The Motion and Control of a Chaplygin Sleigh with Internal Shape in an Ideal Fluid

Transcription

1 The Motion and Control of a Chaplygin Sleigh with Internal Shape in an Ideal Fluid by Christopher Barot A dissertation submitted in partial fulllment of the requirements for the degree of Doctor of Philosophy (Physics) in the University of Michigan 2017 Doctoral Committee: Professor Anthony Bloch, Co-Chair Professor Fred Adams, Co-Chair Professor Silas Alben Professor Cagilyan Kurdak Professor Greg Tarlé

2 c Christopher Barot 2017

3 DEDICATION To my family, especially my mother and aunts who decided when I was very young that I'd be a doctor. This isn't the sort they had in mind, but I think it's pretty good. ii

4 ACKNOWLEDGMENT First and foremost, this work would not have been possible without my advisor Tony Bloch. He worked with me every step of the way and literally wrote the book on nonholonomic mechanics which provided a great deal of background as well as insight on the topics discussed in this work. In addition, I would like to thank Dmitry Zenkov. He provided detailed information on work foundational to this research and also edited sections of it. Further, I'd like to thank the members of my committee: my co-advisor Fred Adams, Silas Alben, Greg Tarlé, and Cagilyan Kurdak, each of whom gave feedback on my dissertation. iii

5 TABLE OF CONTENTS DEDICATION ii ACKNOWLEDGMENT iii LIST OF FIGURES viii LIST OF APPENDICES xiii ABSTRACT xiv Chapter I. Introduction 1 Chapter II. Background Nonholonomic Mechanics Lagrangians on Manifolds The Ehresmann Connection Nonholonomic Constraints iv

6 2.1.4 Symmetry Nonholonomic Systems with Symmetry The Classic Chaplygin Sleigh with Oscillating Mass A Rigid Body in a Kircho Fluid Chapter III. The Chaplygin Sleigh with Oscillator in a Kircho Fluid The Chaplygin Sleigh in a Kircho Fluid with Oscillating Mass The Elliptical Sleigh in a Kircho Fluid with Oscillating Mass Case by Case Analyses An Elliptical Sleigh A Circular Sleigh with Oscillator A Circular Sleigh with Blade Translational Oset, with and Without Oscillator An Elliptical Sleigh with Oscillator An Elliptical Sleigh with Oscillator and with Translational Blade Oset An Elliptical Sleigh with Oscillator and Rotational Blade Oset An Elliptical Sleigh with Oscillator and both Translational and Rotational Blade Osets Simulated Dynamics of the Sleigh in a Kircho Fluid An Elliptical Sleigh A Circular Sleigh with Oscillator v

7 3.4.3 A Circular Sleigh with Blade Translational Oset, with and Without Oscillator An Elliptical Sleigh with Oscillator An Elliptical Sleigh with Oscillator and Translational Blade Oset An Elliptical Sleigh with Oscillator and Rotational Blade Oset An Elliptical Sleigh with Oscillator and Both Translational and Rotational Blade Osets Chapter IV. Controlling the Chaplygin Sleigh in a Kircho Fluid with a Moving Mass Controlling the Motion of the Sleigh Without Rotational Blade Oset Simulated Controlled Motion of the Sleigh Without Rotational Blade Oset With Rotational Blade Oset Simulated Controlled Motion of the Sleigh With Rotational Blade Oset Explicit Solutions Chapter V. Conclusions Future Research Directions Complex Fluids Multiple Bodies Complex Internal Shape Optimal Control vi

8 5.1.5 Controlling ṙ Appendices 104 Bibliography 131 vii

9 LIST OF FIGURES Motion of the blade contact point for a circular sleigh. M = 2, R = 1. Initial conditions: p 1 = 1, p 2 = 1. Run for Time T [ 2, 2] Motion of the blade contact point for a circular sleigh. M = 2, R = 1. Initial conditions: p 1 = 1, p 2 = 1. Run for Time T [ 4, 4] Motion of the blade contact point for an elliptical sleigh. M = 2, A = 1.5, B = 1 Initial conditions: p 1 = 1, p 2 = 1. Run for time T [ 10, 10] Motion of the blade contact point for a circular sleigh with oscillating mass:. M = 2, R = 1, m = 0.2 Initial conditions: r 0 = 1, p 1 = 1, p 2 = 1. Run for time T [ 200, 200] Motion of the blade contact point for a circular sleigh with blade translation oset. M = 2, R = 1, a = 0.1, b = 0.2 Initial conditions: p 1 = 1, p 2 = 1. Run for time T [ 200, 200] Motion of the blade contact point for a circular sleigh with oscillating mass and blade translational oset. M = 2.6, R = 1, m = 0.3, a = 0.3 Initial conditions: r 0 = 1, p 1 = 1, p 2 = 1. Run for time T [ 300, 300] viii

