1 - z w = 1 + z. 1 - w z = 1 + w S 1

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1 Journal of Guidance, Control, and Dynamics Vol. 0, No., May-June 997 Higher Order Cayley Transforms with Applications to Attitude Representations Panagiotis Tsiotras University of Virginia, Charlottesville, VA John L. Junkins y and Hanspeter Schaub z Texas A&M University,College Station, TX Abstract In this paper we generalize previous results on attitude representations using Cayley transforms. First, we show that proper orthogonal matrices, that naturally represent rotations, can be generated a form of \conformal" analytic mappings in the space of matrices. Using a natural parallelism between the elements of the complex plane and the real matrices, we generate higher order Cayley transforms and discuss some of their properties. These higher order Cayley transforms are shown to parameterize proper orthogonal matrices into higher order \Rodrigues" parameters. Introduction The question of the proper choice of coordinates for describing rotations has a very long and exciting history. Starting with the work of Euler and Hamilton a series of dierent parameterizations were introduced several researchers during the past hundred years. We will not delve into these results here since they can be found in any good textbook on attitude representations,. We just mention the work of Stuelpnagel in this area,aswell as the recent survey article Shuster 4 in the special issue in Ref. 5. In this paper we take a slightly more abstract point of view than the previous references. Our main objective isto \unify" some of the existing results in the area of attitude representations. It is hoped that this global view will add to the current understanding of attitude representations. Our motivation stems mainly from the recent results on second order Rodrigues parameters 6{. In particular, in Ref. it was shown that these (Modied) Rodrigues parameters can be generated a second order Cayley transform, the same way the classical Cayley-Rodrigues parameters are generated thecayley transform,9. Viewing the Cayley transform as a bilinear transformation which maps the space of skewsymmetric matrices onto the space of proper orthogonal matrices (and vice versa) one is naturally led to the notion of conformal mappings (a generalization of the bilinear transformation) from the imaginary axis onto the unit circle (and vice versa). We seek to generalize these conformal mappings to matrix spaces. Drawing on the insightful statements Halmos 0 we show thatsuchanintuitive generalization is indeed possible. We are therefore able to generate the Euler parameters, the Rodrigues parameters and the Modi- ed Rodrigues parameters as special cases of such conformal mappings. Higher order Rodrigues parameters can be easily constructed using this approach, although their relevance to applications is still to be determined. We explicitly develop the third and fourth order \Rodrigues parameters" in order to illustrate potential advantages as well as diculties. The Assistant Professor, Department of Mechanical, Aerospace and Nuclear Engineering. Member AIAA. yeppright Professor, Department of Aerospace Engineering. Fellow AIAA. zgraduate Student, Department of Aerospace Engineering. Student member AIAA. question of kinematics of these higher order \Rodrigues parameters" is briey discussed in the last section of the paper. A more in-depth discussion of the kinematics is left for future investigation. The presentation and derivations of the results are kept as formal as possible. The rst part of the paper reviews the standard Cayley transform and it generalizes this transform to higher orders. There is no restriction on the dimension of the matrices involved, i.e., the results hold for nn matrices. In the second part of the paper we apply these results to the case of interest to attitude dynamicists, i.e., the case n. Some notation and terminology is necessary in order to keep the discussion clear and terse. We use the standard mathematical notation SO(n) to denote the space of proper orthogonal matrices of dimension n n. The space of orthogonal matrices is denoted O(n) and it is the set of all (invertible) matrices such thata T A AA T I. Clearly, if A O(n) thendet(a). The qualier \proper" then refers to those orthogonal matrices with positive determinant, that is, SO(n) fa IR nn : AA T I det(a) +g. These matrices represent rotations, while the orthogonal matrices with determinant -involve, in general, reections. The space SO(n) (as well as O(n)) forms a group. We will see later on that one can dene a dierential equation for elements of SO(n). The solutions of this dierential equation form