Aggregative Mechanics of Rigid Body Systems Eng

8/21/2019 Aggregative Mechanics of Rigid Body Systems Eng

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Aggregative mechanics of rigid body

systems

V.A. Konoplev

Sankt-Petersburg

1996


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Contents

Preface ix

1 Kinematics of multibody systems 11.1 Linear space of equipollent systems of line vectors . . . . . . . 1

1.1.1 Equation of the kinematics on the group Lt(R, 6) . . . 101.2 Graph of a tree-like multibody system . . . . . . . . . . . . . . 13

1.2.1 Examples of multibody systems . . . . . . . . . . . . . . 171.3 Kinematic equations of (µ, k − 1; lk)-th kinematic pair . . . . . 22

1.3.1 Examples of equations of kinematic pairs . . . . . . . . 251.4 Kinematics of Hooke-elastic pair . . . . . . . . . . . . . . . . . 271.5 Kinematic equations of lk-th element . . . . . . . . . . . . . . . 29

1.5.1 Examples of kinematic equations of kinematic pairs . . 301.6 Kinematic equations of a tree-like multibody system . . . . . . 33

1.6.1 Examples of constructing configuration matrices . . . . 35

1.6.2 Examples of kinematic equations of kinematic pairs . . 371.6.3 Examples for constructing parastrophic matrices . . . . 38

2 Equations of motion for a multibody system 41

2.1 Equations of motion of an element of a multibody system . . . 412.1.1 Equations of motion of an element of a system carry-

ing dynamically non-balanced and asymmetric rotatingbodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.1.2 Equations of motion of an element of a multibody sys-tem in inertial medium . . . . . . . . . . . . . . . . . . 46

2.2 Equations of motion of a tree-like multibody system . . . . . . 472.2.1 Equations of motion of double pendulum . . . . . . . . 50

2.3 Equations of Hooke-elastic body system motion . . . . . . . . . 53

2.4 Eff ective forms of equations of multibody system motion . . . . 552.5 Determination of preplanned control with the help of forward

and backward sweep methods . . . . . . . . . . . . . . . . . . . 66

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vi Chapter 1

2.6 Algorithm for constructing equations of motion based on for-ward and backward sweep methods . . . . . . . . . . . . . . . . 69

3 Analytic transvective forms of dynamic equations in the Cauchy

form 753.1 Matrix realization of Gauss method . . . . . . . . . . . . . . . . 763.2 Analytic transvective forms of inertia matrix and its inverse . . 763.3 Analytical transvective Cauchy form of motion equations for a

tree-like system . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.4 Computer complexity of the algorithms . . . . . . . . . . . . . 80

4 Diff erential equations of constraints of multibody systems

with the external medium bodies 85

4.1 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.2 Kinematics of pairs ‘the system bodies – the external medium

ones’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.3 Diff erential equations of time-invariant constraints . . . . . . . 914.4 Diff erential equations of time-varying constraints . . . . . . . . 954.5 Diff erential equations of constraints of multibody systems with

loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

References 101

Notation 105


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Preface

The main theme of this book is Computer-Aided Modeling concerned withthe Mechanics of Multibody Systems branch of General Mechanics . It bringstogether eff orts of specialists in the field of mechanics and information scienceto solve the basic problem (see, e.g., Gerdt et al. 1980, Grosheva et al. 1983,Klimov et al. 1989):

to develop theoretical foundations and applied issues of a new field of the human knowledge based on bringing together (‘mutual diff usion’) me-chanics, in particular, the multibody system mechanics, and the branch of information science which is commonly named Computer Algebra .

The above-mentioned problem has been studied in two directions (see Wit-tenburg 1977, Medvedev et al. 1978, Gerdt et al. 1980, Grosheva et al. 1983,Konoplev 1986a, Arais et al. 1987, Velichenko 1988, Klimov et al. 1989):

− the implementation of computer algebra methods in the classical mechan-ics with the aim of developing theoretical and software tools which are

intended to replace the traditionally human work in deriving mechanicalsystems motion equations in Newton-Euler, Lagrange, Appel and Hamil-ton forms;

− the development of new mathematical formalisms in mechanics to pro-duce computer-aided algorithms suitable for the use of computer algebramethods and symbolic computation software.

In recent time the trend has been toward bridging the gap between thesetwo approaches. The book contains the results of the development of a newcomputer-aided mathematical formalism of the multibody system mechanicswhich may be easily implemented with the use of computer algebra tools forsymbolic computations and of standard software for numerical ones.

The efficiency of the represented formalism is determined by the following

facts:− The algorithms are universal in the sense that analytical and numerical

forms of multibody system mechanical models are constructed indepen-

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x Preface

dently of one another, i.e., these algorithms may be used for constructinganalytical and, independently, numerical forms of models bypassing thestage of constructing scalar equations.

− The algorithms provide the efficient solution of direct problems in the

mechanics of multibody systems (with the help of the sweep method).− There are many efficient recurrent procedures.

− There is no need to derive previously analytical forms of any particularfunctionals (such as Lagrangian, Hamiltonian), the Gauss function, theAppel function and so on.

− There is no need to exploit symbolic diff erentiating tools, since the al-gorithms contain no such diff erential operators as Christoff el symbols,three-indexed symbols of Boltzmann, the Jacoby matrix and so on. Alloperations are performed with the use of readily available algebraic tools.

− Where bodies of a system are linked with external bodies, there is no needto derive equations of holonomic and non-holonomic (time-invariant andtime-varying) constraints and computer calculation of the corresponding

Jacoby matrix for these purposes. To this end, only the matrices whichare derived during the construction of motion equations are used.

− The method of indexing that is oriented to multibody systems with tree-like structure is developed.

− The methodology provides the possibility of taking into account the pres-ence of rotating bodies linked to bodies of a system in the case where thereverse linkage to carrier is absent, as well as the influence of an inertialmedium (e.g., water when the flow is potential) by without significantloss of efficiency of algorithms.

− The derivation of Hooke-elastic multibody system motion equations isprovided.

− Methods of deriving an inertia matrix and its inverse in the form of mul-

tiplication of simple matrices (transvective and diagonal) are proposed.− A method of direct derivation of analytical forms of multibody system

motion equations in the Cauchy form is developed without computing aninertia matrix and inverting it.

− Practically full parallel processing is provided.

− Work expenditure is minimized.

The book is written on base of the author’s study of the problems knownas classical in mechanics and control theory. It is organized as follows. InChapter 1 we focus on some general issues that are very important in formingmathematical models in the multibody system mechanics. Many method-ologies for description of rigid multibody systems have been proposed in thepast. One of them is based on application of dual vectors and matrices in rigid

body dynamics: in the last thirty years a growing number of investigationsconcerning this topic have appeared in the literature (see, e.g., Angeles et al.1997). Dual vectors (screws) supply us with the description of a multibody


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Preface xi

system that does not suff er from singularities as in the case, e.g., of Eulerangles. We present the results of our elaboration of a new mathematical for-malism of the multibody system kinematics which is based on the concept of a line vector (Brand 1947, Berezkin 1974). We avoid using the common ter-

minology of screw theory (Dimentberg 1965, Murray et al. 1993, Selig 1996),as the essence of our approach appears as crucially diff erent and more sim-ple. In particular, equipollent systems of line vectors generate 6-dimensionallinear spaces of binary vectors with respect to various reduction centers. Af-ter introducing bases, we arrive at the coordinate space R6 where a groupof motions such as translations and rotations defines isomorphism betweencoordinate representations of these spaces (Konoplev 1987a). This group actsmultiplicatively, as distinct from the analogous group of translations whichacts additively on R3. The multiplicative group of motions permits us to giveefficient tools for defining binary vectors in various frames (for an arbitrarynumber of intermediate motions). This fact is of importance as the maincontent of the multibody system kinematics and dynamics is in constructing

twists and wrenches (kinematical, kinetic and dynamic binary vectors) as inan inertial frame of reference so in frames attached to system bodies.

After introducing the operation of diff erentiating on the group of motions, weobtain an algebraic apparatus for calculating velocities and quasi-velocities of bodies. This permits us to keep the mathematical prerequisites to a minimumand to construct algebraic, physically pictorial models of multibody systemssuitable for computer implementation in symbolic and numerical forms.

A multibody system consists of three levels: kinematical pairs, kinematicalchains of kinematical pairs, and the whole system of kinematic chains (Kono-plev 1986b,c). Preliminary studies have showed that the goal of mathematicalformalization of the multibody system mechanics and, in particular, of thekinematics, together with the efficient apparatus of corresponding computer

calculations, may be achieved with the help of system analysis methodology(modular methodology) (Messarovich et al. 1978, Konoplev 1986a).The methodology of system analysis is efficiently used where equations (ormodules in the corresponding terminology), describing the behavior of a sys-tem on each levels, may be derived from the analogous equations, representingits behavior on the previous level. In such cases, the equations, describing thebehavior of the whole system, appear to be comprised of equations (modules)of previous levels. If there exists an algorithm of module construction for thelowest level system, which possesses an efficient computer implementation,then it turns out that the computer implementation of equation constructionfor the whole system is more efficient then any other algorithm.

