cayley kinematics and the cayley form of dynamic equations

Cayley Kinematics and the Cayley Form of Dynamic EquationsAuthor(s): Andrew J. Sinclair and John E. HurtadoSource: Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 461, No. 2055(Mar., 2005), pp. 761-781Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/30046316 .

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PROCEEDINGS Proc. R. Soc. A (2005) 461, 711-731 -OF-

THE ROYAL doi:10.1098/rspa.2004.1340

SOCIETY A Published online 11 January 2005

Cayley kinematics and the Cayley form of dynamic equations

BY ANDREW J. SINCLAIR AND JOHN E. HURTADO

Texas A&M University, Department of Aerospace Engineering, TAMU-3141, College Station, TX 77843-3141, USA ([email protected])

The Cayley transform and the Cayley-transform kinematic relationships are an important and fascinating set of results that have relevance in N-dimensional orientations and rotations. In this paper these results are used in two significant ways. First, they are used in a new derivation of the matrix form of the generalized Euler equations of motion for N-dimensional rigid bodies. Second, they are used to intimately relate the motion of general mechanical systems to the motion of higher-dimensional rigid bodies. This approach can be used to describe an enormous variety of systems, one example being the representation of general motion of an N-dimensional body as pure rotations of an (N + 1)-dimensional body.

Keywords: Cayley transform; N-dimensional dynamics; rigid-body mechanics

1. Introduction

There have been many studies to extend the principles behind rigid-body mechanics in three-dimensional space to principles that govern motion in higher-dimensional spaces. Some progress has been made in extending the kinematic concepts of orientation, rotation and angular velocity (Bar-Itzhack 1989; Bar-Itzhack & Markley 1990; Junkins & Kim 1993; Bottema & Roth 1979; Mortari 1997, 2001; Schaub et al. 1995), and the dynamic concepts of inertia, momenta and impressed forces (Bloch et al. 2002, 2003; Fedorov & Kozlov 1995; Ratiu 1980; Weyl 1950; Frahm 1875).

The expression that defines the angular-velocity matrix in N-dimensional space is !7(t) = -CCT (Bottema & Roth 1979). The matrix C is an N x N proper orthogonal matrix which can represent a rotation about the origin, and the angular-velocity matrix, 17, is an N x N skew-symmetric matrix with M = N(N - 1)/2 independent elements. The rearranged form C = -12C is sometimes called Poisson's equation. Many properties of the N-dimensional angular-velocity matrix have been established (Bottema & Roth 1979), and perhaps one of the most significant is that only in three- dimensional space can the M independent elements of the angular-velocity matrix be considered as components of an angular-rate vector (i.e. only in three-dimensional space does N = M) (Bar-Itzhack 1989).

Rigid rotations in N-dimensional space take place on an invariant plane (Mor- tari 2001) and are related to the concept of the N-dimensional vector cross-product operation (Mortari 1997). A natural way to parametrize an N-dimensional rigid rotation is with the principal rotation angle and a set of two N-dimensional orthogonal

Received 19 September 2003 © 2005 The Royal Society Accepted 18 March 2004 761



A. J. Sinclair and J. E. Hurtado

vectors that define the invariant (principal) plane (Mortari 2001; Bauer 2002). It is important to note that N-dimensional orientation is different from N-dimensional rotation: an arbitrary orientation in N-dimensional space is a product of a minimum set of rigid rotations and only for the N = 3 case can an arbitrary orientation be realized with one rigid rotation.

The Cayley transform and the Cayley-transform kinematic relationship are another set of results that have relevance in N-dimensional orientations and rotations. These important results have a direct bearing on this paper and will be discussed more fully later.

The governing equations of motion for N-dimensional rigid bodies are often presented in a matrix form. The traditional approach to deriving the equations has been the Hamiltonian method of mechanics (Fedorov & Kozlov 1995; Ratiu 1980; Bloch et al. 2003). That method takes a geometric view and uses concepts in abstract algebra and coordinate-free differential forms to arrive at the matrix Lax pair representation of the N-dimensional generalized Euler equations. The focus in many of these studies has been on different representations of the equations and their integrability (Bloch et al. 2002; Ratiu 1980; Manakov 1976; Mishchenko 1970). Consequently, the systems that have been addressed are those that are not subject to externally applied torques and do not have kinetic-energy functions that depend on the generalized coordinates. In some earlier work (Hurtado & Sinclair 2004) the vector form of the N-dimensional generalized Euler equations was developed by using the Lagrangian method of mechanics. This vector form is easily mapped into the matrix Lax pair form. The vector form of the equations was developed without any a priori selection of orientation parameters for the N-dimensional rigid body.

