accuracy is used to solve the differential algebraic equations of the system. Finally, several numerical examples are studied to verify the mass conservation characteristics of the algorithm under coarse mesh and relatively large time steps, and the pressure calculation results are compared with the experimental results reported in the literature, which shows the stability and accuracy of the algorithms proposed.

Modia3D: Modeling and Simulation of 3D-Systems in Julia

Modia3D is an experimental Julia package to model and simulate 3D mechanical systems. Ideas from modern game engines are used to achieve a highly flexible setup and features of multi-body algorithms are used to get a rigid mathematical formulation and support, for example, of closed kinematic loops. Collision handling is performed on convex geometries with elastic response calculation. A Modia3D model is solved with a variable-step solver. This is important to combine Modia3D with the equation-based modeling system Modia in the future.

On the Computational Methods for Solving the Differential-Algebraic Equations of Motion of Multibody Systems

In this investigation, different computational methods for the analytical development and the computer implementation of the differential-algebraic dynamic equations of rigid multibody systems are examined. The analytical formulations considered in this paper are the Reference Point Coordinate Formulation based on Euler Parameters (RPCF-EP) and the Natural Absolute Coordinate Formulation (NACF). Moreover, the solution approaches of interest for this study are the Augmented Formulation (AF) and the Udwadia–Kalaba Equations (UKE). As shown in this paper, the combination of all the methodologies analyzed in this work leads to general, effective, and efficient multibody algorithms that can be readily implemented in a general-purpose computer code for analyzing the time evolution of mechanical systems constrained by kinematic joints. This study demonstrates that multibody algorithm based on the combination of the NACF with the UKE turned out to be the most effective and efficient computational method. The conclusions drawn in this paper are based on the numerical results obtained for a benchmark multibody system analyzed by means of dynamical simulations.

Hybrid complementarity formulations for robotics applications

AbstractThe focus of this paper is to review hybrid recursive‐complementarity formulations for multibody systems characterized by a large number of bilateral constraints which are frequently encountered in robotics. Here, hybrid implies the use of complementarity contact models with recursive forward dynamics schemes. Such formulations have a common underlying structure which can be applied to multibody systems with a constrained tree‐type topology. These common steps are pointed out. Theoretical formulation is given for systems using three important classes (O(n3), O(n), and O(log (n))) of multibody algorithms. Further, numerical comparison for the efficiency is given for rigid multibody systems. The paper makes recommendations on the choice of hybrid complementarity formulations which would result in the most efficient simulation. The paper further gives a non‐iterative approach to allow the use of explicit higher order integrators for frictionless contacts. The difficulties in extending this approach to allow for frictional contact are also discussed.

Use of General Nonlinear Material Models in Beam Problems: Application to Belts and Rubber Chains

Accurate modeling of many engineering systems requires the integration of multibody system and large deformation finite element algorithms that are based on general constitutive models, account for the coupling between the large rotation and deformation, and allow capturing coupled deformation modes that cannot be captured using beam formulations implemented in existing computational algorithms and computer codes. In this investigation, new three-dimensional nonlinear dynamic rubber chains and belt drives models are developed using the finite element absolute nodal coordinate formulation (ANCF) that allows for a straight forward implementation of general linear and nonlinear material models for structural elements such as beams, plates, and shells. Furthermore, this formulation, which is based on a more general kinematic description, can be used to predict the cross section deformation and its coupling with the extension and bending of the belt drives and rubber chains. The ANCF cross section deformation results are validated by comparison with the results obtained using solid finite elements in the case of a simple tension test problem. The effect of the use of different linear and nonlinear constitutive laws in modeling belt drive mechanisms is also examined in this investigation. The finite element formulation presented in this paper is implemented in a general purpose three-dimensional flexible multibody algorithm that allows for developing detailed models of mechanical systems subject to general loading conditions, nonlinear algebraic constraint equations, and arbitrary large displacements that characterize belt drives and tracked vehicle dynamics. The successful integration of large deformation finite element and multibody system algorithms is shown to be necessary in order to be able to study the dynamics of complex tracked vehicles with rubber chains. A computer simulation of a three-dimensional multibody tracked vehicle model that consists of twenty rigid bodies and two flexible rubber chains is used in order to demonstrate the use of the formulations presented in this investigation.

