Elimination of Constraint Violation and Accuracy Aspects in Numerical Simulation of Multibody Systems

Multibody systems are often modeled as constrained systems, and theconstraint equations are involved in the dynamics formulations. To makethe arising governing equations more tractable, the constraint equationsare differentiated with respect to time, and this results in unstablenumerical solutions which may violate the lower-order constraintequations. In this paper we develop a methodology for numerically exactelimination of the constraint violations, based on appropriatecorrections of the state variables (after each integration step) withoutany modification in the motion equations. While the elimination ofviolation of position constraints may require few iterations, theviolation of velocity constraints is removed in one step. The totalenergy of the system is sometimes treated as another measure of theintegration process inaccuracy. An improved scheme for one-stepelimination of the energy constraint violation is proposed as well. Theconclusion of this paper is, however, that the energy conservation is ofminor importance as concerns the improvement of accuracy of numericalsimulations. Some test calculations are reported.     

Collocation Methods for the Investigation of Periodic Motions of Constrained Multibody Systems

The investigation of periodic motions of constrained multibodysystems requires the numerical solution of differential-algebraicboundary value problems. After briefly surveying the basics of periodicmotion analysis the paper presents an extension of projected collocationmethods [6] to a special class of boundary value problems for multibodysystem equations with position and velocity constraints. These methodscan be applied for computing stable as well as unstable periodicmotions. Furthermore they provide stability information, which can beused to detect bifurcations on periodic branches. The special class ofequations stemming from contact problems like in railroad systems [22]can be handled as well. Numerical experiments with a wheelset modeldemonstrate the performance of the algorithms.     

Constraint Forces and Undetermined Multipliers in Constrained Multibody Systems

This paper addresses the relation between constraining forces, constraint equations, and undetermined multipliers, used in studying constrained multibody systems. A formalism is developed establishing the undetermined multiplier method through the concept of a generalized constraint force, partial velocities, and kinematic constraint equations.     

Energy Preserving Time Integration for Constrained Multibody Systems

This paper describes an energy preserving integration method for thedynamic analysis of nonlinear multibody systems in the presence ofnonlinear constraints. The aspects of finite rotation incrementation,discrete energy conservation and the most common constraint types areexamined in detail. The application is made to several rigid bodyexamples, including an intermittent contact problem.     

Null Space Integration Method for Constrained Multibody Systems with No Constraint Violation

A method for integrating equations of motion of constrained multibodysystems with no constraint violation is presented. A mathematical model,shaped as a differential-algebraic system of index 1, is transformedinto a system of ordinary differential equations using the null-spaceprojection method. Equations of motion are set in a non-minimal form.During integration, violations of constraints are corrected by solvingconstraint equations at the position and velocity level, utilising themetric of the system's configuration space, and projective criterion to thecoordinate partitioning method. The method is applied to dynamicsimulation of 3D constrained biomechanical system. The simulation resultsare evaluated by comparing them to the values of characteristicparameters obtained by kinematic analysis of analyzed motion based onmeasured kinematic data.     

Nonminimal Kane's Equations of Motion for Multibody Dynamical Systems Subject to Nonlinear Nonholonomic Constraints

Nonholonomic constraint equations that are nonlinear in velocities are incorporated with Kane's dynamical equations by utilizing the acceleration form of constraints, resulting in Kane's nonminimal equations of motion, i.e. the equations that involve the full set of generalized accelerations. Together with the kinematical differential equations, these equations form a state-space model that is full-order, separated in the derivatives of the states, and involves no Lagrange multipliers. The method is illustrated by using it to obtain nonminimal equations of motion for the classical Appell–Hamel problem when the constraints are modeled as nonlinear in the velocities. It is shown that this fictitious nonlinearity has a predominant effect on the numerical stability of the dynamical equations, and hence it is possible to use it for improving the accuracy of simulations. Another issue is the dynamics of constraint violations caused by integration errors due to enforcing a differentiated form of the constraint equations. To solve this problem, the acceleration form of the constraint equations is augmented with constraint stabilization terms before using it with the dynamical equations. The procedure is illustrated by stabilizing the constraint equations for a holonomically constrained particle in the gravitational field.     

Simulation of Motion of Constrained Multibody Systems Based on Projection Operator

This paper presents an efficient dynamic formulation for solvingDifferential Algebraic Equations (DAE) by using the notion of orthogonalprojection. Firstly, the constraint equations are expressed explicitlyat acceleration level by using the notion of the orthogonal projection.Secondly, the Lagrangian multiplier is eliminated from the dynamicsequation by the projection operator. Then, the resultant equations areconsolidated into one equation which explicitly correlates theacceleration to the generalized force through a so-called constraint mass matrix. It is proved that the constraint mass matrix isalways invertible and hence the acceleration can be computed in aclosed-form manner even with the presence of redundant constraints or asingular configuration. The equation of motion is given explicitly in arelatively compact form, which can lead to computational efficiency. Italso has a useful physical interpretation, as the component of thegeneralized force contributing to motion dynamics is readily derivedform the formulation. Finally, results obtained from numericalsimulation of motion of a five-bar mechanism is documented.     