Here an N-dimensional rigid body will be defined as a system whose configuration can be completely defined by an N x N proper orthogonal matrix. It will be seen that a wide variety of mechanical systems can be modelled as N-dimensional rigid bodies using this definition, which relaxes some conditions used in earlier work. In this paper, the Lax pair form of the N-dimensional generalized Euler equations of motion are developed following an a priori decision to describe the system using the Cayley orientation and kinematic variables. The Cayley orientation and the kinematic variables are then used to relate the motion of general mechanical systems to the motion of higher-dimensional rigid bodies.

Preliminaries

Index notation is used extensively throughout this paper. The elements of a matrix or tensor, A, are expressed as Aij, and the elements of a vector, a, as ai. The Einstein summation convention is that if any index is repeated twice within a term, then the term represents the summation for every possible value of the index. An index must not be repeated more than twice in a term. Indices that appear only once in each term of an equation are free indices, and the equation is valid for each possible value of the index. The Kronecker delta, 6ij, is equal to unity if i = j and is equal to zero otherwise.

A new numerical relative tensor, Xk, is useful for the developments in this paper. This numerical relative tensor was introduced in some earlier work (Hurtado & Sin- clair 2004) to relate the elements of an N x N skew-symmetric matrix to an M- dimensional vector form:

ik = XiWJ (1.1)

Proc. R. Soc. A (2005)

762



Cayley form 763

Table 1. Summary of properties related to XZk

generating vector to skew-symmetric matrix Uik = XkUj

skew-symmetric matrix to generating vector uj = X Uik

skew-symmetry in the lower indices xk = -xi

upper-index identity XikXk = 23jl

X-6 identity XjkXmnn = Sjm6kn - 6jnSkm

lower-index identity XjkXjn = (N - 1)6n

The upper index j in this expression is summed from 1 to M, whereas the lower indices i and k take on values from 1 to N. In the familiar N = 3 case, Xik becomes the Levi-Civita permutation symbol eijk. The vector w in equation (1.1) is defined as the generating vector of the skew-symmetric matrix A2. The elements of w are specified to be related to the elements of A? in the following form:

0 -wUM WM 0

[2] = 0 --w6 J5 - 4 . (1.2)

w6 0 -w3 w2

S . . --5 W3 0 -W1

W4 -W2 w1 0

There are many ways to specify the independent elements of 2, but the form of equation (1.2) is chosen because it matches the three-dimensional form. This form leads to the following expression for ; k:

S(-1)i+f(i, k) for j = and i k, ik 0 otherwise, (1.3)

where

f(i, k) = (k )2' x (i + k +),

y +(i k- ( i)2), z= y2 - N) + 2 + N +1.

Several properties of Xik are summarized in table 1 (Hurtado & Sinclair 2004).

2. Cayley kinematics

Some of the most important and elegant concepts in N-dimensional kinematics relate to the Cayley transform. This famous relationship provides a unique mapping between proper orthogonal and skew-symmetric matrices (Bottema & Roth 1979):

C = (I - Q)(I + Q)-I = (I + Q)-I(I - Q), (2.1)

Q = (I - C)(I + C)-1 = (I + C)- (I - C). (2.2)





Here, C is an N x N proper orthogonal matrix, whereas Q is an N x N skew- symmetric matrix. The elements of Q are an M-dimensional set of parameters that represent N-dimensional attitude. In fact q, the generating vector of Q, is the vector of extended Rodrigues parameters for N-dimensional spaces (Bar-Itzhack & Markley 1990; Schaub et al. 1995).

In this section, the Cayley transform and the Cayley-transform kinematic relationship will be used to find a general expression for the extended Rodrigues parameter rates, <, in terms of the angular-velocity vector, w. In the derivations that follow, it will be convenient to make the following designations:

I + Q - A+, I- Q - A-, (2.3)

(I + Q)- E_ B+, (I - Q) 1 B-. (2.4)

The Cayley-transform kinematic relationships connect the derivatives of the M independent parameters of Q to the angular-velocity matrix (Junkins & Kim 1993):

2 = 2(1 + Q)-Q(I - Q)1 = 2B+QB-, (2.5)

S= \(I + Q) AI - Q) = A 2A-. (2.6)

Equations (2.5) and (2.6) represent a linear mapping between the generalized velocities, Q, and a set of quasi-velocities, 17, for N-dimensional rotations. Whereas these equations are represented as linear transformations of N x N, skew-symmetric matrices, they can be rewritten in the familiar form in terms of the linear transformation of an M-dimensional vector. In index notation equation (2.6) is written as follows:

Qvp = Ak 2ki A. (2.7)

The generalized-velocity and angular-velocity matrices are mapped into their vector forms using X k:

Xp, = A kX mAlp. (2.8)

Both sides of the above equation are multiplied by i :

xvpXvppqj = xpA, k XWmAlp, (2.9)