Kinematical equations of a kinematical pair, a kinematical chain and the

whole system on the first, second and third levels, respectively, are relationsbetween quasi-velocities and generalized velocities (redundant in the generalcase) and play role of the above-mentioned modules. The computer algorithm


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xii Preface

of deriving the third level module from the first level modules, i.e., kinematicalequations of the whole system, defines the graph of the system, which isrepresented by a two-line array of zeros and units.

The apparatus of the theory developed in the chapter consists of kinematicalequations (kinematical modules) of all levels. The notion of a twist and thegroup of motions of binary vectors form the basis of this theory. We give thecorresponding algorithms in compact, geometrically and physically pictorialmatrix form readily implemental on a computer for symbolic and numericalcalculation. Of course, these equations are equivalent to known vector equa-tions, but, where the number of intermediate motions is greater than three orfour, the vector equations, being written in scalar form, are too cumbersome(Konoplev 1984, 1987a).

At the end of the chapter, we introduce the parastrophic matrix of a multibodysystem. This matrix transforms generalized velocities into quasi-velocities of bodies, providing kinematical equations of the system. The parastrophic ma-trix is a basic mathematical element of the theory. It contains the information

on the system kinematical structure, axes, mechanical and functional config-urations of the system. The parastrophic matrix undoes wrenches of internalreactions in any multibody system subjected to holonomic constraints andextracts control and friction forces, acting on mobility axes of the system.

In Chapter 2, we present the development of the dynamics of multibody sys-tems. As in the case of the kinematics, the system analysis is taken as thebase of methodology: at first, equations of motion of single rigid bodies inlinear space of quasi-velocities are derived (the first-level dynamic modules)in the special matrix form, then equations of motion of kinematical chains(the second-level dynamic modules) and finally equations of motion of thewhole system (the third-level dynamic modules) are obtained.We then derive equations of motion of a multibody system relative to gener-

alized coordinates as a result of eliminating quasi-velocities from the dynamicmodule of the system with the use of kinematical equations (Konoplev 1989a,1990, Konoplev et al. 1991).

We obtain equations of motion for a single rigid body with dynamically un-balanced and asymmetric rotating flywheels placed on it under conditions inwhich carrier motion has no influence on their rotation. The influence of aninertial external medium on system motion is taken into account (e.g., water,provided that the flow is potential) (Konoplev 1987b and 1989b,c).

Deriving dynamic equation of Hooke-elastic bodies motion and constructing(with its use) equations of ‘fast’ and ‘slow’ motions (taking into account cross-eff ects) are also considered in the first part of Chapter 2 (Konoplev et al.1991).

The second part of Chapter 2 is devoted to the problem of work expendi-ture for the algorithms of constructing multibody system equations. Thequadratic, in the degree of freedom, algorithm for inertia matrix calculation


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Preface xiii

and two linear algorithms for the quadratic, in the generalized velocities terms,calculation are obtained. We propose efficient algorithms, yielding the solu-tion for the so called inverse problem of the multibody system dynamics,which consists in the calculation of reactions between bodies of the system

and control inputs in joints, given values of generalized coordinates and veloc-ities. The labor content for each of the algorithms presented is estimated forthe symbolic and numerical forms. To illustrate algorithms implementation,motion equations for a double pendulum with a sliding hanger are derived inthe symbolic form.

The theoretical results presented in the chapter are illustrated by numerousfully considered examples, such as a gyroscope on an inclined base, a flywheelinstalled on a rigid body, a double pendulum with a sliding hanger, a ma-nipulator, a walking machine, and a test bench (Okhotsimsky et al. 1984,Konoplev 1989, Konoplev et al. 1989, 1992a, Belkov 1992). We have studiedthe above-mentioned mechanisms using computer simulation at the requestof industrial firms.

Thus, because the equations of multibody system motion, introduced in Chap-ter 2, contain only constant inertia matrices and the parastrophic matrix, thedefinition of the latter immediately leads to construction of kinematical equa-tions and motion equations (Konoplev 1989a, 1990, Konoplev et al. 1991).

In Chapter 3, we continue to study the problem of reducing the labor contentfor algorithms of motion equation derivation. It is known (Gantmacher 1964,Suprunenko 1972, Skornyakov 1980 and 1983) that the general linear grouphas the system of generators (transvective and diagonal matrices). Here wegive two algorithms for deriving an inertia matrix and its inverse in the formof multiplications of the above-mentioned simplest matrices. This permitsus to bypass constructing an inertia matrix and to define the symbolic andnumerical forms of its inverse immediately. As a result, we derive the motion

equations of a multibody system at once in the Cauchy form (Konoplev 1994,1995).

Estimating the labor content for algorithms for symbolic and numerical formsof matrix representation we see that one of them is the most efficient algorithmwhen the system degree of freedom is greater than 27-30.

In Chapter 4, a new efficient method for constructing diff erential equations of holonomic and non-holonomic, time-invariant and time-varying constraints isdeveloped. It is customary to derive the above-mentioned equations in twostages. First, equations of constraints are written and then the Jacoby matrixof holonomic constraints is derived (Nikulin et al. 1983). With the use of this matrix and that of linear non-holonomic constraints, diff erential equa-tions of constraints are constructed. These equations are used for eliminating

Lagrange multipliers from redundant equations of motion. If there are manyconstraints and the system degree of freedom is great, the construction of these matrices is tedious. We propose a new, in essence, approach to solv-


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xiv Preface

ing this problem. The above-mentioned sequence of operations is replacedwith the inverse one. It turns out that the modular technology of construct-ing multibody system motion equations permits us to derive reaction vectorsof external bodies incidentally during construction of motion equations, in

particular, when determining the parastrophic matrix of the system. It onlyremains to write diff erential equations of constraints with the use of this ma-trix. Thus the problem of cumbersome analytical construction of constraintequations, as well as the Jacoby matrix computation of each step of numericalintegration of motion equations, is completely removed (Konoplev 1989b and1992).

With the help of this methodology, an algorithm for constructing motionequations for a multibody system with internal chain-loops is elaborated.

The book does not review the literature. There are two reasons for this. In thefirst place, it is impossible to do this in full measure, since the list of relevantpublications is enormous and the number of books and papers on multibodysystem mechanics is steadily growing. Secondly, the results connected with

mechanical modeling (the aggregative mechanics) given here does not relyon any works known to the authors. Nevertheless, it is necessary to list theauthors (additional to the above-mentioned Wittenburg 1977, Medvedev et al. 1978, Gerdt et al. 1980, Grosheva et al. 1983, Konoplev 1986a, Araiset al. 1987, Velichenko 1988, Klimov et al. 1989), whose works are uncon-ventional: V.V. Akselrod, A.V. Bansh’ikov, L.A. Burlakova, A.B. Byachkov,L. Chang, M.A. Chubarov, Yu.N. Chelnokov, A.I. Filaretov, F. Freudenstein,H.P. Frisch, A.S. Gorobtsov, T.J. Haug, J.M. Hollrbach, R.L. Huston, R.S.Hwang, V.D. Irtegov, T.R. Kane, A.I. Korzun, G.P. Kulvetis, V.A. Kutergin,E.E. Lavendel, C.S.G. Lee, V.V. Malanin, D. Orin, B. Paul, W. Schiehlen,L.I. Shtejnwolf, A.M. Shulgin, M.W. Walker, O. Wallrapp, P.Y. Willems, J.T.Wang, L. Wang, L. Woo, A.T. Zaremba, among others (we apologize to those

whom we omit to mention).The main part of the book is based on lectures that the author has been givingon elective special courses in two departments of the Baltic State TechnicalUniversity (St Petersburg).

The present volume may successfully be used as an introduction to the me-chanics of rigid bodies, it is distinct, however, in that the flavor is decidedlyone of applied mathematics. It also covers details of various extensions of the theory and recent research results in mechanics. Readers’ familiarity withreal vectors and matrices, with systems of ordinary diff erential and diff erenceequations and with basics of real, complex and functional analysis is presup-posed.

The book will appeal to a wide circle of specialists, including professional

mathematicians interested in modern mathematical formalization of mechan-ics, as well as mechanical engineers, graduates and post-graduate students.They will be able to find numerous efficient algorithms for solving actual


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Preface xv

problems in the field of multibody systems, detailed explanations and exam-ples.

The authors thank Academicians of the Russian Academy of Sciences D.M.Klimov, D.E. Okhotsimsky and Corresponding Member V.F. Juravlev forrepeated critical discussions on the aggregative mechanics issues, and Prof.Yu.F. Golubev and Prof. A.P. Makreev for the support given at decisivemoments for the future of computer-aided methods described in the book.


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Chapter 1

Kinematics of multibody systems

1.1. Linear space of equipollent systems of line vectors

It is well known that a system of forces acting on a rigid body generates thesum and the total moment considered with respect of a certain point calledthe center of reduction. Treating the sum and the moment as an aggregate of two free vectors we may say about a so called binary vector. This notion isvery near to that of a dual vector or screw (Dimentberg 1965, Murray et al.1993, Selig 1996, Angeles 1997) being in the base of analytical theory of screws(screw theory). However the diff erence existing between these notions leads tothat our theory diff ers from screw theory in the same degree as real analysison R2 from complex analysis. In particular, introducing specific notions of theconventional calculus of screws (such as a principal axis, a spiral products, alinear complex, a spiral affinor, analytical functions of dual vectors and so on)is not be necessary, as we need only the main object of the multibody systemkinematics: a multiplicative group of motions (translations and rotations) in

the linear space R6 defining isomorphism between coordinate representationsof binary vector spaces. In a rather natural way, this yields us very convenientcomputer-aided tools for using equipollent systems of line vectors.