On the Equations of Kinematics and Dynamics of Constrained Mechanical Systems

The method of constructing of kinematical and dynamicalequations of mechanical systems, applied to numerical realization, isproposed in this paper. The corresponding difference equations, whichare obtained, give a guarantee of computations with given precision. Theequations of programmed constraints and those of constraintperturbations are defined. The stability of the programmed manifold fornumerical solutions of the kinematical and dynamical equations isobtained by means of corresponding construction of the constraintperturbation equations. The dynamical equations of system withprogrammed constraints are set up in the form of Lagrange equations ingeneralized coordinates. Certain inverse problems of rigid body dynamicsare considered.     

Recursive and Residual Algorithms for the Efficient Numerical Integration of Multi-Body Systems

In this paper a family of methods for multi-body dynamic simulation is introduced. Equations of motion are obtained using a set of Cartesian coordinates and projected onto a set of independent relative coordinates using the concept of velocity transformation. Open-chain systems are solved directly following either a fully recursive or a semi-recursive procedure. Closed-chain systems are solved in two steps; kinematic loops are opened by removing either some kinematic joints or a rigid body, and the resulting open-chain system is solved; closure-of-the-loop conditions are imposed by means of a second velocity transformation. The dynamic formalisms have been developed so as to handle both non-stiff and stiff systems. Non-stiff systems are solved by means of an Adams–Bashforth–Moulton numerical integration scheme, which requires the computation of the function derivatives. Stiff problems are integrated by using either BDF or NDF methods, which require the computation of the residual of the equations of motion and, optionally, the evaluation of the Jacobian matrix. The proposed algorithms have been implemented using an Object-Oriented Programming approach that makes it possible to re-use the source code, keeping programs smaller, cleaner and easier to maintain. Practical examples that illustrate the performance of these implementations are included. These examples have also been solved using a commercial multi-body simulation package and comparative results are included. In most cases, the algorithms here presented outperform those implemented in the commercial package, leading to important savings in terms of total computation times.     

An Implicit Runge–Kutta Method for Integration of Differential Algebraic Equations of Multibody Dynamics

When performing dynamic analysis of a constrained mechanical system, aset of index 3 Differential Algebraic Equations (DAE) describes the timeevolution of the model. This paper presents a state space DAE solutionframework that can embed an arbitrary implicit Ordinary DifferentialEquations (ODE) code for numerical integration of a reduced set of statespace ordinary differential equations. This solution framework isconstructed with the goal of leveraging with minimal effort establishedoff the shelf implicit ODE integrators for efficiently solving the DAEof multibody dynamics. This concept is demonstrated by embedding awell-known public domain singly diagonal implicit Runge–Kutta code inthe framework provided. The resulting L-stable, stiffly accurateimplicit algorithm is shown to be two orders of magnitude faster than astate of the art explicit algorithm when used to simulate a stiffvehicle model.     

Finite Element Method in Dynamics of Flexible Multibody Systems: Modeling of Holonomic Constraints and Energy Conserving Integration Schemes

In this work we discuss an application of the finite elementmethod to modeling of flexible multibody systems employing geometricallyexact structural elements. Two different approaches to handleconstraints, one based on the Lagrange multiplier procedure and anotherbased on the use of release degrees of freedom, are examined in detail.The energy conserving time stepping scheme, which is proved to be wellsuited for integrating stiff differential equations, gouverning themotion of a single flexible link is appropriately modified and extendedto nonlinear dynamics of multibody systems.     

Application of Runge–Kutta–Rosenbrock Methods to the Analysis of Flexible Multibody Systems

Numerical integration methods are discussed for general equations of motion for multibody systems with flexible parts, which are fairly stiff, time-dependent and non-linear. A family of semi-implicit methods, which belong to the class of Runge–Kutta–Rosenbrock methods, with rather weak non-linear stability properties, are developed. These comprise methods of first, second and third order of accuracy that are A-stable and L-stable and hence introduce numerical damping and the filtering of high frequency components. It is shown, both from theory and examples, that it is generally preferable to use deformation mode coordinates to global nodal coordinates as independent variables in the formulation of the equations of motion. The methods are applied to a series of examples consisting of an elastic pendulum, a beam supported by springs, a four-bar mechanism, and a robotic manipulator with collocated control.     