1 i m + W 2- XpXk Ak Ap (2.10)

qi = -xpXmAkAlpWm - Aimwm. (2.11)

The two-index variable Aim represents the components of the matrix A that performs a linear mapping of the angular-velocity vector onto the generalized-velocity vector. Similarly, an expression can be found for the matrix B that performs the inverse mapping; however, this matrix is a function of B+ and B- and is thus less convenient to evaluate. The expression for Aim can be expanded using the definitions of A+ and A-:

A i m+ A Aim - XvpXkl vk lp

= pX m vk + Qvk)1p - Qlp)

X= pX~(kSlp vk - 6 p + lpQvk - QvkQlp). (2.12)


764



Cayley form

This expression can be simplified by analysing the first three terms:

XvpXklvk6lp - XvpXvp 2imi (2.13)

XvpXyklvkQlp -XpXllp, (2.14)

-XpX lp Xk pXkpQvk = XpX vQpl - pX lp. (2.15)

These are substituted back into equation (2.12) to give the final result:

Aim i~ - £XpXklp - ph k kQlp). (2.16)

For the special case N - 3, the equation for the elements of A can be simplified by substituting Eijk for Xjk:

Aim i m E- vipEvmlQlp - vipEkmlQvkQlp). (2.17)

The 'e-6 identity' can be applied to the second term of this equation. Additionally, the fact that Qii equals zero, because Q is skew-symmetric, is used:

EvipEvmlQlp = (SimSpl - 5ilpm)Qlp = -Qim. (2.18)

The third term of equation (2.17) can also be rewritten using the generalized Kro- necker delta (Lovelock & Rund 1989):

V i p

U sp

= 6vk(im6pl - 6pmail) + 6ik(pm(vl - avmapl) + Spk(6vm6il - 3imavl)

= SvkSimSpl - vkapmSil + 6ik6pmavl - aikavmapl + 8pk6vm6il - pkfim5vl•

(2.19)

The third term of equation (2.17) therefore becomes

EvipEkmlQvkQlp = vivm + QmpQip -im vpQvp

= 2Qvivm Q VimvpQ vp. (2.20)

This expression is now rewritten in terms of the generating vector elements qj and the e-6 identity is used once again:

EvipkmlQvkQlp - 2EvriqrEvsmqs - 6imcvrpqrcvspqs

= 2(6rsim - 6rm8is)qrqs - Sim(6rs6pp - 6rp6sp)qrs

= 26imqrqr - 2qiqm - 36imqrqr + 6imqpqp

= -2qiqm. (2.21)

Equations (2.18) and (2.21) are now substituted into equation (2.17) to give the familiar form for the mapping from the angular velocity to the Rodrigues parameter rates (Schaub 1998):

Aim (6im Qim + qiqm). (2.22) Ai - m C m z.m


765




For reference, the complete form of A is given below for three- and four-dimensional bodies:

1 I + q2 qlq2-q3 qq93 +92 [AIM=3 = q92q +q3 1 + q q2q3 - , (2.23)

q939ql - 92 3q2+ql 1 + 9q

1 + q qlq2 3 qlq9193 92

q2ql9 + q3 1 + q 929q3 - q9 1 q3q91 - q2 q3q2 +9q 1 + q

JM=6 2 q4ql 45 q42 q6 q5q2 - qlq6

q5q - 9q4 q3q4 + qlq6 q5q3 + q6 q2 q5 - q3q4 q6q2 q4 q6q3 - 95

9414 - 5 qlq5 + q4 q929q5 - q394

q929q4 - q6 q34 91qq6 q2q6 q4

9295 - 94q6 q3q5 - 96 q3q6 + 95

1 + q24 q45 - q 949q6 - 92

q5q4 +9 1 + 9q q5q6 - 93

q6 4 + q2 q6 5 +93 1 9 q6 (2.24)

It is noteworthy that many researchers reference Cayley's work 'On the motion of rotation of a solid body' (Cayley 1843) as the origin of the Cayley transform and equations (2.1) and (2.2). That paper contains scalar expressions for the case N = 3 which are equivalent to the Cayley transform (Cayley 1843, p. 225) and the Cayley-transform kinematic relationship (Cayley 1843, p. 227). There is no indication, however, that the results apply, nor lead, to the general N-dimensional case, nor is there any indication of the elegant, matrix forms that are central to the Cayley transform and kinematic relationship that are recognized today.

