Keywords: Kinematics; Principal planes; Special orthogonal matrices; Trajectory optimization

Transcription

1 Editorial Manager(tm) for Nonlinear Dynamics Manuscript Draft Manuscript Number: Title: Minimum Distance and Optimal Transformations on SO(N) Article Type: Original research Section/Category: Keywords: Kinematics; Principal planes; Special orthogonal matrices; Trajectory optimization Corresponding Author: Dr. Andrew Sinclair, Corresponding Author's Institution: Auburn University First Author: Andrew Sinclair Order of Authors: Andrew Sinclair; John E Hurtado; John L Junkins Abstract: The group of special (or proper) orthogonal matrices, SO(N), is used throughout engineering mechanics in the analysis and representation of mechanical systems. In this paper, a solution is presented for the optimal transformation between two elements of SO(N). The transformation is assumed to occur during a specified finite time, and a cost function that penalizes the transformation rates is utilized. The optimal transformation is found as a constant-rate rotation in each of the principal planes relating the two elements. Although the kinematics of SO(N) are nonlinear and governed by Poisson's equation, the solution is found to be a linear function of the generalized principal angles. This is made possible by the extension of principal-rotation kinematics from three-dimensional rotations to the general SO(N) group. This extension relates the N-dimensional angular velocity to the derivatives of the principal angles. The cost of the optimal transformation, the square root of the sum of the principal angles squared, also provides a useful measure for the angular distance between two elements of SO(N). Suggested Reviewers: Shane Ross Assistant Professor, Engineering Science and Mechanics, Virginia Polytechnic Institute and State University sdross@vt.edu

2 Craig Woolsey Associate Professor, Aerospace & Ocean Engineering, Virginia Polytechnic Institute and State University Maruthi Akella Associate Professor, Aerospace Engineering & Engineering Mechanics, University of Texas at Austin

3 Manuscript Click here to download Manuscript: SHJ_NonlinDyn.ps Minimum Distance and Optimal Transformations on SO(N) ANDREW J. SINCLAIR 1,, JOHN E. HURTADO 2, and JOHN L. JUNKINS 2 1 Department of Aerospace Engineering, Auburn University, 211 Aerospace Engineering Building, Auburn, AL , USA; 2 Department of Aerospace Engineering, Texas A&M University, TAMU-3141, College Station, TX , USA; Author for correspondence ( sinclair@auburn.edu; fax: ; phone: ) Abstract The group of special (or proper) orthogonal matrices, SO(N), is used throughout engineering mechanics in the analysis and representation of mechanical systems. In this paper, a solution is presented for the optimal transformation between two elements of SO(N). The transformation is assumed to occur during a specified finite time, and a cost function that penalizes the transformation rates is utilized. The optimal transformation is found as a constant-rate rotation in each of the principal planes relating the two elements. Although the kinematics of SO(N) are nonlinear and governed by Poisson s equation, the solution is found to be a linear function of the generalized principal angles. This is made possible by the extension of principal-rotation kinematics from three-dimensional rotations to the general SO(N) group. This extension relates the N-dimensional angular velocity to the derivatives of the principal angles. The cost of the optimal transformation, the square root of the sum of the principal angles squared, 1

4 also provides a useful measure for the angular distance between two elements of SO(N). Keywords: Kinematics, Principal planes, Special orthogonal matrices, Trajectory optimization 1 Introduction Special (or proper) orthogonal matrices, C SO(N), arise frequently in the analysis and representation of dynamic systems. Most commonly the group SO(3) is used to represent the orientation of rigid bodies. By analogy, however, the group SO(N) can be used to describe the configuration of M-degree of freedom systems, where M = N(N 1)/2 [11, 13]. Additionally, the group SO(N) is used frequently in similarity transformations and other decompositions of fundamental quantities related to engineering mechanics. The variety of uses for these matrices motivates the investigation of the following two problems. First, what is the angular distance between two elements of SO(N)? Second, what is the best transition between two orientations belonging to SO(N)? In this paper, these questions will be investigated as the solution to a nonlinear trajectoryoptimization problem subject to the kinematic behavior of the SO(N) group. Of course, many properties of the SO(N) group are generalizations of properties of threedimensional rotations, and it will be shown that using descriptions of principal rotations leads to a solution that is linear in the principal angles. For three-dimensional rotations, Euler s theorem describes any general orientation 2

