LIE GROUP VARIATIONAL INTEGRATORS FOR RIGID BODY PROBLEMS USING QUATERNIONS XUEFENG SHEN AND MELVIN LEOK Abstract. Rigid body dynamics on the rotation group have typically been represented in terms of rotation matrices, unit quaternions, or local coordi- nates, such as Euler angles. Due to the coordinate singularities associated with local coordinate charts, it is common in engineering applications to adopt the unit quaternion representation, and the numerical simulations typically impose the unit length condition using constraints or by normalization at each step. From the perspective of geometric structure-preserving, such approaches are undesirable as they are either computationally less efficient, or interfere with the preservation of other geometric properties of the dynamics. In this paper, we adopt the approach used in constructing Lie group varia- tional integrators for rigid body dynamics on the rotation group to the repre- sentation in terms of unit quaternions. In particular, the rigid body dynamics is lifted to unit quaternions, and the Lie group structure of unit quaternions is used to represent tangent vector intrinsically, thereby avoiding the use of a Lagrange multiplier. A Lie group variational integrator in the unit quaternion representation is derived, and numerical results are presented. 1. Introduction For Lagrange mechanics on the tangent bundle TQ and the Lagrangian L, the theory of variational integrators is well-established [7], and is based on discretizing Hamilton's principle rather than the differential equations of motion. The discrete Lagrangian Ld : Q × Q ! R approximates the action integral over a small time interval, Z h Ld(q0; q1) ≈ ext L(q(t); q_(t))dt; q(0)=q0;q(h)=q1 0 and the discrete Hamilton's principle states that the discrete action sum is station- ary with respect to variations in the discrete solution curve that fixes the endpoints, N−1 X δ Ld(qk; qk+1) = 0; k=0 and this yields the discrete Euler Lagrange equations (1) D2Ld(q0; q1) + D1(q1; q2) = 0: By introducing discrete Legendre transforms, we obtain a variational integrator on the cotangent bundle T ∗Q, ( p = −D L (q ; q ); (2) 0 1 d 0 1 p1 = D2Ld(q0; q1): 1 2 XUEFENG SHEN AND MELVIN LEOK It is easy to easy that (2) recovers (1) when the momentum variables are eliminated, and that (2) is the usual characterization of a symplectic map in terms of a Type I generating function. As such, variational integrators are automatically symplectic, and either form of the discrete equations can be easily implemented in a linear space or in local coordinates. However, the configuration manifold Q is in general not a Euclidean space. Symplectic integrators do not exactly preserve energy, but instead rely on the existence of a modified energy associated for their long-time energy stability properties. Changing the coordinate chart at each time step results in the modified energy that is preserved changing from step to step, which destroys the long-time energy stability typically associated with symplectic integrators. In order to overcome this limitation, it is necessary to consider symplectic integrators that is are well-defined globally on the configuration manifold, and which commute with a change of coordinates. One way to address this issue is to embed Q in Euclidean space. In many cases, the configuration space naturally lives in a higher-dimensional Euclidean space Rn, and is the level set of some constraint function g : Rn ! Rm. So the position and velocity are represented as vectors in the embedding space, and Lagrange multipliers are introduced to represent the constraint forces. When configuration space Q has a high codimension in the Euclidean space, and the constraint function is complicated, the above method is costly to implement. For example, the rotation group SO(3), which is a three-dimensional Lie group, naturally lives in R3×3, a nine-dimensional Euclidean space, with six orthonormality constraints. As opposed to viewing Q extrinsically as an embedded submanifold, when Q is a Lie group, the group structure could be utilized to represent the position and tangent space intrinsically, thus avoiding the introduction of Lagrange multipliers. This idea was used in [4] for the rigid body problem, on configuration space SO(3), and later generalized to the full body problem [5]. Besides the rotation group SO(3), the motion of rigid bodies can also be de- scribed by unit quaternions, which is an extremely popular approach in classical mechanics [2], computer graphics [1, 11], virtual reality [3], and engineering