Energy-momentum conserving integration of multibody dynamics

A rotationless formulation of multibody dynamics is presented, which is especially beneficial to the design of energy-momentum conserving integration schemes. The proposed approach facilitates the stable numerical integration of the differential algebraic equations governing the motion of both open-loop and closed-loop multibody systems. A coordinate augmentation technique for the incorporation of rotational degrees of freedom and associated torques is newly proposed. Subsequent to the discretization, size-reductions are performed to lower the computational costs and improve the numerical conditioning. In this connection, a new approach to the systematic design of discrete null space matrices for closed-loop systems is presented. Two numerical examples are given to evaluate the numerical properties of the proposed algorithms.     

Absolute nodal coordinate plane beam formulation for multibody systems dynamics

A new plane beam dynamic formulation for constrained multibody system dynamics is developed. Flexible multibody system dynamics includes rigid body dynamics and superimposed vibratory motions. The complexity of mechanical system dynamics originates from rotational kinematics, but the natural coordinate formulation does not use rotational coordinates, so that simple dynamic formulation is possible. These methods use only translational coordinates and simple algebraic constraints. A new formulation for plane flexible multibody systems are developed utilizing the curvature of a beam and point masses. Using absolute nodal coordinates, a constant mass matrix is obtained and the elastic force becomes a nonlinear function of the nodal coordinates. In this formulation, no infinitesimal or finite rotation assumptions are used and no assumption on the magnitude of the element rotations is made. The distributed body mass and applied forces are lumped to the point masses. Closed loop mechanical systems consisting of elastic beams can be modeled without constraints since the loop closure constraints can be substituted as beam longitudinal elasticity. A curved beam is modeled automatically. Several numerical examples are presented to show the effectiveness of this method.     

Quadratic and Higher-Order Constraints in Energy-Conserving Formulations of Flexible Multibody Systems

The treatment of constraints is considered here within the framework ofenergy-momentum conserving formulations for flexible multibody systems.Constraint equations of various types are an inherent component of multibodysystems, their treatment being one of the key performance features ofmathematical formulations and numerical solution schemes.Here we employ rotation-free inertial Cartesian coordinates of points tocharacterise such systems, producing a formulation which easily couples rigidbody dynamics with nonlinear finite element techniques for the flexiblebodies. This gives rise to additional internal constraints in rigid bodies topreserve distances. Constraints are enforced via a penalty method, which givesrise to a simple yet powerful formulation. Energy-momentum time integrationschemes enable robust long term simulations for highly nonlinear dynamicproblems.The main contribution of this paper focuses on the integration of constraintequations within energy-momentum conserving numerical schemes. It is shownthat the solution for constraints which may be expressed directly in terms ofquadratic invariants is fairly straightforward. Higher-order constraints mayalso be solved, however in this case for exact conservation an iterativeprocedure is needed in the integration scheme. This approach, together withsome simplified alternatives, is discussed.Representative numerical simulations are presented, comparing the performanceof various integration procedures in long-term simulations of practicalmultibody systems.     

Controllability and motion planning of a multibody Chaplygin's sphere and Chaplygin's top

This paper studies local configuration controllability of multibody systems with nonholonomic constraints. As a nontrivial example of the theory, we consider the dynamics and control of a multibody spherical robot. Internal rotors and sliders are used as the mechanisms for control. Our model is based on equations developed by the second author for certain mechanical systems with nonholonomic constraints, e.g. Chaplygin's sphere and Chaplygin's top in particular, and the multibody framework for unconstrained mechanical systems developed by the first and third authors. Recent methods for determining controllability and path planning for multibody systems with symmetry are extended to treat a class of mechanical systems with nonholonomic constraints. Specific results on the controllability and path planning of the spherical robot model are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Orienting body coordinate frames using Karhunen–Loève expansion for more effective structural simplification

As part of an effort to create an automated modular modeling environment, previous work by the authors developed a junction-inactivity-based structural simplification technique that is particularly suitable for bond-graph models. The technique is highly sensitive to the orientation of the body coordinate frames in multibody systems: improper alignment of body coordinate frames may prohibit a significant simplification. This paper demonstrates how the Karhunen–Loève expansion can be used to automatically detect the existence of and to find the transformation into body coordinate frames that render the bond graph of a multibody system more conducive to simplification. The proposed technique is demonstrated using the simple example of a 3D pendulum constrained to move in a plane, but is applicable to arbitrarily complex multibody dynamics problems. The conclusion is that the Karhunen–Loève expansion successfully complements the junction-inactivity-based structural simplification technique when multibody dynamics are involved in the system, and thus significantly contributes to the development of an automated modular modeling environment.     

