A Geometrical Interpretation of Kane's Equations

A Geometrical Interpretation of Kane's EquationsAuthor(s): Martin LesserSource: Proceedings: Mathematical and Physical Sciences, Vol. 436, No. 1896 (Jan. 8, 1992), pp.69-87Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/52020 .

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A geometrical interpretation of Kane's Equations

BY MARTIN LESSER

Mechanics Department, Royal Institute of Technology, Stockholm S-10044, Sweden

The method for the development of the equations of motion for systems of constrained particles and rigid bodies, developed by T. R. Kane and called Kane's Equations, is discussed from a geometric viewpoint. It is shown that what Kane calls partial velocities and partial angular velocities may be interpreted as components of tangent vectors to the system's configuration manifold. The geometric picture, when attached to Kane's formalism shows that Kane's Equations are projections of the Newton-Euler equations of motion onto a spanning set of the configuration manifold's tangent space. One advantage of Kane's method, is that both non- holonomic and non-conservative systems are easily included in the same formalism. This easily follows from the geometry. It is also shown that by transformation to an

orthogonal spanning set, the equations can be diagonalized in terms of what Kane calls the generalized speeds. A further advantage of the geometric picture lies in the treatment of constraint forces which can be expanded in terms of a spanning set for the orthogonal complement of the configuration tangent space. In all these developments, explicit use is made of a concrete realization of the multidimensional vectors which are called K-vectors for a K-component system. It is argued that the current presentation also provides a clear tutorial route to Kane's method for those schooled in classical analytical mechanics.

1. Introduction

Current technological needs in areas ranging from robotics to spacecraft design requires efficient algorithms for the formulation of equations of motion for mechanical systems consisting of interconnected point masses and rigid bodies. A

procedure developed by Thomas Kane (Kane & Levinson 1985; Kane et al. 1983) has achieved notable success in carrying out this task. Because of the classical nature of the subject, it is not surprising that claims for new developments can lead to considerable controversy. This has indeed occurred in regard to Kane's formulation in terms of what has come to be known as Kane's equations. A sample of this can be found in the references (Banerjee 1987; Desloge 1987, 1989; Levinson 1987). Despite such controversy, this author believes that Kane's procedure is both new and inciteful, especially if it is also associated with a number of auxiliary definitions that Kane has introduced for the task of dynamic analysis. We will briefly return to this point in the last section of this paper. The basic approach taken by Kane is

algorithmic, the algorithm's validity being supported by proof of its equivalence to the Newton-Euler equations and D'Alembert's principle for the behaviour of the constraint forces that act on the mechanical system. The purpose of the present article is to associate Kane's procedure with a geometric picture involving the

configuration manifold's embedding in the system's general coordinate space. The

Proc. R. Soc. Lond. A (1992) 436, 69-87

Printed in Great Britain 69



advantages of such a picture are both computational and insightful. The geometric insight is suggestive of new procedures for optimizing the structure of the equations of motion and of notational apparatus for manipulation of the new objects introduced by Kane. It also shows the connections of Kane's methods with the more classical formulations. It is hoped that this formulation may make it easier for those trained in the traditional manner to adopt Kane's procedure. Thus one of the

objectives of this article is to use the geometrical picture as a foundation for efficiently imparting Kane's methods to professional dynamical analysts. It will also be shown that it is always possible to choose what Kane has called the generalized speeds in such a manner as to obtain a diagonal form of Kane's equations in which each equation only contains the derivative of one of the generalized speeds.

The following section is devoted to setting forth some well known facts in a notation that is consistent both with Kane's formulation and the geometric picture that is used in the sequel. The next section uses this notation to derive Kane's form of the equations of motion for systems of interconnected point particles. We use the geometric picture to introduce the notions of partial velocities, kinematic differential equations and generalized speeds as an aspect of linear transformations in the generalized configuration space of the mechanical system. It is then shown how the formalism is used to deal with linear velocity constraints in both the holonomic and non-holonomic case. This is followed by an extension of the geometrical picture so as to accommodate Kane's equations for the case of systems of interconnected rigid bodies. The article is concluded by an example involving such a system and a short discussion.

In one sense the main points of the article involves the use of notation and the relation of notation to geometry. It is felt that the new notation introduced can be easily appreciated by the classically trained dynamical analyst and does not require that one is conversant with the modern differential geometry which lies below the surface of this presentation. The article is both tutorial and new in that it introduces Kane's method in the above discussed geometrical manner and suggests computational procedures derived from the geometrical picture.

2. Preliminary material

(a) Orthogonal frames The position and attitude of bodies relative to one another and to an inertial frame

of reference can be represented in terms of reference frames fixed with respect to the bodies as the motion progresses. It is convenient to use orthogonal sets of unit vectors for these frames. The notation that we adopt is that (L)b represents a column matrix with three orthonormal vectors (L)bj as components. Thus the index L denotes the frame number, whilej = I ... 3, denotes each of the right-handed orthogonal triad vectors. In a similar manner the representation of a vector, w in a frame L is given by the column matrix of its components (L)Wj = w' (L)b denoted by (L)w. With this notation a vector w can be written as

W = (L)T (L)b = (L)b(L)L), (1)

where the T superscript indicates the transposed matrix. In any problem where only one inertial frame is needed, such a frame will be designated by leaving off the left superscript on the symbols, e.g. bj for an inertially fixed bases vector. This notation provides all of the results needed for orthogonal transformations by straightforward Proc. R. Soc. Lond. A (1992)