5 in terms of a single principal rotation. The principal rotation concept also extends to N-dimensional rotations [1, 2, 7, 9, 10]; however, for higher-dimensional spaces a general orientation requires N/2 principal rotations for even N and (N 1)/2 principal rotations for odd N. In general the number of required rotations can be expressed as L = N/2. These rotations take place on completely orthogonal planes, called the principal planes. For even dimensions these planes completely occupy the space. For odd dimensions, however, one axis is left out of the rotational motion and is referred to as a principal axis. The following section reviews some of the methods that have been developed to describe SO(N) in terms of principal rotations. This view of principal rotations is then extended to develop new results for the kinematic behavior of the rotational group. Using these results, a solution is found for the optimal transition between two elements of SO(N). The resulting optimal cost function is a convenient definition of the angular distance between two elements of SO(N). The two main contributions of the paper are the principal-rotation kinematics and the measure of distance between elements of SO(N). As will be described, the canonical form decomposes the various representation matrices into a block-diagonal form with a 2 2 block associated with each principal plane and a 1 1 block associated with the principal axis if it exists. In illustrating these blocks, a subscript k will be used to indicate the kth 2 2 block on the diagonal of the indicated matrix. 3

6 2 Review of N-dimensional rotations Much of the description of N-dimensional orientations in this section was given by Mortari [7] and Bauer [2]. The current work attempts to follow their notation and conventions as closely as possible with one exception. In discussing a rotated frame both of the above authors define representations of the rotation as the mapping from the rotated frame back to a reference frame. Here the convention will be to give the mapping from the reference frame to the rotated frame. Therefore, many of the definitions given below correspond to the transpose of the rotation matrices given by Mortari and Bauer. 2.1 Rotation matrix The transformation of an N-dimensional vector by a proper orthogonal matrix, C, describes a rotation in N-dimensional space [3]. The following equation describes the transformation from a column matrix parameterizing a vector in a reference coordinate system, the n frame, to a column matrix parameterizing the vector in a rotated coordinate system, the b frame. [r] b = [C] [r] n (1) This matrix C is called a rotation matrix and is the most fundamental representation of N-dimensional rotations. An important similarity transformation of the rotation matrix is the block-diagonal skew-symmetric canonical form for normal matrices [5], referred to here as simply the 4

7 canonical representation. C = P T C P; C = PCP T (2) Here, P is a proper orthogonal matrix, and C is a block-diagonal proper orthogonal matrix. The rows of P are the coordinatization of the principal coordinate vectors in the b frame. [ [P] = [p 1 ] b [p 2 ] b... [p N ] b ] T (3) The matrix C is related to the principal angles. The kth block on the diagonal of C has the following form. C k = cos (φ k + 2πn k ) sin (φ k + 2πn k ) sin (φ k + 2πn k ) cos (φ k + 2πn k ) (4) Here, each angle π φ k π is the value of rotation in the kth principal plane, and the values n k can be any integer. For odd N, the (N, N) element of C forms a 1 1 block and is equal to positive one. The matrix P is itself an N-dimensional rotation matrix that describes the transformation from the rotated frame to a third frame, the principal frame. This coordinate system has coordinate vectors {p 1, p 2,...,p N } which are aligned with the principal planes of the rotation described by C: (p 1, p 2 ), (p 3, p 4 ), etc. The similarity transformation in the second of Eqs. (2) can be described as coordinatizing or viewing C in this principal frame. Note that consistent with his convention mentioned earlier, Bauer defines P as the transpose of the definition given here; thus it is the mapping from the principal to rotated frame. 5

8 2.2 Euler matrix Another representation of N-dimensional rotations that will be useful for the current purposes is the Euler matrix. Whereas this matrix has been used tangentially in previous works [1,7], it will be developed more fully here. The Euler matrix is a skewsymmetric matrix, E, that can be related to the rotation matrix using properties of the matrix exponential and determinant [4]. exp (E) (exp (E)) T = exp (E) exp ( E T) = exp ( E + E T) = exp (0) = I (5) det (exp (E)) = exp (Tr (E)) = exp (0) = +1 (6) Because exp (E) is proper orthogonal, the following relationship to the rotation matrix can be considered the definition of the Euler matrix. C = exp (E) (7) Based on this definition it is possible to relate the Euler matrix to the principal rotations and solve for E. Although it is tempting to simply write E = ln(c), it will be shown that this mapping is not unique because infinitely many solutions for E correspond to any particular orientation. Additionally, the matrix logarithm can suffer from limited range of convergence [4]. Because E is also normal, it can be expressed in the canonical representation. E = P T E P; E = PEP T (8) The first of Eqs. (8) can be substituted into Eq. (7). C = P T exp (E ) P (9) 6

