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.     

Dynamics of Multibody Systems Subjected to Impulsive Constraints

This paper presents a procedure for studying dynamics of multibodysystems subjected to impulsive constraints which may be either holonomicor nonholonomic. The procedure automatically incorporates the effects ofimpulsive constraints through its analysis. The governing equationsthemselves are developed from Kane's equations, using partial velocityvectors and partial angular velocity vectors. Explicit expressions forthe coefficients and terms of the governing equations are presented. Twosubcases are studied: (1) the constraints are instantaneously applied andcontinue to act; and (2) the constraints are instantaneously applied andlifted immediately after completion of impact. The internal impulses ateach joint and constraint impulses associated with the impulsiveconstraints are calculated. The procedure is checked with two examples,whose solutions are established.     

Observability of 3D Motion

This paper examines the inherent difficulties in observing 3D rigid motion from image sequences. It does so without considering a particular estimator. Instead, it presents a statistical analysis of all the possible computational models which can be used for estimating 3D motion from an image sequence. These computational models are classified according to the mathematical constraints that they employ and the characteristics of the imaging sensor (restricted field of view and full field of view). Regarding the mathematical constraints, there exist two principles relating a sequence of images taken by a moving camera. One is the epipolar constraint, applied to motion fields, and the other the positive depth constraint, applied to normal flow fields. 3D motion estimation amounts to optimizing these constraints over the image. A statistical modeling of these constraints leads to functions which are studied with regard to their topographic structure, specifically as regards the errors in the 3D motion parameters at the places representing the minima of the functions. For conventional video cameras possessing a restricted field of view, the analysis shows that for algorithms in both classes which estimate all motion parameters simultaneously, the obtained solution has an error such that the projections of the translational and rotational errors on the image plane are perpendicular to each other. Furthermore, the estimated projection of the translation on the image lies on a line through the origin and the projection of the real translation. The situation is different for a camera with a full (360 degree) field of view (achieved by a panoramic sensor or by a system of conventional cameras). In this case, at the locations of the minima of the above two functions, either the translational or the rotational error becomes zero, while in the case of a restricted field of view both errors are non-zero. Although some ambiguities still remain in the full field of view case, the implication is that visual navigation tasks, such as visual servoing, involving 3D motion estimation are easier to solve by employing panoramic vision. Also, the analysis makes it possible to compare properties of algorithms that first estimate the translation and on the basis of the translational result estimate the rotation, algorithms that do the opposite, and algorithms that estimate all motion parameters simultaneously, thus providing a sound framework for the observability of 3D motion. Finally, the introduced framework points to new avenues for studying the stability of image-based servoing schemes.     

Joint Reaction Forces in Multibody Systems with Redundant Constraints

Redundant constraints are defined as those constraints which can be removed without changing the kinematics of the mechanism. They are usually eliminated from the mathematical model of a multibody system. For a given mechanism the set of redundant constraints can be chosen in many ways. Rigid body systems with redundant constraints do not have a unique solution to the problem of joint reaction forces calculation. If redundant constraints are present in the mechanical system, then the system is statically undetermined. If in the case of dynamics problem the constraints are consistent, all of them are frictionless and we are interested only in positions, velocities and accelerations of the bodies, then the calculation of joint reaction forces is not necessary. In many cases, however, e.g. when we want to take into account friction in joints, the calculation of joint reaction forces cannot be avoided. In some rigid body systems, despite the redundant constraints existence, reaction forces in selected joints can be uniquely determined. The paper presents three methods of finding the constraints for which reaction forces can be uniquely determined using rigid body model. Three different techniques of Jacobian matrix analysis are used.     

Equivalent Control of Constrained Multibody Systems

This paper investigates the title problem from the perspective ofdetermining which forcesystems, although different, will have the same dynamic effectwhen applied to a mechanical system.A procedure is presented for determining how differences incontrol force/moment input are nullifiedby constraints on the system. This in turn provides a basis foroptimal control. The theoreticaldevelopment is illustrated by a series of examples with a model ofa redundant manipulator – a constrained/controlled triple pendulum.     