appli- cations in control [10, 13] and estimation [8, 9] of aerial and space vehicles. In [12], a variational integrator for the rigid body problem was derived in terms of unit quaternions embedded in R4, so as to avoid the complicated constraints in SO(3), but where the unit quaternion constraint is enforced using a Lagrange multiplier. We remark that the unit quaternions S3, is not only the unit sphere in R4, but also a Lie group under quaternion multiplication. In this paper, we will derive a Lie group variational integrator for rigid body problems using unit quaternions, but without the use of unit length constraints. Instead, we use the fact that R3 can be identified with the Lie algebra associated with S3 ⊂ H. This will allow us to obtain a variational integrator for rigid body dynamics in the unit quaternion representation that is globally valid, expresses the relative rotation in terms of three degrees-of-freedom, and avoids the need for ex- plicit constraints. In particular, it yields a numerical method in the unit quaternion representation that exhibits the computational advantages of Lie group variational integrators developed for rotation matrices, thereby making the advantages of geo- metric integrators accessible to engineering applications which are heavily invested in the unit quaternion representation of the rotation group. LIE GROUP VARIATIONAL INTEGRATORS USING QUATERNIONS 3 2. Background 2.1. Rigid body problem. The configuration space of the rigid body is SE(3) = R3 o SO(3), where (x; R) 2 SE(3) describes the position and orientation of the body-fixed frame with respect to an inertial frame. Let ρ 2 R3 denotes the position of mass element in the body-fixed frame, the corresponding inertial position is x + Rρ, and velocityx _ + Rρ_ , then the kinetic energy is given by 1 Z T = kx_ + Rρ_ k2dm 2 B 1 Z Z 1 Z (3) = kx_k2dm + x_ TRρdm_ + kRρ_ k2dm 2 B B 2 B 1 1 = mkx_k2 + tr[RJ_ R_ T]; 2 2 d R T where Jd = B ρρ dm is the nonstandard moment of inertia matrix. In (3), we used the fact that the origin of body-fixed frame is the center of mass of the body, thus Z ρdm = 0: B Since R evolves on SO(3), so RTR = I. By doing differentiation on both sides, we have R_ TR + RTR_ = 0, thus RTR_ is skew symmetric. Define the hat map 3 3×3 3 b· : R ! R by the condition that xyb = x×y for all x; y 2 R . If x = (x1; x2; x3), then xb is given by 0 1 0 −x3 x2 xb = @ x3 0 −x1A ; −x2 x1 0 which is a skew-symmetric matrix. In particular, the hat map defines an isomor- phism between R3 and skew-symmetric matrices. Since RTR_ is skew-symmetric, there exists a unique vector Ω such that RTR_ = Ω.b In other words, (4) R_ = RΩb; where Ω is the angular velocity in the body-fixed frame. This equation has a natural geometric interpretation: SO(3) is a Lie group under matrix multiplication, and its Lie algebra, so(3), consists of all skew-symmetric matrices, 3 so(3) = fΩb j Ω 2 R g: The tangent space at each R can be represented as the pushforward by the left action of R on the Lie algebra, TRSO(3) = (LR)∗so(3): In this way, the tangent bundle T SO(3) is diffeomorphic to SO(3) × so(3) by left- trivialization. With this, we can express the rotational term in the kinetic energy as follows, _ _ T T T tr[RJdR ] = tr[RΩbJdΩb R ] T (5) = tr[ΩbJdΩb ] = ΩTJΩ; 4 XUEFENG SHEN AND MELVIN LEOK where J = tr[Jd]I3 − Jd is standard moment of inertia matrix. Combining (3) and 1 2 1 T (5), we obtain T = 2 mkx_k + 2 Ω JΩ, and the Lagrangian for rigid body problems L : T SE(3) ! R is given by, 1 1 L(x; R; x;_ Ω) = T − V = mkx_k2 + ΩTJΩ − V (x; R): 2 2 2.2. Unit quaternions. Quaternions are a noncommutative division algebra, usu- 3 ally denoted as H, with element q = (qs; ~qv), where qs is the scalar part and ~qv 2 R is the vector part. H inherits the vector space structure and differential structure 4 from R , and has the following multiplication multiplication: given q = (qs; ~qv), p = (ps; ~pv), q · p = (qs · ps − ~qv · ~pv; qs · ~pv + ps · ~qv + ~qv × ~pv): p ∗ ∗ We can also define conjugatation q = (qs; −~qv), norm kqk = qq , and inverse −1 q∗ under quaternion multiplication q = kqk2 . The set of unit quaternions 3 S = fq 2 H j kqk = 1g is diffeomorphic to the unit sphere in R4, and forms a Lie group under quaternion multiplication. The identity element is e = (1; 0; 0; 0), and the Lie algebra 3 3 TeS = f(0; ξ) j ξ 2 R g: The corresponding exponential map is ξ (6) exp(ξ) = cos(jξj); sin(jξj) : jξj 3 For any q0 2 S , the map ξ 7! q0 · exp(ξ) gives a local diffeomorphism between a 3 neighborhood of the origin of R and a neighborhood of q0.