Lie Group Variational Integrators for the Full Body Problem

Transcription

1 Lie Group Variational Integrators for the Full Body Problem arxiv:math/ v2 [math.na] 20 Aug 2005 Taeyoung Lee a,1, Melvin Leok b,1, and N. Harris McClamroch a,2 a Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109, USA b Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA Abstract We develop the equations of motion for full body models that describe the dynamics of rigid bodies, acting under their mutual gravity. The equations are derived using a variational approach where variations are defined on the Lie group of rigid body configurations. Both continuous equations of motion and variational integrators are developed in Lagrangian and Hamiltonian forms, and the reduction from the inertial frame to a relative coordinate system is also carried out. The Lie group variational integrators are shown to be symplectic, to preserve conserved quantities, and to guarantee exact evolution on the configuration space. One of these variational integrators is used to simulate the dynamics of two rigid dumbbell bodies. Key words: Variational integrators, Lie group method, full body problem addresses: tylee@umich.edu (Taeyoung Lee), mleok@umich.edu (Melvin Leok), nhm@engin.umich.edu (N. Harris McClamroch). 1 This research has been supported in part by a grant from the Rackham Graduate School, University of Michigan. 2 This research has been supported in part by NSF under grant ECS Preprint submitted to Elsevier Science 2 February 2008

2 Nomenclature γ i Linear momentum of the ith body in the inertial frame p.12 Γ Relative linear momentum p.17 J i Standard moment of inertia matrix of the ith body p.9 J di Nonstandard moment of inertia matrix of the ith body p.9 J R J dr Standard moment of inertia matrix of the first body with respect to the second body fixed frame Nonstandard moment of inertia matrix of the first body with respect to the second body fixed frame p.14 p.14 m i Mass of the ith body p.9 m Reduced mass for two bodies of mass m 1 and m 2, m = m 1m 2 m 1 +m 2 p.17 M i Gravity gradient moment on the ith body p.12 Ω i Angular velocity of the ith body in its body fixed frame p.9 Ω Angular velocity of the first body expressed in the second body fixed frame p.13 Ω Ω = (Ω 1,Ω 2,,Ω n ) p.9 Π i Angular momentum of the ith body in its body fixed frame p.12 Π Angular momentum of the first body expressed in the second body fixed frame p.17 R i Rotation matrix from the ith body fixed frame to the inertial frame p.8 R Relative attitude of the first body with respect to the second body p.13 R R = (R 1,R 2,,R n ) p.9 v i Velocity of the mass center of the ith body in the inertial frame p.12 V Relative velocity of the first body with respect to the second body in the second body fixed frame p.13 V 2 Velocity of the second body in the second body fixed frame p.13 x i Position of the mass center of the ith body in the inertial frame p.8 X Relative position of the first body with respect to the second body expressed in the second body fixed frame p.13 X 2 Position of the second body in the second body fixed frame p.13 x x = (x 1,x 2,,x n ) p.9 2

3 1 Introduction 1.1 Overview The full body problem studies the dynamics of rigid bodies interacting under their mutual potential, and the mutual potential of distributed rigid bodies depends on both the position and the attitude of the bodies. Therefore, the translational and the rotational dynamics are coupled in the full body problem. The full body problem arises in numerous engineering and scientific fields. For example, in astrodynamics, the trajectory of a large spacecraft around the Earth is affected by the attitude of the spacecraft, and the dynamics of a binary asteroid pair is characterized by the non-spherical mass distributions of the bodies. In chemistry, the full rigid body model is used to study molecular dynamics. The importance of the full body problem is summarized in [7], along with a preliminary discussion from the point of view of geometric mechanics. The full two body problem was studied by Maciejewski [12], and he presented equations of motion in inertial and relative coordinates and discussed the existence of relative equilibria in the system. However, he does not derive the equations of motion, nor does he discuss the reconstruction equations that allow the recovery of the inertial dynamics from the relative dynamics. Scheeres derived a stability condition for the full two body problem [20], and he studied the planar stability of an ellipsoid-sphere model [21]. Recently, interest in the full body problem has increased, as it is estimated that up to 16% of near-earth asteroids are binaries [13]. Spacecraft motion about binary asteroids have been discussed using the restricted three body model [22], [2], and the four body model [23]. Conservation laws are important for studying the dynamics of the full body problem, because they describe qualitative characteristics of the system dynamics. The representation used for the attitude of the bodies should be globally defined since the complicated dynamics of such systems would require frequent coordinate changes when using representations that are only defined locally. General numerical integration methods, such as the Runge-Kutta scheme, do not preserve first integrals nor the exact geometry of the full body dynamics [4]. A more careful analysis of computational methods used to study the full body problem is crucial. Variational integrators and Lie group methods provide a systematic method of constructing structure-preserving numerical integrators [4]. The idea of the variational approach is to discretize Hamilton s principle rather than the continuous equations of motion [17]. The numerical integrator obtained from the discrete Hamilton s principle exhibits excellent energy properties [3], conserves first integrals associated with symmetries by a discrete version of Noether s theorem, and preserves the symplectic structure. Many interesting differential equations, including full body dynamics, evolve on a Lie group. Lie group methods consist of numerical integrators that preserve the geometry of the configuration space by automatically remaining on the Lie group [5]. Moser and Vesselov [18], Wendlandt and Marsden [25] developed numerical integrators for a free rigid body by imposing an orthogonal constraint on the attitude variables and 3

4 by using unit quaternions, respectively. The idea of using the Lie group structure and the exponential map to numerically compute rigid body dynamics arises in the work of Simo, Tarnow, and Wong [24], and in the work by Krysl [8]. A Lie group approach is explicitly adopted by Lee, Leok, and McClamroch in the context of a variational integrator for rigid body attitude dynamics with a potential dependent on the attitude, namely the 3D pendulum dynamics, in [9]. The motion of full rigid bodies depends essentially on the mutual gravitational potential, which in turn depends only on the relative positions and the relative attitudes of the bodies. Marsden et al. introduce discrete Euler-Poincaré and Lie-Poisson equations in [14] and [15]. They reduce the discrete dynamics on a Lie group to the dynamics on the corresponding Lie algebra. Sanyal, Shen and McClamroch develop variational integrators for mechanical systems with configuration dependent inertia and they perform discrete Routh reduction in [19]. A more intrinsic development of discrete Routh reduction is given in [10] and [6]. 1.2 Contributions The purpose of this paper is to provide a complete set of equations of motion for the full body problem. The equations of motion are categorized by three independent properties: continuous / discrete time, inertial / relative coordinates, and Lagrangian / Hamiltonian forms. Therefore, a total of eight types of equations of motion for the full body problem are given in this paper. The relationships between these equations of motion are shown in Fig. 1, and are further summarized in Fig. 7. Continuous time (Sec. 3) Inertial coordinates (Sec. 3.1) Relative coordinates (Sec. 3.2) Lagrangian (3.1.1) Hamiltonian (3.1.2) Lagrangian (3.2.1) Hamiltonian (3.2.2) Discrete time (Sec. 4) Inertial coordinates (Sec. 4.1) Relative coordinates (Sec. 4.2) Lagrangian (4.1.1) Hamiltonian (4.1.2) Lagrangian (4.2.1) Hamiltonian (4.2.2) Fig. 1. Eight types of equations of motion for the full body problem Continuous equations of motion for the full body problem are given in [12] without any formal derivation of the equations. We show, in this paper, that the equations of motion for the full body problem can be derived from Hamilton s principle. The proper form for the variations of Lie group elements in the configuration space lead to a systematic derivation of the equations of motion. 4