trajectories (one-parameter subgroups) on SO(n) and this dierentiable structure makes SO(n) actually a Lie group (i.e. a group with a dierentiable manifold structure). The space of n n skew-symmetric matrices will be denoted so(n) That is, so(n) fa IR nn : A ;A T g. The space so(n) is actually the tangent vector space to SO(n) at the identity. Finally, following the standard mathematical language, we use the symbols C, IR, iir and S to denote the complex numbers, the real numbers, the imaginary numbers, and the numbers with absolute value one (i.e., the numbers on the unit circle), respectively. The symbol sp() denotes the spectrum of a matrix, i.e., the set of its eigenvalues. The Cayley Transform Cayley's transformation parameterizes a proper orthogonal matrix C as a function of a skew-symmetric matrix Q. 5

2 It is, therefore, a map : so(n)! SO(n). The classical Cayley transform 9 is given C (Q) (I ; Q)(I + Q) ; (I + Q) ; (I ; Q) () Since Q is skew-symmetric all its eigenvalues are pure imaginary. Thus, all the eigenvalues of the matrix I + Q are nonzero and the inverse in Eq. () exists. The Cayley transform is therefore well-dened for all skew-symmetric matrices. The inverse transformation is identical and is given Q ; (C) (C) (I ; C)(I + C) ; (I + C) ; (I ; C) () The inverse transformation is not dened when C has an eigenvalue at ;, because in this case det(i + C) 0. Since C is orthogonal, all its eigenvalues lie on the unit circle S f(x x ) IR : x + x g. Therefore sp(c) S, and the transformation in Eq. () requires that ; 6 sp(c). One can easily show that C SO(n) if Q so(n) and thus, the Cayley transformation is injective (one-to-one) and surjective(onto) between the set of skew-symmetric matrices andthesetofproper orthogonal matrices with no eigenvalue at ;. Cayley Transforms as Conformal Mappings The three most important subsets of the complex numbers are the real numbers IR, the imaginary numbers, and the numbers with absolute value one (i.e., the numbers on the unit circle S.). Trivially, these sets are subsets of the complex plane C. There is a very elegant analog between these three subsets of the complex plane and the n n matrices 0, i.e., the elements of IR nn. This analog can be easily understood and appreciated as follows: An elementary result in matrix algebra states that every n n matrix with real elements can be decomposed into the sum of a symmetric and a skew-symmetric matrix. For example, any A IR nn can be written as A (A + A T ) +(A ; A T ). The rst matrix in this equation is symmetric and the second matrix is skew-symmetric. Symmetric matrices always have real eigenvalues and skew-symmetric matrices have always imaginary eigenvalues. Recall now that a complex number can always be decomposed into the sum of a real and an imaginary part. This parallelism between complex numbers and matrices allows one to treat the symmetric matrices as the \real numbers" and the skew-symmetric matrices as the \imaginary numbers" in the set of IR nn matrices 0. In addition, recall that an orthogonal matrix in IR nn has all its eigenvalues on the unit circle. Drawing the previous parallelism even further we can therefore treat the orthogonal matrices as the \elements on the unit circle" in the space IR nn. Similar statements can be made for the case of n n matrices with complex entries (elements of C nn ), where now hermitian, skew-hermitian and unitary matrices have tobe used instead of symmetric, skew-symmetric and orthogonal matrices, respectively. We intend to use this heuristic correspondence between complex numbers and n n matrices in order to motivate and generalize the Cayley transform to higher order. Before we proceed, we briey review some elements from complex function theory,. First, recall that a (complex) function is analytic in an open set if it has a derivative ateach point in that set. In particular, f is analytic at a point z 0 if it is analytic in a neighborhood of z 0. Moreover, analytic functions have (uniformly) convergent power series expansions. A transformation w f(z) where w z Cissaidtobe conformal at a point z 0 if f is analytic there and f 0 (z 0) 6 0. A conformal mapping is actually conformal at each point in a neighborhood of z 0, since the analyticity off at z 0 implies analyticity in a neighborhood of z 0. Moreover, since f 0 is continuous at z 0,itfollows that there is also a neighborhood of z 0 with f 0 (z) 6 0forallz in this neighborhood. It is a trivial consequence of the above denition that the composition of conformal mappings is also a conformal mapping. A signicant special class of conformal mappings in the complex plane is the class of linear fractional transformations (also called