9 Next, consider the following form for the blocks on the diagonal of E. E k = 0 φ k + 2πn k φ k 2πn k 0 = (φ k + 2πn k )J (10) The exponential of this form is the canonical form of the rotation matrix. exp (E k) = sin (φ k + 2πn k )J + cos (φ k + 2πn k ) I = C k (11) Therefore, the canonical form of E is made of blocks as shown in Eq. (10), the matrix P is the canonical transformation for both C and E, and C = exp(e ). The matrix E is shown below for odd N. [E ] = 0 φ 1 + 2πn φ 1 2πn φ L + 2πn L φ L 2πn L (12) For even N, the form is identical except the omission of the row and column of zeros. Again, the sign convention above is chosen to be consistent with C in Eq. (4). Because of the ambiguity in the integers n, infinitely many values of E and E exist which correspond to any particular C. 3 Kinematics of principal rotations In the previous section several methods to describe N-dimensional rotations in terms of the principal planes and angles were reviewed. In this section new results are presented 7

10 that relate the N-dimensional angular velocity and the derivatives of the principal planes and angles. This results in kinematic differential equations for P and φ k [12], but here the focus will be on φ k. Traditionally, the kinematic evolution of N-dimensional rotations has not been related to the principal rotations. Instead, Poisson s equation for the derivative of C is used directly. Ċ = ΩC ; Ω = CC T (13) Here, Ω is the N-dimensional skew-symmetric angular-velocity matrix. Whereas Poisson s equation holds for any value of N, for three-dimensional rotations there also exists relationships between the angular velocity and the derivatives of the principal angle, φ, and principal axis, â [6]. Following an analogous approach, the derivatives of the principal angles and principal planes of higher dimensions can also be related to the angular velocity [12]. For the current purposes, it will be sufficient to examine the principal angles. The angular velocity and the principal-angle derivatives can be related using the canonical form of C. The derivative of the canonical form is taken as follows. Ċ = P T C P + P T d dt (C ) P + P T C P (14) As mentioned, P itself is a rotation matrix describing the transformation from the rotated to principal frame. Its derivative is related to the angular velocity of the principal frame relative to the rotated frame. This skew-symmetric matrix is defined as Ψ = PP T, and P satisfies the Poisson equation P = ΨP. This is used to 8

11 rewrite the derivative of C. ( ) d Ċ = P T dt (C ) + ΨC C Ψ P (15) The angular velocity Ω can now be written in terms of the canonical form C and its derivative d dt (C ). ( ) Ω = CC d T = P T dt (C ) + ΨC C Ψ PP T C T P = P T ( d dt (C ) C T + Ψ C ΨC T ) P (16) The blocks on the diagonal of C have the form shown in Eq. (4), and the derivative of C will of course also be block diagonal with blocks of the following form. [ ] d dt (C ) = φ sin (φ k + 2πn k ) cos (φ k + 2πn k ) k (17) k cos (φ k + 2πn k ) sin (φ k + 2πn k ) For odd N, the (N, N) element of d dt (C ) forms a 1 1 block and is equal to zero. Because both C and d dt (C ) are block diagonal, the product d dt (C ) C T will also be block diagonal with blocks of the following form. [ ] d dt (C ) C T k = [ ] d dt (C ) k [ ] C T = k 0 φk φ k 0 (18) Again, for odd N the (N, N) element of the product is zero. The remaining terms in parentheses on the right-hand side of Eq. (16) are skew symmetric but in general not block diagonal. To develop the relationship between the angular velocity and the principal angle derivatives, however, the blocks on the diagonal of these terms will be investigated. Because C is block diagonal, the third term has the following form where 9

12 the 2πn k terms in C have been dropped for convenience. [ C ΨC T] = k C kψ k C k T cos (φ k ) sin (φ k ) 0 Ψ 2k 1,2k cos (φ k ) sin (φ k ) = sin (φ k ) cos (φ k ) Ψ 2k 1,2k 0 sin (φ k ) cos (φ k ) = 0 Ψ 2k 1,2k Ψ 2k 1,2k 0 (19) The blocks on the diagonal of C ΨC T are identical to the blocks on the diagonal of Ψ. Therefore, the only contribution to the blocks on the diagonal of Eq. (16) comes from Eq. (18). [ PΩP T ] k = 0 φk φ k 0 (20) Thus, the blocks on the diagonal of the angular velocity viewed in the principal frame are simply related to the derivative of the principal angles. This relation is a new result and is a generalization of the three-dimensional concept φ = a T ω [6]. The minus signs in these components are artifacts of the convention chosen to define the angular-velocity matrix as Ω = CC T. These minus signs are entirely equivalent to the convention in defining the (1, 2) component of the three-dimensional angular-velocity matrix as ω 3. Of course, this choice is made to make angular-velocity matrix multiplication equivalent to the angular-velocity vector cross product. These conventions are maintained for N-dimensions even though the angular-velocity vector and cross product lose their physical significance. In the following section, this result for the components of angular velocity lying in the principal planes will be used to study the optimal transition between two elements of SO(N). 10