5 This paper develops discrete variational equations of motion for the full body model following a similar variational approach but carried out within a discrete time framework. Since numerical integrators are derived from the discrete Hamilton s principle, they exhibit symplectic and momentum preserving properties, and good energy behavior. They are also constructed to conserve the Lie group geometry on the configuration space. Numerical simulation results for the interaction of two rigid dumbbell models are given. This paper is organized as follows. The basic idea of the variational integrator is given in section 2. The continuous equations of motion and variational integrators are developed in section 3 and 4. Numerical simulation results are given in section 5. An appendix contains several technical details that are utilized in the main development. 2 Background 2.1 Hamilton s principle and variational integrators The procedure to derive the Euler-Lagrange equations and Hamilton s equations from Hamilton s principle is shown in Fig. 2. Configuration Space (q, q) TQ Configuration Space (q k,q k+1 ) Q Q Lagrangian L(q, q) Discrete Lagrangian L d (q k,q k+1 ) Action Integral G = t f t 0 L(q, q)dt Action Sum G d = L d (q k,q k+1 ) Variation δg = d dǫ Gǫ = 0 Legendre transform. p = FL(q, q) Variation δg d = d dǫ Gǫ d = 0 Legendre transform. p k = FL(q, q) k Euler-Lagrange Eqn. d L dt q L q = 0 Hamilton s Eqn. q = H p, ṗ = H q Dis. E-L Eqn. D qk L dk 1 +D qk L dk = 0 Dis. Hamilton s Eqn. p k = D qk L dk, p k+1 = D qk+1 L dk Fig. 2. Procedures to derive the continuous and discrete equations of motion When deriving the equations of motion, we first choose generalized coordinates q, and a corresponding configuration space Q. We then construct a Lagrangian from the kinetic and the potential energy. An action integral G = t f t 0 L(q, q)dt is defined as the path integral of the Lagrangian along a time-parameterized trajectory. After taking the variation of the 5

6 action integral, and requiring it to be stationary, we obtain the Euler-Lagrange equations. If we use the Legendre transformation defined as p δ q = FL(q, q) δ q, = d dǫ L(q, q + ǫδ q), (1) ǫ=0 where δ q T q Q, then we obtain Hamilton s equations in terms of momenta variables p = FL(q, q) T Q. These equations are equivalent to the Euler-Lagrange equations [16]. There are two issues that arise in applying this procedure to the full body problem. The first is that the configuration space for each rigid body is the semi-direct product, SE(3) = R 3 s SO(3), where SO(3) is the Lie group of orthogonal matrices with determinant 1. Therefore, variations should be carefully chosen such that they respect the geometry of the configuration space. For example, a varied rotation matrix R ǫ SO(3) can be expressed as R ǫ = Re ǫη, where ǫ R, and η so(3) is a variation represented by a skew-symmetric matrix. The variation of the rotation matrix δr is the part of R ǫ that is linear in ǫ: R ǫ = R + ǫδr + O(ǫ 2 ). More precisely, δr is given by δr = d dǫ R ǫ = Rη. (2) ǫ=0 The second issue is that reduced variables can be used to obtain equations of motion expressed in relative coordinates. The variations of the reduced variables are constrained as they must arise from the variations of the unreduced variables while respecting the geometry of the configuration space. The proper variations of Lie group elements and reduced quantities are computed while deriving the continuous equations of motion. Generally, numerical integrators are obtained by approximating the continuous Euler- Lagrange equation using a finite difference rule such as q k = (q k+1 q k )/h, where q k denotes the value of q(t) at t = hk for an integration step size h R and an integer k. A variational integrator is derived by following a procedure shown in the right column of Fig. 2. When deriving a variational integrator, the velocity phase space (q, q) TQ of the continuous Lagrangian is replaced by (q k,q k+1 ) Q Q, and the discrete Lagrangian L d is chosen such that it approximates a segment of the action integral L d (q k,q k+1 ) h 0 L(q k,k+1 (t), q k,k+1 (t)) dt, where q k,k+1 (t) is the solution of the Euler-Lagrange equation satisfying boundary conditions q k,k+1 (0) = q k and q k,k+1 (h) = q k+1. Then, the discrete action sum G d = L d (q k,q k+1 ) approximates the action integral G. Taking the variations of the action sum, we obtain the discrete Euler Lagrange equation D qk L d (q k 1,q k ) + D qk L d (q k,q k+1 ) = 0, 6

7 where D qk L d denotes the partial derivative of L d with respect to q k. This yields a discrete Lagrangian map F Ld : (q k 1,q k ) (q k,q k+1 ). Using a discrete analogue of the Legendre transformation, referred to as a discrete fiber derivative FL d : Q Q T Q, variational integrators can be expressed in Hamiltonian form as p k = D qk L d (q k,q k+1 ), (3) p k+1 = D qk+1 L d (q k,q k+1 ). (4) This yields a discrete Hamiltonian map F Ld : (q k,p k ) (q k+1,p k+1 ). A complete development of variational integrators can be found in [17]. 2.2 Notation Variables in the inertial and the body fixed coordinates are denoted by lower-case and capital letters, respectively. A subscript i is used for variables related to the ith body. The relative variables have no subscript and the kth discrete variables have the second level subscript k. The letters x,v,ω and R are used to denote the position, velocity, angular velocity and rotation matrix, respectively. The trace of A R n n is denoted by tr[a] = n [A] ii, i=1 where [A] ii is the i,ith element of A. It can be shown that tr[ab] = tr[ba] = tr [ B T A T] = tr [ A T B T], (5) tr [ A T B ] 3 = [A] pq [B] pq, (6) p,q=1 tr[pq] = 0, (7) for matrices A,B R n n, a skew-symmetric matrix P = P T R n n, and a symmetric matrix Q = Q T R n n. The map S : R 3 R 3 3 is defined by the condition that S(x)y = x y for x,y R 3. For x = (x 1,x 2,x 3 ) R 3, S(x) is given by It can be shown that 0 x 3 x 2 S(x) = x 3 0 x 1. x 2 x 1 0 S(x) T = S(x), (8) S(x y) = S(x)S(y) S(y)S(x) = yx T xy T, (9) 7

8 for x,y R 3 and R SO(3). S(Rx) = RS(x)R T, (10) S(x) T S(x) = ( x T x ) I 3 3 xx T = tr [ xx T] I 3 3 xx T, (11) Using homogeneous coordinates, we can represent an element of SE(3) as follows: R x SE(3) 0 1 for x R 3 and R SO(3). Then, an action on SE(3) is given by the usual matrix multiplication in R 4 4. For example R 1 x 1 R 2 x 2 = for x 1,x 2 R 3 and R 1,R 2 SO(3). R 1R 2 R 1 x 2 + x The action of an element of SE(3) on R 3 can be expressed using a matrix-vector product, once we identify R 3 with R 3 {1} R 4. In particular, we see from that x 2 Rx 2 + x 1. R x 1 x 2 = Rx 2 + x Continuous time full body models In this section, the continuous time equations of motion for the full body problem are derived in inertial and relative coordinates, and are expressed in both Lagrangian and in Hamiltonian form. We define O e 1 e 2 e 3 as an inertial frame, and O Bi E i1 E i2 E i3 as a body fixed frame for the ith body, B i. The origin of the ith body fixed frame is located at the center of mass of body B i. The configuration space of the ith rigid body is SE(3) = R 3 s SO(3). We denote the position of the center of mass of B i in the inertial frame by x i R 3, and we denote the attitude of B i by R i SO(3), which is a rotation matrix from the ith body fixed frame to the inertial frame. 3.1 Inertial coordinates Lagrangian: To derive the equations of motion, we first construct a Lagrangian for the full body problem. Given (x i,r i ) SE(3), the inertial position of a mass element of B i is given 8