Nonlinear dynamics of three-dimensional belt drives using the finite-element method

In this paper, new nonlinear dynamic formulations for belt drives based on the three-dimensional absolute nodal coordinate formulation are developed. Two large deformation three-dimensional finite elements are used to develop two different belt-drive models that have different numbers of degrees of freedom and different modes of deformation. Both three-dimensional finite elements are based on a nonlinear elasticity theory that accounts for geometric nonlinearities due to large deformation and rotations. The first element is a thin-plate element that is based on the Kirchhoff plate assumptions and captures both membrane and bending stiffness effects. The other three-dimensional element used in this investigation is a cable element obtained from a more general three-dimensional beam element by eliminating degrees of freedom which are not significant in some cable and belt applications. Both finite elements used in this investigation allow for systematic inclusion or exclusion of the bending stiffness, thereby enabling systematic examination of the effect of bending on the nonlinear dynamics of belt drives. The finite-element formulations developed in this paper are implemented in a general purpose three-dimensional flexible multibody algorithm that allows for developing more detailed models of mechanical systems that include belt drives subject to general loading conditions, nonlinear algebraic constraints, and arbitrary large displacements. The use of the formulations developed in this investigation is demonstrated using two-roller belt-drive system. The results obtained using the two finite-element formulations are compared and the convergence of the two finite-element solutions is examined.

The Theory and Application of a Classical Eulerian Multibody Code

The topic of multibody analysis deals with the automatic generation and subsequent solution of the equations of motion for a system of interconnected bodies. Many academic and industrial computer programs have been developed to carry out this task. However, although some of these programs can obtain the equations of motion in fully symbolic form, it is believed that all the existing programs for multibody analysis solve these equations numerically. The idea behind the research which underpins this paper is to select and implement a symbolic version of an existing multibody algorithm and then to integrate it with a recently developed solver which can obtain approximate analytical solutions to the equations of motion. The present paper deals with the selection of this multibody method. The theory behind the chosen method (the Roberson and Schwertassek algorithm) is described in some detail but also in a much more concise form than can be found in the literature. In addition, a fully worked example of the application of the multibody algorithm to a practical physical problem is given. Such examples are rare in the literature, and so it is intended that this paper can serve as a basis for enhanced didactic practice in this traditionally difficult subject area.

A State?Time Formulation for Dynamic Systems Simulation Using Massively Parallel Computing Resources

A novel state–time (ST) formulation for the simulation and analysis of the dynamic behavior of complex multibody systems is presented. The method proposes a computationally fast algorithm which is better able to fully exploit anticipated future immensely parallel computing resources (e.g. pecta flop machines and beyond) than existing multibody algorithms. The intent of the algorithm is to yield significantly reduced simulation turnaround time in situations where massively parallel (>106 processors) computing resources are available to it. It is shown that as a consequence of such a ST discretization scheme, the system of governing equations yields a set of loosely coupled nonlinear algebraic equations which is at most quadratic in the ST variables, with significant linear components. As such, it is well-suited in structure for nonlinear algebraic equations solvers. The linear-quadratic (LQ) structure of these equations further permits the use of a special solution scheme, which is expected to yield superior performance relative to more traditional Newton–Raphson type schemes when applied to large general systems.

A Parallel Logarithmic Order Algorithm for General Multibody System Dynamics

A novel optimal order optimal resource parallel multibody algorithm with general system applicability is derived directly from the sequential recursive \(\mathcal{O}(n)\) methods and the most recent developments in recursive constraint treatments. This new Recursive Coordinate Reduction Parallelism (RCRP) is the first optimal order \({\text{(}}\mathcal{O}(\log _2 n))\) parallel direct method with a sequential implementation that is exactly the efficient \({\text{(}}\mathcal{O}{\text{(}}n)\) algorithm. Consequently, the RCRP sets new benchmarks for performance over a wide range of problem size and parallel resources. Comparisons to existing methods also demonstrate that the RCRP is presently the best general parallel method.