Vectorial objects

Physical magnitudes characterized with the help of one real number are calledscalars. We may easy give examples of scalars: time, angle, length, volume,electrical resistance, mass, work, etc . Some of them can be supplied withsign but it is not essential. On the other hand there are many magnitudessuch that their description cannot be introduced by means of one number.Magnitudes such as velocity, acceleration and force are called vectorial. Inthe simplest case a vector is introduced as a line segment in which the initialpoint is distinguished from the terminal one, the former being also called apoint of application. Points are classified as zero vectors.

1


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2 Chapter 1

There are various categories of vectors. A vector of the first category – abounded vector – is given above with fixing its initial and terminal points.This way of (affine) definition does not use the notion of length. With thehelp of some metric a bounded vector can be characterized by the following

elements:

− an initial point;− a direction;− a length (norm).

The classical example of bounded vectors can be vectors of some physical field,e.g., the Earth’s gravitational one.

A vector of the second category – a line vector – is defined by the fol-lowing elements:

− a straight line along which this vector is directed;− a sense;− a length (norm).

In other words, a line vector is defined as a class of equivalence: the set of bounded vectors having the same line of action, sense and length. The usualmechanical example of a line vector is force: it is characterized by its value,the line of action and sense while its point of application is arbitrary on thisline (see the axioms of statics in the Introduction).

At last, a vector of the third category – a free vector – is given by thefollowing elements:

− a direction;− a length (norm).

In other words, any free vector is defined as a class of equivalence: the setof bounded vectors having the same direction, sense and length. The classicmechanical example of a free vector is translation of a rigid body from onestate into another: its points are replaced in the same direction and at thesame distance, i.e., it is characterized by its value and the line of action whileits point of application is not of importance.

In every separate case of using vectorial objects their category must be estab-lished.

Systems of line vectors

From the mathematical point of view such variety of vectorial objects forcesus to work out specific rules in order to handle with them. We assume thatthe fundamentals of geometrical theory of free and line vectors is known (see,e.g., Berezkin 1974). In particular, free vectors submit to the linear space

axiomatics (it is assumed to be known). Any three linear independent vectorscan be used as a basis for introducing a coordinate space that permits us tomake a step from geometry ideas to algebraic ones. With fixing of a certain


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Kinematics of multibody systems 3

origin we may introduce the corresponding frame and treat bounded vectorsas diff erence between radius vectors of their initial and terminal points.

However this mathematical apparatus gives nothing for line vectors as, ingeneral, we have no idea what sum of two line vectors is. At the same timeit is easy to see that any force system satisfies to the linear space axioms(recall that, in the multibody system mechanics, any force is a line vector):we may stretch or shrink all vectors of any system with the help of a certainreal multiplier α and form the union (or diff erence) of any two systems.

Let Λ be the set of all line vectors. Consider two systems hα = {λk ⊂ Λ, k ∈K(α)} and hβ = {λk ⊂ Λ, k ∈ K(β )} of line vectors where K(α) and K(β )are some sets of naturales.

Definition 1.1 The systems hα and hβ, K(α) 6= K(β ), of line vectors are said to be equipollent

hα ∼ hβ

if they determine the same sum x and total moment µ (with respect to the same point of reduction).

The pair of vectors x and µ can be used for constructing various models of theclass of equipollent systems. E.g., in the conventional calculus of screws, thispair defines a so called dual vector or screw x + ωµ where ω is such operatorthat ω2 = 0 (Dimentberg 1965, Angeles 1997). Below a simpler computer-aided model is constructed (we do not aim to discuss here screw calculus aswe wish to keep the mathematical prerequisites to a minimum).

Notation 1.1 Henceforth

− D3 is 3-dimensional space of points and V3 is 3-dimensional space of free vectors associated with it;

− a given line vector λ ∈ Λ, x ∈ V3 is the free vector having the same length, direction and sense as λ;− a given a ∈ D3, lxa is the action line of λ (i.e., the straight line passing through the point a in parallel to the vector x), xa is the bounded vector generated by reducing x to the point a;− a point O σ ∈ D3 is chosen as a so called center of reduction;

− (−−−→O σ, a) is the bounded vector with the terminal point a (here it is possible

that a = O σ);− raσ is the free vector with the same length, direction and sense as

(−−−→O σ, a).

Define the moment (free vector) of xa with respect to O σ

µxaσ = raσ × x (1.1)


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4 Chapter 1

where the symbol × means the vector product of free vectors.

As a line vector is defined as the class of equivalence for bounded vectors beingsituated on the corresponding straight line, we may reduce the free vector xto any point l ∈ lx

a. It is well known that its moment (with respect to O σ) is

equal to µxaσ, i.e.,

rlσ × x = µxaσ (1.2)

where rlσ is the free vector with the same length, direction and sense as (−−→O σ, l).

This fact does not depend on choosing l ∈ lxa. That is why we have reasonsthe free vectors x and µxaσ to be used for describing the line vector λ (beingsituated on the line lxa of action).

Thus we see that line vectors can be also defined as a class of equivalence forbounded vectors having the same line of action, sense, length and the samemoment with respect of any point (Dimentberg 1965).

Definition 1.2 We shall say that a line vector λ is reduced to the center O σ

if the aggregate (the ordered couple of Pl¨ ucker vectors)

lxaσ = col {x, µxaσ} (1.3)

is computed.

We defined lxaσ as an element of the Cartesian product of V3 × V3 (while itbelongs to V3 +̇V3, too) as later on we shall use other elements of V3 × V3.

Proposition 1.1 The set of aggregates is not a linear space.

Proof Indeed, consider a free vector y ∈ V3 and take two points a and bsuch that µyaσ 6= µ

ybσ. Then there are two aggregates (generated by the free

vectors y and −y with the parallel action lines lya and l−yb ) such that their

sum is col {y − y, µyaσ − µy

bσ} where the first component is equal zero, and

the second component is not equal zero. But according to relation (1.1) it isimpossible (the second component must be equal zero, too). Thus this sumis no aggregate. 2

Consider the set Λ of all lines vectors and construct the corresponding setof aggregates (with respect to the given center O σ). Due to the fact thatµxaσ = µ

xlσ for any l ∈ l

xa, for the sake of brevity we shall often omit the index

a in the notation of aggregates and their line of action.

Definition 1.3 Let hα = {λk ⊂ Λ, k ∈ K(α)} be a system of line vectors where K(α) is some set of naturales. Then the sum

hσα = Xk∈K(α)

lxσk (1.4)

(of the aggregates de fi ned with respect to the given center O σ) is called a binary vector.


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In particular, a binary vector is an aggregate if the aggregates composing thisvector have action lines intersecting each other in one point a: lxaσ + l

yaσ =

col {x + y, raσ × (x + y)} = lx+yaσ as the set of aggregates with action lines,

intersecting each other in one point, is a linear space.

Proposition 1.2 The set Hσ of all binary vectors is a 6-dimensional linear space.

Proof The amount of all sums of aggregates, i.e., binary vectors, is a sumof aggregates, i.e., a binary vector, too. As to the set dimension it is easy tochoose six (orthonormal) vectors being its basic ones. 2

Thus we may treat any aggregate as a special case of the binary vector gen-erated by one line vector.

Proposition 1.3 The set H of equipollent systems of line vectors is isomor-phic to the linear space Hσ.

Let us finish this section with pointing out the simplest system in a givenequipollent system of line vector. This question is of importance in mechan-ics. In order to give the answer we define the following binary vectors andaggregates.

Definition 1.4 Let us take a free vector x ∈ V3 and two arbitrary points aand b ∈ D3 such that µxaσ 6= µ

xbσ. Then the binary vector hσ being sum of two

aggregates lxaσ and l−xbσ (with respect to the same reduction center O σ ∈ D3)

is called a couple, and the distance h between the lines lxa and lxb is called the

couple arm.

It is easy to see that

hσ = col {03, rab × x}

where rab is the free vector with the same length, direction and sense as (−→b, a);

the symbol 03 means 3-dimensional (free or coordinate) null vector.Note that a couple is invariant with respect to the choice of a reduction center,as it is generated by the free vector x ∈ V3 and depends on the arm of couple.

Definition 1.5 An aggregate and a binary vector de fi ned by

lxσ = col {x, 03}, hσα = col {X

k∈K(α)

xk, 03}, x 6= 0,X

xk 6= 0

are called degenerate.

Note that


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6 Chapter 1

− an aggregate lxσ is degenerate if its line of action goes through the centerO σ ∈ D3, i.e., the degeneration property depends on the reductioncenter;

− it is not necessary for a degenerate binary vector to be always a sum of

degenerate aggregates.Let us present the binary vector (1.4) in the form

hσα = col {h(1), h(2)σ } = col {h

(1), 03} + col {03, h(2)σ }

where first component h(1) ∈ V3 is called the sum of the system (or the sum

of the vectors xk), and the second component h(2)σ ∈ V3 is called the total

moment with respect to the center O σ.