M. Lesser 70



A geometrical interpretation of Kane's Equations 71

application of matrix multiplication rules appropriately modified for vector scalar products when the matrices have vector components. Thus the unit dyad has the representation

U= (L)bT (L)b (2)

while the unit matrix is given by

U =(L)b (L)bT (3)

where in the last equation we intend the matrix outer product with the vector scalar product between components. The relation between frames is found by using the appropriate representation of the unit dyad, thus

(L)b = (L)b U = (L)b (S)bT (S)b = R(b. (4)

The direction cosine matrix RLs = (L)b (S)bT (5)

defined by the above relation, possesses all the usual properties of an orthogonal matrix, as can be verified by direct use of the above notation. Thus it is evident from the notation that the inverse is given by the transpose, so that

R s= RSL. (6)

(b) Differentiation of vectors

Kane stresses the importance of keeping track of which frame is fixed in any differentiation operation by attaching a superscript to the differentiation symbol. In this paper we designate this by the notation

L(L)d/dt] w = (L)bT d (L)w/dt = (d (L)w/dt)T (L)b. (7)

The relation between the derivative of vectors with respect to different frames is then

conveniently derived by use of properties of the unit dyad as represented in the

appropriate frames. Thus if L and M represent two frames, one has the relation

[(L)d/dt] w = [(Md/dt] w + {[(L)d/dt] (M)bT} (b ' w. (8)

This leads to the definition of the angular velocity dyad, 3

(L)I(M) = {[(L)d/dt] (M)bj} (M)bj, (9) j=1

which, because of its easily shown antisymmetry, can be expressed as the vectorial

angular velocity

(L)W(M) = (M)b {[(L)d/dt] (M)b}' (M)b3 + (M)b2{[(L)d/dt] (M)b3} (M)bl + (M)b {[(L)d/dt] (M)bl} (M)b2. (10)

This last result leads to the well known equation

[(L)d/dt] w = [(M)d/dt] w+ (L)(M) x w. (11)

(c) Darboux vectors

The idea of angular velocity can be generalized to the situation where the triad vectors depend on one or more parameters which themselves may be functions of time. This leads to the concept of a Darboux vector (Stoker 1969), which is a vector

Proc. R. Soc. Lond. A (1992)



associated with a given triad and parameter. It is helpful to introduce a notation for the operation which yields the Darboux vector associated with a particular parameter and triad. Thus we define the partial slant operation as:

(M)b(/L)/q = (M)b {[(L)8/q] } (m)b2}. ()b3 + (m)b2 {[/(L)/dq] ( )b3}. (M)b

+ (2 b3{[(La/cq] (M)b,} (Mb2. (12)

The total slant operation applies when we want the Darboux vector associated with the single independent parameter case, so that the total derivatives appear in the above formula and we use a double slant in the notation. Thus the angular velocity is given by

(L))(M) (M)b(L)/t (13)

and if parameters qj depend on a single parameter t, simple application of the chain rule gives the result for an arbitrary vector w that

dt = 2 {-< q + ()b(L)/qjX W} q - + (M)b(L)/t x W. (14)

The Darboux vector provides a tool for the derivation of the theory of curves or surfaces by considering a moving triad which depends on one or two coordinates. Calculation of the rates of change of the triad vectors, with respect to appropriate coordinates using the above results, lead to the classical equations of curve or surface theory. For example it is an immediate consequence of this that a space curve is determined by the curvature and torsion expressed as a function of the curve parameter. We make use of the slant operation and the Darboux vector in describing the geometric interpretation of Kane's equations for systems of rigid bodies.

(d) Configuration geometry and K-vectors

For the purposes of this article a point mechanism is taken as a collection of position vectors from a distinguished origin to a set of points, where the vectors are given as a function of a set of parameters, denoted by q = {q1, ..., qj}. Each point may have a non-negative scalar number, its mass, associated with it. It is assumed that the points are interconnected, i.e. constrained. The reason it is necessary to consider massless points is that forces may be applied at for example the centre of a rigid massless rod connecting two mass points. The constraints may be purely geometric or specified as a linear relation between the velocities of the mechanism's points. If the latter are not integrable we have a mechanism with linear non-holonomic constraints, otherwise the mechanism is holonomic. In setting up the theory we start with the holonomic case. One of the advantages of Kane's formulation is the ease in which it deals with the linear non-holonomic case, and in the sequel we will see how this relates to the geometric interpretation of the theory.

For the purely holonomic case the mechanism's spatial configuration is specified in terms of n parameters or generalized coordinates. For a K point mechanism this means that the configuration of the mechanism can be thought of as a point in a 3K- dimensional euclidean space. Because of the constraints, this configuration point is confined to an n dimensional surface which is embedded in the 3K-dimensional space. The geometric picture that we associate with Kane's equations is related to the picture of this embedding. To aid this picture it is useful to have terminology, some of which is quite standard. Thus the n-dimensional surface is called the configuration surface. The tangent hyperplane at any point on the surface is called the configuration Proc. R. Soc. Lond. A (1992)

72 M't. L?es8er




tangent space. The full 3K-dimensional space in which the configuration space is embedded will be called the mechanism's coordinate space. At any point on the configuration surface. we can represent a vector in the coordinate space as the sum of two vectors, a vector in the tangent space and another in the (m = 3K-n)- dimensional space which is the local orthogonal complement with respect to the full coordinate space.