bilinear transformations) dened w az + b (ad ; bc 6 0) () cz + d An important property of the linear fractional transformations is that they always transform circles and lines into circles and lines. In particular, in this paper we are interested in conformal transformations of the form in Eq. () which map the unit circle on the imaginary axis and vice versa. One such transformation is given w f(z), where f(z) ; z +z It is an easy exercise to show that if z then jwj, that is, w S and thus, w is on the unit circle. Conversely, if w S then the inverse transformation z f ; (w) given f ; (w) ; w (5) +w implies that the real part of z is zero and thus, z. The inverse transformation in Eq. (5) is dened everywhere except at w ;. The point w ; is mapped to innity (see Fig. ). In fact, the map in Eq. (4) introduces a one-to-one transformation f :!S nf;g. Im - z w + z - w z + w S Fig. Bilinear transformation. Let us nowintroduce the conformal mapping g n : S! S dened g n(w) w n n ::: (6) The function g n is a mapping from the unit circle onto the unit circle. This transformation is only locally injective. Therefore the inverse of g n exists only locally. Given e i S the solution of the equation w n (n :::) yields that (4) w e i(+k)n k 0 ::: n; (7) Equation (7) shows that, in general, the equation w n has more than one solution. This result will turn out to be benecial in Section 5 when we discuss the application of higher order Cayley-transforms to attitude representations, 59

3 because these roots can be used to avoid the inherent singularities of three-dimensional parameterizations of SO(). For k 0 in Eq. (7) we get that w e i(n). We willcall this the principal nth root of. The composition of the maps f and g n is the function h n :!S dened h n g n f, that is ; z n h n(z) () +z which maps the imaginary axis onto the unit circle. Similarly to g n, this map is only locally invertible. A local inverse is obtained, for example, setting k 0 in Eq. (7), in which case we have that z ;ei(n) +e i(n) (recall that h n(z) e i ). 4 Higher Order Cayley Transforms One of the most celebrated results in matrix algebra is the Cayley-Hamilton theorem. This theorem states that a matrix satises its own characteristic polynomial. An important consequence of this theorem is that, given any matrix A IR nn and an analytic function F (z) inside a disk of radius r in the complex plane, one can unambiguously dene the matrix-valued function F (A) if the eigenvalues of A lie inside the P disk of radius r. In other words, if P F is given F (z) i0 izi (jzj r) then F (A) i0 iai and the previous series converges assuming that j jjr, where j sp(a) forj ::: n. Therefore, the matrix F (A) is well-dened. Moreover, the eigenvalues of the matrix F (A) are F ( j)(j ::: n) (Refs. 4,5). Consider now the conformal mapping f from Eq. (4) which maps the imaginary axis on the unit circle. This function is analytic everywhere. According to the previous discussion, the matrix f(q) (I ; Q)(I + Q) ; (I + Q) ; (I ; Q) (9) is well-dened for Q so(n) and, actually, C f(q) SO(n). Comparison between the previous equation and Eq. () reveals that the Cayley transform can be viewed as a special case of a conformal mapping in the space of matrices. We have seen that there is a natural correspondence between and so(n), as well as between S and SO(n). (We caution the the mathematically inclined reader to take these statements in the context of the discussion in Section. We do not claim that this correspondence carries any more weight than providing one qualitative motivation for the generalization of certain complex analytic results to analogous results in the space of matrices). Following Eq. () we can also dene a series of transformations h n : so(n)! SO(n) h n(q) (I ; Q) n (I + Q) ;n (I + Q) ;n (I ; Q) n (0) where Q is a skew-symmetric matrix. It should be clear now that C h n(q) is a proper orthogonal matrix, i.e., C SO(n). We shall refer to the family of maps h n(q) in Eq. (0) as Higher Order Cayley Transforms. The consequences of such a generalization in attitude representations will become apparent in the next section. For now, let us concentrate on the inverse map h ; n : SO(n)! so(n). Since h n g n f one obtains h ; n f ; g n ;. The function f ; is given Eq. (5) which, when applied to a proper orthogonal matrix Q with no eigenvalue at ;, gives the inverse of the classical (or rst order) Cayley transform as in Eq. (). The map g n ; : SO(n)! SO(n) on the other hand requires the nth root of an orthogonal matrix. First, we showthatg n ; is well-dened in the sense that the nth root of a (proper) orthogonal matrix with no eigenvalue at ; is also a (proper) orthogonal matrix with no eigenvalue at ;. This will also prove that the composition of maps g n ; and f ; is well-dened since the range of g n ; is in the domain of f ;. To this end, consider an orthogonal matrix C SO(n) such that 6 ; for all sp(c). Since the matrix C is normal it can be decomposed as C UU for some unitary matrix U where blockdiag( ::: (n;) +) if n is odd, blockdiag( ::: n )ifn is even, and j diag(e i j e ;i j ). The diagonal elements of the matrix arethe eigenvalues of C. The principal kth root of the matrix C is then given W U k U where W k C and k blockdiag( k k ::: k +) if n is (n;) odd, k blockdiag( k k ::: k ) if n is even, n and k j diag(e i( j k) e ;i( j k) ). Since e i j 6 ; for all j ::: n (n ; ) the angles j 6 0 deg and thus also jk 6 0 deg for k ::: and thus e i( j k) 6 ;. Notice that in order to keep W proper we always choose the positive rootoftheeigenvalue +. 5 Attitude Representations In this section we concentrate on the ramications of the previously developed results to attitude representations. Our motivation for investigating Cayley transforms in the rst place, stems from the fact that proper orthogonal matrices represent rotations. In particular, SO() is the con- guration space of all three-dimensional rotations. In other words, every element of SO() represents a physical rotation between two reference frames in IR and conversely, every rotation can be represented an element in SO(). As a reference frame, viz. a body, rotates freely in the three-dimensional space, the corresponding rotation matrix C traces a curve inso() such thatc(t) SO() for all t 0. The dierential equation characterizing this trajectory on SO() is given _C [!] C () where, given a vector! (!!! ) IR, the matrix [!] is dened " 0! ;! # [!] ;! 0! ()! ;! 0 In the sequel we apply the results of the previous section in order to parameterize the rotation group. In particular, the series of conformal mappings from Eq. (0) provides a family of parameters on SO(). Before undertaking this task we investigate another important conformal mapping. 5. The Exponential Map and the Euler Parameters Linear fractional transformations are not the only class of conformal mappings from the imaginary axis onto the unit circle. The exponential map, dened w exp(z) e z () A unitary matrix satises UU U U I, where U denotes the complex conjugate transpose of the matrix U. 50

4 also maps (actually the strip ;i z i) onto S. Clearly, if z i then jwj. The inverse transformation is z log w i ( +n) n 0 ::: (4) and is dened only locally. We can therefore dene the exponential map from the space of skew-symmetric matrices to the space of proper orthogonal matrices. This exponential map is dened, as usual, C e Q X n0 n! Qn (5) and the series converges for every Q. One can easily show that C thus dened is indeed proper orthogonal. For the three-dimensional case, the matrix Q so() can be parameterized Q []. As before, given a vector ( ) T IR we use the notation [] to denote the skew-symmetric matrix in Eq. (). Euler's formula 4 yields where kk. vector e [] I +sin [] +(; cos ) [] (6) Normalizing the vector we get a unit ^e kk (7) Euler's theorem states that any rotation can be represented a nite rotation (principal rotation) about a single axis (principal axis). That is, the principal axis and the principal angle suce to determine the rotation matrix. From a mathematical perspective this amounts to parameterizing elements in SO() the principal axis and the principal angle. By letting the principal axis be along the direction of theunitvector ^e and letting the principal angle be as above, Eq. (6) shows how this parameterization is achieved. Clearly, C( ^e) e [^e] () Moreover, introducing the Euler parameter vector q (q 0 q q q ) T q 0 cos qi ^ei sin i (9) and substituting in Eq. (6) one obtains the well-known formula for the rotation matrix in terms of the Euler parameters 4 C(q) (q 0 ; ~q T ~q)i +~q~q T +q 0[~q] (0) where ~q (q q q ) T IR is the vector part of the Euler parameters. Therefore, the Euler parameter representation, as well as the Euler axis/angle representation are obtained generalizing the conformal mapping in Eq. () to the space of matrices. Notice from Eq. (0) that C(q) C(;q) and both q and ;q can be used to describe the same physical orientation. This fact can be used to construct alternative, or \shadow", sets of kinematic parameters obtained via the Cayley transforms. 