It is easy to see that the inner product (h(1), h(2)σ ) does not depend on the

point of reduction. With its help we state the following assertion (see, e.g.,Banach 1951, Berezkin 1974).

Proposition 1.4 Depending on whether the inner product (h(1), h(2)σ ) is dif-

ferent from zero or equal to zero, every α-system of line vectors is equipollent to one of the systems:

− the aggregate col {h(1), 03} and a couple having the moment h(2)σ ;

− two (skew) aggregates such that one of them has the action line passing through the center O σ;− one aggregate;− the zero vector.

In general, one must not suppose that the simplest system is unique. For thesake of simplicity, classes of equipollent systems of line vectors are usuallyidentified with their simplest representatives.

Group of motions

Relation (1.1) can be used in order to define the two-parametric set of linearoperators M σa : V3 → V3 acting by the law

µxaσ = M σa x (1.5)

i.e., for any vector x ∈ V3, we have the free vector µxaσ being equal to the

moment of x reduced to the point a with respect to the center O σ (see theindices of µxaσ). Thus the aggregate (1.3) assumes the form

l

x

aσ =· I

M σa¸

x (1.6)

where I is the identity operator in V3.


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Choosing diff erent reduction centers, we may define diff erent operators gener-ating the (1.6)-kind aggregates and the corresponding linear spaces of binaryvectors.Let us consider two linear spaces Hσ and Hτ of binary vectors that generated

with respect to points O σ and O τ ∈ D3, respectively. It is easy to see thatthese spaces are isomorphic, i.e., there is the biunique correspondence definedby the linear operator

T στ =

· I OM στ I

¸ (1.7)

acting from Hτ to Hσ (here the operator M στ is defined as in relation (1.5);

O is the null operator in V3).It is clear that after introducing bases in Hσ and Hτ we may give the coordi-nate representation of operator (1.7) in the space R6 (it is clear that all thesespaces are isomorphic).


− eσ = {eσ1 , eσ2 , e

σ3} is an orthonormal basis in V3;

− a reduction center O σ ∈ D3 is chosen as the origin of the Cartesian frame Eσ = (O σ, e

σ);− the basis vectors of Eσ generate the coordinate space R3 and the fol-lowing vectorial matrix

keσk = keσ1 | eσ2 | e

σ3k (1.8)

i.e., the matrix with entries being vectors (coordinate columns);− xσ = col {xσ1 , x

σ2 , x

σ3} ∈ R3 is the column of coordinates of any x ∈ V3,

i.e., x = keσk xσ;− the cap over the coordinate column, e.g., xσ means the passage from xσ

to the following skew-symmetric matrix

x̂σ =

0 −xσ3 xσ2

xσ3 0 −xσ1

−xσ2 xσ1 0

(1.9)

Proposition 1.5 (Konoplev 1987a) Let an aggregate lxaσ be de fi ned (with re-spect to the reduction center O σ ∈ D3 with the help of a certain free vector x ∈ V3 reduced to some point a ∈ D3). Then there is the following coordinate representation of lxaσ in R6

lxσaσ = Gσa x

σ (1.10)

where

Gσa =

· E M σσa

¸, M σσa = r̂

σaσ (1.11)


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8 Chapter 1

rσaσ ∈ R3 is the coordinate column (see the outer superscript) of the radius

vector (−−−→O σ, a) in the frame Eσ (see the superscript); E is 3 × 3-dimensional

identity matrix (see also Berezkin 1974).

Introduce a new Cartesian frame Eτ = (O τ , eτ ) with some orthonormal basis

eτ . We may always define the rotation matrix cστ such that

keτ k = keσk cστ , xσ = cστ x

τ (1.12)

(xτ ∈ R3 is the column of coordinates of x in the basis eτ ; | det cστ | = 1).


− oστ ∈ V3 is a vector of translation of Eτ (see the subscript) to Eσ(see the superscript), oσστ is its coordinate column in e

σ (see the outer superscript), and ôσστ is the skew-symmetric matrix induced by o

σστ (see

notation (1.9));

− Hσσ and Hτ τ are the coordinate representations of Hσ and Hτ generated by the bases eσ and eτ , respectively.

Proposition 1.6 (Konoplev 1987a and 1989a,b) The isomorphism Lστ : Hτ τ →

Hσσ (= Lστ H

τ τ ) is de fi ned by the following matrix

Lστ = T σστ C

στ (1.13)

where

T σστ =

· E Oôσστ E

¸, C στ = diag {c

στ , c

στ } (1.14)

are matrices of translation and rotation (induced by the translation with the vector oστ and the rotation c

στ ), respectively; O is 3×3-dimensional null matrix.

Proof Suppose lxτ τ k = Gτ τ x

τ k is the coordinate representation of an aggregate

from relation (1.4). Then (see also relations (1.10) and (1.11))

Lστ lxτ τ k = L

στ G

τ τ x

τ k = T

σστ C

στ G

τ τ x

τ k = T

σστ C

στ G

τ τ c

σ,T τ c

στ x

τ k =

T σστ Gστ x

σk = G

σσ x

σk = l

xσσk

Taking in account that Lστ is a linear operator and going in the both sides of this equality to sums of the (1.4)-kind, we easily obtain the desired result. 2

From the foregoing proposition follows at once that the next assertion holds.

Proposition 1.7 The linear operator T στ : Hτ → Hσ has the coordinate rep-resentation T σστ in the basis e

σ.


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Proposition 1.8 The set of all motions Lστ : Hτ τ → H

σσ is the multiplicative

group

L(R, 6) = {Lστ : Lστ = T

σστ C

στ } (1.15)

where σ and τ parameters determining which points the system Λ of all line vectors is reduced to.

Proof It is true because Lστ Lτ

p = T σστ C

στ T

ττ p C

τ p = T

σστ C

στ T

ττ p C

σ,T τ C

στ C

τ p =

T σστ T τσ

p C σ

p = T σσ

p C σ

p = Lσ

p and (Lστ )−1 = (T σστ C

στ )−1 = C σ,T τ (T

σστ )

−1 =

C τ σT τσσ C

τ ,T σ C

τ σ = T

ττ σ C

τ σ = L

τ σ ∈ L(R, 6). 2

Remark 1.1 We may treat two objects to be equivalent if there is an Euclid-ean motion transforming one of them into another (Sternberg 1964). It means that the set of all spaces of binary vectors can be considered as the class of equivalence de fi ning an ‘abstract’ binary vector for a given system of line vec-

tors. In this sense we may say about computation of a binary vector at a certain point or about its transform from one point at another.

In the multibody system mechanics, we may treat any instantaneous complexmotion as a screw motion along a certain line (this line is a so called screwaxis) (see, e.g., Banach 1951). This case is characterized by the property thatthe first component of some binary vector

hτα = col {h(1), h(2)τ }

is parallel to the second one. If this is not the case we may always to choosea new reduction center O σ such that

col {h(1), ph(1)} = T στ

col {h(1), h(2)τ

} (1.16)

where p is some real number which we shall define below.Indeed, from relation (1.16) follows that there is the relation

ph(1) = oστ × h(1) + h(2)τ

Using the inner product between this vector and h(1) we have

p = (h(1), h

(2)τ )

(h(1), h(1))

On the other hand, the vector product between ph(1) and h(2)τ yields

oστ = h(1) × h

(2)τ

(h(1), h(1))


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10 Chapter 1

Proposition 1.9 (Dimentberg 1965) The set of all reduction points O σ such

that the vector h(2)σ is parallel to h(1) is the straight line given by

o

σ

τ =

h(1) × h(2)τ

(h(1), h(1)) + s h

(1)

where s is a scalar parameter.

1.1.1. Equation of the kinematics on the group Lt(R, 6)

In general, we shall distinguish two kinds of instantaneous motion of frames:translation and rotation. Let a matrix cστ = c

στ (t) define the rotation of

the basis eτ with respect to the immobile basis eσ considered in the basis eσ

(| det cστ | = 1) (see (1.8) and (1.12)). Denote by Rt ⊂ R3 the one-dimensionalsubspace generated by the unit (time-varying) eigenvector corresponding tothe real (time-invariant) eigenvalue of the matrix cστ . It is clear that the

origin O τ of the rotating frame Eτ is on Rt. Let P be the two-dimensionalsubspace being orthogonal to Rt and passing through the point O τ . Suppose

aτ = (−−−→O τ , a) is the radius vector of a ∈ P. Let us also denote by aτσ and aττ

the coordinate columns of aτ in the bases eσ and eτ of the immovable androtating frames, respectively. Then from aτσ = cστ a

ττ follows that (see (1.12))

aτσ. = cσ.τ aττ = cσ.τ c

σ,T τ a

τσ (1.17)

where (recall) we use convention (0.2) for any diff erentiable function c = c(t).

