Dynamics of a 3D Elastic String Pendulum

Joint 48th IEEE Conference on Decision and Control and ThA06.5 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009

Taeyoung Lee, Melvin Leok∗, and N. Harris McClamroch†

Abstract— This paper presents an analytical model and a We show that the governing equations of motion can be geometric numerical integrator for a rigid body connected to an developed according to Hamilton’s variational principle. elastic string, acting under a gravitational potential. Since the The second part of this paper deals with a geometric point where the string is attached to the rigid body is displaced from the center of mass of the rigid body, there exist nonlinear numerical integrator for the 3D elastic string pendulum. Ge- coupling effects between the string deformation and the rigid ometric numerical integration is concerned with developing body dynamics. A geometric numerical integrator, referred to as numerical integrators that preserve geometric features of a a Lie group variational integrator, is developed to numerically system, such as invariants, symmetry, and reversibility [9]. preserve the Hamiltonian structure of the presented model A geometric numerical integrator, referred to as a Lie group and its Lie group configuration manifold. These properties are illustrated by a numerical simulation. variational integrator, has been developed for a Hamiltonian system on an arbitrary Lie group in [10]. I.INTRODUCTION A 3D elastic string pendulum is a Hamiltonian system, and its configuration manifold is expressed as the product of the The dynamics of a body connected to a string are of special orthogonal group SO(3) and the space of connected relevance in several engineering problems such as cable curve segments on R3. This paper develops a Lie group cranes, towed underwater vehicles, and tethered spacecraft. variational integrator for a 3D elastic string pendulum based It has been shown that gravitational forces acting along a on the results presented in [10]. The proposed geometric string can alter the tension of the string and significantly numerical integrator preserves symplecticity and momentum influence the connected rigid body dynamics [1]. Therefore, maps, and exhibits desirable energy conservation properties. it is important to accurately model the string dynamics, the It also respects the Lie group structure of the configuration dynamics of the body, and their interaction. manifold, and avoids the singularities and computational Several dynamic and numerical models have been devel- complexities associated with the use of local coordinates. oped. Lumped mass models, where the string is spatially In summary, this paper develops an analytical model and discretized into connected point masses, were developed a geometric numerical integrator for a 3D elastic string pen- in [2], [3], [4]. Finite difference methods in both the spatial dulum. These provide a mathematical model and a reliable domain and the time domain were applied in [5], [6]. Finite numerical simulation tool that characterizes the nonlinear element discretizations of the weak form of the equations of coupling between the string dynamics and the rigid body motion were applied in [6], [7]. dynamics accurately. It can be used to study non-local, large The goal of this paper is to develop an analytical model maneuvers of the 3D elastic string pendulum accurately over and a numerical simulation tool for a rigid body connected to a long time period. a string acting under a gravitational potential. This dynamic model is referred to as a 3D elastic string pendulum. It II. 3D ELASTIC STRING PENDULUM generalizes the 3D pendulum model introduced in [8] to include the effects of string deformations, and extends the Consider a rigid body that is attached to an elastic string. model of a string pendulum with a point mass bob [6]. The other end of the string is fixed to a pivot point. We This paper provides a realistic and accurate analytical assume that the rigid body can freely translate and rotate in model for the 3D elastic string pendulum. We assume that a three dimensional space, and the string is extensible and the point where the string is attached to the rigid body flexible. The bending stiffness of the string is not considered is displaced from the center of mass of the rigid body as the diameter of the string is assumed to be negligible so that there exist nonlinear coupling effects between the compared to its length. The point where the string is attached string deformation dynamics and the rigid body dynamics. to the rigid body is displaced from the center of mass of the rigid body so that the dynamics of the rigid body is coupled Taeyoung Lee, Mechanical and Aerospace Engineering, Florida Institute to the string deformations and displacements. This is referred of Technology, Melbourne, FL 39201 [email protected] to as a 3D elastic string pendulum. Melvin Leok, Mathematics, Purdue University, West Lafayette, IN 47907 [email protected] We choose a global reference frame and a body-fixed N. Harris McClamroch, Aerospace Engineering, University of Michigan, frame. The origin of the body-fixed frame is located at the Ann Arbor, MI 48109 [email protected] end of the string where the string is attached to the rigid body. ∗This research has been supported in part by NSF under grants DMS- 0714223, DMS-0726263, DMS-0747659. Since the string is extensible, we need to distinguish between †This research has been supported in part by NSF under grant CMS- the arc length for the stretched deformed configuration and 0555797. the arc length for the unstretched reference configuration.