Spatial Formulation of Elastic Multibody Systems with Impulsive Constraints

The problem of modeling the transient dynamics ofthree-dimensional multibody mechanical systems which encounter impulsiveexcitations during their functional usage is addressed. The dynamicbehavior is represented by a nonlinear dynamic model comprising a mixedset of reference and local elastic coordinates. The finite-elementmethod is employed to represent the local deformations ofthree-dimensional beam-like elastic components by either a finite set ofnodal coordinates or a truncated set of modal coordinates. Thefinite-element formulation will permit beam elements with variablegeometry. The governing equations of motion of the three-dimensionalmultibody configurations will be derived using the Lagrangianconstrained formulation. The generalized impulse-momentum-balance methodis extended to accommodate the persistent type of the impulsiveconstraints. The developed formulation is implemented into a multibodysimulation program that assembles the equations of motion and proceedswith its solution. Numerical examples are presented to demonstrate theapplicability of the developed method and to display its potential ingaining more insight into the dynamic behavior of such systems.     

Some Open Problems in Dynamic Analysis of Flexible Multibody Systems

Some open questions arising in the dynamical formulation ofsystems of hinge-connected flexible bodies are discussed. The first one deals with the choice of the floating reference frame associated to abody undergoing large rigid body motions but small elastic deformations.The second one concerns the so-called geometric stiffening (orcentrifugal stiffening) effects. The last problem is concerned with theeventual appearance of higher-order terms in the kinetic energy of thesystem for large rates and large accelerations.     

三维几何约束系统的等价性分析

针对过约束、完整约束和欠约束三维几何约束系统的求解问题,提出了等价性分析方法.该方法基于三维几何约束系统的内在等价性,充分挖掘几何领域知识,依据拆解约束闭环、缩减约束闭环和析出约束闭环等原则,采用等价约束替换来处理几何约束闭环问题,优化几何约束图的结构,实现几何约束系统的优化分解.最后用多个实例验证了该方法的正确性和有...     

三维陈述式约束模型的自动生成

由于缺乏有效的自动生成约束模型的方法，三维变量化技术的进一步发展受到制约，通过转换边界表示和提取特征中的本质约束信息，自动生成三维陈述式约束模型，以InteSolid2.0为基础，实现了原型系统。     

A 3D Finite Element Method for Flexible Multibody Systems

An efficient finite element (FE) formulation for the simulation of multibody systems is derived from Hamilton's principle. According to the classical assumptions of multibody systems, a large rotation formulation has been chosen, where large rotations and large displacements, but only small deformations of the single bodies are taken into account. The strain tensor is linearized with respect to a co-rotated frame. The present approach uses absolute coordinates for the degrees of freedom and forms an alternative to the floating frame of reference formulation that is based on relative coordinates and describes deformation with respect to a co-rotated frame. Due to the modified strain tensor, the present formulation distinguishes significantly from standard nodal based nonlinear FE methods. Constraints are defined in integral form for every pair of surfaces of two bodies. This leads to a small number of constraint equations and avoids artificial stress singularities. The resulting mass and stiffness matrices are constant apart from a transformation based on a single rotation matrix for each body. The particular structure of this transformation allows to prevent from the usually expensive factorization of the system Jacobian within implicit time--integration methods. The present method has been implemented and tested with the FE-package NGSolve and specific 3D examples are verified with a standard beam formulation.     

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.     

基于概率PCA模型的三维人体运动分类

提出基于概率主成分分析的三维人体运动自动识别与分类算法.它根据不同类别的人体运动应各有自己代表性的特征集,采用概率PCA方法建立各类动作的高斯分布模型;然后构建基于最小错误率贝叶斯决策理论的多分类器,实现对未知的动作序列(或具有代表性的帧)进行多分类决策.该方法具有概率模型的优点,适合高维数据处理;同时,这种方法能够提取运动数据的内在特征,较好地消除了运动数据在时间轴上的差异带来的问题,从而准确地对运动数据进行分类,实验结果证明了本文方法的有效性.     