We call this m-dimensional space the complementary configuration tangent space or more simply, the complementary space. If the constraints are independent of time, the configuration surface is fixed in the coordinate space. If the constraints depend on time, the configuration surface moves in the coordinate space, that is, it may be considered as a one parameter family of surfaces. The kinematic history of the mechanism in the time-independent case may be represented as a trajectory confined to the fixed configuration surface. In the time-dependent case the trajectory is not, parallel to any of the instantaneous configuration surfaces.

It is common in the formulation of equations of motion in mechanics to ignore as far as possible the complementary space and to confine attention to the configuration space. For the purpose of associating a geometric picture with Kane's formulation, it is useful to retain the full coordinate space in an explicit manner. For this purpose we define a K--vector as a column matrix of size K, each component of which is a euclidean vector. The configuration of a K point mechanism is then given by a K- vector consisting of K position vectors, each dependent on n generalized coordinates, qj and the time. Clearly this provides an explicit description of the instantaneous configuration surface as embedded in the 3K-dimensional coordinate space. The K- vectors also inherit a natural scalar product from the scalar or dot product. We take as the dot product for two K-vectors the sum of the dot products of the component vectors. To state this explicitly we indicate K vectors by the notation:

r r<1(q, t)] r'(q,t)= (15)

r<K(q,t)

that is, we adopt the convention of denoting the entire K-vector by dropping the index from the superscripted left bracket. The inherited scalar product will be denoted by a fat dot, thus

K

v< w< = E v<L w<L. (16) L=1

Thus two K-vectors are orthogonal if their dot product vanishes. The norm or

magnitude of a K-vector is taken as the square root of its dot product with itself. The tangent space at a point of the configuration surface is spanned by any n

independent vectors that are tangent to the surface at the point in question. Simple geometric considerations shows that

= - (a/aqj)r< (17)

provides a set of n tangent vectors, which under suitable conditions will be linearly independent. Recall our convention that without left superscripts, the differentiation of a vector is to be taken as being with respect to an inertial or some special reference frame. As these tangent vectors are derivable by differentiation with respect to the

generalized coordinates, which can also be considered to be coordinates on the




configuration surface, they provide a so-called coordinate bases for the tangent space. Arbitrary linear combinations of the tangent vectors can be taken at each point of the configuration surface. In general these will not be derivable from coordinates in the same way as the coordinate bases, i.e. there will be no coordinate system on the surface in terms of which such an arbitrary bases can be obtained. We will use the notation Tf for a coordinate bases tangent vector, and f/' for an arbitrary bases tangent vector, which may or may not be part of a coordinate bases. As the dimension of the full coordinate space is 3K = n+m, we can find a set of m independent K-vectors which are orthogonal to all of the bases tangent vectors. These vectors, which span the complementary space, will be denoted by 7'. There are many well-known ways to accomplish this task, one of which will be employed in an example. In general it will not be assumed that the bases vectors, for either the configuration tangent space or the complementary space, are orthogonal or even normalized. The application of simple linear algebraic transformation theory to these bases vectors will be used both in the interpretation of Kane's formalism and as a means of increasing the usefulness of the formalism.

3. Kane's Equations and D'Alembert's Principle

(a) Constraint force D'Alembert's Principle is in part a reasonable assumption about the behaviour of

the forces that maintain the constraints. In the case of time independent constraints, it is relatively trivial in that it assumes that the constraint forces will be orthogonal to the path of each particle in the mechanism. In the case of time-dependent constraint, it is assumed that the instantaneous constraint will be orthogonal to a path that lies in the instantaneous configuration surface. In effect this is the so-called principle of virtual work. In a more physical light, the assumption is that whatever physical processes produce the force in the materials responsible for the constraining action, they act on a very short timescale in contrast to the classical mechanical motion, hence they have no sense of the past or future position of the configuration. Thus it is very reasonable to assume that they act orthogonally to the instantaneous surface and not to the actual trajectory. These facts should be well known, however, they are emphasized here, as in many presentations of mechanics the principle of virtual work is introduced as a means of elimination of constraint forces without any physical motivation for its validity. It can be argued that the principle of virtual work, as normally used, is not clear or necessary, and that it is better to use the clearer physical and mathematical concept of orthogonality of constraint forces to the instantaneous configuration surface. Thus we take as the foundation of classical particle and rigid body mechanics, the equations of Newton and Euler and D'Alembert's principle of constraint. With suitable notation Kane's equations are an immediate consequence of these. The notation introduced above provides a simple expression for this principle. Thus let the total force acting on a particle system be denoted by the total applied force K-vector, R <, composed of the K force vectors R<L

acting on each point of the mechanism. This vector can be split into two components, the constraint force and the applied force, thus

R< = R< +R . (I8) D'Alembert's principle of constraint is then simply stated as

Rc .J = 0, (19) Proc. R. Soc. Lond. A (1992)

74 M. Lesser




where flJ is a tangent vector to the instantaneous configuration manifold. If we can find n independent tangent vectors, we have n independent conditions on the force.