5. Rodrigues Parameters Since the Euler parameters satisfy the additional constraint q 0 + q + q + q, one is naturally led to consider the elimination of this constraint, thus reducing the number of coordinates from four to three. The Rodrigues parameters achieve this dening j qj q 0 j () The three parameters then provide a threedimensional parameterization of SO(). The inverse transformation of Eq. () is given j q 0 q ( + ^ ) j j () ( + ^ ) where ^ T + +. The Rodrigues parameters are related to the principal axis and angle through the equation tan ^e () The rotation matrix in terms of the Rodrigues parameters can be easily computed using Eq. (0) and Eq. (). C() +^ ; ( ; ^ )I + T +[] (4) It is remarkable the fact that the previous parameterization of SO() can also be achieved means of the Cayley transformation in Eq. (). If we introduce the skew-symmetric matrix R ;[], the transformation C (I ; R)(I + R) ; (I + R) ; (I ; R) (5) produces exactly the matrix in Eq. (4). Therefore the classical Cayley-Rodrigues parameters representation is obtained generalizing the conformal mapping in Eq. (4) to the space of matrices. 5. Modied Rodrigues Parameters The normalization in Eq. () is not the only possible one. A more judicious normalization for eliminating the Euler parameter constraint is through stereographic projection,,6,7. Using this approach, the new variables j : qj +q 0 j (6) provide another set of parameters on SO(). These parameters are referred to in the literature as the Modied Rodrigues parameters 4 and have distinct advantages over the classical Rodrigues parameters. In particular, while the Rodrigues parameters do not allow eigenaxis rotations of more than 0 deg, the Modied Rodrigues parameters allow for eigenaxis rotations of upto 60 deg 7,,6{. This can be immediately deduced the corresponding relationship between and the principal axis and angle tan ^e (7) 4 which iswell-behaved for 0 <. Since both q and ;q describe the same physical orientation (recall the discussion at the end of Section 5.), a second set of parameters dened s j : ; qj j () ; q 0 5

5 referred to as the \shadow" set 7, can be used to describe the same physical orientation. These parameters are also given s ; ^e (9) tan(4) The transformation between and s is given 7 s ; ^ (0) where ^ T + + tan (4). The rotation matrix associated with the Modied Rodrigues Parameters is given 4,6, C() I +4 ( ; ^ ) ( + ^ ) []+ ( + ^ ) [] () In Ref. it was shown that these parameters can also be dened a Cayley transformation of second order. That is, if S ;[] then the transformation C (I ; S) (I + S) ; (I + S) ; (I ; S) () produces exactly the matrix in Eq. (). Notice that the inverse of the transformation () is not unique and it requires the square root of an orthogonal matrix. Given C SO() we need to nd a matrix W such that C W. Once a matrix W is calculated, the skew-symmetric matrix S containing the Modied Rodrigues parameters is computed from S (I ; W )(I + W ) ; (I + W ) ; (I ; W ) () Reference outlines this approach. To every orthogonal matrix corresponds a principal angle and a principal direction according to Eq. (). From Eqs. () and C W one therefore has that W e ()[^e] (4) and W has half the principal angle of C. It should be apparent now how the Modied Rodrigues parameters double the domain of validity of the parameterization taking the square of the classical Cayley transform. This observation motivates the search of higher dimensional Cayley transforms for attitude representations. Such transformations are expected to increase the domain of validity even further. This is the topic of the next section. 5.4 Higher Order Rodrigues Parameters According to the discussion in the previous section one expects that higher order Cayley transformations will increase the domain of validity of the corresponding parameters. The main task of this section is to derive these higher order parameters and nd their connections to the Rodrigues parameters, the Modied parameters and the Euler parameters. To this end, consider rst the fourth order Cayley transform dened C (I ; T ) 4 (I + T ) ;4 (5) for some skew-symmetric matrix T ;[]. We know that the matrix C is (proper) orthogonal. Let ( ) T IR be the vector of these parameters. Our purpose is to establish connections between the \attitude" parameters and the other classical attitude parameters such as the Euler parameters of the Modied Rodrigues parameters. Recall from the results of Section that if F is analytic function, then the eigenvalues of the matrix F (A) are given F ( j) where j are the eigenvalues of A. It is an easy exercise to show that the eigenvalues of the skew-symmetric matrix T are given f0 i ( + + ) g. Similarly, the eigenvalues of the matrix S are given f0 i ( + + ) g. Let denote an eigenvalue of T and an eigenvalue of S. Comparing Eqs. () and (5) one sees that the matrices S and T are related (I ; S)(I + S) ; (I ; T ) (I + T ) ; (6) This suggests that and are related ; + ; + (7) or ( + ) + () + Solving for and substituting the