To the current authors' knowledge, these features first appear in Cayley's work 'Sur quelques proprietes des determinants gauches' (Cayley 1846). For reasons that are not mentioned in that paper, Cayley investigates some properties of what he calls 'left systems'. A left system is defined as a square collection of quantities, Aij, that satisfy the following relationships:

j J #J' (2.25)

Aii = 1, no sum on i. J

The indices i and j range from 1 to N. Note that the quantities Aij are essentially the elements of an N x N matrix, A, which may be expressed as A = I - Q, where Q is a skew-symmetric matrix. He goes on to introduce the 'inverse left system', A, which satisfies ATA - KI, where K is the determinant of A. While investigating some transformation properties of left systems, Cayley discovered his famous transform (Cayley 1846, p. 120):

On a done le theorme suivant: Les coefficients propres a la transformation de coordonn6es rectangulaires, peuvent etre exprimes rationnelle- ment au moyen de quantites arbitraires Ars, soumises aux conditions Ars = -Asr, [r s]; Arr = 1. Pour les developper, il faut d'abord former


766



Cayley form

le determinant K di ce systime, puis le systime inverse A,,... et ecrire Ka,rs = 2As [r = s]; Karr,, = 2Arr -K; ce qui donne le systime cherch&.

Cayley's coefficients ars are equivalent to the elements of C, and the relationship in Cayley's theorem can be written KC = 2A - KI. Using ATA = KI, the relationship is seen to become KC = AA. When A is written as A = I - Q, the inverse system is A = K(I + Q) -, which leads to C = (I - Q)(I + Q)-1. From this result, it is straightforward to write the inverse form of the Cayley transform relating skew- symmetric Q to the proper orthogonal matrix C. Note that although Cayley does not use matrix notation, he does use what is recognized today as a form of index notation to manipulate system quantities (i.e. matrix elements). A generalization of the Cayley transform in matrix notation appears in Cayley's works 'Sur la transformation d'une fonction quadratique en elle meme par des substitutions lineaires' (Cayley 1855, p. 288) and 'A Memoir on the automorphic linear transformation of a bipartite quadric function' (Cayley 1858, p. 44). These papers consider the transformation of the variables of quadratic functions and not just the rotation of an orthogonal coordinate system. In the following section, the Cayley kinematics are used (as generalized coordinates and motion variables) to directly derive the equations for rotational motion of an N-dimensional body.

3. Tensor form of Lagrange's equations

Previously, the tensor form of the N-dimensional rotational equations of motion have been derived by generalizing Euler's equations via a Hamiltonian approach (Bloch et al. 2002; Fedorov & Kozlov 1995; Ratiu 1980) or by mapping the vector form of Lagrange's equations in terms of the Hamel coefficients (Hurtado & Sinclair 2004). In this section a new approach is presented using Lagrange's equations and Cayley kinematics to directly derive the tensor form of the equations of motion.

Lagrange's equations of motion in terms of the generalized coordinates and velocities are given by the following familiar form:

d (T0 OTo dt k 4= fiJ.q (3.1)

In this equation To = To(q, q) is a function of the generalized-coordinate and velocity vectors and is equal to the kinetic energy of the system. The generalized forces, fi, are associated with the generalized coordinates.

This equation can be rewritten in terms of the generalized coordinate and velocity matrices. To do this the skew-symmetric matrix elements are substituted using the relationships given below:

q_ 1X^kQ k. q_ 1 xkQk (3.2) qi = -Xjk jk, Qi Xjk jk. (3.2)

Substituting equations (3.2) into the expression for To for any given system will define a new kinetic-energy function To To(Q, Q). Whereas equations (3.2) give the vector elements in terms of the matrix elements, the inverse mapping can also be considered:

Qjk Xjk41, Qjk Xjkql. (3.3)


767




The partial derivatives of these equations can be used to find a matrix form of Lagrange's equations:

Q =kjk Oqp Xkli = Xirk, (3.4)

Oq = XJk- qi X i

=j Xk Xjk l i X (3.5)

Although the elements of Q, Q and 2 satisfy the skew-symmetry constraint, in the following derivations one has the choice of enforcing the constraint or not. Either option will produce the correct, final equation of motion as long as it is applied consistently (see the appendix). For convenience, the constraint is not enforced in the following derivations, and in essence, the elements of these skew-symmetric matrices are treated as independent. This allows the partial derivatives of the kinetic-energy function with respect to the matrix elements to be written using the following chain rules without a priori considering the form of the constraint:

To _ To Qj k i (To3 Oqi OQjk oqi XijkQjk

T0 _ 00 O k (3.7) O9i OQjk O9i Xjk Qjk )

Substituting these expressions into Lagrange's equations gives

dt Xjk )jk - Xjk fjk fi, (3.8)

d (To> T 1 iF

d 8To 8To d( - = 'oF . (3.10) dt oqij Oij 2

Here, F represents the skew-symmetric matrix of generalized forces. Equation (3.10) is the matrix form of Lagrange's equations in terms of the generalized coordinates and velocities. The factor of one-half appears on the right-hand side because the contribution to To due to the generalized coordinate qi (or the generalized velocity qi) is shared in To equally between the corresponding Qjk and Qkj (or Qjk and Qkj) elements.