(b) Generalized active and inertial force The newtonian equations of motion are conveniently expressed in terms of the

momentum K-vector, defined as the K vector made up of the momentum at each point of the mechanism. Thus = <L (20)

where ML is the mass of the Lth particle and v<L is the vectorial velocity of the particle relative to an inertial frame. Newton's equation is then given by

-P'<^R +R +R< =. (21)

Using the constraint relation, we project this equation onto the tangent space of the point which describes the configuration of the mechanism. This gives

[-P< +R<].^fj =0, (22) one equation for each independent tangent vector. In his formulation, Kane does not use the language of K-vectors as discussed above. However, he defines two quantities, the generalized active force and the generalized inertial force, which he uses to represent the above relation. Thus let

Fj = R< . fi,<, (23) and Fj= -P' <.^, (24) be the generalized active and inertia forces associated with the tangent bases set fi'. When required, the tangent bases set used can be noted by attaching a left superscript to the defined quantities. The form of the constraint free equations of motion F* +F = 0 (25) is referred to as Kane's Equations; however, to really use this nae, we must still is referred to as Kane's Equations; however, to really use this name, we must still discuss another concept emphasized by Kane in his formulation, the notion of generalized speeds. We approach this subject in a somewhat different manner than Kane's algorithmical arguments, making direct use of our geometrical picture.

(c) Determination of constraint forces To complete this part of the discussion, we examine the equations for the

determination of the constraint forces. There are two cases to consider, one in which the active forces are independent of the constraint force, the other, of which dry friction is a prime example, in which the active forces depend on the constraint forces. As the constraint force is in the complementary space, we can represent it in terms of a linear combination of any set of K-vectors which span this space. Denote such a set, as above, by yj' so that

m

R< = sjy'. (26) j=1

Now using this representation for the constraint force, simply project the equations of motion onto the m K-vectors that span the complementary space, giving the following m equations for the parameters sj:

m -P< . y < + R yj,< + E? sYi . Y, = 0 (27)

i=l


710



It is convenient to extend Kane's terminology and define the generalized inertial constraint force, GJ, as the first term in the above, the generalized applied constraint force, Gj, as the second term, and the complementary bases coupling coefficients as

aY'n Z<,<(28)

so that the equation for the constraint force expansion coefficients can be expressed in the form

m G* +G+ siaj 0. (29)

i=l

When the active forces are independent of the constraint forces, this set of equations can be solved after one obtains the solution of Kane's equations; however, in cases such as dry friction where some of the active forces are functions of the constraint forces the two sets of equations must be solved together.

(d) Equivalent principles D'Alembert's principle of constraint may be formulated in a number of ways.

L. A. Pars (Pars 1965) provides an extensive discussion of this subject in which the basic form is termed the fundamental equation. In the so-called second form, the principle of constraint is expressed in terms of finite variations of velocities which are consistent with the instantaneous constraints. These are sometimes called virtual velocities. An infinitesimal form of this is due to Jourdain. This is also sometimes called the principle of virtual power and in much of the German literature is referred to as Jourdain's principle. The tangent vector methods used in the remainder of this article are in a sense equivalent to this principle, which has been taken by some authors as a means of deriving forms similar to Kane's equations. An example of this is in Kruezer & Schiehlen (1990). The classical Gibbs-Appell formulation is easily derived from Pars third fundamental equation, which can be interpreted in terms of virtual accelerations.

4. Kinematic equations

(a) Tangent K-vectors and partial velocities

The determination of tangent vectors to the configuration surface lies at the heart of Kane's approach to mechanics. A major strength of the method is that such vectors can be obtained directly by the inspection of velocity expressions for the various points of the mechanism. Such velocity expressions may be found by standard techniques of kinematical analysis. If one starts with a particular choice of generalized coordinates and proceeds to velocities of points in terms of them, it is natural that time derivatives of the coordinates will make an appearance. In general the equations of motion will contain all the pertinent generalized coordinates, qj, and their time derivatives, qj. Simple application of the chain rule shows that velocity expressions depend linearly on the time derivatives of the generalized coordinates, thus

v'<= r< +tr<, (30) i=1 qi - t

where v< is the velocity K-vector consisting of the column matrix of velocity components of each point. Note also the use of our convention that a derivative of


76 M. Lesser




a vector without an indication of frame means that the derivative operation is with respect to the inertial frame. With the notation Tz for the coordinate bases tangent vectors and

= ar</t, (31)

this velocity expression takes the form n

v< = E , 4 +, *T (32) i=1

Thus it is obvious that if we, by any means, obtain an expression for the K vector velocity, the K vector coefficients of the time derivatives of the generalized coordinates provide us with a set of coordinate bases tangent vectors! Kane's calls the ordinary vectors which are the component vectors of these K vectors partial velocity. Thus T2<L is the partial velocity vector associated with the point L and the generalized velocity 4q. If one stays with coordinate bases tangent vectors it is relatively straightforward to proceed from Kane's equations to the various forms of Lagrange's equations, however even if one keeps this restriction, Kane's formulation has certain practical calculational advantages.

(b) Transformations of tangent vectors and generalized speeds The consideration of linear transformations to new sets of tangent bases vectors

lends power and flexibility to the dynamic analysis. Note that in this theory, we do not assume orthonormality of these vectors. The most general linear transformation of such a transformation is given by

n

fli< = Z PT,7 , (33) j=l

where the notation shows explicitly that the transformation coefficients are from the T to the / bases. While it is convenient to think of the T bases as being a coordinate bases, the general transformation applies to any set of bases vectors. The key to what follows is the consequences of the transformation on the velocity expression (32). To clearly see the geometric relationships in the coordinate space, we introduce notational conventions similar to the ones used in the discussion of bases vector transformations in three-dimensional physical space discussed previously. Thus let the symbol T stand for the matrix column composed of tangent K-vectors, q for a column of n generalized coordinates, q for a column of generalized velocities and so forth. The matrix array of transformation coefficients from the T to the f/ representation is represented by IT '. Superscript T indicates transposition as before, and it is assumed that the inverse transformation from f to T exists so that

flT [Ti = U. (34)