9 by x i + R i ρ i, where ρ i R 3 denotes the position of the mass element in the body fixed frame. Then, the kinetic energy of B i can be written as T i = 1 ẋ i + 2 Ṙiρ i 2 dm i. B i Using the fact that B i ρ i dm i = 0 and the kinematic equation Ṙi = R i S(Ω i ), the kinetic energy T i can be rewritten as T i (ẋ i,ω i ) = 1 ẋ i 2 + S(Ω i )ρ i 2 dm i, 2 B i = 1 2 m i ẋ i tr[ S(Ω i )J di S(Ω i ) T], (12) where m i R is the total mass of B i, Ω i R 3 is the angular velocity of B i in the body fixed frame, and J di = B i ρ i ρ T i dm i R 3 3 is a nonstandard moment of inertia matrix. Using (11), it can be shown that the standard moment of inertia matrix J i = B i S(ρ i ) T S(ρ i )dm i R 3 3 is related to J di by J i = tr[j di ]I 3 3 J di, and that the following condition holds for any Ω i R 3 S(J i Ω i ) = S(Ω i )J di + J di S(Ω i ). (13) We first derive equations using the nonstandard moment of inertia matrix J di, and then express them in terms of the standard moment of inertia J i by using (13). The gravitational potential energy U : SE(3) n R is given by U(x 1,x 2,,x n,r 1,R 2,,R n ) = 1 2 n i,j=1 i j where G is the universal gravitational constant. Then, the Lagrangian for n full bodies can be written as B i B j Gdm i dm j x i + R i ρ i x j R j ρ j, (14) L(x,ẋ,R,Ω) = n i=1 1 2 m i ẋ i tr[ S(Ω i )J di S(Ω i ) T] U(x,R), (15) where bold type letters denote ordered n-tuples of variables. For example, x (R 3 ) n, R SO(3) n, and Ω (R 3 ) n are defined as x = (x 1,x 2,,x n ), R = (R 1,R 2,,R n ), and Ω = (Ω 1,Ω 2,,Ω n ), respectively. Variations of variables: Since the configuration space is SE(3) n, the variations should be carefully chosen such that they respect the geometry of the configuration space. The variations of x i : [t 0,t f ] R 3 and ẋ i : [t 0,t f ] R 3 are trivial, namely x ǫ i = x i + ǫδx i + O(ǫ 2 ), ẋ ǫ i = ẋ i + ǫδẋ i + O(ǫ 2 ), 9

10 where δx i : [t 0,t f ] R 3, δẋ i : [t 0,t f ] R 3 vanish at the initial time t 0 and at the final time t f. The variation of R i : [t 0,t f ] SO(3), as given in (2), is δr i = R i η i, (16) where η i : [t 0,t f ] so(3) is a variation with values represented by a skew-symmetric matrix (ηi T = η i ) vanishing at t 0 and t f. The variation of Ω i can be computed from the kinematic equation Ṙi = R i S(Ω i ) and (16) to be S(δΩ i ) = d dǫ Ri ǫt Ṙi ǫ = δrt i Ṙ i + Ri T δṙi, ǫ=0 = η i S(Ω i ) + S(Ω i )η i + η i. (17) Equations of motion: Lagrangian form If we take variations of the Lagrangian using (16) and (17), we obtain the equations of motion from Hamilton s principle. We first take the variation of the kinetic energy of B i. δt i = d dǫ T i (ẋ i + ǫδẋ i,ω i + ǫδω i ), ǫ=0 = m i ẋ T i δẋ i tr[ S(δΩ i )J di S(Ω i ) T + S(Ω i )J di S(δΩ i ) T]. Substituting (17) into the above equation and using (5), we obtain δt i = m i ẋ T i δẋ i tr[ η i {J di S(Ω i ) + S(Ω i )J di }] tr[η i {S(Ω i ) {J di S(Ω i ) + S(Ω i )J di } {J di S(Ω i ) + S(Ω i )J di } S(Ω i )}]. Using (9) and (13), δt i is given by δt i = m i ẋ T i δẋ i tr[ η is(j i Ω i ) + η i S(Ω i J i Ω i )]. (18) The variation of the potential energy is given by δu = d dǫ U(x + ǫδx,r + ǫδr), ǫ=0 where δx = (δx 1,δx 2,,δx n ), δr = (δr 1,δR 2,,δR n ). Then, δu can be written as n 3 δu = U 3 U [δx i ] p + [R i η i ] pq, [x i ] p [R i ] pq = i=1 p=1 n ( U i=1 x i T δx i tr [ η i Ri T p,q=1 U R i where [A] pq denotes the p,qth element of a matrix A, and U U [x i ] p, [ U R i ] pq = ]), (19) x i, U R i are given by [ U x i ] p = U [R i ] pq, respectively. The variation of the Lagrangian has the form δl = n δt i δu, (20) i=1 10

11 which can be written more explicitly by using (18) and (19). The action integral is defined to be i=1 G = tf The variation of the action integral can be written as n tf δg = m i ẋ T i δẋ i U t 0 x i Using integration by parts, n t f δg = m i ẋ T i δx i 1 i=1 t 0 2 tr[η is(j i Ω i )] n tf + U x i i=1 t 0 t 0 L(x,ẋ,R,Ω)dt. (21) Tδx i tr [ η i S(J i Ω i ) + η i { S(Ω i J i Ω i ) + 2R T i t f tf + t 0 t 0 Tδx i tr [η i { S(Ω i J i Ω i ) + 2R T i }] U dt. R i m i ẍ T i δx i + 1 [ ] 2 tr η i S(J i Ωi ) dt }] U dt. R i Using the fact that δx i and η i vanish at t 0 and t f, the first two terms of the above equation vanish. Then, δg is given by n tf { δg = δx T i m i ẍ i + U } + 12 { }] [η x tr i S(J i Ωi + Ω i J i Ω i ) + 2Ri T U dt. i R i i=1 t 0 From Hamilton s principle, δg should be zero for all possible variations δx i : [t 0,t f ] R 3 and η i : [t 0,t f ] so(3). Therefore, the expression in the first brace should be zero. Furthermore, since η i is skew symmetric, we have by (7), that the expression in the second brace should be symmetric. Then, we obtain the continuous equations of motion as S(J i Ωi + Ω i J i Ω i ) + 2R T i Using (8), we can simplify (22) to be m i ẍ i = U, x i U = S(J i Ωi + Ω i J i Ω i ) T + 2 U T R. (22) R i R i S(J i Ωi + Ω i J i Ω i ) = U T R Ri T R i U R i. Note that the right hand side expression in the above equation is also skew symmetric. Then, the moment due to the gravitational potential on the ith body, M i R 3 is given by S(M i ) = U T R Ri i Ri T U R i. This moment M i can be expressed more explicitly as the following computation shows. S(M i ) = U T R i Ri T R i U, R i = ut ri 1 u T ri 2 u T ri 3 r i1 r i2 rt i 1 ri T 2 ri T 3 u ri1 u ri2, r i3 u ri3 11