The matrix form of Lagrange's equations in terms of generalized coordinates and quasi-velocities will now be developed. To do this, another kinetic-energy function, T = T1 (Q, 7), is defined by substituting the Cayley-transform kinematic relationship into To. The partial derivatives of To can be expressed in terms of T by using the chain rule:

aTo aT1 aT a2kl

aTo a_ akkl

OQ = OQ O2k OQ (3.11)

S k(3.12) gij 0 kl 0j"


768



Cayley form

These expansions are substituted into equation (3.10):

I (&Tl& 1 + &T1 -k) - F - (3.13) dt 8- kl d ij - Qij d kl dQij 2

d ( Ti 92kl & 1 k-l 1 T + -dt df nkl &Qn, I k 2 z F(3.14) dt 9^kl dQij Wkl Qij 2 Qij

In order to expand the terms on the left-hand side of this equation, the derivatives of the elements of !2 must be considered. Thus, equation (2.5) is rewritten in index notation:

9k = 2Bk VQPBl. (3.15)

In order to develop the first term of equation (3.14) the derivative of 2kl with respect to Qij is considered:

,kj = 2B Q pB- = 2Bkiv5jpB~ =- 2BkB . (3.16)

The first term of equation (3.14) is therefore given by

d 8Ti 8dk0 l d 9T dt \92k9 Qij - 29kI ki l

d T1 -T-

= 2dt ( k) BB + 2 kl ( + + B + BB ). (3.17)

Clearly, the tensors B+ and B- must be investigated. These are found using the definitions in equations (2.3) and (2.4):

A+B+ -= I. (3.18)

Taking a derivative gives

A+B+ + A+B+ = 0, (3.19)

B+ = -B+A+B+. (3.20)

From the definition of A+, however, it is true that A+ = Q:

B+ = -B+QB+. (3.21)

By applying similar steps to the product A-B-, the derivative of B- can be found:

B- = B-QB-. (3.22)

These expressions for B+ and B- are substituted back into equation (3.17):

d (T1 9kl d (T1i I + R+8-1 +BBQ B). dt 9kl aij) = dt J B + 2 BkQSBrsiBJs + kijrs B)

(3.23) Now the second term of equation (3.14) is considered. First, the derivative of Sk

with respect to Qij is given by

8kl &B+ . OBp Q = 2 QB c + 2BkvQP pi (3.24)


769




This result is used to rewrite the second term of equation (3.14):

a O kl -2 T ( Bkv pB + B9t Q . (3.25)

802kl Qij k^ij pl kv vP Qij

The derivatives of B+ and B- with respect to the elements of Q must be investigated:

A~+B+ = r, (3.26)

dA+ B+ rf B+ Ak+ k= 0, (3.27) Qij B+ +r A Qij

v = -B rB + .(3.28) Qij sr ij kv

The derivatives of Ak can be found using the definition of A+:

OA+ _Qrk_ r - irjk. (3.29)

83Qij oij

Based on this, the derivative of B+ is given by

8B -- = -B+SirjkB = B+B. (3.30) ij si j

By applying similar steps to B- the derivatives of its elements are found:

Bd-

= B B-. (3.31)

Substituting equations (3.30) and (3.31) into equation (3.25) gives the following for the second term of equation (3.14):

9T Qkl ij= 2 l (- B B JBQPB + B,,vpB iB ). (3.32)

Finally, equations (3.23) and (3.32) can be substituted back into equation (3.14):

2 T, BB + 20k (-B rsBBjB + B BJrQrsBi)

- 2B&lvQ + B v BPi ) = F + i. (3.33)

The second and third terms of the left-hand side can be grouped:

d 8Ti\

+ 2T Br (-BB Bj + BB BB- + BBBs - B B B)

± T, - -Fi+ , (3.34) -2 Z a ij


770



Cayley form

2 T1. B BB +2 B i +Bj js+r is rij + 2 ),Q (-B BrB + BB B B+BT, - B+ B B )

0T =Fzj + (3.35)

dt 8kl jl

+ T1 Aa abr(B (Bs + gB) - Bs(B j + B))

SFij + (3.36)

Both sides of the above equation are now multiplied by A A to allow for future simplifications:

aT1A - (BBA+ +- (+r 2ab8 ddAdA Bki +z(B-s +B>- BB(B± Bi ))

±F+ )IA+A- d T ij id cj.