Using this notation the velocity expression (32) may be written as

v< =T+ = q + T T + T<, (35)

while the transformation equations take the form

p = /T r, (36) r = *#Tft. (37)

Substitution of the transformation equation into the velocity expression then gives the relation

< = 3TTf 4 + Tz. (38)


77



To interpret this relationship, it is useful to think of the velocity K-vector as pointing out of a plane representing the configuration tangent hypersurface. Let 17 and H7 represent projection operators which decompose a K-vector into components parallel and orthogonal to the plane, so that

v< = 1v< +H 1 v. (39)

Such projection operators can be obtained explicitly in terms of the bases sets for the configuration and complementary tangent spaces. Now in general the term T< will have a non-vanishing projection parallel to the tangent hyperplane. This projection can be represented in terms of any bases set, so for example

H. T< = pTfh. (40)

We are free to decompose these expansion coefficients into an arbitrary sum of the form

Ah = f+ Pg. (41)

Note that this decomposition is not unique. Also note that

H1, v< = H7 Tt, (42)

as the out of plane K-vector velocity component is not contained in a sum of expressions involving the tangent vectors. Therefore we can write

v< = fT[T,Tr4 +f ] + [fT g + 7, T<]. (43)

If we define the generalized speeds as

u = [T f rq +f ], (44)

and let flt = [PTg + v,T t ], (45)

we can write the velocity relation in the form

v< = Tu + <. (46)

Also, using the invertibility of the transformation equation, we can express the generalized speeds as functions of the generalized velocities. This gives the relation known as the kinematic differential equations, i.e.

q = T TTu-- _ A/ TTf. (47)

It is now possible to make several crucial observations. First of all the transformation equations apply to any sets of tangent K vector bases, the bases involved need not be derivable from coordinates. Secondly, we are free to choose any convenient set of generalized speed expressions which lead to consistent velocity expressions. This means that we can pick off the set of tangent bases vectors associated with the generalized speeds by inspection, just as in the coordinate case. Thirdly, we can choose generalized coordinates and generalized speeds that are convenient for the problem at hand and then use a kinematic analysis to find the relationship, that is, the kinematic differential equations (47).

The special case in which one takes the generalized speeds equal to the generalized velocities, that is, in which u = q, will seldom yield the most convenient form of the equations of motion. The freedom of choosing generalized speeds thus provides a powerful aid in the dynamical analysis. This capability also exist in the Gibbs-Appell formulation of mechanics; however, Kane's approach with the added insights of the


MI. Le8eser 78




geometric picture presented here appears to provide a superior and more easily applied set of tools in attacking individual problems. This is a somewhat subjective assessment which requires further study.

(c) Linear velocity constraints

The purpose of this section is to briefly outline Kane's approach to mechanisms with velocity constraints and to see how they fit into the geometric picture and the transformation of tangent bases K-vectors. Velocity constraints can arise when we have some implicit condition among the generalized coordinates and it is convenient to represent the condition in differentiated form. Thus the total time derivative of a condition, f(q, t) = 0, gives a linear relation among the qj. A more intrinsic case occurs in the application of rolling constraints which lead to a non-integrable relation among the time derivatives of the generalized coordinates or the generalized speeds. Kane calls both cases non-holonomic, however we will apply this term only to the latter situation.

To discuss the treatment of linear relations among the generalized speeds, we start with the special case of generalized velocities and a related coordinate K-vector bases. Assume that the generalized coordinates are so labelled that we have k = n-p relations in terms of the first p generalized velocities for the last k of these quantities. To give an explicit representation of this in matrix form, we use the notation that a['J J represents a rectangular array with components a,m with m running from k to s and n going from i to j, while b[l j] represents a column with index running from i to j. A linear relation between generalized velocities then takes the form

Lp+l, n] aKP+1' nl [j, n] + b[p+l, n] (48)

where the matrix a and the column b may be functions of q and t, but not of q. The

velocity expression (32) can be expressed in the form

/) --TT[1, p] 4[1, p] + 7lp+l, n] [p+l, n] + t (49)

so that substitution of the velocity constraint relation leads to the result

?< = ( ~T 4r1T I p] all P T b 4- r< v' = (Tl p] +, ' TP+l, n p+l, n]) 41l, p]] + Tp+l,n] b[p+l,n] + t . (50)

The first p generalized velocities may now be treated as independent quantities associated with the p bases K-vectors given by the relation

/?T ?-7-T T -1, P]I P1] , pI = '{I + T[+1, n] a[p+l,n] (51)

These new tangent K-vectors may now be used to obtain p equations of Kane's form. In effect the appropriate tangent space is a p-dimensional subspace of the n- dimensional geometric configuration tangent space. In the case where the velocity relations arise from the differentiation of relations among the generalized coordinates or equivalently when they are integrable, we have the so-called holonomic situation. This is equivalent to the fact that it is then possible to find a coordinate tangent bases set of K-vectors for the p-dimensional subspace. In the non-holonomic case the lack of integrability is equivalent to the impossibility of finding such a coordinate tangent bases for this subspace. Thus this ties together the well-known facts about holonomic and non-holonomic systems with the geometric interpretation of Kane's formalism. The above arguments are easily transferred to the case where we use the generalized speeds which are not generalized velocities, i.e. when