人眼定位的边缘强度信息积分投影方法

为了提高积分投影方法对人眼定位的准确性和适用范围,提出了一种在极坐标系下对边缘强度信息进行积分投影的改进方法。基于肤色特征确定出人脸区域,采用Kirsch算子建立边缘强度信息图像,对不同极角方向进行积分投影,确定出人眼角度方向,对人眼所在角度方向的边缘强度进行微分累加运算确定出人眼的极径,从而实现人眼的极坐标定位。实验结果表明,该方法能够有效地提高各种姿态人脸图像中人眼定位的准确性,尤其对于旋转人脸图像的人眼定位,该算法具有良好的鲁棒性。     

An Implementation Method for Constrained Flexible Multibody Dynamics Using a Virtual Body and Joint

A convenient implementation method for constrained flexiblemultibody dynamics is presented by introducing a virtual rigid body andjoint. The general purpose program for rigid and flexible multibodydynamics consists of three major parts of a set of inertia modules, aset of force modules, and a set of joint modules. Whenever a new forceor joint module is added to the general purpose program, the modules forthe rigid body dynamics are not reusable for the flexible body dynamics.Consequently, the corresponding modules for the flexible body dynamicsmust be formulated and programmed again. Since the flexible bodydynamics handles more degrees of freedom than the rigid body dynamicsdoes, implementation of the module is generally complicated and prone tocoding mistakes. In order to overcome these difficulties, a virtualrigid body is introduced at every joint and force reference frames. Newkinematic admissibility conditions are imposed on two-body referenceframes of virtual and original bodies by introducing a virtual flexiblebody joint. There are some computational overheads due to the additionalbodies and joints. However, since computation time is mainly dependenton the frequency of flexible body dynamics, the computational overheadof the presented method is not a critical problem, while implementationconvenience is dramatically improved.     

并行投影融合隐式凸可行性模型的WSN定位仿真

为了解决当前无线传感器网络(Wireless Sensor Network,WSN)未知节点定位精度低,对初始节点要求高等难题,提出了并行加权投影融合隐式凸可行性的WSN协同定位机制.基于数学模型,分析了WSN定位问题,并将其转换成隐式凸可行性问题;再引入权重理论和并行投影思想,将其嵌入到隐式凸可行性问题中,设计了基于加权凸集投影的WSN协同分布式定位算法;构造了隐式凸可行性问题的数学模型,对属性进行了分析与证明.最后对改进机制进行了测试,结果表明:与当前WSN定位算法相比,改进机制具有更高的定位精度,经验积累分布函数值最大,收敛速度最快,呈单调收敛.     

An Efficient Method for Synthesis of Planar Multibody Systems Including Shape of Bodies as Design Variables

A point contact joint has been developed and implemented in a joint coordinate based planar multibody dynamics analysis program that also supports revolute and translational joints. Further, a segment library for the definition of the contours of the point contact joints has been integrated in the code and as a result any desired contour shape may be defined. The sensitivities of the basic physical variables of a multibody system, i.e., the positions, velocities, accelerations and reactions of the system with respect to the automatically identified independent design variables may be determined analytically, allowing design problems where the shape of the bodies are of interest to be handled in both a general and efficient way.     

An Integrated Runge–Kutta Root Finding Method for Reliable Collision Detection in Multibody Systems

Described in this paper is an integrated approach for reliable detection of state events occurring during numerical integration of the equations of motion of multibody systems. The method combines an explicit Runge–Kutta 4/5 Dormand scheme with continuous dense output extension with a polynomial root detection algorithm warranting root detection for large time steps enclosing several roots. The method is implemented in C++ and integrated within an object-oriented code for simulation of the equations of motion of mechanical systems including collisions. Numerical comparisons with standard methods are shown for a number of test examples, displaying the robustness of the method.     

信息系统集成方法研究

信息系统集成所要解决的问题是把业务相关的独立信息系统整合起来,为用户提供一个统一的功能强大的集成信息系统。文章首先对信息系统的层次结构进行了说明,在此基础上探讨了信息系统集成的层次、模型、以及不同层次的集成方法。     

An Optimized Iterative Method for Numerical Solution of Large Systems of Equations Based on the Extremal Property of Zeroes of Chebyshev Polynomials

An iterative method suitable for numerical solution of large systems of equations is presented. An extremal property of the Chebyshev polynomials is established, providing a logical foundation for the proposed procedure. A modification of the method is applicable for evaluation of the maximal eigenvalue of a matrix with real eigenvalues and of the associated eigenvector.     

Asymptotic stability of block boundary value methods for delay differential-algebraic equations

Block boundary value methods are applied to solve a class of delay differential-algebraic equations. We focus on the asymptotic stability of the numerical methods for linear delay differential-algebraic equations with multiple delays. It is shown that A-stable block boundary value methods satisfying a restrictive condition can preserve the asymptotic stability of the analytical solution. Numerical experiments further confirm the effectiveness and stability of the methods.