Joint 48th IEEE Conference on Decision and Control and ThA06.5 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009 Dynamics of a 3D Elastic String Pendulum Taeyoung Lee, Melvin Leok∗, and N. Harris McClamrochy 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 yThis 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. 978-1-4244-3872-3/09/$25.00 ©2009 IEEE 3347 ThA06.5 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 2 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(^ ρc + ρ)k dM(ρ) Fig. 1. 3D Elastic String Pendulum B 2 1 1 = Mr_(l) · r_(l) + Ω · JΩ + Mr_(l) · RΩ^ρ ; (3) 2 2 c Define where B denotes the region enclosed by the rigid body R l 2 R Total length of the unstretched string surface, and we used the following properties: B ρ dM = 0; R ^ 2 s 2 [0; l] Length of the string from the pivot to a ma- xy^ = −yx^ ; 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) 2 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) 2 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 2 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 2 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 2 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 2 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 2 + Mass of the rigid body R where E and A denote the Young’s modulus and the sectional J 2 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 2 [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 1 3 1 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) = fR 2 R j R R = I; det[R] = 1g. pendulum is given by III. CONTINUOUS-TIME ANALYTICAL MODEL L = Tstr − Vstr + Trb − Vrb: (6) In this section, we develop continuous-time equations B.