A DAE Approach to Flexible Multibody Dynamics

The present work deals with the dynamics of multibody systems consisting ofrigid bodies and beams. Nonlinear finite element methods are used to devise a frame-indifferent spacediscretization of the underlying geometrically exact beam theory. Both rigid bodies and semi-discrete beams are viewed as finite-dimensional dynamical systems with holonomic constraints. The equations of motion pertaining to the constrained mechanical systems under considerationtake the form of Differential Algebraic Equations (DAEs).The DAEs are discretized directly by applying a Galerkin-based method.It is shown that the proposed DAE approach provides a unified framework for the integration of flexible multibody dynamics.     

Non-linear dynamics of flexible multibody systems

In this work we set to examine several important issues pertinent to currently very active research area of the finite element modeling of flexible multibody system dynamics. To that end, we first briefly introduce three different model problems in non-linear dynamics of flexible 3D solid, a rigid body and 3D geometrically exact beam, which covers the vast majority of representative models for the particular components of a multibody system. The finite element semi-discretization for these models is presented along with the time-discretization performed by the mid-point scheme. In extending the proposed methodology to modeling of flexible multibody systems, we also present how to build a systematic representation of any kind of joint connecting two multibody components, a typical case of holonomic contraint, as a linear superposition of elementary constraints. We also indicate by a chosen model of rolling contact, an example of non-holonomic constraint, that the latter can also be included within the proposed framework. An important aspect regarding the reduction of computational cost while retaining the consistency of the model is also addressed in terms of systematic use of the rigid component hypothesis, mass lumping and the appropriate application of the explicit-implicit time-integration scheme to the problem on hand. Several numerical simulations dealing with non-linear dynamics of flexible multibody systems undergoing large overall motion are presented to further illustrate the potential of presented methodology. Closing remarks are given to summarize the recent achievements and point out several directions for future research.     

Task-level approaches for the control of constrained multibody systems

This paper presents a task-level control methodology for the general class of holonomically constrained multibody systems. As a point of departure, the general formulation of constrained dynamical systems is reviewed with respect to multiplier and minimization approaches. Subsequently, the operational space framework is considered and the underlying symmetry between constrained dynamics and operational space control is discussed. Motivated by this symmetry, approaches for constrained task-level control are presented which cast the general formulation of constrained multibody systems into a task space setting using the operational space framework. This provides a means of exploiting task-level control structures, native to operational space control, within the context of constrained systems. This allows us to naturally synthesize dynamic compensation for a multibody system, that properly accounts for the system constraints while performing a control task. A set of examples illustrate this control implementation. Additionally, the inclusion of flexible bodies in this approach is addressed.     

Real-time inverse dynamics control of parallel manipulators using general-purpose multibody software

This work deals with the problem of computing the inverse dynamics of complex constrained mechanical systems for real-time control applications. The main goal is the control of robotic systems using model-based schemes in which the inverse model itself is obtained using a general purpose multibody software, exploiting the redundant coordinate formalism. The resulting control scheme is essentially equivalent to a classical computed torque control, commonly used in robotics applications. This work proposes to use modern general-purpose multibody software to compute the inverse dynamics of complex rigid mechanisms in an efficient way, so that it suits the requirements of realistic real-time applications as well. This task can be very difficult, since it involves a higher number of equations than the relative coordinates approach. The latter is believed to be less general, and may suffer from topology limitations. The use of specialized linear algebra solvers makes this kind of control algorithms usable in real-time for mechanism models of realistic complexity. Numerical results from the simulation of practical applications are presented, consisting in a “delta” robot and a bio-mimetic 11 degrees of freedom manipulator controlled using the same software and the same algorithm.     