A. Lagrangian Kinetic energy: The total kinetic energy is composed of the kinetic energy of the string Tstr and the kinetic energy s(s, t) of the rigid body Trb. Let r˙(s, t) be the partial derivative of s r(s, t) with respect to t. The kinetic energy of the string is r(s, t) Z l 1 2 P Tstr = µ kr˙(s)k ds. (2) 0 2 P Let ρ ∈ R3 be the vector from the center of mass of the rigid R(t) body to a mass element of the rigid body represented in the body fixed frame. The location of the mass element is given ρc ρc by r(l)+R(ρc +ρ) in the global reference frame. Therefore, the kinetic energy of the rigid body can be written as Reference configuration Deformed configuration Z 1 2 Trb = kr˙(l) + RΩ(ˆ ρc + ρ)k dM(ρ) Fig. 1. 3D Elastic String Pendulum B 2 1 1 = Mr˙(l) · r˙(l) + Ω · JΩ + Mr˙(l) · RΩˆρ , (3) 2 2 c Define where B denotes the region enclosed by the rigid body R l ∈ R Total length of the unstretched string surface, and we used the following properties: B ρ dM = 0; R ∧ 2 s ∈ [0, l] Length of the string from the pivot to a ma- xyˆ = −yxˆ ; J = − B((ρ + ρc) ) dM. terial point P for the unstretched reference Potential Energy: The strain of the string at a material configuration point located at r(s) is given by + s(s, t) ∈ R Length of the string from the pivot to a ∆s(s) − ∆s material point P for the stretched deformed = lim = s0(s) − 1, ∆s→0 ∆s configuration 3 where ()0 denote the partial derivative with respect to . The r(s, t) ∈ R Vector from the pivot to a material point P s in the global reference frame tangent vector at the material point is given by R ∈ SO(3) Rotation matrix from the body-fixed frame ∂r(s) ∂r(s) ∂s r0(s) et = = = . to the reference frame ∂s ∂s ∂s(s) s0(s) Ω ∈ R3 Angular velocity of the rigid body repre- Since this tangent vector has the unit length, we have s0(s) = sented in the body-fixed frame 0 0 3 kr (s)k. Therefore, the strain is given by = kr (s)k − 1. ρc ∈ R Vector from the origin of the body fixed frame to the center of mass of the rigid body The potential energy of the string is composed of the elastic represented in the body fixed frame potential and the gravitational potential: + Z l µ ∈ R Mass density of the string per unit un- 1 0 2 stretched length Vstr = EA(kr (s)k − 1) − µgr(s) · e3 ds, (4) 0 2 M ∈ + Mass of the rigid body R where E and A denote the Young’s modulus and the sectional J ∈ 3×3 Inertia matrix of the rigid body represented R area of the string, respectively, and the unit vector e in the body fixed frame 3 represents the gravity direction. A configuration of this system can be described by the Since the location of the center of mass of the rigid body locations of all the material points of the string, r(s, t) for is r(l) + Rρc in the global reference frame, the gravitational s ∈ [0, l], and the attitude of the rigid body R(t) with respect potential energy of the rigid body is to the reference frame. So, the configuration manifold is ∞ 3 ∞ 3 V = −Mg(r(l) + Rρ ) · e . (5) G = C ([0, l], R ) × SO(3), where C ([0, l], R ) denotes rb c 3 3 the space of smooth connected curve segments on R and From (2)-(5), the Lagrangian of the 3D elastic string 3×3 T SO(3) = {R ∈ R | R R = I, det[R] = 1}. pendulum is given by