12 = ( u T ri 1 r i1 r T i 1 u ri1 ) + ( u T ri2 r i2 r T i 2 u ri2 ) + ( u T ri3 r i3 r T i 3 u ri3 ), where r ip,u rip R 1 3 are the pth row vectors of R i and U R i, respectively. Substituting x = r T i p, y = u T ri p into (9), we obtain S(M i ) = S(r i1 u ri1 ) + S(r i2 u ri2 ) + S(r i3 u ri3 ), = S(r i1 u ri1 + r i2 u ri2 + r i3 u ri3 ), (23) Then, the gravitational moment M i is given by M i = r i1 u ri1 + r i2 u ri2 + r i3 u ri3. (24) In summary, the continuous equations of motion for the full body problem, in Lagrangian form, can be written for i (1,2,,n) as where the translational velocity v i R 3 is defined as v i = ẋ i. v i = 1 U, m i x i (25) J i Ωi + Ω i J i Ω i = M i, (26) ẋ i = v i, (27) Ṙ i = R i S(Ω i ), (28) Equations of motion: Hamiltonian form We denote the linear and angular momentum of the ith body B i by γ i R 3, and Π i R 3, respectively. They are related to the linear and angular velocities by γ i = m i v i, and Π i = J i Ω i. Then, the equations of motion can be rewritten in terms of the momenta variables. The continuous equations of motion for the full body problem, in Hamiltonian form, can be written for i (1,2,,n) as γ i = U, x i (29) Π i + Ω i Π i = M i, (30) ẋ i = γ i, m i (31) Ṙ i = R i S(Ω i ). (32) 3.2 Relative coordinates The motion of the full rigid bodies depends only on the relative positions and the relative attitudes of the bodies. This is a consequence of the property that the gravitational potential can be expressed using only these relative variables. Physically, this is related to the fact that the total linear momentum and the total angular momentum about the mass 12

13 center of the bodies are conserved. Mathematically, the Lagrangian is invariant under a left action of an element of SE(3). So, it is natural to express the equations of motion in one of the body fixed frames. In this section, the equations of motion for the full two body problem are derived in relative coordinates. This result can be readily generalized to the n body problem. Reduction of variables: In [12], the reduction is carried out in stages, by first reducing position variables in R 3, and then reducing attitude variables in SO(3). This is equivalent to directly reducing the position and the attitude variables in SE(3) in a single step, which is a result that can be explained by the general theory of Lagrangian reduction by stages [1]. The reduced position and the reduced attitude variables are the relative position and the relative attitude of the first body with respect to the second body. In other words, the variables are reduced by applying the inverse of (R 2,x 2 ) SE(3) given by (R T 2, RT 2 x 2) SE(3), to the following homogeneous transformations: RT 2 RT 2 x 2 R 1 x 1, R 2 x 2 = RT 2 R 1 R2 T(x 1 x 2 ), RT 2 R 2 R2 T(x 2 x 2 ), = RT 2 R 1 R2 T(x 1 x 2 ), I (33) This motivates the definition of the reduced variables as X = R T 2 (x 1 x 2 ), (34) R = R T 2 R 1, (35) where X R 3 is the relative position of the first body with respect to the second body expressed in the second body fixed frame, and R SO(3) is the relative attitude of the first body with respect to the second body. The corresponding linear and angular velocities are also defined as V = R2 T (ẋ 1 ẋ 2 ), (36) Ω = RΩ 1, (37) where V R 3 represents the relative velocity of the first body with respect to the second body in the second body fixed frame, and Ω R 3 is the angular velocity of the first body expressed in the second body fixed frame. Here, the capital letters denote variables expressed in the second body fixed frame. For convenience, we denote the inertial position and the inertial velocity of the second body, expressed in the second body fixed frame by X 2,V 2 R 3 : X 2 = R T 2 x 2, (38) V 2 = R T 2 ẋ2. (39) Reduced Lagrangian: The equations of motion in relative coordinates are derived in the same way used to derive the equations in the inertial frame. We first construct a reduced 13

14 Lagrangian. The reduced Lagrangian l is obtained by expressing the original Lagrangian (15) in terms of the reduced variables. The kinetic energy is given by T 1 + T 2 = 1 2 m 1 ẋ m 2 ẋ tr[ S(Ω 1 )J d1 S(Ω 1 ) T] tr[ S(Ω 2 )J d2 S(Ω 2 ) T], = 1 2 m 1 V + V m 2 V tr[ S(Ω)J dr S(Ω) T] tr[ S(Ω 2 )J d2 S(Ω 2 ) T], where (10) is used, and J dr R 3 3 is defined as J dr = RJ d1 R T, which is an expression of the nonstandard moment of inertia matrix of the first body with respect to the second body fixed frame. Note that J dr is not a constant matrix. Using (10), it can be shown that J dr also satisfies a property similar to (13), namely S(J R Ω) = S(Ω)J dr + J dr S(Ω), (40) where J R = RJ 1 R T R 3 3 is the standard moment of inertia matrix of the first body with respect to the second body fixed frame. Using (14), the gravitational potential U of the full two bodies is given by Gdm 1 dm 2 U(x 1,x 2,R 1,R 2 ) = B 2 x 1 + R 1 ρ 1 x 2 R 2 ρ 2, B 1 and it is invariant under an action of an element of SE(3). Therefore, the gravitational potential can be written as a function of the relative variables only. By applying the inverse of (R 2,x 2 ) SE(3) as given in (33), we obtain U(x 1,x 2,R 1,R 2 ) = U(R2 T (x 1 x 2 ),0,R2 T R 1,I 3 3 ), Gdm 1 dm = 2 B 1 B 2 R T 2 (x 1 x 2 ) + R2 TR 1ρ 1 I 3 3 ρ 2, Gdm 1 dm 2 = B 2 X + Rρ 1 ρ 2, B 1 U(X,R). Here we abuse notation slightly by using the same letter U to denote the gravitational potential as a function of the relative variables. Then, the reduced Lagrangian l is given by l(r,x,ω,v,ω 2,V 2 ) = 1 2 m 1 V + V m 2 V tr[ S(Ω)J dr S(Ω) T] tr[ S(Ω 2 )J d2 S(Ω 2 ) T] U(X,R). (41) Variations of reduced variables: The variations of the reduced variables must be restricted to those that can arise from the variations of the original variables. For example, the variation of the relative attitude R is given by δr = d dǫ R2 ǫt R1, ǫ ǫ=0 14