[p+l, n] = all, , n] U1 + b[+, (52) Proc. Rn. Soc. L+,ond. A (1, 92), n

Proc. R. Sec. Lond. A (1992)

79



5. Extension to rigid bodies

(a) Tangent vector fields The application of D'Alemberts Principle to rigid bodies requires the determination

of a tangent vector at each point of each rigid body in the mechanism. Let the position vector from a chosen point in an inertial frame to the mass centre of each

body be designated by r<L, i.e. by the same notation used for particle positions above. The position of each point in the rigid body from the inertial origin will be noted by r<L>, while the distance from the centre of mass to the point will be taken as rL>. The corresponding K-vectors will be indicated, as before, by dropping the index. Again we determine a set of tangent K-vectors by considering the velocity expression for all the points of the mechanism. Thus

r<> = r< +r', (53)

which by taking the derivative with respect to the inertial system and using the results of our preliminary discussion and the fact that in the reference frame attached to each body, the distance vector between the mass centre and the field point is constant during a motion, gives the result that

v<> = v> +()xr>. (54)

In this last equation, we have taken the cross product operation between two K- vectors as being defined to be the K-vector with components composed of the component cross product. From our discussion of Darboux vectors, it should also be clear that

(L) = (L)bt. (55)

To write the Newton-Euler equations for the rigid body system, we assume that the total force and the mass density on body L is given by f<L(rL>) and p<L(rL>)

respectively. Note that these are functions of position in the frame fixed to each body and that they may also depend on time, though we have suppressed this dependence for clarity. The moment of inertia dyad for each body about its centre of mass, which will depend on the relevant density distribution, is denoted by I'L while the body's angular momentum is given by

H<L = I<L. (L) (56)

Thus the Newton-Euler equations take the form

P<L f <L d7 (57) L)

H<L = rL' xf<L dVL. (58) L)

The velocity relationship for rigid bodies, equation (54) can be expressed in terms of Darboux vectors associated with the generalized coordinates by the use of the chain rule. In terms of coordinate bases tangent vectors, this gives the relation for the body L that

n n

v<L> = J j+T<L + +L)b (L)bi/t x rL>. (59)

j=Proc. R. Soc. Lond. A (1992) Proc. R. Soc. Lond. A (1992)

80 M. Lesser




To bring this into some agreement with Kane's notation, let

^L) -

^(L)bqj, (60) (L) = (L)bt. (61)

Kane calls the first set of quantities above, partial angular velocities while the coordinate tangent vectors are called partial velocities. Thus the final form of the velocity expression at an arbitrary point of body L is given by

n < = (T L + L) x rL) + T<L + )(x L) xrL (62) v<L> = C ("3 3 (62)

j=1

thus a tangent K-vector at an arbitrary point of the body is given by

Z<> > = T + o )xr>. (63)

Again, note that these vectors can be found from velocity expressions in terms of coefficients of the various g. As this tangent vector is a function of the position of the point considered in the body, we will call the associated K-vector for the system of bodies a tangent vector field.

(b) D'Alembert's Principle and Kane's Equations D'Alembert's Principle, for a system of rigid bodies, can now be stated in terms of

the total applied force density and suitable tangent K-vector fields. Let

f< =fa< +f<, (64)

represent the decomposition of the applied force density into the portions due to the applied forces and the forces of constraint. D'Alembert's Principle then states that

JffcT > d = 0, (65)

where the integration is carried out over each of the bodies making up the system, i.e. over each of the components of the full K-vector. It is now straightforward to eliminate the constraint forces from the Newton-Euler form of the equations of motion in the K-vector form. Thus use the big dot product on the momentum and

angular momentum equation with the partial velocity and partial angular velocity K-vector associated with the coordinate q2 and sum the result, i.e.

P<.,< +HI<' (f< T7 +r> xf<. o))dV - f(s .,<d + xs

= f(f' >.,g>)dV.

Therefore if D'Alembert's Principle holds, we can replacef< in the above byf', i.e. the constraint forces give a vanishing contribution to the above integral. The applied force and applied torque K-vectors are defined by

R = d,f (66)

T= r> xf. (67)


81



The constraint-force-independent form of the equations of motion then become:

R <? + T ' *T ( = P T' * + H'< . + (68)

Kane as in the particle mechanism case above, uses the notation Fj for the left side of the above equation and -F, for the right side. Again, he calls these generalized active force and the generalized inertia force of the mechanism. With this

interpretation we thus again arrive at the form called Kane's Equations. By a suitable extension of our K-vector notation we can write these equations in

exactly the same form as used in the particle mechanism case. This allows the exact same interpretations in regard to transformations of K-vector tangent bases and the introduction of generalized speeds and the kinematic differential equations. Thus we define the Lth applied force pair as:

R<L {R<L Ta<L} (69)

and the body momentum pair as

p<= {p<L, H<L (70)

The tangent vector pair is taken as

TL ={T<L, (L)}. (71)

The pair dot product of two pairs is defined by the sum of the component dot products so that R <L. <L - R<L'<L+ T<L. (L) (72) a f a i Ta j

which allows us to write the constraint force free equations of motion in the form

-P<' * < + R< * = 0. (73)

The first term is Kane's generalized inertial force, and the second is the generalized active force associated with the jth generalized coordinate. Note that the coordinate space is now 6K-dimensional. It is an obvious extension of the previous discussion to show that the new objects obey the same transformation rules for tangent vectors and that we can define generalized speeds, non-coordinate based tangent vectors and the like. Thus exactly the same formalism and geometric interpretation of Kane's approach applies. This will be demonstrated more explicitly by means of the example of the next section.