III.CONTINUOUS-TIME ANALYTICAL MODEL L = Tstr − Vstr + Trb − Vrb. (6) In this section, we develop continuous-time equations B. Euler-Lagrange Equations t of motion for a 3D elastic string pendulum. The attitude Let the action integral be G = R f L dt. It is composed kinematics equation of the rigid body is given by t0 of two parts, Gstr and Grb, contributed by the string and R˙ = RΩˆ, (1) by the rigid body, respectively. According to the Hamilton’s principle, the variation of the action integral is equal to where the hat map ˆ· : R3 → so(3) is deﬁned by the condition zero for ﬁxed boundary conditions, which yields the Euler- that xyˆ = x × y for any x, y ∈ R3. Lagrange equations of the 3D elastic string pendulum.

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By repeatedly applying integration by parts, the variation IV. LIE GROUP VARIATIONAL INTEGRATOR of Gstr can be written as The Euler-Lagrange equations developed in the previous

Z tf 0 Z l section provide an analytical model for a 3D elastic string kr (l)k − 1 0 h δGstr = −EA r (l) · δr(l) + − µr¨(s) pendulum. However, the standard ﬁnite difference approxi- kr0(l)k t0 0 mations or ﬁnite element approximations of those equations 0 0 kr (s)k − 1 0 i using a general purpose numerical integrator may not pre- + µg e3 + EA r (s) · δr(s) ds dt. (7) kr0(s)k serve the geometric properties of the system accurately [9]. Lie group variational integrators provide a system- (See [6] for details.) atic method of developing geometric numerical integra- Next, we found the variation of Grb. It can be written as tors for Lagrangian/Hamiltonian systems evolving on a Lie

Z tf group [10]. As they are derived from a discrete analogue h ˆ i δGrb = Mr˙(l) + MRΩρc · δr˙(l) + Mge3 · δr(l) of Hamilton’s principle, they preserve symplecticity and the t0 momentum map, and it exhibits good total energy behavior. T + JΩ + MρˆcR r˙(l) · δΩ They also preserve the Lie group structure as they update a + Mr˙(l) · δRΩˆρc + Mge3 · δRρc dt. (8) group element using the group operation. These properties are critical for accurate and efficient simulations of rigid The variation of a rotation matrix can be written as body dynamics [12]. In this section, we develop a Lie group variational inte- d d δR = R = R exp ηˆ = Rηˆ grator for a 3D elastic string pendulum. d =0 d =0 A. Finite Element Model for η ∈ 3 [11]. The corresponding variation of the angular R We discretize the string by one-dimensional line el- velocity is obtained from the kinematics equation (1): N ements. Thus, the unstretched length of each element is l d T ∧ u = N . A natural coordinate ζ ∈ [0, 1] in the a-th element δΩˆ = (R ) R˙ = (η ˙ + Ω × η) . 1 is defined by ζ = (s − u(a − 1)). Let S0,S1 be shape d =0 u functions given by S0(ζ) = 1 − ζ, and S1(ζ) = ζ. The Substituting these into (8) and applying integration by parts, position vectors for the end nodes of the a-th element are we obtain given by rk,a, rk,a+1 when t = kh for a fixed time step h.

Z tf Using this ﬁnite element model, the position vector r(s, t) h 2 i δGrb = −M r¨(l) − RρˆcΩ˙ + RΩˆ ρc − ge3 · δr(l) of a material point in the a-th element is approximated by t0 h T T i r(s, t) = S0(ζ)rk,a + S1(ζ)rk,a+1 ≡ rk,a(ζ). (13) + −JΩ˙ − MρˆcR r¨(l) + MρˆcΩˆR r˙(l) · η˙