Diff erentiating the relation cσ,T τ cστ = E we have c

σ,T τ c

σ.τ = −(c

σ.τ )

T cστ . Hencecσ,T τ c

σ.τ = −((c

σ,T.τ c

στ )

T )T = −(cσ,T τ cσ.τ )

T , i.e., the matrix cσ.τ cσ,T τ is skew-

symmetric. Therefore there exists a free vector w = wστ = keσk wσστ =

keτ k wστ τ such that

ŵσστ = cσ.τ c

σ,T τ , ŵ

στ τ = c

σ,T τ c

σ.τ (1.18)

where wσστ and wστ τ are the coordinate columns of w

στ in the bases e

σ and eτ ,respectively.

It is easy to see that the vector wστ defines the instantaneous angular velocityof the basis eτ with respect to the basis eσ whereas the left-hand side of re-lation (1.17) is the coordinate representation of the instantaneous translationvelocity of a ∈ P.

Note that the right-hand side of relation (1.17) is coordinate representationof the vector product

vστ = wστ × a

τ (1.19)

Thus geometrically we may determine the instantaneous angular velocity asa line vector with the line of action which is defined by the free vectors wστ and vστ .


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Let us consider the case where a couple of instantaneous angular velocities wand −w defines the rotations of Eσ and Eτ , respectively. The origins of Eσand Eτ are on the action lines of w and −w. Let P be the two-dimensionalsubspace orthogonal to w . Suppose there are the points aσ and aτ where the

action lines of w and −w intersect with P. Introduce the radius vector of a ∈ P: aσ = (−−→aσ, a) and a

τ = (−−→aτ , a).With the help of relation (1.19) we may compute the translation velocities ineach rotation

vσ = w × aσ, vτ = −w × a

τ

and the full velocity of the point a

vσ + vτ = w × (aσ − aτ )

where the diff erence aσ − aτ depends only on the position of the axes of rotation as it is equal to (−−−→aσ, aτ ).

Proposition 1.10 (Berezkin 1974) Any couple of instantaneous rotation is equivalent to one instantaneous translation.

Henceforth we shall suppose that the frame Eτ takes part only in instanta-neous rotational motions.

Definition 1.6 Let Lαi = {lwik, w

στ k ∈ V3, k ∈ K(α)} be a system of aggre-

gates de fi ning the instantaneous angular velocities of the basis eτ with respect to eσ in the rotation of Eτ with respect to the fi xed axes l

wk (here the index

i = σ or i = τ points out the frame which the system of line vectors is reduced to). Then the sum

W στ i = Xk∈K(α)

lwik (1.20)

is called the twist of motion of the frame Eτ (see the fi rst subscript in (1.20))with respect to Eσ (see the superscript in (1.20)) reduced to the origin of Ei(see the second subscript in (1.20)) (see Angeles 1997).

We may define twists in the same frame which origin corresponding systemsof line vectors is reduced to, e.g., W στ i. When it is Eτ , that is almost alwaysthe case, the notation becomes simpler: W σττ is replaced with W

στ . In this

case the subindex from the coordinate notation has a double meaning beingin the same time the index of the moving frame and the index of the framewhich the twist is reduced to. Furthermore we shall use the general notationonly in the cases where an ambiguity could occur.

Proposition 1.11 The second component of W στ is the instantaneous velocity vστ ∈ V3 of translation of Eτ with respect to Eσ in the instant t.


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12 Chapter 1

Proof Let cστ k be the rotation matrices of Eτ with the fixed axes lwk and let

Lατ = {lwτ k, k ∈ K(α)} be the corresponding systems of aggregates. Due to

relation (1.19) the second component of the aggregate lwτ k is the instantaneousvelocity vστ k ∈ V3 of translation of the point O τ (the origin of Eτ ) with respect

to Eσ. Going to summation in (1.20), we obtain for the second component of W στ : v

στ =

Pk∈K(α) v

στ k, that proves our proposition. 2

Due to the supposition concerning the participation of the frames only in in-stantaneous rotational motions, the following conclusions are a direct corollaryof the above proposition:

− the instantaneous translation velocity is a free vector fixing the state of the axis of rotation of the frame;

− this velocity can be represented in the basis eτ as a linear combination of coordinate columns of the instantaneous angular velocities with constantmatrix coefficients âττ k : v

στ τ =

Pk∈K(α) â

ττ k w

στ τ k ;

− twist (1.20) can also be represented in the coordinate form (in Eτ )

W στ τ =

Xk∈K(α)

lwτ τ k = col {w

στ τ , v

στ τ } (1.21)

where wστ τ = P

k∈K(α) wστ τ k , v

στ τ =

Pk∈K(α) v

στ τ k .

As the twist W στ defines the instantaneous rotation and translation, we maysay about the matrices T σστ (t) and C

στ (t) and the corresponding one-parameter

group Lt(R, 6) = {Lστ (t) : Lστ (t) = T

σστ (t)C

στ (t), t ∈ R}. Now we shall state

our main assertion.

Proposition 1.12 (Konoplev 1987a and 1989a) The equation of kinematics on the one-parameter group Lt(R, 6) has the following form

Lσ.τ = Lστ Φ

στ τ , Φ

στ τ = · ŵ

στ τ O

v̂στ

τ ŵστ

τ ¸ (1.22)

Proof With a little algebra we have

Lσ.τ = (T σστ C

στ )

· = T σσ.τ C στ + T

σστ C

σ·τ = (T

σσ.τ + T

σστ C

σ·τ C

σ,T τ )C

στ =

(T σσ.τ + T σστ [ ŵ

σστ ])C

στ = T

σστ (T

σσ.τ + [ ŵ

σστ ])C

στ =

T σστ C στ C

σ,T τ Φ

σστ C

στ = T

σστ C

στ Φ

στ τ = L

στ Φ

στ τ

where

[ ŵσστ ] = diag{ŵσστ , ŵ

σστ } (1.23)

2

Equation (1.22) permits us to introduce the operation of diff erentiation in var-ious spaces of binary vectors that can be easily realized numerically (Konoplev1987a).


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Proposition 1.13 The di ff erentiation operations in the (moving and immov-able) frames Eτ and Eσ, respectively, are connected by the following relation

hσ.σ = Lστ (h

τ .τ +Φ

στ τ h

τ τ )

where hτ τ ∈ Hτ τ and h

σσ ∈ H

σσ are the coordinate representations of a pair of

binary vectors generated by the same system of line vectors.

Proof From hσσ = Lστ h

τ τ and relation (1.22) follows that h

σ.σ = L

στ h

τ .τ +

Lσ.τ hτ τ = L

στ h

τ .τ + L

στ Φ

στ τ h

τ τ . 2

Remark 1.2 In the case where the matrix Φστ τ is constant we may easily de fi ne the exponential form of the rotation matrix

cστ = exp{ŵσστ t} = w

σστ w

σσ,T τ + cos{t}(1−w

σστ w

σσ,T τ ) + sin{t}ŵ

σστ

with using the properties of the skew-symmetrical matrix ŵσστ :

( ŵσστ )2k+1 = (−1)kŵσστ , (ŵσστ )2k = (−1)k(1−wσστ wσσ,T τ )

After that we may de fi ne the other entries of the block-triangular matrix Lστ as some polynomial of trigonometrical functions with the help of the Cauchy formula (Bellman et al. 1963).

1.2. Graph of a tree-like multibody system

Let us consider a system of n rigid bodies {Gk}, k = 1, n, and the corre-sponding graph which is defined with giving its elements (Fig. 1.1).Introduce the following notations and concepts (Konoplev 1984, 1985a, 1987aand 1989a,c).


− Ek = (O k, ek) is a Cartesian frame with the origin O k ∈ D3 and with

the basis ek = {ek1, ek2, e

k3} in V3;

− ak is the radius-vector of a point a of a rigid body Gk, akk is the column

of its coordinates in the basis ek;− N is a set of natural numbers.

It is said that the frame Ek = (O k, ek) is attached to the rigid body Gk if akk is constant.

Notation 1.5 Henceforth E lk is lk-th element of the graph where the index lstands for the trunk of the tree, and the index k stands for the corresponding level, l, k ∈ N (see Fig. 1.1).


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14 Chapter 1

Figure 1.1. Graph of a multibody system with tree-like structure

Let us assume that the graph is oriented from the root E 10 to its nodes bya sequence of indices l and k, l being said to be the principal index in thissequence.

Defi

nition 1.7 We shall use the following sets of the graph elements:− the set (lk)+ of accessibility, i.e., all elements of the graph that can be reached from E lk by motion up along the tree (in the direction in which indices increase) ( E lk ∈ (lk)+), for the tree-like graph this set is a subtree (with the root (lk)) of the main tree;− the set (lk)− of counter accessibility, i.e., all elements of the graph that can be reached from E lk by motion down along the tree (in the direction in which indices decrease), for the tree-like graph, the set of counter ac-cessibility is a kinematic chain, the fi rst and last elements in this chain being E 10 and E lk, respectively ( E lk ∈ (lk)−);− the set (lk)+ of right incidence of E lk, i.e., of all elements of the graph that can be reached from E lk for one step up along the tree (E lk /∈ (lk)

+);− the set (lk)− left incidence of E lk, i.e., all elements of the graph that

can be reached from E lk for one step down along the tree (E lk /∈ (lk)−);− two elements E µ,k−1 and E lk are called (µ, k − 1; lk)-th kinematic pair if E lk ∈ (µ, k− 1)

+ for µ ≤ l.