expressions for and in the previous equation one obtains that i ( + + ) i ( + + ) ; ; ; Upon squaring this expression one obtains (9) ( ; ; ; (40) ) This equation suggests that and are related j j j (4) ; ^ where ^ + +. Arbitrarily, and without loss of generality, we choose the solution with the plus sign. Substitution in S and computing C from Eq. () veries the expression in Eq. (4). The relation between and q is obtained observing that j ; ^ qj j (4) +q 0 After some calculations one obtains that ; ^ Using now Eq. (4) one nally obtains that q j p p + +q0 p (4) +q0 j p j (44) +q 0 ( + q 0) Conversely, starting from ; ^ +q 0 (45) +^ and using Eq. (4) one obtains that the Euler parameters are given in terms of the parameters from and ( ; ^ ) q j 4 j j (46) ( + ^ ) ; ^ q 0 ; ( ; 6^ +^ 4 ) (47) +^ ( + ^ ) where ^ 4 (^ ). Letting W (I ; T )(I + T ) ; and since C W 4 one obtains that W e (4)[^e] (4) 5

6 where is the principal angle of C. Moreover, using the denition of the Euler parameters from Eq. (9) one obtains the following result for the parameters sin() ^e (49) +cos() cos(4) where ^e is the unit vector along the principal axis. Keeping the plus sign, Eq. (49) can be further reduced to the simple formula + tan ^e (;4 <<4) (50) From Eq. (50) it is apparent that is proportional to the principal rotation axis, like the classical and the Modied Rodrigues parameters, where now the proportionality factor is f() tan(). Equation (50) is reassuring, since it proves that the parameters indeed behave as \higher order" Rodrigues parameters which can be used to \linearize" the domain of validity of the kinematic parameterization. By this, we mean that Eq. (50) behaves almost linearly as a function of the principal angle (especially in the region ; ) see also Fig.. τ π + tan π tan f( φ) -Pi Pi Pi Pi 4 Pi 5 Pi τ S τ + τ S + τ Fig. Comparison of original and \shadow" parameters. If we choose the minus sign in Eq. (49) we obtain that ; ; τ S τ + ^e (0 <<) (5) tan() Moreover, reversing the signs of the Euler parameters in Eq. (44), one obtains that the parameters have a unique set of \shadow" parameters like the Modied Rodrigues parameters 7. These parameters are obtained setting s ; sin() ^e (5) ; cos() sin(4) It can be easily veried that the corresponding \shadow" parameters reduce to and s + tan() ; ^e (; <<6) (5) tan() + ; s + tan() ^e (;6 <<) (54) ; tan() As the original parameters approach +, the associated \shadow" parameters s approach zero and vice versa. The φ general transformation between the original and the \shadow"setisgiven s ; ; ^ ^ +(+^ )^ (55) where ^ (^ ). Equations (50),(5),(5) and (54) can be used in order to compute the four distinct roots of Eq. (5). Note also that Eqs. (50),(5),(5) and (54) can be also written in the form tan ; k 4 ^e k 0 (56) respectively. The \shadow" parameter set s is shown side-side with the original parameters in Fig.. The \shadow" set is plotted in grey color. Figure also shows that parameters are indeed very linear for small rotations within 0 deg. As with the Modied Rodrigues parameters (and other stereographic parameters 7 ), these \shadow" parameters represent the same physical orientation as the original set and abide the same dierential kinematic equation. They could be used to avoid the problems of approaching the 70 deg principal rotation. By switching to the \shadow" trajectory, all numerical problems would be avoided. Having, however, a principal rotation range of 70 deg is really more than needed. Limiting the principal rotations to be within 0 deg would suce and be much more attractive. As the magnitude of approaches tan() then one would simply switch the to their \shadow" set. Having kk tan() corresponds to q 0 0. From Eq. (44) one can then see that at this point, the two sets of parameters are related ; s. The combined set of original and \shadow" parameters would provide a set of attitude coordinates which are\very linear" with respect to the principal rotation angle, more so even than the Modied Rodrigues parameters. We note in passing that the previous approach can be easily extended to any Cayley transform of order k, since Eqs. (6) and (7) can be used iteratively. For the third order Cayley transform we have that C (I ; P ) (I + P ) ; (I + P ) ; (I ; P ) (57) where P ;[p] andp (p p p ) T IR the corresponding parameters. If and p denote the respective eigenvalues of the skew-symmetric matrices R and P then, using Eqs. (5) and (57), they must be related ; + ; p + p (5) Upon expanding the previous equality and solving for one obtains p( + p) (59) + p The previous equation suggests that j and p j are related j pj ( ; p ; p ; p ) ; (p + p + j (60) p ) In order to get the relation of p to the Euler parameter vector one can set p j ( ; p ; p ; p ) ; (p + p + p ) qj q 0 (6) 5