Modeling and analysis of planar rigid multibody systems with translational clearance joints based on the non-smooth dynamics approach

The main purpose of this paper is to present and discuss a methodology for a dynamic modeling and analysis of rigid multibody systems with translational clearance joints. The methodology is based on the non-smooth dynamics approach, in which the interaction of the elements that constitute a translational clearance joint is modeled with multiple frictional unilateral constraints. In the following, the most fundamental issues of the non-smooth dynamics theory are revised. The dynamics of rigid multibody systems are stated as an equality of measures, which are formulated at the velocity-impulse level. The equations of motion are complemented with constitutive laws for the normal and tangential directions. In this work, the unilateral constraints are described by a set-valued force law of the type of Signorini’s condition, while the frictional contacts are characterized by a set-valued force law of the type of Coulomb’s law for dry friction. The resulting contact-impact problem is formulated and solved as a linear complementarity problem, which is embedded in the Moreau time-stepping method. Finally, the classical slider-crank mechanism is considered as a demonstrative application example and numerical results are presented. The results obtained show that the existence of clearance joints in the modeling of multibody systems influences their dynamics response.     

Analysis of Large Flexible Body Deformation in Multibody Systems Using Absolute Coordinates

To consider large deformation problems in multibody system simulations afinite element approach, called absolute nodal coordinate.formulation,has been proposed. In this formulation absolute nodal coordinates andtheir material derivatives are applied to represent both deformation andrigid body motion. The choice of nodal variables allows a fullynonlinear representation of rigid body motion and can provide the exactrigid body inertia in the case of large rotations. The methodology isespecially suited for but not limited to modeling of beams, cables andshells in multibody dynamics.This paper summarizes the absolute nodal coordinate formulation for a 3D Euler–Bernoulli beam model, in particular the definition of nodal variables, corresponding generalized elastic and inertia forces and equations of motion. The element stiffness matrix is a nonlinear function of the nodal variables even in the case of linearized strain/displacement relations. Nonlinear strain/displacement relations can be calculated from the global displacements using quadrature formulae.Computational examples are given which demonstrate the capabilities of the applied methodology. Consequences of the choice of shape.functions on the representation of internal forces are discussed. Linearized strain/displacement modeling is compared to the nonlinear approach and significant advantages of the latter, when using the absolute nodal coordinate formulation, are outlined.     

Flexible Multibody Dynamics: Review of Past and Recent Developments

In this paper, a review of past and recent developments in the dynamics of flexible multibody systems is presented. The objective is to review some of the basic approaches used in the computer aided kinematic and dynamic analysis of flexible mechanical systems, and to identify future directions in this research area. Among the formulations reviewed in this paper are the floating frame of reference formulation, the finite element incremental methods, large rotation vector formulations, the finite segment method, and the linear theory of elastodynamics. Linearization of the flexible multibody equations that results from the use of the incremental finite element formulations is discussed. Because of space limitations, it is impossible to list all the contributions made in this important area. The reader, however, can find more references by consulting the list of articles and books cited at the end of the paper. Furthermore, the numerical procedures used for solving the differential and algebraic equations of flexible multibody systems are not discussed in this paper since these procedures are similar to the techniques used in rigid body dynamics. More details about these numerical procedures as well as the roots and perspectives of multibody system dynamics are discussed in a companion review by Schiehlen [79]. Future research areas in flexible multibody dynamics are identified as establishing the relationship between different formulations, contact and impact dynamics, control-structure interaction, use of modal identification and experimental methods in flexible multibody simulations, application of flexible multibody techniques to computer graphics, numerical issues, and large deformation problem. Establishing the relationship between different flexible multibody formulations is an important issue since there is a need to clearly define the assumptions and approximations underlying each formulation. This will allow us to establish guidelines and criteria that define the limitations of each approach used in flexible multibody dynamics. This task can now be accomplished by using the absolute nodal coordinate formulation which was recently introduced for the large deformation analysis of flexible multibody systems.     