We shall consider two variants of indexing the graph for a system of rigidbodies. In the first way, the elements of the graph will be the frames takingparts in one of the simplest motions Ek−1 → Ek (either translation with adirectional vector being one of the vectors from ek−1, or rotation with a fixedvector which is from the basis ek−1 again). In the second variant, the elementsof the graph will be rigid bodies taking parts in some of the simplest motionswith respect to the frames attached to them.

According to the stated above, if indices mark the frames taking parts in

one of the simplest motions, then the motion of an arbitrary kinematic pair(µ, k − 1; lk) is completely determined by the generalized coordinates q lki ∈R1, i = 1, 6. Besides, if the motion E µ,k−1 → E lk turns out to be the


26/121


translation oµ,k−1lk : O µ,k−1 → O lk with the magnitude olki and the direction

elki ∈ elk when q lki = o

lki , i = 1, 3, and if the motion E µ,k−1 → E lk turns

to be the rotation cµ,k−1lk : eµ,k−1 → elk with a fixed unit vector elki−3 ∈

elk and an angle θlki then q lki = θ

li for i = 4, 6. One non-constant (later

on we shall consider functional) motion in (µ, k − 1; lk)-th kinematic pair(either translation olki , i = 1, 3, or rotation with the angle θ

lki , i = 4, 6)

can be preceded by several (from 0 to 6) constant (later on we shall considerconstructive) translations plki , i = 1, 3 and rotations with angles φ

lki , i =

4, 6. If indices stand for the corresponding rigid bodies, then motion of any(µ, k − 1; lk)-th kinematic pair is determined by at least two and at mostsix generalized coordinates q lk = col {q l1, q

l2, . . . , q

l6}, where q

lki are defined as

above. The same is valid in the case of preliminary translations and rotations.

For any (µ, k−1; lk)-th kinematic pair let us introduce an intermediate Carte-sian frame Elkk = (O lkk, elkk), obtained by translation of the frame Eµ,k−1with the column of constructive translation pµ,k−1lkk ≡ p

lkk ≡ col { plk1 , plk2 , p

lk3 }

and rotation of the basis eµ,k−1 in the basis elkk with the constructive an-

gles φlkα , φlkβ , φlkγ , where α, β , γ = 4, 6. When the indices of angles in onekinematic pair are the same, we shall replace the symbol φ with the sym-bol ψ, φlkα = ψ

lkγ , in the case of α = γ . For example, (φ

lk3 , φ

lk2 , φ

lk3 ) =

(φlk3 ,φlk2 ,ψ

lk3 ).

Later, the frame Elkk = (O lkk, elkk) will be called constructive for the kine-

matic pair (µ, k − 1; lk). The transition to the constructive frame Elkk isimposed not only by the presence of constructive motions in a direct (con-structive) sense but also by the change of frames in some interdisciplinaryproblems which arise due to the application of this technique. For example,in the study of motion of an airplane with respect to a ship, it is necessaryto introduce a transition from the ship frame to the plane one, i.e . fromone console (the unit vector e3 is directed in the same direction in which the

cantilever is oriented) to another when the positions of the cantilevers can bearbitrary, etc .

Since the methods and algorithms which are to be developed here should berealized numerically by means of computers let us make our approach concern-ing functional and constructive motions in a given kinematic pair (µ, k−1; lk)more formal.


− the elements E lk of the graph are Cartesian frames Elk which participate in several (from 0 to 6) constructive motions and in a functional motion (de fi ned by the generalized coordinates q lki , i = 1, 6, where q

lki = o

lki , i =

1, 3, or q lki = θlki , i = 4, 6) with respect to the constructive frame Elkk;

− plk = pµ,k−

1;µ,k−

1lkk = col { plk1 , plk2 , plk3 } is the coordinate column of con-structive translation pµ,k−1lkk of Elkk in Eµ,k−1, represented in the basis eµ,k−1;


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16 Chapter 1

− φlk = φµ,k−1lkk = col {φlkα ,φ

lkβ , φ

lkγ } is the column of constructive angles

of orientation of the basis elkk in the basis eµ,k−1.

Proposition 1.14 The relative position of the elements of (µ, k − 1; lk)-th

kinematic pair is determined by twelve functions, combined in two sextuples of functions

Rµ,k−1lk = col {Rµ,k−1lkk , R

lkklk } (1.24)

where

Rµ,k−1lkk = col { pµ,k−1;µ,k−1lkk , φ

µ,k−1lkk } (1.25)

can contain up to six functions (constants) di ff erent from 0;

Rlkklk = col {0, . . . , 0, q lki , 0, . . . , 0} (1.26)

can contain only one function di ff erent from zero.

Definition 1.8 The two sextuples of constants Rµ,k−1lkk and functions Rlkklk

generating (1.24) are called constructive and functional con fi gurations of the kinematic pair (µ, k − 1; lk), respectively.

The union of all functional configurations corresponding to all kinematic pairs,usually are called a configuration of the system as usually.

Let the elements E lk of the graph be rigid bodies Glk with the frames Elkattached to them which take parts in some (from 0 to 6) of the constructiveand in some (from 2 to 6) of the functional motions (defined by the generalizedcoordinates q lki , i = 1, 6, where q

lki = o

lki , i = 1, 3, or q

lki = θ

lki , i = 4, 6) with

respect to Elkk.


− olk = col {olk1 , olk2 , o

lk3 } is the coordinate column of the free vector of

functional translations of Elk in the constructive frame Elkk, olk ≡ olkklk ;

− θlk = col {θlk4 , θlk5 , θ

lk6 } is the column of functional angles of orientation

of the basis elk in the basis elkk of the constructive frame Elkk, θlk ≡ θlkklk .

Proposition 1.15 The relative position of the elements of (µ, k − 1; lk)-th kinematic pair is de fi ned by con fi guration (1.24) where

Rµ,k−1lkk = col { plk, φlk}, Rlkklk = col {o

lk, θ lk} (1.27)

the constructive con fi guration Rµ,k−1lkk being able to have up to six constants

di ff erent from 0, while the functional con fi guration Rlkklk can have from 2 to 6 functions di ff erent from 0 (for the case of only one generalized coordinate see (1.24)).


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1.2.1. Examples of multibody systems

Henceforth we use the notation 0 = (0, 0, 0) for 3-dimensional null row.

Gyroscope on a tilted base

Mark with indices (see Fig. 1.2) frames taking parts in one of the simplestfunctional motions.

Figure 1.2. Gyroscope on a tilted base and its graph

Henceforth all mobility axes of the multibody system are assumed to be re-duced at the origin of motion. In the case under consideration there are

R0

1

= col {R0

1k

, R1k

1

}, R0

1k

= (0, 0,φ1

5

,φ1

6

)T , R1k

1

= (0, 0, θ1

4

, 0)T

R12 = col {R12k, R

2k2 }, R

12k = 0, R

2k2 = (0, 0, θ

25, 0)

T

R23 = col {R23k, R

3k3 }, R

23k = 0, R

3k3 = (0, 0, 0, θ

36)

T

Flywheel on ( lk)-body

For this example we have

Rlks = col {Rlksk, R

sks }, R

lksk = ( p

s1, p

s2, p

s3,φ

s4,φ

s5, 0)

T , Rsks = (0, 0, 0, θs6)

T

where ps = col { ps1, ps2, ps3} is the constructive position column of the flywheelin Elk; φ

s4, φ

s5 are the constructive angles of the flywheel orientation in e

lk;θs6 is the angle of the flywheel rotation in e

sk (see Fig. 1.3).


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18 Chapter 1

Figure 1.3. Flywheel on lk -th body

Double pendulum with sliding hanger

Let us enumerate the frames which participate in the simplest motions, namely:E1, E2, E3 (we skip here the first index l due to the uniqueness) (see Fig.1.4).The motion of the kinematic pairs (0, 1), (1, 2) and (2, 3) is defined by thefunctional translation E2 with the magnitude o12 along the basic vector e

02 = e

12

from the basis e0 of the frame E0 = (O 0, eo), by the functional rotation with

the fixed basic vectors e12 = e22 from the basis e

2 of E2 with respect to E1on the angle θ25, and also by the constructive translation with the fixed unitvectors e21 = e

31 and by the functional rotation with the fixed unit vector e

33

from the basis e3 of E3 with respect to E2 on the angle θ36. So we obtain:

q 12 = o12, q

25 = θ

25, q

36 = θ

36 , and the configuration of the system assumes the

form

R01 = col {R01k, R

1k1 }, R

01k = 0, R

1k1 = (0, o

12, 0, 0)

T

R12 = col {R12k, R

2k2 }, R

12k = 0, R

2k2 = (0, 0, θ

25, 0)

T

R23 = col {R23k, R

3k3 }, R

23k = ( p

31, 0, 0, 0)

T , R3k3 = (0, 0, 0, θ36)

T

The sets of accessibility and contour accessibility and the sets of left and rightincidence assume the form:

(o)+ = {E0, E1, E2, E3}, (1)+ = {E1, E2, E3}, (2)+ = {E2, E3}

(3)+ = {E3}, (o)− = {E0}, (1)− = {E0, E1}

(2)−

= {E0, E1, E2}, (3)−

= {E0, E1, E2, E3}

(o)+ = {E1}, (1)

+ = {E2}, (2)+ = {E3}, (3)

+ = ∅

(o)− = ∅, (1)− = {E0}, (2)− = {E1}, (3)

− = {E2}


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Re-numbering the rigid bodies of the system (Fig. 1.4), we get two kine-matic pairs (0, 1), (1, 2) which motions are defined by the same general-ized coordinates as in the previous case, but in the slightly modified form:q 1 = col {o12, θ

15}, q

26 = θ

36.