15 Substituting (16) into the above equation, = δr T 2 R 1 + R T 2 δr 1. δr = η 2 R T 2 R 1 + R T 2 R 1 η 1, = η 2 R + ηr, where a reduced variation η so(3) is defined as η = Rη 1 R T. The variations of other reduced variables can be obtained in a similar way. The detailed derivations are given in A.1, and we summarize the results as follows: δr = ηr η 2 R, (42) δx = χ η 2 X, (43) S(δΩ) = η S(Ω)η + ηs(ω) + S(Ω)η 2 η 2 S(Ω) + S(Ω 2 )η ηs(ω 2 ), (44) δv = χ + S(Ω 2 )χ η 2 V, (45) S(δΩ 2 ) = η 2 + S(Ω 2 )η 2 η 2 S(Ω 2 ), (46) δv 2 = χ 2 + S(Ω 2 )χ 2 η 2 V 2, (47) where χ,χ 2 : [t 0,t f ] R 3 and η,η 2 : [t 0,t f ] so(3) are variations that vanish at the end points. These Lie group variations are the key elements required to obtain the equations of motion in relative coordinates Equations of motion: Lagrangian form The reduced equations of motion can be computed from the reduced Lagrangian using the reduced Hamilton s principle. By taking the variation of the reduced Lagrangian (41) using the constrained variations given by (42) through (47), we can obtain the equations of motion in the relative coordinates. Following a similar process to the derivation of δt i,δu as in (18) and (19), the variation of the reduced Lagrangian δl can be obtained as δl = χ T [m 1 (V + V 2 )] χ T [m 1 Ω 2 (V + V 2 )] + χ T 2 [m 1(V + V 2 ) + m 2 V 2 ] χ T 2 [m 1Ω 2 (V + V 2 ) + m 2 Ω 2 V 2 ] tr[ ηs(j RΩ) + ηs(ω 2 J R Ω)] tr[ η 2S(J 2 Ω 2 ) + η 2 S(Ω 2 J 2 Ω 2 )] χ T U [ X + tr η 2 X U T ] + tr[ η 2 R U T ηr U T ], (48) X R R where we used the identities (9), (13) and (40), and the constrained variations (42) through (47). The action integral in terms of the reduced Lagrangian is G = tf t 0 l(r,x,ω,v,ω 2,V 2 )dt. (49) 15

16 Using integration by parts together with the fact that χ,χ 2,η and η 2 vanish at t 0 and t f, the variation of the action integral can be expressed from (48) as tf δg = t 0 tf χ T χ T 2 t 0 tf t 0 tf t 0 { m 1 ( V + V 2 ) + m 1 Ω 2 (V + V 2 ) + U } dt X {m 1 ( V + V } 2 ) + m 2 V2 + m 1 Ω 2 (V + V 2 ) + m 2 Ω 2 V 2 [ tr η { S( (J R Ω) + Ω 2 J R Ω) 2R U R { tr [η 2 S(J 2 Ω2 + Ω 2 J 2 Ω 2 ) + 2X U T X T }] dt + 2R U T }] dt. R dt From the reduced Hamilton s principle, δg = 0 for all possible variations χ,χ 2 : [t 0,t f ] R 3 and η,η 2 : [t 0,t f ] so(3) that vanish at t 0 and t f. Therefore, in the above equation, the expressions in the first two braces should be zero and the expressions in the last two braces should be symmetric since η,η 2 are skew symmetric. Then, we obtain the following equations of motion, m 1 ( V + V 2 ) + m 1 Ω 2 (V + V 2 ) = U X, (50) m 2 V2 + m 2 Ω 2 V 2 = U X, (51) S( (J R Ω) + Ω 2 J R Ω) = S(M), S(J 2 Ω2 + Ω 2 J 2 Ω 2 ) = U X XT X U T + S(M), (52) X where M R 3 is defined by the relation S(M) = U to the derivation of (23), M can be written as R RT R U R T. By a procedure analogous M = r 1 u r1 + r 2 u r2 + r 3 u r3, (53) where r p,u rp R 3 are the pth column vectors of R and U R, respectively. Equation (50) can be simplified using (51) as V + Ω 2 V = m 1 + m 2 m 1 m 2 U X. For reconstruction of the motion of the second body, it is natural to express the motion of the second body in the inertial frame. Since V 2 = ṘT 2 ẋ2 + R T 2 ẍ2 = S(Ω 2 )V + R T 2 v 2, (51) can be written as m 2 R T 2 v 2 = U X. Equation (52) can be simplified using the property U X XT X U T X = S(X U X ) from (9). The kinematics equations for Ṙ and Ẋ can be derived in a similar way. 16

17 In summary, the continuous equations of relative motion for the full two body problem, in Lagrangian form, can be written as V + Ω 2 V = 1 U m X, (54) (J R Ω) + Ω 2 J R Ω = M, (55) J 2 Ω2 + Ω 2 J 2 Ω 2 = X U + M, (56) X Ẋ + Ω 2 X = V, (57) Ṙ = S(Ω)R S(Ω 2 )R, (58) where m = m 1m 2 m 1 +m 2. The following equations can be used for reconstruction of the motion of the second body in the inertial frame: v 2 = 1 U R 2 m 2 X, (59) ẋ 2 = v 2, (60) R 2 = R 2 S(Ω 2 ). (61) These equations are equivalent to those given in [12]. However, (59) is not given in [12]. Equations (54) though (61) give a complete set of equations for the reduced dynamics and reconstruction. Furthermore, they are derived systematically in the context of geometric mechanics using proper variational formulas given in (42) through (47). This result can be readily generalized for n bodies Equations of motion: Hamiltonian form Define the linear momenta Γ,γ 2 R 3, and the angular momenta Π,Π 2 R 3 as Γ = mv, γ 2 = mv 2, Π = J R Ω = RJ 1 Ω 1, Π 2 = J 2 Ω 2. Then, the equations of motion can be rewritten in terms of these momenta variables. The continuous equations of relative motion for the full two body problem, in Hamiltonian form, can be written as Γ + Ω 2 Γ = U X, (62) Π + Ω 2 Π = M, (63) Π 2 + Ω 2 Π 2 = X U X + M, (64) Ẋ + Ω 2 X = Γ m, (65) Ṙ = S(Ω)R S(Ω 2 )R, (66) 17

18 where m = m 1m 2 m 1 +m 2. The following equations can be used to reconstruct the motion of the second body in the inertial frame: U γ 2 = R 2 X, (67) ẋ 2 = γ 2, (68) m 2 R 2 = R 2 S(Ω 2 ). (69) 4 Variational integrators A variational integrator discretizes Hamilton s principle rather than the continuous equations of motion. Taking variations of the discretization of the action integral leads to the discrete Euler-Lagrange or discrete Hamilton s equations. The discrete Euler-Lagrange equations can be interpreted as a discrete Lagrangian map that updates the variables in the configuration space, which are the positions and the attitudes of the bodies. A discrete Legendre transformation relates the configuration variables with the linear and angular momenta variables, and yields a discrete Hamiltonian map, which is equivalent to the discrete Lagrangian map. In this section, we derive both a Lagrangian and Hamiltonian form of variational integrators for the full body problem in inertial and relative coordinates. The second level subscript k denotes the value of variables at t = kh + t 0 for an integration step size h R and an integer k. The integer N satisfies t f = kn + t 0, so N is the number of time-steps of length h to go from the initial time t 0 to the final time t f. 4.1 Inertial coordinates Discrete Lagrangian: In continuous time, the structure of the kinematics equations (28), (58) and (61) ensure that R i, R and R 2 evolve on SO(3) automatically. Here, we introduce a new variable F ik SO(3) defined such that R ik+1 = R ik F ik, i.e. F ik = R T i k R ik+1. (70) Thus, F ik represents the relative attitude between two integration steps, and by requiring that F ik SO(3), we guarantee that R ik evolves in SO(3). Using the kinematic equation Ṙi = R i S(Ω i ), the skew-symmetric matrix S(Ω k ) can be approximated as S(Ω ik ) = R T i k Ṙ ik R T i k R ik+1 R ik h = 1 h (F i k I 3 3 ). (71) The velocity, ẋ ik can be approximated simply by (x ik+1 x ik )/h. Using these approximations of the angular and linear velocity, the kinetic energy of the ith body given in (12) 18