6. An example We now consider a simple example with the purposes of showing the formalism at

work, how one chooses the points to be considered in the construction of K-vectors, the extensions needed to deal with rigid body problems and how one may choose generalized velocities which lead to decoupled equations of motion. While a major advantage of the formalism is its usefulness in treating complex systems, the above purposes are best served by the consideration of a relatively simple example, thus we consider the two-dimensional mechanism shown in figure 1. It consists of a massless rod of length 21 free to rotate about the joint at point 9, which is also taken as the origin of an inertial reference frame. A planar body is connected to the rod at point BP at a distance I from point 0. The centre of mass of the body is located at point Y a distance s from the pivot point -. A point mass M1 is located at the point 2 at the end of the rod. As the problem is basically planar, we use n to indicate a unit vector pointing in the direction of all rotation axis. The point mass is taken as body 1 and the distributed mass as body 2. The component of the inertia dyad for rotation about


M. Lesser 82




bl

Figure 1. Example rigid body system.

the mass centre is taken as I, while the total mass is M2. For simplicity it is assumed that the line ?9 is a principle axis. It is also assumed that we have a torsion spring of strength k at point O, and that the mechanism is placed in a uniform gravitational field of constant acceleration g. The orthogonal unit vectors, b1, b2 are fixed in the frame of the rod, while the unit vectors cl, c2 are oriented along principle axis as shown. The orthonormal set nj are fixed in the inertial frame.

The dimension of the K-vector employed depends on our interest in constraint forces. In this example we ignore the constraints and base our analysis on the three points 9, J and Y. The first point involves a point mass, the second a distributed mass. The last point, 9 must be considered because of the fact that a force is applied to the system by the torsion spring located at this point. We take as generalized coordinates of this holonomic mechanism the angles q1 and q2, where cosql = blqn1, and cos q2 -- cl 'n1. For uniformity of treatment we will treat all points of the mechanism as body pairs, i.e. rigid bodies, though of course the point mass and the location of the torsion spring are rather degenerate cases. In the preliminary of set- up we use the generalized velocities qj as generalized speeds, so that the kinematical differential equations are simply

i = q (74) The extended position K-vector is given by

[ {2b1, b} < = {lb + scl, c} , (75)

{lb, b}

which leads to the extended K-vector velocity

{21ub2, u 1 n}

v'= \{lu b2 +su2 c2, n} . (76) {lui b1, u1 in}

By inspection of the coefficient vectors for the generalized speeds Uj e see that two coordinate-based tangent vectors are given by

{2lb2, n}] = {lb2,O} , (77)

L {lb, n}


83



and T = {s2n}. (78) {0, 0}

To complete the formulation we must form the extended momentum and applied force K-vectors. The derivative of each of the components of the momentum K- vector must be found with respect to the inertial frame n. This is a rather cumbersome expression, and as we are more interested here in the procedure than in the specific problem we simply list the momentum and applied force K-vectors. Thus

{21M u- b2' O}

p< = {M2tlul b2 +XM su c, lun} (79) {0, O0}

- {-Ml gn2, O} and R = {-Mi gn2, k(l-q)n} (80)

{0, -k(q-q2) n}

The generalized active and inertial force are then calculated using the above tangent vectors.

To conclude this example, a procedure is sketched for how one can arrive at new generalized speeds for which each equation of motion will involve only one of the derivatives of the set of generalized speeds, in other words for which Kanc's equations will be diagonal in the derivatives of the new generalized speeds. This is simply done by orthogonalizing the tangent bases vectors under a suitable metric. In the simplest situation Ml = M21 = I, which can then all be taken as unity without loss of generality. Thus let

1 /= II; / I I i 11, (8 1 )

where the norm: [ If - l1 1 T1. (82)

The new second independent bases vector is simply obtained by subtracting off the projection of the old bases vector, i.e. by Gram-Schmidt orthogonalization, thus

< = <-( .f . (83) X2 = Z2 -2 1 1 al (83 If we are content with the new vectors being orthogonal but not normal we can stop at this point, i.e. we need not normalize the last result. With the notation

a 111, (84)

2= 2 fi (85)

we find that e = a1 18, (86)

2< = a21 + . (87)

To find the appropriate generalized speeds for which the new tangent vectors are natural consider the velocity K-vector expression for a field point, i.e.

v< > - (T< + -C) X r') l + (T< + Co) X r') 2: (88)

which we write in the shortened form

v<> = 1{~ I r>}+U2{T Ir }. (89)


84 M. Lesser



A yeometrical interpretation of Kane's Equations

Now simply substitute the new orthogonal tangent vectors in this expression and examine the coefficients, thus

< > (1 a + u2 a2) {l< I r>} u, { Ir>}, (90)

which gives the new generalized speeds

W1 = u1a + U2 a2, (91)

2 = u2. (92)

To obtain the kinematic differential equations in standard form, solve for the u and use their relation to the q. The resulting equations of motion will be diagonal because derivatives of the generalized speeds arise from the momentum vector which has the form of a sum of generalized speeds times tangent vectors. The derivative of this will give a sum of derivatives of generalized speeds times tangent vectors plus terms linear in the generalized speeds. Therefore when the big dot product is taken, all the orthogonal vector terms will be eliminated, leaving only one derivative of a generalized speed in the equation associated with each tangent vector. This is true in general and demonstrates that except in unusual circumstances, it is always possible to find a set of generalized speeds which will lead to a diagonal form of Kane's equations of motion. Of course there will be a price to pay in the increase in the complexity of the kinematic differential equations. In a sense, the orthogonalization procedure offers a means of complexity sharing between Kane's Equations and the kinematic differential equations.