Multibody Dynamics Modeling of Variable Length Cable Systems

This paper presents a procedure for studying the dynamics ofvariable length cable systems. Such systems commonly occur in deploymentand retrieval (pay-out and reel-in) in cable towing systems such as inship and marine applications.The cable is modeled as a chainand treated as a multibody system. The chain links in turn are modeledas lumped masses. The pay-out/reel-in process is modeled with variablelength links near the towing point.Application in marine systems are presented and discussed.     

Dynamic Response Calculation of Spatial Elastic Multibody Systems with High-Frequency Excitation

The objective of this paper is to establish a computationalscheme for dynamic response calculations of a three-dimensionalmultibody mechanical system with impulsive forces, which give rise tohigh-frequency excitations. The finite-element method is employed torepresent the local deformations of three-dimensional beam-like elasticcomponents by either a finite set of nodal coordinates or a truncatedset of modal coordinates. A reduced-order model is obtained by invokinga modal transformation. Both planar and complex modal reduction schemesare established. The developed formulation is implemented into amultibody simulation program that assembles the equations of motion andproceeds with its solution. The computational scheme permits a change inthe basis of the modal space in order to regulate the admittance ofhigher frequencies and to accommodate any change in the kinematicconfiguration. Numerical examples are presented to demonstrate theapplicability of the developed computational scheme.     

On Derivation of Motion Equations for Systems with Non-Holonomic High-Order Program Constraints

The paper develops and discusses the generalization of modeling methods for systems with non-holonomic constraints. The classification of constraints has been revisited and a concept of program constraints introduced. High-order non-holonomic constraints (HONC), as presented in examples, are the generalization of the constraint concept and may, as a constraint class, include many of motion requirements that are put upon mechanical systems. Generalized program motion equations (GPME) that have been derived in the paper can be applied to systems with HONC. Concepts of virtual displacements and a generalized variational principle for high-order constraints are presented. Classical modeling methods for non-holonomic systems based on Lagrange equations with multipliers, Maggi, Appell–Gibbs, Boltzman–Hamel, Chaplygin and others are peculiar cases of GPME. The theory has been illustrated with examples of high-order constraints. Motion equations have been derived for a system subjected to a constraint that programmed a trajectory curvature profile. Efficiency, advantages and disadvantages of GPME have been discussed.     

3维GIS空间建模方法评述

3维(3D)空间建模方法是3DGIS研究的核心和难点之一,是建立3DGIS的基础。迄今为止,3D空间建模方法多种多样,建模对象不同,建模方法也随之不同。根据3D空间建模研究的对象,以地球表面为界分为地理空间建模和地学空间建模两大类,并通过现有3D空间建模方法的对比研究,提出了现有3D空间建模研究的关键问题。     

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.     

Dynamics of Flexible Multibody Systems with Non-Holonomic Constraints: A Finite Element Approach

In this article it is shown how non-holonomic constraints can beincluded in the formulation of the dynamic equations of flexiblemultibody systems. The equations are given in state space formwith the degrees of freedom, their derivatives and the kinematiccoordinates as state variables, which circumvents the use ofLagrangian multipliers. With these independent state variables forthe system the derivation of the linearized equations of motion isstraightforward. The incorporation of the method in a finiteelement based program for flexible multibody systems is discussed.The method is illustrated by three examples, which show, amongother things, how the linearized equations can be used to analysethe stability of a nominal steady motion.     

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.     

与历史无关的三维参数化模型构造方法

针对解析曲面构成的B-rep模型,提出了一种陈述性三维几何约束模型的构造方法．基于位置和姿态自由度解耦思想给出了几何实体的外形描述,借助空间矢量正交或平行条件定义了基本约束单元,并推导出常见工程约束的代数方程．讨论了陈述式约束模型自动构造过程,开发了三维模型变形设计原型系统,并验证该方法的可行性。