19 can be approximated as ( 1 T i (ẋ i,ω i ) T i h (x i k+1 x ik ), 1 ) h (F i k I 3 3 ), = 1 2h 2m i xik+1 x ik h 2tr[ (F ik I 3 3 )J di (F ik I 3 3 ) T], = 1 2h 2m i x ik+1 x ik h 2tr[(I 3 3 F ik )J di ], where (5) is used. A discrete Lagrangian L d (x k,x k+1,r k,f k ) is constructed such that it approximates a segment of the action integral (21), L d = h 2 L ( x k, 1 h (x k+1 x k ),R k, 1 h (F k I) ) = + h 2 L ( x k+1, 1 h (x k+1 x k ),R k+1, 1 h (F k I) ), n 1 2h m i xik+1 x ik h tr[(i 3 3 F ik )J di ] i=1 h 2 U(x,R ) h k k 2 U(x,R ), (72) k+1 k+1 where x k (R 3 ) n, R k SO(3) n, and F k (R 3 ) n, and I (R 3 3 ) n are defined as x k = (x 1k,x 2k,,x nk ), R k = (R 1k,R 2k,,R nk ), F k = (F 1k,F 2k,,F nk ), and I = (I 3 3,I 3 3,,I 3 3 ), respectively. This discrete Lagrangian is self-adjoint [4], and self-adjoint numerical integration methods have even order, so we are guaranteed that the resulting integration method is at least second-order accurate. Variations of discrete variables: The variations of the discrete variables are chosen to respect the geometry of the configuration space SE(3). The variation of x ik is given by x ǫ i k = x ik + ǫδx ik + O(ǫ 2 ), where δx ik R 3 and vanishes at k = 0 and k = N. The variation of R ik is given by δr ik = R ik η ik, (73) where η ik so(3) is a variation represented by a skew-symmetric matrix and vanishes at k = 0 and k = N. The variation of F ik can be computed from the definition F ik = R T i k R ik+1 to give δf ik = δr T i k R ik+1 + R T i k δr ik+1, = η ik R T i k R ik+1 + R T i k R ik+1 η ik+1, = η ik F ik + F ik η ik+1. (74) 19

20 4.1.1 Discrete equations of motion: Lagrangian form To obtain the discrete equations of motion in Lagrangian form, we compute the variation of the discrete Lagrangian from (19), (73) and (74), to give δl d = n 1 h m i(x ik+1 x ik ) T (δx ik+1 δx ik ) + 1 h tr[( ) ] η ik F ik F ik η ik+1 Jdi i=1 ( h T U k δx ik + U ) T k+1 δx ik+1 + h [ ] 2 x ik x ik+1 2 tr η ik Ri T U k k + η ik+1 R T U k+1 i R k+1, ik R ik+1 (75) where U k = U(x k,r k ) denotes the value of the potential at t = kh + t 0. Define the action sum as G d = N 1 k=0 L d (x k,x k+1,r k,f k ). (76) The discrete action sum G d approximates the action integral (21), because the discrete Lagrangian approximates a segment of the action integral. Substituting (75) into (76), the variation of the action sum is given by δg d = N 1 k=0 i=1 n { 1 δx T i k+1 h m i(x ik+1 x ik ) h } U k+1 2 x ik+1 { + δx T i k 1 h m i(x ik+1 x ik ) h } U k 2 x { ik + tr [η ik+1 1 h J d i F ik + h }] U 2 RT k+1 i k+1 R ik+1 [ { 1 + tr η ik h F i k J di + h }] U 2 RT k i k. R ik Using the fact that δx ik and η ik vanish at k = 0 and k = N, we can reindex the summation, which is the discrete analogue of integration by parts, to yield δg d = N 1 k=1 i=1 n { } 1 δx ik h m ( ) U i xik+1 2x ik + x ik 1 + h k x ik [ { 1 ( ) + tr η ik Fik J di J di F ik 1 + hr T U k h ik R ik }]. Hamilton s principle states that δg d should be zero for all possible variations δx ik R 3 and η ik so(3) that vanish at the endpoints. Therefore, the expression in the first brace should be zero, and since η ik is skew-symmetric, the expression in the second brace should be symmetric. Thus, we obtain the discrete equations of motion for the full body problem, in Lagrangian form, for i (1,2,,n) as 1 ( ) U xik+1 2x ik + x ik 1 = h k, (77) h x ik 20

21 1 ( ) F ik+1 J di J di Fi T h k+1 J di F ik + Fi T k J di = hs(m ik+1 ), (78) R ik+1 = R ik F ik, (79) where M ik R 3 is defined in (24) as M ik = r i1 u ri1 + r i2 u ri2 + r i3 u ri3, (80) where r ip,u rip R 1 3 are pth row vectors of R ik and U k R ik, respectively. Given the initial conditions (x i0,r i0,x i1,r i1 ), we can obtain x i2 from (77). Then, F i0 is computed from (79), and F i1 can be obtained by solving the implicit equation (78). Finally, R i2 is found from (79). This yields an update map (x i0,r i0,x i1,r i1 ) (x i1,r i1,x i2,r i2 ), and this process can be repeated Discrete equations of motion: Hamiltonian form As discussed above, equations (77) through (79) defines a discrete Lagrangian map that updates x ik and R ik. The discrete Legendre transformation given in (3) and (4) relates the configuration variables x ik, R ik and the corresponding momenta. This induces a discrete Hamiltonian map that is equivalent to the discrete Lagrangian map. The discrete Hamiltonian map is particularly convenient if the initial conditions are given in terms of the positions and momenta at the initial time, (x i0,v i0,r i0,ω i0 ). Before deriving the variational integrator in Hamiltonian form, consider the momenta conjugate to x i and R i, namely P vi R 3 and P Ωi R 3 3. From the definition (1), F vi L is obtained by taking the derivative of L, given in (15), with ẋ i while holding other variables fixed. δẋ T i P vi = F vi L(x,ẋ,R,Ω), = d dǫ L(x,ẋ + ǫδẋ i,r,ω), ǫ=0 = d dǫ T i (ẋ i + ǫδẋ i,ω i ), ǫ=0 = δẋ T i (m iẋ i ), where δẋ i (R 3 ) n denotes (0,0,,δẋ i,,0), and T i is given in (12). Then, we obtain which is equal to the linear momentum of B i. Similarly, tr [ S(δΩ i ) T ] P Ωi = FΩi L(x,ẋ,R,Ω), = d dǫ T i (ẋ i,ω i + ǫδω i ), ǫ=0 P vi = m i v i = γ i, (81) = 1 2 tr[ S(δΩ i )J di S(Ω i ) T + S(Ω i )J di S(δΩ i ) T], = 1 2 tr[ S(δΩ i ) T S(J i Ω i ) ], 21