10 3.4.7 Motion of the blade contact point for an elliptical sleigh with oscillating mass. M = 2.6, A = 1.5, B = 1, m = 0.3. Initial conditions: r 0 = 1, p 1 = 1, p 2 = 1. Run for time T [ 400, 400] Motion of the blade contact point for an elliptical sleigh with oscillating mass and translational blade oset. M = 2.6, A = 1.5, B = 1, m = 0.4, a = Initial conditions: r 0 = 1, p 1 = 0.5, p 2 = 0.5. Run for time T [ 500, 500] Angular momentum of the blade contact point for an elliptical sleigh with oscillating mass and translational blade oset. M = 2.6, A = 1.5, B = 1, m = 0.4, a = Initial conditions: r 0 = 1, p 1 = 0.5, p 2 = 0.5. Run for time T [ 500, 500] Translational momentum of the blade contact point for an elliptical sleigh with oscillating mass and translational blade oset. M = 2.6, A = 1.5, B = 1, m = 0.4, a = 0.2. Initial conditions: r 0 = 1, p 1 = 0.5, p 2 = 0.5. Run for time T [ 500, 500] Motion of the blade contact point for an elliptical sleigh with oscillating mass and rotational blade oset. M = 2.6, A = 1.5, B = 1, m = 0.6, α = 0.2. Initial conditions: r 0 = 1, p 1 = 0.5, p 2 = 0.5. T [ 200, 200] Motion of the blade contact point for an elliptical sleigh with oscillating mass and rotational blade oset (future trajectory only). M = 2.6, A = 1.5, B = 1, m = 0.6, α = 0.2. Initial conditions: r 0 = 1, p 1 = 0.5, p 2 = 0.5. T [0, 200] Further future motion of the blade contact point for an elliptical sleigh with oscillating mass and rotational blade oset. M = 2.6, A = 1.5, B = 1, m = 0.6, α = 0.2. Initial conditions: r 0 = 1, p 1 = 0.5, p 2 = 0.5. T [ 200, 400] 67 ix

11 3.4.14Large time Motion of the blade contact point for an elliptical sleigh with oscillating mass and rotational blade oset. M = 2.6, A = 1.5, B = 1, m = 0.6, α = 0.2. Initial conditions: r 0 = 1, p 1 = 0.5, p 2 = 0.5. T [800, 2000] Motion of the blade contact point for an elliptical sleigh with oscillating mass and rotational blade oset. M = 2, A = 2, B = 1, m = 1, α = 1. Initial conditions: r 0 = 1, p 1 = 1, p 2 = 2. T [ 200, 200] Positive time motion of the blade contact point for an elliptical sleigh with oscillating mass and rotational blade oset. M = 2, A = 2, B = 1, m = 1, α = 1. Initial conditions: r 0 = 1, p 1 = 1, p 2 = 2. T [0, 200] Large time motion of the blade contact point for an elliptical sleigh with oscillating mass and rotational blade oset. M = 2, A = 2, B = 1, m = 1, α = 1. Initial conditions: r 0 = 1, p 1 = 1, p 2 = 2. T [200, 1000] Motion of the blade contact point for an elliptical sleigh with oscillating mass and translational as well as rotational blade oset:. M = 2, A = 2, B = 1, m = 1, a = 1, b = 0, α = π. Initial conditions: r 4 0 = 1, p 1 = 1, p 2 = 1. Run for time T [ 200, 200] Positive time motion of the blade contact point for an elliptical sleigh with oscillating mass and translational as well as rotational blade oset. M = 2, A = 2, B = 1, m = 1, a = 1, b = 0, α = π. Initial conditions: r 4 0 = 1, p 1 = 1, p 2 = 1. Run for time T [0, 200] Motion of the blade contact point for an elliptical sleigh with adjustable mass and translational blade oset. M = 2, A = 2, B = 1, m = 1, α = 0, a = 0.2. Initial conditions: r 0 = 1, p 1 = 1, p 2 = 0. T [ 200, 200] x

12 4.2.2 Large time motion of the blade contact point for an elliptical sleigh with adjustable mass and translational blade oset. M = 2, A = 2, B = 1, m = 1, α = 0, a = 0.2. Initial conditions: r 0 = 1, p 1 = 1, p 2 = 0. T [ 2000, 2000] Evolution of the momenta of the blade contact point for an elliptical sleigh with adjustable mass and translational blade oset. M = 2, A = 2, B = 1, m = 1, α = 0, a = 0.2. Initial conditions: r 0 = 1, p 1 = 1, p 2 = 0. T [ 200, 200] Motion of the blade contact point for an elliptical sleigh with adjustable mass and translational blade oset M = 2, A = 2, B = 1, m = 1, α = 0, a = 0.2, initial conditions: r 0 = 1, p 1 = 1, p 2 = 0 and additional possible future motion with r 0 = 2. T [ 200, 200] Complete motion of the blade contact point for an elliptical sleigh with adjustable mass and translational blade oset. M = 2, A = 2, B = 1, m = 1, α = 0, a = 0.2. Initial conditions: p 1 = 1, p 2 = 0 and r = 1 for t < 0 r = 2 for t > 0. T [ 200, 200] Dynamically controlled complete motion of the blade contact point for an elliptical sleigh with adjustable mass and translational blade oset. M = 2, A = 2, B = 1, m = 1, α = 0, a = 0.2. r = 1 for 0 < t < 10, r = 0.4 for 10 < t < 30, r = 2 for 30 < t < 100, r = 0.4 for 100 < t < 250, and r = 4 for 250 < t < 300. Initial conditions: p 1 = 1, p 2 = 1. T [0, 300] xi