(3.37)

Both terms on the left-hand side of the above equation can be simplified. It is convenient to start with the first term:

BBfA~A = .d.c (3.38)

This result is used to rewrite the first term of equation (3.37):

2- OdA - 2kdIcl- ( dt \9kl i id c3 dt \Ofk/

dt dTc) (3.39)

The second term of equation (3.37) is now examined:

AA A A bR (B + B() - - - B + B ))

= A sA~ (kdA Br (Bj, + B ) - 6ciAB (Bj + Br ))

= AaAg (6kdB(5 + Ccs) - 6c1Bksdr + Cdr)). (3.40)


771



772 A. J. Sinclair and J. E. Hurtado

The elements of C + I can be found using the Cayley transform:

C + I = (I- Q)(I + Q)-1 + I

= (I + Q)- - Q(I + Q)- + I

= (I + Q)- - Q(I + Q)- + (I + Q)(I + Q)1

= 2(I + Q)- = 2B+. (3.41)

This result can be used to further simplify equation (3.40):

A AcAA (B Brl (B js+ BJ) - Br- + RB ))

= AAr(26kdBrBc - 26,iB+Bd)

= 2A aAb(6kdBiB - 6clBB+). (3.42)

This result is used to rewrite the second term of equation (3.37):

1ab A+AA A (Bk r+Br (Bis + B+) B B (B+ ± Br)) OTt + _ (- B B^ + 5dB^) +

= 2S2ab AaA (kdBrlB - 6 BclskB ). (3.43)

The skew-symmetry of f is used in the following manner:

-kl cl Bsk dr lk Cl k B dr

aT, = - kB B. (3.44)

This expression is substituted into equation (3.43):

aT A ) _ + + ) S Aab A A(BBk rl (Bjs + j?) j;(Br B ))

a T = 2fab lA/ A (6kdBrlB+ - kBB)

8T = 2 7ab (5kd6caSbl + Sck6laSbd)

2 akl

= l2c ~ + 2ld - 2dl 0 Jec

adt at -2 cl- 2 a dl. (3.45)

Equations (3.39) and (3.45) can now be substituted back into equation (3.37):

d (Ti Ti aT 1(1 T dt 1 + /cl - 2di = 2 Fij + AA . (3.46) dt 89dc j dl ci 2 2 8Qij c3

The above equation is the matrix form of Lagrange's equations for generalized coordinates and the N x N angular-velocity matrix (or Cayley quasi-velocity




Cayley form

matrix). For convenience the right-hand side is defined as 1Gdc, where G is related to the generating vector g, whose elements can be shown to be gk = Ark(fr+(T1/qr)). The equivalent vector representation of the equations is

dt ± (XfcJ - X jic) - k. (3.47)

To express these equations in the Lax pair form, the derivative of T1 with respect to the angular-velocity components is defined as the angular-momentum vector 1:

OT1 & =k. (3.48)

9wk

The partial derivative of the kinetic energy with respect to (i is expressed using the chain rule:

OT^1 T X . (3.49) 9wk -9-ij 9Wjk - 9Xi ij

The angular-momentum vector is used to generate the angular-momentum matrix L:

lk - i i4. (3.50)

This gives

i Lij. (3.51)

Equation (3.46) can now be rewritten in terms of the angular-momentum matrix:

d t( Ldc) + 1Ldi 2cl - L 2dl - 1Gdc. (3.52)

In matrix notation this result gives the following Lax pair form:

L = (L - L2L) + G = [L, 17] + G. (3.53)

4. Cayley quasi-velocities and the Cayley form

An M-degrees-of-freedom mechanical system can be intimately related to an N- dimensional rigid body through the Cayley kinematic equations. Recall the Cayley kinematic equations:

forward relationship, 2 = 2(1 + Q)-Q(I - Q)-; (4.1)

inverse relationship, Q -= ( + Q)f2(I- Q). (4.2)

And recall also the forward Cayley-transform expression and Poisson's equation for C:

forward relationship, C = (I - Q)(I + Q)-1 = (I + Q)-(I - Q); (4.3)

Poisson's equation, C = -f2C. (4.4)

These lead to the following remarkable result.


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Table 2. The Cayley form of the dynamic equations

M-dimensional vector forms

kinematics qi = Aijj

dynamics d(OT1/Owk)/dt + Xi(Xc~Ocj - X.jic)GJ9I/ Wr) = gk

N x N matrix forms

kinematics Q - (I + Q)f2(I - Q) dynamics L = [L, ] + G

If the M-dimensional (M = N(N - 1)/2) generalized-coordinate vector, q, of an M-degrees-of-freedom mechanical system is considered the generating vector of an N x N skew-symmetric matrix Q, and if Q and its derivative are used to define the quasi-velocities 0I via the forward Cay- ley kinematic equation, then Q generates a proper N x N orthogonal matrix, C, via the forward Cayley transform that evolves according to C = - C. Furthermore, the motion of the system is governed by the following matrix differential equations:

Q = (I + Q)(I - Q), L = [L, n] + G.

Consequently, the motion of an M-degrees-of-freedom mechanical system can be viewed as the rotational motion of an N-dimensional rigid body. When q is used in this manner (i.e. a generating vector), its elements are called the extended Rodrigues parameters; the elements of 12 are given the name Cayley quasi-velocities; and this treatment is called the Cayley form.