A Non-Incremental Nonlinear Finite Element Solution for Cable Problems

In this investigation, a nonlinear finite element method for the large deformation and rotation of cable problems is presented. This method is based on finite element absolute nodal coordinate formulation that guarantees the continuity of all displacement gradients and leads to a constant mass matrix. The classical cable theory is first reviewed and the assumptions used in this linear theory are defined in order to demonstrate the basic differences between the linear theory and the nonlinear finite element formulation proposed in this paper for cable applications. The elastic cable forces in the absolute nodal coordinate formulation are obtained using a general continuum mechanics approach that accounts for the effect of all geometric nonlinearities. It is shown in this investigation that the use of the general continuum mechanics approach leads to a simpler and more efficient formulation as compared to the use of the assumptions of the linear theory that employs a local finite element coordinate system. The results obtained using the absolute nodal coordinate formulation show a good agreement with the results obtained using the classical cable theory when linear cable problems are considered. In particular it is shown that the use of perturbation methods to linearize the finite element equations of motion leads to modal characteristics results that are in a good agreement with the linear theory. The results of this investigation obtained using explicit numerical integration also show the potential of the proposed finite element formulation in the nonlinear analysis of cables that experience large rotations and deformations. The generalization of the procedure presented in this paper to three-dimensional cable problems is demonstrated and the computer implementation in multibody algorithms is discussed.

Dynamic modeling in the simulation, optimization, and control of bipedal and quadrupedal robots

AbstractFundamental principles and recent methods for investigating the nonlinear dynamics of legged robot motions with respect to control, stability, and design are discussed. One of them is the still challenging problem of producing dynamically stable gaits. The generation of fast walking or running motions requires methods and algorithms adept at handling the nonlinear dynamical effects and stability issues which arise. Reduced, recursive multibody algorithms, a numerical optimal control method, and new stability and energy performance indices are presented which are well‐suited for this purpose. Difficulties and open problems are discussed along with numerical investigations into the proposed gait generation scheme. Our analysis considers both bipedal and quadrupedal gaits.

INCREASING STABILITY IN DYNAMIC GAITS USING NUMERICAL OPTIMIZATION

Optimal gait planning is applied in this work to the problem of improving stability in quadruped locomotion. In many settings, it is desired to operate legged machines at high performance levels where rapid velocities and a changing environment make stability of utmost concern. Since gait planning still remains a vital component of legged system control design, an efficient method of determining periodic paths is presented which optimize a dynamic stability criterion. Efficient recursive multibody algorithms are used with numerical optimal control software to solve the minimax performance stability criteria.

A Survey of Rail Vehicle Track Simulations and Flexible Multibody Dynamics

The important development in rolling contact theory withthe recent advances made in the field of flexible multibody dynamics canserve as the foundation for developing new analysis and designcomputational methods for railroad vehicle/track systems. Multibody computational methods can be used tosimulate the dynamic effects due to the structural flexibility of thevehicle components and the track, multiple coupled railway carsnegotiating variable track geometry, derailments resulting from gagewidening, three-dimensional wheel/rail contact, vehicle trackinteraction, and vehicle dynamics under high operating speeds. Asdiscussed in this review paper, based on the new development in rollingcontact theory, one can predict the contact forces between the wheel andthe rail when realistic kinematic and dynamic descriptions are providedat the contact points. Basic quantities such as the longitudinal,lateral and spin creepages are essential for the accurate prediction ofthe contact forces. Nonetheless, in many investigations, linearized orpartially linearized kinematic expressions are used to define thecreepages. Using new multibody algorithms which allowfor the analysis of nonlinear models, the linearization schemescurrently employed in railroad vehicle/track research can be evaluated.Other related issues considered in thissurvey article include wheel/track interaction models, safetyconsideration, experimental verification of simulation results, andapplications which can significantly benefit from adopting the moregeneral computational flexible multibody techniques. As oneof the main objectives of this survey article is to shed more light onthe limitations of some of the current simulation tools used by railroadindustry and at the same time demonstrate the benefits of adoptingcomputational flexible multibody methodologies in railroad vehicle/tracksimulations, some future research areas are identified in the Summaryand Conclusions section.