Figure 1.4. Double pendulum with sliding hanger

The system configuration is again determined by the same four functions asin the previous case, but their form is modified

R01 = col {R01k, R1k1 }, R01k = 0, R1k1 = (0, o12, 0, 0, θ25, 0)T

R12 = col {R12k, R

2k2 }, R

12k = ( p

21, 0, 0, 0)

T , R2k2 = (0, 0, 0, θ26)

T

Manipulator

Manipulator is a technical system with many joints and 6-degrees of freedomrigid bodies (Fig. 1.5). After re-numbering the frames which participate inevery simplest motion, we obtain

q 14 = θ14, q

25 = θ

25, q

35 = θ

35, q

44 = θ

44, q

55 = θ

55, q

66 = θ

66

Then the configuration of the system assumes the form (Konoplev 1986b,c,Zaremba et al. 1991)

R01 = col {R01k, R

1k1 }, R

01k = ( p

11, 0, 0, 0)

T , R1k1 = (0, θ14, 0, 0)

T


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20 Chapter 1

R12 = col {R12k, R

2k2 }, R

11k = 0, R

1k2 = (0, 0, θ

25, 0)

T

R23 = col {R23k, R

3k3 }, R

23k = (0, p

32, 0, 0)

T , R3k3 = (0, 0, θ35, 0)

T

R34 = col {R34k, R

4k4 }, R

34k = (0, p

42, 0, 0)

T , R4k4 = (0, θ44, 0, 0)

T

R45 = col {R45k, R5k5 }, R45k = 0, R5k5 = (0, 0, θ55, 0)T R56 = col {R

56k, R

6k6 }, R

56k = 0, R

6k6 = (0, 0, 0, θ

66)

T

Figure 1.5. Six-component manipulator

Walking machine

The walking machine with the kinematic scheme given on Fig. 1.6 has 24excessive generalized coordinates.

Figure 1.6. Walking machine


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If the elements of the graph are the joints of this system, then the configurationof the system becomes

R1011 = col {R1011k, R

11k11 }, R

1011k = 0

R11k11 = (o111 , o112 , o113 , θ114 , θ115 , θ116 )T

For the i-th leg we have (i = 1, 6):

R11i2 = col {R11i2k, R

i2ki2 }, R

i2i3 = col {R

i2i3k, R

i3ki3 }, R

i3i4 = col {R

i3i4k, R

i4ki4 }

R11i2k = ( pi21 , p

i22 , p

i23 , 0)

T , Ri2ki2 = (0, θi24 , 0, 0)

T

Ri2i3k = ( pi31 , p

i32 , p

i33 , 0)

T , Ri3ki3 = (0, 0, θi35 , 0)

T

Ri3i4k = ( pi41 , p

i42 , p

i43 , 0)

T , Ri4ki4 = (0, 0, θi45 , 0)

T

Test bench for imitating car motion

The test bench is constructed to imitate free motion of a given machine or asystem of machines in laboratory environment (Fig. 1.7).

Figure 1.7. Test bench

This system consists of 25 rigid bodies. It has 48 excessive generalized co-ordinates, which are restrained by 42 holonomic constraints. The test benchconstruction consists of one platform and 6 supports every of which involving

4 bodies. Every support has 7 degrees of freedom. After numbering of theframes which participate in every simplest motion, (in contrast to the case of the double pendulum) there are configurations:


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22 Chapter 1

- for the platform

R1011 = col {R1011k, R

11k11 }, R

1011k = 0, R

11k11 = (o

111 , 0, 0, 0)

T

R1112 = col {R1112k, R

12k12 }, R

1112k = 0, R

12k12 = (0, o

122 , 0, 0)

T

R1213 = col {R1213k, R13k13 }, R1213k = 0, R13k13 = (0, 0, o133 , 0)T

R1314 = col {R1314k, R

14k14 }, R

1314k = 0, R

14k14 = (0, θ

144 , 0, 0)

T

R1415 = col {R1415k, R

15k15 }, R

1415k = 0, R

15k15 = (0, 0, θ

155 , 0)

T

R1516 = col {R1516k, R

16k16 }, R

1516k = 0, R

16k16 = (0, 0, 0, θ

166 )

T

- for i-th support (i = 1, 6)

R16i7 = col {R16i7k, R

i7ki7 }, R

16i7k = ( p

i71 , p

i72 , p

i73 , 0)

T

Ri7ki7 = (0, θi74 , 0, 0)

T , Ri7i8 = col {Ri7i8k, R

i8ki8 }, R

i7i8k = 0

R18ki8 = (0, 0, 0, θi86 )


i9ki9 }, R

i8i9k = (0, p

i92 , 0, 0)

T

Ri9ki9 = (0, θi94 , 0, 0)


i10ki10 }, R

i9i10k = 0

Ri10ki10 = (0, 0, θ

i105 , 0)

T

, Ri10i11 = col {R

i10i11k, R

i11ki11 }, R

i10i11k = 0

Ri11ki11 = (0, 0, 0, θi116 )


i11ki12 }

Ri11i11k = (0, pi122 , 0, 0)

T , Ri11ki12 = (0, 0, 0, θi126 )

T

Ri12i13 = col {Ri12i13k, R

i13ki13 }, R

i12i13k = (0, p

i132 , 0, 0)

T

Ri13ki13 = (0, oi132 , 0, 0)

T

1.3. Kinematic equations of (µ, k − 1; lk)-th kinematic pair

Kinematic equations give the connection between the coordinates of twist of (µ, k− 1; lk)-th kinematic pair and the generalized coordinates and velocities

of this pair. Let us begin with deriving the main kinematic equalities of thistheory (Gantmacher 1964, Diedonne 1972, Konoplev 1986b, Velichenko 1988).To establish them we need the following notion.

Definition 1.9 Let W στ = col {wστ , v

στ } be the twist of motion of the frame

Eτ with respect to Eσ. Then the vector

V στ = col {vστ , w

στ }

is called the quasi-velocity of Eτ with respect to Eσ.

Proposition 1.16 Let

− Ek take part in a motion with respect to Es, and E p take part in a

motion with respect to Ek;− W sk , W k

p , W s

p be the twists of the kinematic pairs (Es, Ek), (Ek, E p)and (Es, E p) (see (1.20), (1.21)).


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Then the following kinematic equalities hold

W sp p = L pkW

skk + W

kp p , V

sp p = L

k,T p V

skk + V

kp p (1.28)

where − L pk is a motion from group (1.15);− W skk , W

kp p , W

sp p and V

skk , V

kp p , V

sp p are the coordinate representations

of the twists and the corresponding quasi-velocities.

Proof 1. After reducing all twists to E p, we get that W sp

pp = W sp

kp + W kp

pp .After that, to prove the first equality it is enough to apply transform (1.13). 2.To prove the second equality, let us introduce 6×6-dimensional matrix ² whosemain diagonal is formed by two 3 × 3-dimensional null matrixes, and whosesecond diagonal is formed by two 3 × 3-dimensional identity matrixes. Since²² = E : we get easily that ²W sp p = ²L

pk²²W

skk + ²W

kp p where ²L

pk² = L

k,T p . 2

The term main emphasizes the fundamental meaning of equality (1.28): the

deriving of the kinematic equations for subsystems of the basic system is doingwith its help.

Proposition 1.17 (Konoplev 1984 and 1986b,c) Let

− the con fi guration of the pair (µ, k − 1; lk) be given by relation (1.24);

− Lµ,k−1lk = Lµ,k−1lkk L

lkklk be a transformation of the coordinate columns of

screws, generated by con fi guration (1.24), where the transformation

Lµ,k−1lkk = T µ,k−1

lkk C µ,k−1lkk

is de fi ned by the constructive con fi guration

Rµ,k−1lkk = col { pµ,k−1lkk , φ

µ,k−1lkk }

p

µ,k−1

lkk ≡

p

µ,k−1;µ,k−1

lk , φ

µ,k−1

lkk ≡

φ

µ,k−1

lk

and the transformation Llkklk = T lkk

lk C lkklk is induced by the functional

con fi guration Rlkklk = col {0, . . . , 0, q lki , 0, . . . , 0}:

T µ,k−1lkk =

· E O

ˆ pµ,k−1lkk E

¸, C µ,k−1lkk = diag {c

µ,k−1lkk , c

µ,k−1lkk }

cµ,k−1lkk = cα−3(φlkα )cβ−3(φ

lkβ )cγ −3(φ

lkγ )

ci−3(φlki ) = E + ê

lki−3 sinφ

lki + ê

lki−3(1− cosφ

lki )

2

Llkklk =

½ T lkklk if i = 1, 3C lkklk otherwise

T lkklk =· E O

ôlkklk E ¸

, C lkklk = diag {clkklk , clkklk }

olkklk = col {0, . . . 0, olki , 0, . . . 0}, c

lkklk = ci(θ

lki+3), i = 1, 3


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24 Chapter 1

Then the kinematic equation of (µ, k − 1; lk)-th kinematic pair is

V µ,k−1;lklk = Rlkk.lk = f

lki q

lk.i , q

lk.i =

½ olk.i if i = 1, 3θlk.i if i = 4, 6

(1.29)

where f lki is 6-dimensional (unit) column such that its i-th entry is equal to 1( i = 1, 6), the other ones are equal to zero.