The equivalent body is defined as an N-dimensional rigid body because the generalized coordinates that define the extended Rodrigues parameters completely describe the configuration of the system. This is an extension of the kinematic definition of a three-dimensional rigid body. Previous researchers have also extended certain dynamic properties relating to three-dimensional rigid bodies in their definitions of an N-dimensional rigid body (Bloch et al. 2002; Fedorov & Kozlov 1995; Ratiu 1980). These definitions, however, preclude the study of systems whose Lagrangian functions depend on the generalized coordinates. The governing equations presented herein for an M-degrees-of-freedom mechanical system or, equivalently, an N-dimensional rigid body are summarized in table 2. The numerical relative tensor ^k can be used to map the equations from one form to the other.

One application of the Cayley form is the representation of the general motion of an N-dimensional body as the pure rotation of an (N + 1)-dimensional body. The general motion of an N-dimensional body consists of M rotational and N transla- tional degrees of freedom:

= (N + 1)N

= (N + 1)((N + 1)- 1). (4.5)


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Cayley form

Figure 1. Planar rigid body.

This total number of degrees of freedom is therefore equal to the number of rotational degrees of freedom for an (N + 1)-dimensional body, and the Cayley form can be used to relate these two motions. In the following section the general motion of a two- dimensional body and pure rotations of an equivalent three-dimensional body are presented in detail.

5. Planar-motion example

An enormous variety of mechanical systems can be studied using the Cayley form. One of the easiest to visualize, however, is the general motion of a two-dimensional body. This planar motion consists of three degrees of freedom (see figure 1) and can therefore be related to the rotational motion of a three-dimensional body. The generalized coordinates for the problem are [q]- [0 x y] . To analyse this problem in the Cayley form, the matrix form of the kinematic equations is used to solve for the vector form. The generalized coordinates and Cayley quasi-velocities are arranged into the skew-symmetric matrices Q and 7 as follows:

0 -y x 0 -Gw3 LJ2

[Q]= y 0 -0 , []= K3 0 -wl . (5.1) -X 0 0 -wC2 01 0

From the Cayley kinematic relationship the linear mapping between the generalized- velocity and quasi-velocity vectors is found:

[1 +02 Ox y Oy+x [q= [A[w]= Ox + y 1+x2 xy - 0 [wa]. (5.2)

2 0y - x xy±+O 1 + y2

The form of equation (5.2) is identical to the relationship between the three- dimensional Rodrigues parameter rates and the angular velocity.

The dynamic equations can now be developed using the vector form in equation (3.47). The kinetic energy of the system in terms of the generalized velocities is described by the following, where I is the rotational inertia of the body and m is the body's mass:

To0(q) = 1i02 + 1mT(2 + y2) qTJq. (5.3)


775




16

14

12

10

8

6

4

2

0

-2

8 x y

0 2 4 6 8 10

generalised

coordinates

time

Figure 2. Generalized coordinates.

The matrix J is now the system mass matrix, given by

~I 0]

[J] = 0 m 0 . (5.4) 00 m

Using equation (5.2) the kinetic energy can now be expressed as a function of the generalized coordinates and the Cayley quasi-velocities:

Ti(q, w) - wT(ATJA)w. (5.5)

The Cayley form of Lagrange's equations can now be applied to T1. First, T1 is rewritten using index notation and the various derivatives are then computed:

Tl = wiAiiJimAmjj, (5.6)

= -- AimAmj Alim Wi, (5.7)

OT1 _-- = Air JmAmjj, (5.8) &wr

d ( Ti) dt k =a ) Air JimAmjW + Air JimAmjj + Air4JmAmjj. (5.9)

The derivatives of A are computed as follows:

9Air 9Air Air -= os Atwt. (5.10)

9qs s qs

The generalized forces in terms of the generalized velocities are equal to the moment and force components applied to the body:

[f]= [M F, F]T.


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Cayley form

1.5

1.0

0.5

0

-0.5

-1.0

d6/dt dx/dt dy/dt

0 2 4 6 8 10

GENERALISED

VELICITIES

CAYLEY

QUASI-VELOCITIES

time

Figure 3. Generalized velocities.

1.2

1.0

0.8

0.6

0.4

0.2

0

-0.2' 0 2 4 6

(01

(02 (03

time 8 10

Figure 4. Cayley quasi-velocities.

Using these generalized forces and equations (5.7)-(5.10), the equations of motion can be assembled and solved for the components Wjj.