h T T i The partial derivative with respect to s is given by + −MρˆcΩˆR r˙(l) + MgρˆcR e3 − ΩˆJΩ · η dt, (9) 0 ∂r(s, t) ∂ζ 1 0 r (s, t) = = (rk,a+1 − rk,a) ≡ r . (14) where we repeatedly use the property: y ·xzˆ =zy ˆ ·x for any ∂ζ ∂s u k,a 3 x, y, z ∈ R . The partial derivative with respect to t is approximated by From (7) and (9), the variation of the action integral is 1 given by δG = δGstr + δGrb, and it is equal to zero for any r˙(s, t) = (S0(ζ)∆rk,a + S1(ζ)∆rk,a+1) ≡ vk,a(ζ), (15) variation according to Hamilton’s principle. This yields the h following Euler-Lagrange equations: where the Delta-operator represents the change over one time step, i.e. ∆rk,a = rk+1,a − rk,a. 0 ∂ kr (s, t)k − 1 0 µr¨(s, t) − µg e3 − EA r (s, t) = 0, B. Discrete-Lagrangian ∂s kr0(s, t)k (10) Using these finite element model, a configuration of the 2 discretized 3D elastic pendulum at t = kh is described M r¨(l, t)−RρˆcΩ˙ + RΩˆ ρc − ge3 by gk = (rk,1, . . . , rk,N+1,Rk), and the corresponding 0 (11) 3 N+1 kr (l, t)k − 1 0 configuration manifold is G = (R ) × SO(3). + EA r (l, t) = 0, kr0(l, t)k We define a discrete-time kinematics equation as follows. 3 T T Define fk = (∆rk,1,..., ∆rk,N+1,Fk) ∈ G for ∆rk,a ∈ JΩ˙ + ΩˆJΩ + MρˆcR r¨(l, t) − MgρˆcR e3 = 0. (12) R and Fk ∈ SO(3) such that gk+1 = gkfk: Conserved quantities: The total energy, given by E = (rk+1,1, . . . , rk+1,N+1,Rk+1) T + V + T + V , is preserved. As the Lagrangian is str str rb rb = ( + ∆ + ∆ ) (16) invariant under the rotation about the gravity direction, the rk,1 rk,1, . . . , rk,N+1 rk,N+1,RkFk . total angular momentum about the gravity direction is con- Therefore, fk represents the relative update between two R l served. It is given by π3 = { 0 µrˆ(s)r ˙(s) ds + Mrˆ(l)(r ˙(l) + integration steps. This ensures that the Lie group structures RΩˆρc) − Mrˆ˙(l)Rρc + RJΩ}· e3. are preserved since gk is updated by the Lie group action.