7. Conclusions

The procedure developed by Kane for the formulation of equations of motion for systems of interconnected rigid bodies, while related to standard techniques, appears to have a number of practical advantages over such techniques. These advantages are related to the special emphasis given to the so-called partial velocities and partial angular velocities, which along with what are called generalized speeds, are used to implement D'Alembert's Principle of constraint. In this paper it has been shown that the partial velocities and partial angular velocities, which are three-dimensional euclidean vectors, are components of multidimensional tangent vectors to the configuration manifold of the system and that Kane's Equation are simply the

projections of the Newton-Euler equations of motion onto these tangent vectors. This interpretation of Kane's formalism in terms of the geometry of the configuration manifold was made explicit in terms of what were called K-vectors, which are vectors in the embedding coordinate space of the configuration manifold. There are two distinct advantages in adding this interpretation to Kane's algorithmic approach.

The first involves the fact that the full coordinate space can be decomposed into the tangent space to the configuration manifold and its orthogonal complement. Thus constraint forces can be clearly and explicitly represented in terms of K-vectors which span the orthogonal complement. This eliminates the need to introduce fictitious generalized speeds, as done by Kane, to obtain representations for the constraint force. The fictitious generalized speed technique is a useful tool where one has insight into the problem and should be used by the experienced investigator; however, the complete representation of constraint forces in terms of the spanning set of the orthogonal complement of the tangent space provided a fail safe procedure Proc. R. Soc. Lond. A (1992)

85



that has a clear geometric interpretation. Wang & Huston (1987) have also discussed this point using undetermined multipliers for the determination of constraint forces. Their method is equivalent to the one discussed in this paper, but without the

geometric interpretation. The second advantage is that we developed a procedure for the diagonalization of

the equations of motion by orthogonalization of the spanning set for the tangent space. By considering the transformation of the basic expansion for the K-vector velocity, we saw how to obtain sets of generalized speeds which would lead to separate equations of motion for each 2j. The practical utility of this procedure is still to be determined; however, it is important to see how the coupling of the geometric picture with Kane's ideas can suggest interesting ways of manipulating the equations of motion.

In some sense there is nothing new in an intrinsic way in many of the manipulations of points of view that mechanics has undergone. A number of articles, referred to in the introduction to this paper, discuss the equivalence of Kane's method to other formulations. Of course if it is correct it must be equivalent to any formulation which handles the same class of problems, such as those involving non- holonomic systems. This does not address the fact that some points of view are more suggestive, less costly in calculational effort or easier to understand and teach than others. In the final analysis this is a subjective matter. The term Kane's equations really refers to a particular point of view in choosing various constructs useful in formulating and manipulating mechanical constructs. The fact that it has gained considerable acceptance in the technological mechanics community provides a good reason for understanding it and its relations to other formulations. I feel that Kane's approach provides many advantages in understanding, calculation and teaching. It provides a common theory and approach to problems involving non-conservative forces, difficult geometries and non-holonomic constraints. The geometric point of view taken in this paper adds to all these features.

I thank Dr Nicholas Apazidis and Dr Hanno Essen of the Mechanics Department of the Royal Institute of Technology for their continued interest and comments. Thanks are also due to the Swedish Board for Technical Development (STU) for financial support for these efforts.

References

Banerjee, A. K. 1987 Comment on 'Relationship between Kane's Equations and Gibbs-Appell Equations'. J. Guidance Control Dynamics 10, 596-597.

Desloge, E. A. 1987 Relationship between Kane's Equations and the Gibbs-Appell Equations. J. Guidance Control Dynamics 10, 120-122.

Desloge, E. A. 1989 Efficacy of the Gibbs-Appell method for generating equations of motion for complex systems. J. Guidance Control Dynamics 12, 114-116.

Kreuzer, E. & Schiehlen, W. 1990 NEWEUL-software for the generation of symbolical equations of motion. In Multibody systems handbook (ed. W. Schiehlen). Springer-Verlag.

Kane, T. R. & Levinson, D. A. 1985 Dynamics: theory and applications. McGraw-Hill. Kane, T. R., Likins, P. W. & Levinson, D. A. 1983 Spacecraft dynamics. McGraw-Hill. Levinson, D. A. 1987 Comment on 'Relationship between Kane's Equations and Gibbs-Appell

Equations'. J. Guidance Control Dynamics 10, 593. Levinson, D. A. & Kane, T. R. 1990 AUTOLEV - a new approach to multibody dynamics. In

Multibody systems handbook (ed. W. Schiehlen). Springer-Verlag. Pars, L. A. 1965 A treatise on analytical dynamics. Heinemann.


86 M. Lesser




Schiehlen, W. (ed.) 1990 IMultibody systems handbook. Springer-Verlag. Stoker, J. J. 1969 Differential geometry. Wiley Interscience.

Wang, J. T. & lHuston, R. L. 1987 Kane's Equations with undetermined multipliers - application to constrained multibody systems. Trans. ASME J. appl. Mech. 54, 424-429.

Received 26 March 1991; revised 26 July 1991; accepted 2 September 1991




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A Geometrical Interpretation of Kane's Equations