7 and solve for^p p + p + p. After some algebraic calculations, it is not dicult to show that, in fact, (^p +) ( ; ^p ) q 0 (6) Solution of the previous equation for ^p requires the solution of a cubic equation. Once ^p is known however, it can be substituted into Eq. (6) to get the desired result. Actually, from Eqs. (6) and (6) we have that q 0 ; ^p ( + ^p ) q j pj( ; ^p ) j (6) ( + ^p ) Letting W (I ; P )(I + P ) ; then since C W one obtains that W e ()[^e] (64) where is the principal angle of C. A straightforward, but tedious calculation shows that the parameters p are related to the principal axis and angle through p tan ^e (; <<) (65) 6 Similarly to Eqs. (50)-(54), multiple solutions using the \shadow set" can also be derived, and are left to the interested reader. 6 Kinematics The kinematic equations in terms of the parameters can be computed as follows. From Eqs. () and (5) we have that _C d dt [(I ; T )4 ](I + T ) ;4 +(I ; T ) 4 d dt [(I + T );4 ] [!](I ; T ) 4 (I + T ) ;4 (66) or that d dt [(I ; T )4 ] ; C(T ) d dt [(I + T )4 ][!](I ; T ) 4 (67) where we have used the fact that da ; dt ;A ; (dadt)a ; P for any square matrix A. Using also the fact that da n n; dt j0 Aj (d Adt)A n;j; and performing the dierentiations in the left-hand-side of Eq. (67), one obtains a set of nine linear equations in terms of _ _. and _. Similarly, the right-hand-side of Eq. (67) is linear in terms of!!!. Choosing three (independent) equations out of these nine, we get a linear system of the form V () _ U()! (6) Solving for _ we nally get that the kinematic equations for the orientation parameters are given d dt V ; () U()! G()! (69) where the matrix G() is given G() ( ; ^ ) T ; 4( ; ^ )[] ( ; ^ ) +( ; 6^ +^ 4 )I (70) These kinematic equations are not as simple as the corresponding kinematic equations for the Rodrigues or the Modied Rodrigues parameters,6,7. The limiting behavior of these equations as ^! will be investigated next. We will show that Eq. (69) is actually well-dened and the apparent singularity at ^, equivalently at, is removable. To this end, denote kk the Frobenious norm of a real matrix A, i.e.,kak trace(a T A). For the kinematic equations in Eq. (69) then, after some laborious but straightforward calculations, one obtains kg()k trace[g() T G()] trace 4( ; ^ ) ^ T 64( ; ^ ) +4(; ^ )( ; 6^ +^ 4 ) T (7) +(; 6^ +^ 4 ) I ; 6( ; ^ ) [] From the denition of the matrix [] we have that [] T ; ^ I. Substituting in Eq. (7) and noticing that trace( T )^, Eq. (7) reduces to thus, kg()k ( ; ^ 4 ) 64 ( ; ^ ) ( + ^ ) 64 (7) lim kg()k ^! 4 < (7) The last equation implies that the behavior of the parameters is well-conditioned at. In addition, because of the near-linear behavior between and the magnitude of, for small principal angles, Eq. (69) is expected to behave ina more \linear-like" fashion than either the Cayley-Rodrigues or the Modied Rodrigues parameters. Similarly, for the third order Cayley parameters, one can derive the following kinematic equations ( ; ^p )pp T ; ( ; ^p )[p] dp dt 6( ; ^p ) + ( ; ^p )I! (74) These equations can be derived starting from Eqs. () and (57) and using similar arguments as before. A similar analysis as before shows that the limiting behavior of this system as ^p! p is well-dened and no singularity is encountered during integration. This is also veried through numerical simulations in the next section. In fact, for all the parameters p and the singularity of the kinematics is entirely due to the same mechanism, as for the classical Cayley-Rodrigues parameters. Namely, the kinematic differential equations themselves are well dened and continuous functions (as opposed to the Eulerian angle case) but the quadratic and higher order polynomial nonlinearities induce the possibility of nite escape times. 7 Numerical Example In order to demonstrate the potential benets of the previous kinematic parameters we present the results of the following simulations. We integrated Eqs. (69) and (74) as well as the corresponding kinematic equations in terms of the Cayley-Rodrigues () and the Modied Rodrigues parameters () starting from the zero orientation and subject to the constant angular velocity vector! (0:5 0:4 ;0:) (radsec). This corresponds to a linearly increasing value of the principal angle. The results of the simulations are shown in Fig.. This gure actually shows 54