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A discrete Lagrangian Ld(gk, fk): G × G → R is an gk+1 = gkfk, (21) approximation of the Jacobi solution of the Hamilton–Jacobi equation, which is given by the integral of the Lagrangian where TL : TG → TG is the tangent map of the left along the exact solution of the Euler-Lagrange equations over translation, Df represents the derivative with respect to f, ∗ R h −1 ˙ and Ad : G × g∗ → g∗ is co-Ad operator [13]. Using this a single time step, Ld(gk, fk) ≈ 0 L(˜g(t), g˜ (t)g˜(t)) dt, where g˜(t) : [0, h] → G satisfies Euler-Lagrange equations result, we develop a Lie group variational integrator for a 3D elastic string pendulum on G = ( 3)N+1 × SO(3). with boundary conditions g˜(0) = gk, g˜(h) = gkfk. The R resulting discrete-time Lagrangian system approximates the Derivatives of the discrete Lagrangian: The derivatives Euler-Lagrange equations to the same order of accuracy as of the discrete Lagrangian of the a-th element, given by (17), the discrete Lagrangian approximates the Jacobi solution. with respect to ∆rk,a and ∆rk,a+1 are given by Substituting (13)-(15) into (6), the contribution of the a-th 1 1 h element to the discrete Lagrangian is chosen as follows. D∆r Ld = m(∆rk,a + ∆rk,a+1) + mge3 k,a k,a 3h 2 4 Z 1 1 2 h e Ldk,a = µ kvk,a(ζ)k udζ + ∇Vk+1,a, 0 h 2 Z 1 1 1 1 h h 0 2 D∆r Ld = m(∆rk,a+1 + ∆rk,a) + mge3 − EA( r − 1) − µg rk,a(ζ) · e3 udζ k,a+1 k,a 2 2 k,a 3h 2 4 0 h Z 1 − ∇ e h 1 0 2 Vk+1,a. (22) − EA( rk+1,a − 1) − µg rk+1,a(ζ) · e3 udζ. 2 2 0 2 e kxk−u 3 This is given by where ∇Vk,a = κ kxk x for x = rk,a+1 − rk,a ∈ R . 1 1 Then, from (19), the derivative of the discrete Lagrangian Ld = m∆rk,a · ∆rk,a + m∆rk,a · ∆rk,a+1 with respect to ∆r , for a ∈ {2,...,N}, is given by k,a 6h 6h k,a 1 + m∆rk,a+1 · ∆rk,a+1 D∆rk,a Ldk = D∆rk,a Ldk,a + D∆rk,a Ldk,a−1 6h 1 h = m(∆rk,a−1 + 4∆rk,a + ∆rk,a+1) + mg(2rk,a + 2rk,a+1 + ∆rk,a + ∆rk,a+1) · e3 6 4 h h h e h e 1 2 + + ∇ − ∇ (23) − hκ(kr − r k − u) mge3 Vk+1,a Vk+1,a−1. 4 k,a+1 k,a 2 2 2 1 2 Similarly, the derivative of the discrete Lagrangian with − h krk,a+1 + ∆rk,a+1 − rk,a − ∆rk,ak − u) ), (17) 4 respect to rk,a, for a ∈ {2,...,N}, is given by EA where m = µu, κ = u . So, the contribution of the string PN h e e to the discrete Lagrangian is L = L . The Drk,a Ldk = hmge3 + (∇V + ∇V ) dk,str a=1 dk,a 2 k,a k+1,a contribution of the rigid body to the discrete Lagrangian is h e e chosen as follows. − (∇V + ∇V ). (24) 2 k,a−1 k+1,a−1 1 1 Ld = M∆rk,N+1 · ∆rk,N+1 + tr[(I − Fk)Jd] k,rb 2h h Next, we find the derivatives of the discrete Lagrangian M with respect to ∆rk,N+1 and rk,N+1. They are contributed + ∆rk,N+1 · Rk(Fk − I)ρc h by the N-th string element and the rigid body, and they can h be obtained from (18) and (22) as follows. + Mg (rk,N+1 + Rkρc) · e3 2 1 m 1 h D∆r Ld = (M + )∆rk,N+1 + m∆rk,N + Mg (r + ∆r + R F ρ ) · e , (18) k,N+1 k h 3 6h 2 k,N+1 k,N+1 k k c 3 M h m h e 3×3 + Rk(Fk − I)ρc + (M + )ge3 − ∇V , where Jd ∈ R is a nonstandard inertia matrix defined by h 2 2 2 k+1,N 1 Jd = 2 tr[J] I3×3 − J, as introduced in [11]. (25) From (17), (18), the discrete Lagrangian of the 3D elastic m h h D L = h(M + )ge − ∇V e − ∇V e . string pendulum is as follows. rk,N+1 dk 2 3 2 k,N 2 k+1,N N (26) X L (g , f ) = L (g , f ) + L (g , f ). (19) dk k k dk,a k k dk,rb k k Now, we find the derivatives of the discrete Lagrangian a=1 with respect to F and R . From (18), we have C. Discrete-time Euler-Lagrange Equations k k The following discrete-time Euler-Lagrange equations, re- 1 M DF Ld · δFk = tr[−δFkJd] + ∆rk,N+1 · RkδFkρc ferred to as a Lie group variational integrator, have been de- k k h h veloped for Lagrangian systems on arbitrary Lie groups [10]: h + MgRkδFkρc · e3 ∗ ∗ ∗ 2 T L · D −Ad −1 · (T L · D ) e fk−1 fk−1 Ldk−1 f e fk fk Ldk 1 k (20) ∗ = tr[−δFkJd] + Ak · δFkρc, + TeLgk · Dgk Ldk = 0, h