22 where (5) and (13) are used. Now, we obtain { tr [S(δΩ i ) T P Ωi 1 }] 2 S(J iω i ) = 0. Since S(Ω i ) is skew-symmetric, the expression in the braces should be symmetric. This implies that P Ωi P T Ω i = S(J i Ω i ) = S(Π i ). (82) Equations (81) and (82) give expressions for the momenta conjugate to x i and R i. Consider the discrete Legendre transformations given in (3) and (4). Then, δx T i k D xi,k L d (x k,x k+1,r k,f k ) = d dǫ L d (x k + ǫδx ik,x k+1,r k,f k ), ǫ=0 = δx T i k [ 1 h m i(x ik+1 x ik ) + h 2 where δx ik (R 3 ) n denotes (0,0,,δx ik,,0). Therefore, we have U k x ik ], (83) D xi,k L d (x k,x k+1,r k,f k ) = 1 h m i(x ik+1 x ik ) h 2 U k x ik. (84) From the discrete Legendre transformation given in (3), P vi,k = D xi,k L d. Using (81) and (84), we obtain γ ik = 1 h m i(x ik+1 x ik ) + h 2 U k x ik. (85) Using the discrete Legendre transformation given in (4), P vi,k+1 = D xi,k+1 L d, we can derive the following equation similarly: γ ik+1 = 1 h m i(x ik+1 x ik ) h 2 U k+1 x ik+1. (86) Equations (85) and (86) define the variational integrator in Hamiltonian form for the translational motion. Now, consider the rotational motion. We have tr [ T η ik D Ri,k L ] [ { 1 d = tr η ik h F i k J di + h }] U 2 RT k i k, (87) R ik where the right side is obtained by taking the variation of L d with respect to R ik, while holding other variables fixed. Since η ik is skew-symmetric, D Ri,k L d + D Ri,k L d T = 1 h where M ii R 3 is defined in (80). ( Fik J di J di F T i k ) h 2 S(M i k ), (88) 22

23 From the discrete Legendre transformation given in (3), P Ωi,k = D Ri,k L d, we obtain the following equation by using (82) and (88), S(Π ik ) = 1 h ( Fik J di J di F T i k ) h 2 S(M i k ). (89) Using the discrete Legendre transformation given in (4), P Ωi,k+1 = D Ri,k+1 L d, we can obtain the following equation: S(Π ik+1 ) = 1 h F T i k ( Fik J di J di F T i k ) Fik + h 2 S(M i k+1 ). (90) By using (10) and substituting (89), we can reduce (90) to the following equation in vector form. Π ik+1 = F T i k Π ik + h 2 F T i k M ik + h 2 M i k+1. (91) Equations (89) and (91) define the variational integrator in Hamiltonian form for the rotational motion. In summary, using (85), (86), (89) and (91), the discrete equations of motion for the full body problem, in Hamiltonian form, can be written for i (1,2,,n) as x ik+1 = x ik + h γ ik h2 U k, m i 2m i x ik (92) γ ik+1 = γ ik h U k h U k+1, 2 x ik 2 x ik+1 (93) hs(π ik + h 2 M i k ) = F ik J di J di F T i k, (94) Π ik+1 = F T i k Π ik + h 2 F T i k M ik + h 2 M i k+1., (95) R ik+1 = R ik F ik. (96) Given (x i0,γ i0,r i0,π i0 ), we can find x i1 from (92). Solving the implicit equation (94) yields F i0, and R i1 is computed from (96). Then, (93) and (95) gives γ i1, and Π i1. This defines the discrete Hamiltonian map, (x i0,γ i0,r i0,π i0 ) (x i1,γ i1,r i1,π i1 ), and this process can be repeated. 4.2 Relative coordinates In this section, we derive the variational integrator for the full two body problem in relative coordinates by following the procedure given before. This result can be readily generalized to n bodies. Reduction of discrete variables: The discrete reduced variables are defined in the same way as the continuous reduced variables, which are given in (34) through (39). We introduce F k SO(3) such that R k+1 = R T 2 k+1 R 1k+1 = F T 2 k F k R k. i.e. F k = R k F 1k R T k. (97) 23

24 Discrete reduced Lagrangian: The discrete reduced Lagrangian is obtained by expressing the original discrete Lagrangian given in (72) in terms of the discrete reduced variables. From the definition of the discrete reduced variables given in (34) and (38), we have x 1k+1 x 1k = R 2k+1 (X k+1 + X 2k+1 ) R 2k (X k + X 2k ), = R 2k { F2k (X k+1 + X 2k+1 ) (X k + X 2k ) }, (98) x 2k+1 x 2k = R 2k { F2k X 2k+1 X 2k }. (99) From (71), S(Ω 1k ) and S(Ω 2k ) are expressed as S(Ω 1k ) = 1 h (F 1 k I 3 3 ), = 1 h RT k (F k I 3 3) R k, (100) S(Ω 2k ) = 1 h (F 2 k I 3 3 ). (101) Substituting (98) through (101) into (72), we obtain the discrete reduced Lagrangian. l dk = l d (X k,x k+1,x 2k,X 2k+1,R k,f k,f 2k ) = 1 2h m 1 F2k (X k+1 + X 2k+1 ) (X k + X 2k ) h m 2 F2k X 2k+1 X 2k h tr[(i 3 3 F k )J drk ] + 1 h tr[(i 3 3 F 2k )J d2 ] h 2 U(X k,r k ) h 2 U(X k+1,r k+1 ), (102) where J drk R 3 3 is defined to be J drk = R k J d1 R T, which gives the nonstandard moment k of inertia matrix of the first body with respect to the second body fixed frame at t = kh+t 0. Variations of discrete reduced variables: The variations of the discrete reduced variables can be derived from those of the original variables. The variations of R k, X k, and F 2k are the same as given in (42), (43), and (74), respectively. The variation of F k is computed in A.2. In summary, the variations of discrete reduced variables are given by δr k = η k R k η 2k R k, (103) δx k = χ k η 2k X k, (104) δf k = η 2k F k + F 2k η k+1 F T 2 k F k + F k ( η k + η 2k ), (105) δx 2k = χ 2k η 2k X 2k, (106) δf 2k = η 2k F 2k + F 2k η 2k+1. (107) These Lie group variations are the main elements required to derive the variational integrator equations Discrete equations of motion: Lagrangian form As before, we can obtain the discrete equations of motion in Lagrangian form by computing the variation of the discrete reduced Lagrangian which, by using (103) through (107), is 24