8 0 COMPARISON OF ORIENTATION PARAMETERS plotted in Fig ρ σ p τ Conclusions φ (rad) We have extended the classical Cayley transform which maps skew-symmetric matrices to proper orthogonal matrices to higher orders. The approach is based on the observation that Cayley transforms can be viewed as generalized conformal (bilinear) mappings in the space of matrices. The Euler parameters, the Rodrigues parameters and the Modi- ed Rodrigues parameters follow as special cases of this approach. In addition, we have generated a family of higher order \Rodrigues parameters" which could be used as parameters for the rotation group. It still remains, however, to determine the applicability of these higher order parameters in realistic attitude problems Fig. Orientation parameter comparison. σ COMPARISON OF ORIENTATION PARAMETERS τ φ (rad) COMPARISON OF ORIENTATION PARAMETERS 0. σ 0.6 τ φ (rad) Fig. 4 Orientation parameters and their \shadow" sets. only the rst components of the kinematic parameter vectors, as the other two components exhibit similar behavior. As it is evident from this gure, the classical and the Modied Rodrigues parameters encounter the singularity earlier that the and the p parameters. Also the p parameters become singular earlier than the parameters. We note, however, that since discontinuities in the parameter description are typically acceptable in applications, the Modied Rodrigues parameters can be made to avoid the singularity altogether simply switching to their \shadow" set. The same also holds for the parameters via Eq. (55) or the p parameters. Figure 4 shows the simulation where the parameters and are allowed to switch to their respective \shadow" sets. Although the points of switching are arbitrary and can be chosen according to the particular application, a reasonable choice is to switch when the parameters and the corresponding \shadow" set have opposite signs. This ensures continuity of the magnitude. From Eqs. (0) and (55) this occurs when k, k :::. This is the situation depicted in Fig. 4. The parameters are shown in solid line, and the parameters are shown in dashed line. Since the classical Rodrigues parameters do not have an associated \shadow" set (better, the shadow set coincides with the original parameters), only the and parameters are Acknowledgement: The work of the rst author was supported in part the National Science Foundation under CAREER Grant CMS-964 and in part the University of Virginia SEAS Dean's Fund. References Hughes, P. C.,Spacecraft Attitude Dynamics, John Wiley & Sons, New York, 96, pp. 5{. Markley, F. L., \Parameterizations of the Attitude," in Spacecraft Attitude Determination and Control, Wertz, J. R. (editor), D. Reidel Publishing Company, Dordrecht, Holland, 97, pp. 40{40. Stuelpnagel, J., \On the Parameterization of the Three- Dimensional Rotation Group," SIAM Review, Vol. 6, No. 4, 964, pp. 4{40. 4 Shuster, M. D., \A Survey of Attitude Representations," Journal of the Astronautical Sciences, Vol. 4, No. 4, 99, pp. 49{57. 5 Special Issue on Attitude Representations, Journal of the Astronautical Sciences, Vol. 4, No. 4, Wiener, T. F., Theoretical Analysis of Gimballess Inertial Reference Equipment Using Delta-Modulated Instruments. PhD thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts, March Marandi, S. R. and Modi, V., \A Preferred Coordinate System and the Associated Orientation Representation in Attitude Dynamics," Acta Astronautica, Vol. 5, No., 97, pp. {4. Schaub, H., Tsiotras, P., and Junkins, J. L., \Principal Rotation Representations of Proper N N Orthogonal Matrices," International Journal of Engineering Science, Vol., No. 5, 995, pp. 77{95. 9 Junkins, J. L. and Kim, Y., Introduction to Dynamics and Control of Flexible Structures, AIAA, New York, 99, pp. 5{57. 0 Halmos, P. R., Finite Dimensional Vector Spaces, Vol. 7 of Annals of Mathematics Studies, Princeton University Press, Princeton, New Jersey, 95, pp. 05{. Curtis, M. L., Matrix Groups. Springer-Verlag, New York, 979, pp Churchill, R. V. and Brown, J. W., Complex Variables and Applications, McGraw Hill, NewYork, 990. Conway, J. B., Functions of One Complex Variable, Springer Verlag, New York,

9 4 Horn, R. and Johnson, C. R., Matrix Analysis. Cambridge, United Kingdom, Cambridge University Press, 95, p Brogan, W. L., Modern Control Theory, rd ed., Prentice Hall, New Jersey, 99, p Tsiotras, P., \On New Parameterizations of the Rotation Group in Attitude Kinematics," Technical Report, Dept. of Aeronautics & Astronautics, Purdue University, West Lafayette, IN, January Schaub, H. and Junkins, J. L., \Stereographic Orientation Parameters for Attitude Dynamics: A Generalization of the Rodrigues Parameters," in Journal of the Astronautical Sciences, Vol. 44, No., Jan.-Mar. 996, pp. {9. Tsiotras, P., \New Control Laws for the Attitude Stabilization of Rigid Bodies," in th IFAC Symposium on Automatic Control in Aerospace, pp. 6{, Sept. 4-7, 994. Palo Alto, CA. 56