13 4 Optimal kinematic maneuvers Using the results of the previous two sections, the optimal transformation between two elements of SO(N) is now investigated. It is well known that the minimum angular distance between two orientations in three dimensions is the principal angle associated with the rotation matrix relating them. A rigid-body rotational maneuver about the corresponding Euler axis through the principal angle is called an eigenaxis rotation. Clearly, a related solution is anticipated in the N-dimensional case. Of course, infinitely many transformations can relate two boundary conditions in SO(N). To obtain a single transformation, a cost function can be defined to determine an optimal solution. A cost function that penalizes the transformational rates used in the maneuver is considered. Ċdt = C (t + dt) C (t) dc (21) J 1 = T 0 dc = T 0 Ċ dt (22) Here, indicates the Frobenius matrix norm: A tr (AA T ). This function can be used to evaluate the cost or distance of a particular transitional path between two orientations. In Eq. (22), the initial time is zero, and the final time is T. Without loss of generality, the boundary conditions are chosen such that the rotated frame is initially aligned with the reference frame, C (0) = I, and an arbitrary final orientation is chosen, C (T) = F. The matrix Ċ obeys Poisson s equation. Ċ = ΩC (23) 11

14 This kinematic equation can be used in Eq. (22) to obtain the following. J 1 = T 0 ΩC dt = T 0 tr(ωωt )dt (24) The minimum distance between the two orientations is now given as the minimization of Eq. (24) subject to the boundary conditions and kinematic equations, Eq. (23). It is convenient to disregard the square root function in Eq. (24) and investigate a slightly modified problem. J 2 = 1 2 T 0 tr ( ΩΩ T) dt (25) The minimization of Eq. (25) is now sought subject to the kinematic equations and boundary conditions. Because Eqs. (24) and (25) are related via a monotone transformation of the integrand, they share the same minimizer. The Hamiltonian for the problem is easily found by introducing a matrix of costates, λ. H = 1 2 tr( ΩΩ T) tr ( λ T ΩC ) (26) From the Hamiltonian, the first-order necessary conditions are computed using properties of the trace. Ċ = H λ = ΩC (27) λ = H C = ΩT λ (28) 0 = H Ω = 1 2 Ω Ω λct (29) The third condition gives the following. Ω = λc T (30) 12

15 Next, consider the derivative of Eq. (30) together with the first two conditions, Eqs. (27) and (28). Ω = λc T + λċt = Ω T λc T λc T Ω T (31) Equation (30) itself can now be used, as well as the skew-symmetry of Ω. Ω = Ω T Ω ΩΩ T = Ω T Ω Ω T Ω = 0 (32) Therefore, the optimal angular velocity is a constant. Equations (26 32) demonstrate the optimality of constant angular velocity, relying only on kinematic principles. Alternative demonstrations focus on kinetic principles, like Hamilton s principle of motion. Equation (25) can be recognized as the action integral for an N-dimensional rigid body with identity mass matrix. According to Hamilton s principle, the equations of motion, given by Ratiu [8], give a stationary value for the action integral. For identity mass matrix, these equations simplify to Ω = 0. For constant angular velocity, the Poisson kinematic equations have the following solution. C (t) = exp ( Ωt) C (0) (33) The value of the optimal angular velocity can be related to the boundary conditions of the problem. F = exp ( ΩT) (34) The implication of this is that along the optimal solution, the matrix ΩT is an Euler matrix of the final orientation. It was observed earlier that infinitely many Euler 13