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M T h T where Ak = h Rk ∆rk,N+1 + 2 MgRk e3. The variation of ˆ 3 Fk can be written as δFk = Fkζk for ζk ∈ R . Therefore, this can be written as ˆ ∗ DFk Ldk · (Fkζk) = (TI LFk · DFk Ldk ) · ζk 1 h ˆ i ˆ = tr −FkζkJd + Ak · Fkζkρc. h By repeatedly applying the following property of the trace operator, tr[AB] = tr[BA] = tr[AT BT ] for any A, B ∈ 3×3 ˆ R , the ﬁrst term can be written as tr[−FkζkJd] = (a) t ∈ [0, 1.25] (b) t ∈ [1.25, 2.5] ˆ ˆ T 1 ˆ T tr[−ζkJdFk] = tr[ζkFk Jd] = − 2 tr[ζk(JdF0 − Fk Jd)]. T 1 Using the property of the hat map, x y = − 2 tr[ˆxyˆ] for any 3 T ∨ x, y ∈ R , this can be further written as ((JdFk −Fk Jd) )· 3 ζk, where the vee map ∨ : so(3) → R is the inverse of the hat map. As y · xzˆ =zy ˆ · x for any x, y, z ∈ R3, the second T ˆ T term can be written as Fk Ak · ζkρc =ρ ˆcFk Ak · ζk. Using these, we obtain

∗ 1 T ∨ T T LF · DF Ld = (JdFk − F Jd) +ρ ˆcF Ak. (27) I k k k h k k ∗ T ∨ (c) t ∈ [2.5, 3.75] (d) t ∈ [3.75, 5] The co-Ad operator on SO(3), AdF T p = F p = (F pFˆ ) 3 ∗ for p ∈ (R ) , yields Fig. 2. Snapshots of a 3D elastic string pendulum maneuver. Strain energy 1 distribution is illustrated by color shading (an animation is available at ∗ ∗ T ∨ http://my.ﬁt.edu/˜taeyoung). AdF T · (TI LFk · DFk Ldk ) = (FkJd − JdFk ) + F[kρcAk. k h (28)