25 given by δl dk = 1 h χt k h χt k [ m1 (X k+1 + X 2k+1 ) m 1 F T 2 k (X k + X 2k ) ] [ m1 (X k + X 2k ) m 1 F 2k (X k+1 + X 2k+1 )] + 1 h χt 2 k+1 [ m1 (X k+1 + X 2k+1 ) m 1 F T 2 k (X k + X 2k ) + m 2 X 2k+1 m 2 F T 2 k X 2k ] + 1 h χt 2 k [ m1 (X k + X 2k ) m 1 F 2k (X k+1 + X 2k+1 ) + m 2 X 2k m 2 F 2k X 2k+1 ] 1 h tr[ η k+1 F2 T ] 1 k F k J drk F 2k + h tr[η F J k k dr k ] 1 h tr[ ] 1 η 2k+1 J d2 F 2k + h tr[η 2 k F 2k J d2 ] h U 2 χt k + h [ T k X k 2 tr U ] [ η 2k X k k h U X k 2 χt k+1 + h k+1 X k+1 2 tr U ] T η 2k+1 X k+1 k+1 X k+1 + h [ T T 2 tr U η 2k R k U ] [ k η R k R k k + h T k R k 2 tr U η 2k+1 R k+1 U ] T k+1 η R k+1 R k+1 k+1. k+1 R k+1 (108) The action sum expressed in terms of the discrete reduced Lagrangian has the form G d = N 1 k=0 l d (X k,x k+1,x 2k,X 2k+1,R k,f k,f 2k ). (109) The discrete action sum G d approximates the action integral (49), because the discrete Lagrangian approximates a piece of the integral. Using the fact that the variations χ k,χ 2k,η k,η 2k vanish at k = 0 and k = N, the variation of the discrete action sum can be expressed as N 1 { 1 δg d = h χt m k 1 F2 T k 1 (X k 1 + X 2k 1 ) + 2m 1 (X k + X 2k ) k=1 m 1 F 2k (X k+1 + X 2k+1 ) h 2 U } k X k N 1 { 1 + h χt 2 k m 1 F2 T k 1 (X k 1 + X 2k 1 ) + 2m 1 (X k + X 2k ) m 1 F 2k (X k+1 + X 2k+1 ) k=1 } m 2 F2 T k 1 X 2k 1 + 2m 2 X 2k m 2 F 2k X 2k+1 N 1 + k=1 N 1 + k=1 [ { 1 ( ) tr η k F2 T h k 1 F k 1 R k 1 J d1 R T F k 1 2 k 1 + F k R k J d1 R T U hr k k k R k [ { 1 ( tr η 2k Jd2 F h + F ) T T U 2k 1 2 k J d2 + k U }] hxk + hr k X k. k R k From Hamilton s principle, δg d should be zero for all possible variations χ k,χ 2k R 3 and η k,η 2k so(3) which vanish at the endpoints. Therefore, in the above equation, the expressions in the first two braces should be zero, and the expressions in the last two braces should be symmetric since η k,η 2k are skew-symmetric. After some simplification, we obtain T }] 25

26 the discrete equations of relative motion for the full two body problem, in Lagrangian form, as F 2k X k+1 2X k + F2 T k 1 X k 1 = h2 U k, (110) m X k F k+1 J drk+1 J drk+1 F T = F T ( k+1 2 Fk k J drk J drk F T ) F2k h 2 S(M k k+1 ), (111) F 2k+1 J d2 J d2 F2 T k+1 = F2 T ( F2k k J d2 J d2 F2 T ) F2k k + h 2 X k+1 U + h 2 S(M k+1 ), X k+1 (112) R k+1 = F T 2 k F k R k, (113) R 2k+1 = R 2k F 2k. (114) It is natural to express equations of motion for the second body in the inertial frame. x 2k+1 2x 2k + x 2k 1 = h2 m 2 R k U k X k. (115) Given (X 0,R 0,R 20,X 1,R 1,R 21 ), we can determine F 0 and F 20 from (113) and (114). Solving the implicit equations (111) and (112) gives F 1 and F 21. Then X 2, R 2 and R 22 are found from (110), (113) and (114), respectively. This yields the discrete Lagrangian map (X 0,R 0,R 20,X 1,R 1,R 21 ) (X 1,R 1,R 21,X 2,R 2,R 22 ) and this process can be repeated. We can separately reconstruct x 2k using (115) Discrete equations of motion: Hamiltonian form Using the discrete Legendre transformation, we can obtain the Hamiltonian map, in terms of reduced variables, that is equivalent to the Lagrangian map given in (110) through (115). We will only sketch the procedure as it is analogous to the approach of the previous section. First, we find expressions for the conjugate momenta variables corresponding to (81) and (82). We compute the discrete Legendre transformation by taking the variation of the discrete reduced Lagrangian as in (83) and (87). Then, we obtain the discrete equations of motion in Hamiltonian form using (3) and (4). The discrete equations of relative motion for the full two body problem, in Hamiltonian form, can be written as ( X k+1 = F2 T k X k + h Γ k m h2 2m ) U k Γ k+1 = F T 2 k ( Γ k h 2 X k Π k+1 = F2 T k (Π k h 2 M k ( Π 2k+1 = F2 T k Π k + h 2 X U + h k X k 2 M k ), (116) U k X k U k+1, (117) X k+1 h 2 ) h 2 M, (118) k+1 ) + h 2 X k+1 U X k+1 + h 2 M k+1, (119) R k+1 = F2 T k F k R k, (120) ( hs Π k h ) 2 M = F k k J drk J drk F T, (121) k 26

27 ( hs Π 2k + h 2 X U + h ) k X k 2 M = F k 2k J d2 J d2 F2 T k. (122) It is natural to express equations of motion for the second body in the inertial frame for reconstruction: x 2k+1 = x 2k + h γ 2 k + h2 U R k m 2 2m k, 2 X k (123) γ 2k+1 = γ 2k + h 2 R U k + h k X k 2 R U k+1, k+1 X k+1 (124) R 2k+1 = R 2k F 2k. (125) Given (R 0,X 0,Π 0,Γ 0,Π 20 ), we can determine F 0 and F 20 by solving the implicit equations (121) and (122). Then, X 1 and R 1 are found from (116) and (120), respectively. After that, we can compute Γ 1, Π 1, and Π 21 from (117), (118) and (119). This yields a discrete Hamiltonian map (R 0,X 0,Π 0,Γ 0,Π 20 ) (R 1,X 1,Π 1,Γ 1,Π 21 ), and this process can be repeated. x 2k, γ 2k and R 2k can be updated separately using (123), (124) and (125), respectively, for reconstruction. 4.3 Numerical considerations Properties of the variational integrators: Variational integrators exhibit a discrete analogue of Noether s theorem [17], and symmetries of the discrete Lagrangian result in conservation of the corresponding momentum maps. Our choice of discrete Lagrangian is such that it inherits the symmetries of the continuous Lagrangian. Therefore, all the conserved momenta in the continuous dynamics are preserved by the discrete dynamics. The proposed variational integrators are expressed in terms of Lie group computations [5]. During each integration step, F ik SO(3) is obtained by solving an implicit equation, and R ik is updated by multiplication with F ik. Since SO(3) is closed under matrix multiplication, the attitude matrix R ik+1 remains in SO(3). We make this more explicit in section 4.4 by expressing F ik as the exponential function of an element of the Lie algebra so(3). An adjoint integration method is the inverse map of the original method with reversed time-step. An integration method is called self-adjoint or symmetric if it is identical with its adjoint; a self-adjoint method always has even order. Our discrete Lagrangian is chosen to be self-adjoint, and therefore the corresponding variational integrators are second-order accurate. Higher-order methods: While the numerical methods we present in this paper are secondorder, it is possible to apply the symmetric composition methods, introduced in [26], to construct higher-order versions of the Lie group variational integrators introduced here. Given a basic numerical method represented by the flow map Φ h, the composition method is obtained by applying the basic method using different step sizes, Ψ h = Φ λsh... Φ λ1 h, 27