Figures 2-4 show simulation results from the integration of these equations of motion and the Cayley kinematic equations, using mass property values of I = m = 1. The initial conditions, forces and moment were chosen such that the body translated with constant speed through a circular path of unit radius over a period of ten time units. During this period the body also performed two complete rotations. This motion is depicted in figure 5. Figures 2 and 3 show the solutions for the generalized coordinates and velocities obtained using a traditional Lagrangian


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2.5

t=5 t=0

x

t\ =10

t = 7.5

Figure 5. Example planar motion.

formulation. Figure 4 shows the results obtained for the quasi-velocities using the Cayley form. Of course, the solution for the generalized coordinates matched that obtained from the traditional approach. The implication of the Cayley form is that there is an equivalent rotational motion of a three-dimensional body that corresponds to the motion of the two-dimensional body. This equivalent motion is described by Rodrigues-parameter and angular-velocity trajectories equal to the solutions shown in figures 2 and 4, respectively.

6. Discussion

This paper has focused on the Cayley transform and the Cayley-transform kinematic relationship, and their use in developing the matrix form of the generalized Euler equations of motion for N-dimensional rigid bodies. Related to these equations are the more general Euler-Poincard equations, which govern the motion of left-invariant Lagrangian systems corresponding to general Lie groups (Bloch et al. 2003, p. 134; Marsden & Ratiu 1999). The Hamel coefficients for an N-dimensional rigid body (Hurtado & Sinclair 2004) were not explicitly encountered in the devel- opment because of the a priori decision to describe the system using the Cayley orientation and kinematic variables.

The Cayley transform and the Cayley-transform kinematic relationship allowed the realization that the motion of a general M-degrees-of-freedom (M = N(N - 1)/2) mechanical system is related to the rotational motion of an N-dimensional rigid body. The definition of an N-dimensional rigid body used here is kinematics based and hinges on three facts:


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Cayley form

(i) the extended Rodrigues parameters, which completely describe the configuration of the system, parametrize an N x N proper orthogonal matrix;

(ii) an N x N proper orthogonal matrix can be used to relate the orientation of one N-dimensional reference frame to another; and

(iii) the orientation of a rigid body can be defined as the orientation of a reference frame that is embedded within it.

A useful relationship that was developed was the mapping of the skew-symmetric matrix Q, which comprises the extended Rodrigues parameters for N-dimensional spaces, to the matrix A that appears in the equation q = Aw. This mapping allows the equations that govern the rotational motion of an N-dimensional rigid body to be written in an M-dimensional vector form.

The benefits and implications of the relationship between M-degrees-of-freedom mechanical systems and N-dimensional rigid bodies are still being investigated. Fur- ther research is needed to investigate the extent to which the elegant tools that have been developed for analysing, controlling and approximating the motion of three- dimensional rigid bodies can be modified and applied to N-dimensional rigid bodies. The use of the Cayley form will then allow these tools to be applied to a wide variety of physical systems. Further research is also needed to investigate the 'in-between' M situation, i.e. M = N(N - 1)/2. One straightforward approach is to pad the vector of extended Rodrigues parameters with additional fictitious coordinates until M = N(N - 1)/2. The additional fictitious coordinates then represent constrained degrees of freedom.

Appendix A.

In the derivation of the N-dimensional rotational equations of motion contained in the text above, the elements of the skew-symmetric matrices Q, Q and 2 were treated as independent. This allowed partial derivatives with respect to these elements to be written in the following manner:

To To kQjk i OTo

9qi QQjk 9qi = jQjk '

= iv jp. (A 2) a ij

Another option in performing the derivation is to treat the elements of the skew- symmetric matrices as dependent by considering the form of the skew-symmetric constraint. Of course, the constraint specifies that for each non-zero element of a skew-symmetric matrix there will be an equal and opposite element. This means that the partial derivative of To with respect to Qjk will produce two equal terms: one corresponding to the contribution of Qjk and the other corresponding to the contribution of Qkj. To maintain the desired result a factor of one-half is needed.


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The partial derivatives thus take on the new form given below:

&To 1 To &Qjk 1 i aTo =qi 29Qjk 9qi 2X Qjk A

Si6ip - 5ipj. (A 4)

The above equation produces +1 if i = v and j = p or -1 if i = p and j = v. Consider the three-dimensional example To = and 9To/93 = 3. This will produce the following form for To:

To = 2 + 21i (A 5)

If the elements are treated as independent, clearly the partial derivative with respect to q3 will be

0To aTo aQ12 aTo 921

9q3 0 9Q12 93 9Q21 93

-= 212 2Q21

= q3. (A 6)

If the elements are treated as dependent, then the same result will be achieved, as shown below:

1 To 012 1 a0o 9Q21 2 aQ12 a3 2aQ21 aq3

= - 120 221 ) + , ± 212 21

= ((12 - 211) _1 2 (- 2 2)1

= q3. (A 7)

Therefore, either method can be used to achieve the N-dimensional rotational equations of motion given in the text.

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