Similarly, we can derive the derivative of the discrete For given (gk−1, fk−1), gk is explicitly computed by (33) Lagrangian with respect to Rk as follows. and (34). The update fk is computed by a fixed point iteration for Fk: we select an initial guess of Fk; ∆rk,a is obtained ∗ M ∧ T T LR · DR Ld = ((Fk − I)ρc) R ∆rk,N+1 by solving (30) and (31), which requires the inversion of a I k k k h k h h fixed 3N ×3N matrix; a new Fk is computed by solving the + Mgρˆ RT e + MgF[ρ RT e . 2 c k 3 2 k c k 3 implicit equation (32); these are repeated until Fk converges. (29) When solving the implicit equation (32), we first express F as Cay(ξˆ ), where ξ ∈ 3, by using the Cayley Discrete-time Euler-Lagrange Equations: Substituting k k k R transform and the hat map, and apply Newton’s iteration (See (23)-(29) into (20)-(21), we obtain discrete-time Euler- Section 3.3.8 in [10]). This yields a Lagrangian flow map, Lagrange equations for a 3D elastic string pendulum as (g , f ) 7→ (g , f ), which is applied iteratively. follows. k−1 k−1 k k V. NUMERICAL EXAMPLE 1 2 2 2 m(∆ rk,a−1 + 4∆ rk,a + ∆ rk,a+1) 6h (30) We now demonstrate the computational properties of the e e − hmge3 + h∇Vk,a−1 − h∇Vk,a = 0, Lie group variational integrator developed in the previous 1 m 2 1 2 e section by considering a numerical example. The material (M + )∆ rk,N+1 + m∆ rk,N + h∇V h 3 6h k,N properties of the string are chosen to represent a rubber string 1 m as follows [6]: µ = 0.025 kg/m, l = 1 m, EA = 40 N. The + M(RkFk − 2Rk + Rk−1)ρc − h(M + )ge3 = 0, h 2 rigid body is chosen as an elliptic cylinder with a semimajor (31) axis 0.06 m, a semiminor axis 0.04 m, and a height 0.1 m. 1 Its mass and location of the center of mass are given by (F J − J F T − J F + F T J )∨ k d d k d k−1 k−1 d M = 0.1 kg, ρ = [0.04, 0.01, 0.05] m. h (32) c M T 2 T Initially, the string is aligned to the horizontal e1 + ρˆcR ∆ rk,N+1 − hMgρˆcR e3 = 0, h k k axis at rest, and the rigid body has an initial velocity rk+1,a = rk,a + ∆rk,a, (33) [0, 0.2, −0.5] m/s. We use N = 20 elements. Simulation time is T = 5 seconds, and time step is h = 0.0001 second. Rk+1 = RkFk. (34) Fig. 2 illustrates the resulting maneuver of the 3D elastic 2 where ∆ rk,a = ∆rk,a −∆rk−1,a = rk+1,a −2rk,a +rk−1,a, string pendulum. This shows the nontrivial coupling between l EA e kxk−u u = N , m = µu, κ = u , and ∇Vk,a = κ kxk x for x = the string deformations and the rigid body dynamics. rk,a+1 − rk,a. Equation (30) is satisfied for a ∈ {2,...,N}, Fig. 3 shows the corresponding energy transfer, total and (33) is satisfied for a ∈ {2,...,N + 1}. For any k, the energy, total angular momentum deviation, orthogonality vector rk,1 = 0 since the pivot is fixed. errors of rotation matrices, velocities of the rigid body, and

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1.5 an analytical model that is defined globally on the Lie 0.056 1 group configuration manifold, and the Lie group variational 0.5 0.054 integrator preserves the geometric features of the system, 0 thereby yielding a reliable numerical simulation tool for 0.052 −0.5 complex maneuvers over a long time period. 0.05 These can be extended to include the effects of control −1 inputs by using the discrete Lagrange-d’Alembert principle −1.5 0.048 0 1 2 3 4 5 0 1 2 3 4 5 t t [15], which modifies the discrete Hamilton’s principle by (a) Energy transfer (kinetic energy of (b) Total energy taking into account the virtual work of the external control the rigid body: red, kinetic energy of inputs. When applied to an optimal control problem, this the string: black, gravitational poten- allows us to find optimal maneuvers accurately and effi- tial: green, elastic potential: blue) ciently, as there is no artificial numerical dissipation induced −9 −13 x 10 x 10 1 2 by the computational method. Furthermore, optimal large- angle rotational maneuvers can be easily obtained without 0.5 1.5 the singularities and complexity associated with local param- 0 eterizations, since the configuration is represented globally 1 on the Lie group [16]. −0.5