Proof In order to simplify the proof let us re-number by the integers0, 1, . . . , 7 the frames that participate in every simplest constructive and func-tional motions of (µ, k − 1; lk)-th kinematic pair. Hence, the motion of (µ, k − 1; lk)-th kinematic pair will be a superposition of the simplest mo-tions E0 → E1 → . . . → E7, E0 = Eµ,k−1, E7 = Elk, and the first six of them p11, p

22, p

33, φ

4α, φ

5β , and φ

6γ are constructive, while the 7-th q

7i , i = 1, 6,

is functional.From the second equation of (1.28) follows that

V µ,k−1;lk

lk = V 077 = L

1,T 7 V

011 + V

177 = L

1,T 7 V

011 + L

2,T 7 V

122 + V

277 =

L1,T 7 V 011 + L

2,T 7 V

122 + L

3,T 7 V

2,33 + L

4,T 7 V

344 + L

5,T 7 V

455 +

L6,T 7 V 566 + L

7,T 7 V

677 V

677 ; V

066 = 0

V 677 =

½ col {c6,T 7 e

7i o

7.i ; 0

T } = col {e7i o7.i , 0

T } if i = 1, 3col {0T , e7i θ

7.i } otherwise

and finally V µ,k−1;lklk = col {v677 , w

677 } = f

7i q

7.i = f

lki q

lk.i due to V

n,n+1n+1 =

0, n = 0, 5. 2

Let the configuration of (µ, k − 1; lk)-th kinematic pair be given in the form(1.24)-(1.27).


− Lµ,k−1lk = Lµ,k−1lkk L

lkklk is transformation of the coordinates of twists in-

duced by the con fi guration Rµ,k−1lk = col {Rµ,k−1lkk , R

lkklk } where the trans-

formation Lµ,k−1lkk = T µ,k−1

lkk C µ,k−1lkk is determined by the constructive con-

fi guration Rµ,k−1lkk = col { plk, φlk} and the transformation Llkklk = T

lkklk C

lkklk

is determined by the functional one Rlkklk = col {olk, θlk};

− q lk is the column of generalized coordinates of (µ, k−1; lk) -th kinematic pair (non-null elements of the functional con fi guration Rlkklk );− ²lkklk is 3 × 3-dimensional matrix of the kind

²lkklk = kcT 3 (θ

lk6 )c

T 2 (θ

lk5 )e

lk1 | c

T 3 (θ

lk6 )e

lk2 | e

lk3 k

− clkk

lk = c1(θlk

4 )c2(θlk

5 )c3(θlk

6 ) be the orientation matrix for elk

in the constructive frame Elkk;

M lkklk = diag {clkk,T lk , ²

lkklk } (1.30)


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is the matrix of transition from the generalized velocities q lk.i of (µ, k −

1; lk)-th kinematic pair to the quasi-velocities V µ,k−1;lklk of the same kine-matic pair.

Proposition 1.18 (Konoplev 1984 and 1986b,c) The kinematic equation of (µ, k − 1; lk)-th kinematic pair is

V µ,k−1;lklk = M lkklk R

lkk.lk = M

lkklk kf

lkkq lk. (1.31)

where kf lkk is the mobility axes matrix for (µ, k−1; lk)-th kinematic pair that is constituted by the 6-dimensional unit columns f lki (each of them has 1 at its i-th position, i = 1, 6, the other ones being equal to zero).

Proof As in the previous case, to simplify the proof let us re-number bythe integers 0, 1, . . . , 12, the frames that participate in the simplest con-structive and functional motions of (µ, k − 1; lk)-th kinematic pair. Then

motion of (µ, k − 1; lk)-th kinematic pair will be superposition of twelve sim-plest motions E0, E1, E2, . . . , E12, where E0 = Eµ,k−1, E12 = Elk, thefirst six of them p11, p

22, p

33, φ

4α, φ

5β and φ

6γ are constructive and the second

six o71, o82, o

93, θ

104 , θ

115 and θ

126 are functional. Taking in account the second

equality from (1.28) we obtain

V µ,k−1,klk = V 0,1212 = L

1,T 12 V

011 + L

2,T 12 V

122 + L

6,T 12 V

566 + L

7,T 12 V

677 +

L8,T 12 V 788 + L

9,T 12 V

899 + L

10,T 12 V

9,1010 + L

11,T 12 V

10,1111 +

L12,T 12 V 11,1212 = L

7,T 12 col {v

677 , 0

T } + L8,T 12 col {v788 , 0

T } +

L9,T 12 col {v899 , 0

T } + L10,T 12 col {0T , w9,1010 } +

L11,T 12 col {0T , w10,1111 } + L

12,T 12 col {0

T , w11,1212 } =

C 7,T 12 T 7,T 12 col {v677 , 0T } + C 8,T 12 T 8,T 12 col {v788 , 0T } +

C 9,T 12 col {v899 , 0

T } + C 10,T 12 col {0T , w9,1010 } +

C 11,T 12 col {0T , w10,1111 } + col {0

T , w11,1212 } =

col {0T , c11,T 12 c10,T 11 w

9,1010 } + C

10,T 12 col {v

697 + v

798 + v

899 , 0

T } +

col {0T , c11,T 12 w10,1111 } + col {0

T , w11,1212 } =

col {c9,T 12 v699 , 0

T } + col {0T , c11,T 12 c10,T 11 e

101 θ

10.4 } +

col {0T , c11,T 12 e112 θ

11.5 } + col {0

T , e123 θ12.6 } =

col {0T , ²912col {θ10.4 , θ

11.5 , θ

12.6 }} + col {c

9,T 12 v

699 , 0

T } =

col {c6,T 12 col {o7.1 , o

8.2 , o

9.3 }, 0

T } +

col {0T

, ²612col {θ

10.4 , θ

11.5 , θ

12.6 }} =

M 612(o71, o

82, o

93, θ

10.4 , θ

11.5 , θ

12.6 )

T = M lkklk kf lkkq lk. 2


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26 Chapter 1

1.3.1. Examples of equations of kinematic pairs

Gyroscope (Fig. 1.2)

There areV 011 = M

1k1 f

14 θ

1.4 = Ef

14 θ

1.4 , V

122 = M

2k2 f

25 θ

2.5 = f

25 θ

2.5

V 233 = M 3k3 f

36 θ

3.6 = f

26 θ

3.6

Flywheel (Fig. 1.3)

There is

V 0ss = f s6θ

s.6

Double pendulum (Fig. 1.4)

There are two variants to write down the equations. The first one is

V 011 = M 1k1 f

12 o

1.2 = f

12 o

1.2 , V

122 = M

2k2 f

25 θ

2.5 = f

25 θ

2.5

V 233 = M 3k3 f

36 θ

3.6 = f

36 θ

3.6

And the second variant is

V 011 = kf 12 | f

15kcol {o

1.2 , θ

1.5 } = kf

1kq 1., kf 1k = kf 12 | f 15k

V 122 = M 2k2 f

25 θ

2.5 = f

36 θ

3.6 , q

1. = col {o1.2 , θ1.5 }, q

2.6 = θ

3.6

Manipulator (Fig. 1.5)

There are

V 011 = M 1k1 f

14 θ

1.4 = f

14 θ

1.4 , V

122 = M

2k2 f

25 θ

2.5 = f

25 θ

2.5 , . . . ,

V 566 = M 6k6 f

66 θ

6.6 = f

66 θ

6.6

Walking machine (Fig. 1.6)

There is

V 10,1111 = M 11k11 kf

11kq 11.

where for the matrix M 11k11 we get

c11k,T 11 = cT 3 (θ

116 )c

T 2 (θ

115 )c

T 1 (θ

114 )

²

11k

11 = kcT

3 (θ11

6 )c

T

2 (θ11

5 )e

11

1 | cT

3 (θ11

6 )e

11

2 | e11

3 kf 11 = E, q 11 = (o111 , o

112 , o

113 , θ

114 , θ

115 , θ

116 )

T

V 11,i2i2 = M i2ki2 f

i24 θ

i2.4 = f

i24 θ

i2.4 , . . . , V

i3,i4i4 = M

i4ki4 f

i45 θ

i4.5 = f

i45 θ

i4.5


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Test bench (Fig. 1.7)

For the platform from variant 1 is fulfilled

Aggregative Mechanics of Rigid Body Systems Eng

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