16 matrices exist for any particular orientation. The optimal angular velocity can be written using the canonical form of the Euler matrix, which was presented in Eqs. (8) and (12). For odd N, this is shown below. [Ω] = 1 T [P]T 0 φ 1 + 2πn φ 1 2πn [P] (35) φ L + 2πn L φ L 2πn L For even N, the form is identical omitting the row and columns of zeros. Therefore, infinitely many angular velocities satisfy the first-order optimality conditions, and it remains to be shown that selecting n k = 0 gives the global minimum of the cost function. Because the optimal angular velocity has been found to be a constant, the cost function evaluated for the optimal solution can be rewritten as follows. J 2 = T 2 tr( Ω T Ω ) (36) In the previous section it was observed that applying the canonical transformation P of the rotation matrix to the angular velocity does not in general produce a block-diagonal form. Because the optimal angular velocity is proportional to the Euler matrix of F, however, Eq. (35) shows that in this case PΩP T will be block diagonal. This will be defined as Ω. J 2 = T 2 tr( P T Ω T Ω P ) = T 2 tr ( Ω T Ω ) (37) 14

17 The product Ω T Ω, however, is a diagonal matrix, and its trace can be used to write the cost function as shown below. J 2 = 1 2T [ (φ1 + 2πn 1 ) (φ L + 2πn L ) 2] (38) Therefore, finding the optimal-angular velocity is reduced to minimizing each of the parenthetical terms above. This is clearly done by selecting each n k equal to zero. This gives the final result for the optimal angular velocity. [Ω] = 1 T [P]T 0 φ φ [P] (39) φ L φ L Considering the results from the previous section, the optimal maneuver is a rotation in each of the principal planes relating the initial and final orientations with a rotational rate of φ k /T. It is the coordinatization in the principal planes that allows the solution to the nonlinear trajectory-optimization problem to be expressed in this form, which is linear in the principal angles. As mentioned, this solution for the minimization of J 2 must also minimize J 1. The optimal cost for J 1 can be evaluated using the found angular velocity. J 1 = T tr (ΩΩ T ) = φ φ 2 L (40) This demonstrates that the minimum angular distance between two N-dimensional orientations is indeed the square root of the sum of the squares of the principal angles. 15

18 5 Conclusion In this paper, the kinematic interpretation of principal rotations was extended to higher dimensional elements of SO(N). This result led to a solution for the optimal transition between two elements of this group. The optimal cost of the resulting solution provides a convenient, intuitive measure for the angular distance between such matrices. It is anticipated that these results will prove useful in a wide variety of mechanics fields where elements of SO(N) are used in the analysis and comparison of engineering systems. Critically, the posed trajectory-optimization problem was subject to the nonlinear kinematics of the SO(N) group described by Poisson s equation. Judicious selection for the representation and coordinatization of the control variables, however, resulted in a physically meaningful solution that is linear in the orientation variables. This was made possible through the use of the Euler matrix, canonical form, and principal plane concepts. These transformations allowed the presentation of a solution of linear form to the nonlinear problem. References [1] Itzhack Y. Bar-Itzhack. Extension of Euler s theorem to n-dimensional spaces. IEEE Transactions on Aerospace and Electronic Systems, 25(6): , [2] Robert Bauer. Euler s theorem on rigid body displacements generalized to n dimensions. The Journal of the Astronautical Sciences, 50(3): ,

19 [3] O. Bottema and B. Roth. Theoretical Kinematics. North-Holland Publishing Company, Amsterdam, Chapters 1 and 2. [4] Morton L. Curtis. Matrix Groups. Springer-Verlag, New York, [5] Roger A. Horn and Charles R. Johnson. Matrix Analysis. Cambridge University Press, pp [6] Peter C. Hughes. Spacecraft Attitude Dynamics. John Wiley & Sons, New York, pp [7] Daniele Mortari. On the rigid rotation concept in n-dimensional spaces. Journal of the Astronautical Sciences, 49(3): , [8] T. Ratiu. The motion of the free n-dimensional rigid body. Indiana University Mathematics Journal, 29(4): , [9] Hanspeter Schaub, Panagiotis Tsiotras, and John L. Junkins. Principal rotation representations of proper n n orthogonal matrices. International Journal of Engineering Science, 33(15): , [10] P. H. Schoute. Le déplacement le plus général dans l espace à n dimensions. Annales de l Ecole Polytechnique de Delft, 7: , [11] A. J. Sinclair and J. E. Hurtado. Cayley kinematics and the Cayley form of dynamic equations. Proceedings of the Royal Society of London Series A, 461(2055): ,

20 [12] A. J. Sinclair, J. E. Hurtado, and J. L. Junkins. Kinematics of N-dimensional principal rotations. In AAS/AIAA Spaceflight Mechanics Meeting, Copper Mountain, Colorado, January Paper AAS [13] A. J. Sinclair, J. E. Hurtado, and J. L. Junkins. Application of the Cayley form to general spacecraft motion. Journal of Guidance, Control, and Dynamics, 29(2): ,