0.5 −1 REFERENCES [1] T. McLain and S. Rock, “Experimental measurement of rov tether −1.5 0 0 1 2 3 4 5 0 1 2 3 4 5 tension,” in Proceedings of Intervention/ROV 92, 1992. t t [2] D. Chapman, “Towed cable behaviour during ship turning manoeu- (c) Deviation of the total angular mo- (d) Orthogonality error of rotation vers,” Ocean Engineering, vol. 11, no. 4, pp. 327–361, 1984. mentum about the gravity direction matrices kI − RT Rk [3] R. Driscoll, R. Lueck, and M. Nahon, “Development and validation of a lumped-mass dynamics model of a deep sea ROV system,” Applied 50 1.25 Ocean Research, vol. 22, no. 3, pp. 169–182, 2000. 1.2 [4] T. Walton and H. Polacheck, “Calculation of transient motion of 0 submerged cables,” Mathematics of Computation, vol. 14, no. 69, pp. 1.15 27–46, 1960. −50 0 1 2 3 4 5 1.1 [5] C. Koh, Y. Zhang, and S. Quek, “Low-tension cable dynamics: Nu- 5 merical and experimental studies,” Journal of Engineering Mechanics, 1.05 vol. 125, no. 3, pp. 347–354, 1999. 0 [6] A. Kuhn, W. Seiner, J. Zemann, D. Dinevski, and H. Troger, “A com- 1 parison of various mathematical formulations and numerical solution −5 0.95 methods for the large amplitude oscillations of a string pendulum,” 0 1 2 3 4 5 0 1 2 3 4 5 t t Applied Mathematics and Computation, vol. 67, pp. 227–264, 1995. (e) Velocity / angular velocity of the (f) Stretched length of the string [7] B. Buckham, F. Driscoll, and M. Nahon, “Development of a ﬁnite rigid body (second components) element cable model for use in low-tension dynamics simulation,” Journal of Applied Mechanics, vol. 71, pp. 476–485, 2004. Fig. 3. Numerical simulation of a 3D elastic string pendulum [8] J. Shen, A. Sanyal, N. Chaturvedi, D. Bernstein, and N. H. McClam- roch, “Dynamics and control of a 3D pendulum,” in Proceedings of IEEE Conference on Decision and Control, Dec. 2004, pp. 323–328. [9] E. Hairer, C. Lubich, and G. Wanner, Geometric numerical integration, the stretched length of the string. As shown in Fig. 3(b), the ser. Springer Series in Computational Mechanics 31. Springer, 2000. [10] T. Lee, “Computational geometric mechanics and control of rigid computed total energy of the Lie group variational integrator bodies,” Ph.D. dissertation, University of Michigan, 2008. oscillates near the initial value, but there is no systematic [11] T. Lee, M. Leok, and N. H. McClamroch, “A Lie group variational drift over long time periods. This is due to the symplectic integrator for the attitude dynamics of a rigid body with application to the 3D pendulum,” in Proceedings of the IEEE Conference on Control property of the numerical solutions [14]. The Lie group Application, 2005, pp. 962–967. variational integrator preserves the momentum map as in Fig. [12] ——, “Lie group variational integrators for the full body problem in 3(c), and it also preserves the orthogonal structure of rotation orbital mechanics,” Celestial Mechanics and Dynamical Astronomy, vol. 98, no. 2, pp. 121–144, June 2007. matrices accurately. The orthogonality errors, measured by [13] J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, T −13 kI − R Rk, are less than 2 × 10 in Fig. 3(d). 2nd ed., ser. Texts in Applied Mathematics. Springer-Verlag, 1999, These show that the Lie group variational integrator pre- vol. 17. [14] E. Hairer, “Backward analysis of numerical integrators and symplectic serves the geometric characteristic of the 3D elastic string methods,” Annals of Numerical Mathematics, vol. 1, no. 1-4, pp. 107– pendulum accurately even for the presented complex maneu- 132, 1994. ver that exhibits nontrivial energy transfer between different [15] C. Kane, J. Marsden, M. Ortiz, and M. West, “Variational integrators and the Newmark algorithm for conservative and dissipative me- dynamic modes. chanical systems,” International Journal for Numerical Methods in Engineering, vol. 49, no. 10, pp. 1295–1325, 2000. VI.CONCLUSIONS [16] T. Lee, M. Leok, and N. H. McClamroch, “Computational geometric optimal control of rigid bodies,” Communications in Information and We have developed continuous-time equations of motion Systems, special issue dedicated to R. W. Brockett, vol. 8, no. 4, pp. and a geometric numerical integrator, referred to as a Lie 445–472, 2008. group variational integrator, for a 3D elastic string pendulum. The continuous-time equations of motion provide

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