An Augmented Formulation for Mechanical Systems with Non-Generalized Coordinates: Application to Rigid Body Contact Problems

In computational multibody algorithms, the kinematic constraintequations that describe mechanical joints and specified motiontrajectories must be satisfied at the position, velocity andacceleration levels. For most commonly used constraint equations, onlyfirst and second partial derivatives of position vectors with respect tothe generalized coordinates are required in order to define theconstraint Jacobian matrix and the first and second derivatives of theconstraints with respect to time. When the kinematic and dynamicequations of the multibody systems are formulated in terms of a mixedset of generalized and non-generalized coordinates, higher partialderivatives with respect to these non-generalized coordinates arerequired, and the neglect of these derivatives can lead to significanterrors. In this paper, the implementation of a contact model in generalmultibody algorithms is presented as an example of mechanical systemswith non-generalized coordinates. The kinematic equations that describethe contact between two surfaces of two bodies in the multibody systemare formulated in terms of the system generalized coordinates and thesurface parameters. Each contact surface is defined using twoindependent parameters that completely define the tangent and normalvectors at an arbitrary point on the body surface. In the contact modeldeveloped in this study, the points of contact are searched for on lineduring the dynamic simulation by solving the nonlinear differential andalgebraic equations of the constrained multibody system. It isdemonstrated in this paper that in the case of a point contact andregular surfaces, there is only one independent generalized contactconstraint force despite the fact that five constraint equations areused to enforce the contact conditions.

APPLICATION OF MULTIBODY METHODOLOGY TO ROTATING SHAFT PROBLEMS

The equations of motion presented in the literature for a circular shaft rotating about its centroidal axis assume a constant angular velocity and include a single coupling effect resulting from the Coriolis inertia force component of the gyroscopic moment. The objective of this investigation is to address the limitations of the classical equations of motion and provide a comprehensive model for a shaft rotating about its axis at an arbitrary angular velocity. The general equations of motion for a flexible body are derived through the application of the principle of virtual work in dynamics and are tailored to the specific case of the rotating shaft problem. The equations are shown to include both the Coriolis and centrifugal inertia forces, and the effect of the inertia terms on the system dynamic stability is demonstrated. The effect of the rotary inertia on the axial and transverse deformations is formulated and the coupling terms are obtained. The case of a shaft rotating with a non-constant angular velocity is also examined and the effect of the angular acceleration on the stability of the shaft is addressed. The generality of the approach presented in this study is further demonstrated by considering the dynamics of a rotating shaft subject to a base excitation. The coupling between the base motion and the deformation of the shaft is examined numerically and the effect of the support motion on the dynamics of the shaft is discussed for both a low level and high level disturbance. The results presented in this investigation demonstrate that the general flexible body formulation can be used to study rotating shafts. As a consequence, general purpose flexible multibody computer algorithms can be used to systematically solve more general rotating shaft problems.

Wittenburg's formulation of multibody dynamics equations from a graph-theoretic perspective

In the past 20 years, many multibody or “self-formulating” computer programs have been developed for the simulation of dynamic mechanical systems. These programs are based on algorithms that enable the computer to formulate, automatically, the equations of motion for a specified system, given only a description of the system as input; numerical methods can be applied to the resulting differential-algebraic equations in order to determine the time response of the system. Whether explicitly or implicitly, all of these multibody algorithms use concepts from linear graph theory in some form or another. The “vector-network technique” represents a direct application of graph-theoretic methods to dynamic mechanical systems. In contrast, Wittenburg's well-known formalism for creating the equations of motion appears, at first glance, to use linear graph theory for the sole purpose of representing the system topology. The goal of this paper is to examine in detail the relationship between these two methodologies. In particular, the authors will establish the equivalence of the equations derived using Wittenburg's formulation, and those arising out of a formal graph-theoretic approach.