力学与控制科学

基于对力学与控制科学的相似性的分析,阐述了4个有意义的力学系统控制问题:① 具非完整约束的非线性动力学系统的控制;②具非单一平衡位置的力学系统及其总体性质;③ 多体动力学与控制;④具分布参数的力学系统.最后给出了一个具启示性的看法.     

Developments of multibody system dynamics: computer simulations and experiments

It is an exceptional success when multibody dynamics researchers Multibody System Dynamics journal one of the most highly ranked journals in the last 10 years. In the inaugural issue, Professor Schiehlen wrote an interesting article explaining the roots and perspectives of multibody system dynamics. Professor Shabana also wrote an interesting article to review developments in flexible multibody dynamics. The application possibilities of multibody system dynamics have grown wider and deeper, with many application examples being introduced with multibody techniques in the past 10 years. In this paper, the development of multibody dynamics is briefly reviewed and several applications of multibody dynamics are described according to the author’s research results. Simulation examples are compared to physical experiments, which show reasonableness and accuracy of the multibody formulation applied to real problems. Computer simulations using the absolute nodal coordinate formulation (ANCF) were also compared to physical experiments; therefore, the validity of ANCF for large-displacement and large-deformation problems was shown. Physical experiments for large deformation problems include beam, plate, chain, and strip. Other research topics currently being carried out in the author’s laboratory are also briefly explained. Commemorative Contribution.     

Complex Flexible Multibody Systems with Application to Vehicle Dynamics

A formulation to describe the linear elastodynamics offlexible multibody systems is presented in this paper. By using a lumpedmass formulation the flexible body mass is represented by a collectionof point masses with rotational inertia. Furthermore, the bodydeformations are described with respect to a body-fixed coordinateframe. The coupling between the flexible body deformation and its rigidbody motion is completely preserved independently of the methods used todescribe the body flexibility. In particular, if the finite elementmethod is chosen for this purpose only the standard finite elementparameters obtained from any commercial finite element code are used inthe methodology. In this manner, not only the analyst can use any typeof finite elements in the multibody model but the same finite elementmodel can be used to evaluate the structural integrity of any systemcomponent also. To deal with complex-shaped structural models offlexible bodies it is necessary to reduce the number of generalizedcoordinates to a reasonable dimension. This is achieved with thecomponent mode synthesis at the cost of specializing the formulation toflexible multibody models experiencing linear elastic deformations only.Structural damping is introduced to achieve better numerical performancewithout compromising the quality of the results. The motions of therigid body and flexible body reference frames are described usingCartesian coordinates. The kinematic constraints between the differentsystem components are evaluated in terms of this set of generalizedcoordinates. The equations of motion of the flexible multibody systemare solved by using the augmented Lagrangean method and a sparse matrixsolver. Finally, the methodology is applied to model a vehicle with acomplex flexible chassis, simulated in typical handling scenarios. Theresults of the simulations are discussed in terms of their numericalprecision and efficiency.     

Nonholonomic Multibody Dynamics

The paper deals with the nonholonomic multibody system dynamics from apoint of view which is caused by some actual applications in high-tecareas like high-speed train technology or biomechanics of somedisciplines in high-performance sports. Obviously, looking at suchproblems, there are very close connections between classical analyticaldynamics, differential geometry and modern control theory. But theseconnections cannot be used to get new composed results in solvingcomplicated problems of multibody system dynamics because correspondingsoftware tools are not enough in tune with each other. This paper willgive some ideas for developing a unified basis for modeling, simulationand control of nonholonomic multibody systems.First, a derivative-free approach for generating Lagrangian motionequations of multibody systems with kinematical tree structure as wellas for constrained multibody systems is given. This has been done byusing differential-geometric concepts in a Riemannian space. Secondly,the well-known theorem of Frobenius is considered with respect to itsclassical interpretation by the so-called object of nonholonomy as wellas by its modern interpretation in the nonlinear control theory usingLie-brackets. The ideas are illustrated by the classical rollingcondition and edge condition on double-curved surfaces. Specialnumerical problems in simulation of multibody systems subject toadditional kinematic constraints are discussed. Finally threeapplications are given.