13 4.2.7 Dynamically controlled evolution of momenta of the blade contact point for an elliptical sleigh with adjustable mass and translational blade oset. M = 2, A = 2, B = 1, m = 1, α = 0, a = 0.2. r = 1 for 0 < t < 10, r = 0.4 for 10 < t < 30, r = 2 for 30 < t < 100, r = 0.4 for 100 < t < 250, and r = 4 for 250 < t < 300. Initial conditions: p 1 = 1, p 2 = 1. T [0, 800] Motion of the blade contact point for an elliptical sleigh with adjustable mass and translational blade oset. M = 2, A = 2, B = 1, m = 1, r = 1, α = π 4, a = 1. Initial conditions: p 1 = 1, p 2 = 1. T [ 200, 200] Motion of the blade contact point for an elliptical sleigh with adjustable mass and translational blade oset as well as alternate future trajectory. M = 2, A = 2, B = 1, m = 1, r = 1, r(t > 0) = 2, α = π, a = 1. Initial conditions: 4 p 1 = 1, p 2 = 1. T [ 200, 200] Dynamically controlled motion of the blade contact point for an elliptical sleigh with adjustable mass and translational blade oset. M = 2, A = 2, B = 1, m = 1, r = 1, r(t > 0) = 2 2 2, α = π, a = 1. Initial conditions: 4 p 1 = 0, p 2 = 1. T [ 200, 200] xii

14 LIST OF APPENDICES APPENDIX A: Mathematica Calculations APPENDIX B:MathematicaSimulation Code xiii

15 ABSTRACT In this dissertation we will examine a nonholonomic system with Lie group symmetry: the Chaplygin sleigh coupled to an oscillator moving through a potential uid in two dimensions. This example is chosen to illustrate several general features. The sleigh system in the plane has SE(2) symmetry. This group symmetry will be used to separate the dynamics of the system into those along the group directions and those not. The oscillator motion is not along the group and so acts as an additional conguration space coordinate that plays the role of internal shape. The potential uid serves as an interactive environment for the sleigh. The interaction between the uid and sleigh depends not only on the sleigh body shape and size but also on its motion. The motion of the sleigh causes motion in the surrounding uid and vice-versa. Since the sleigh body is coupled to the oscillator, the oscillator will have indirect interaction with the uid. This oscillator serves as internal shape and interacts with the external environment of the sleigh through its coupling to the sleigh body and the nonholonomic constraint; it will be shown that this interaction can produce a variety of types of motion depending on the sleigh geometry. In particular, when the internal shape of the system is actively controlled, it will be proven that the sleigh can be steered through the plane towards any desired position. In this way the sleigh-uid-oscillator system will demonstrate how a rigid body can be steered through an interactive environment by controlling things xiv

16 wholly within the body itself and without use of external thrust. xv

17 CHAPTER I Introduction Nonholonomic mechanics is concerned with the motion of systems subject to nonintegrable constraints on the system velocities rather than integrable constraints, which are called holonomic. This classication dates back to Heinrich Hertz ( [1]) in The constraints are nonintegrable in the sense that they cannot be expressed as derivatives of constraints dealing with the system positions. Such constraints occur explicitly in physical systems exhibiting rolling contact or sliding contact. These constraints limit the allowed velocities in a way that depends on the system conguration at the particular moment. By contrast, constraints on conguration space can also come from symmetries which give rise to conserved quantities. For example, angular momentum constraints, which are integrable constraints on the phase space, can be seen as nonintegrable constraints on the conguration space ( [2]). These systems evolve according to the Lagrange-d'Alembert equations which are discussed in detail in section 2.1. Historically there has been a great deal of confusion over the proper equations of motion that govern nonholonomic mechanical systems. A large part of that confusion has centered on the question of whether or not there is a variational principle from which to derive the equation of motion ( [2]), and when should one impose the nonholonomic constraints, before or after 1

18 taking variations. The Lagrange-d'Alembert principle states that the correct method for mechanical systems is to impose the constraints after taking variations, and although this has been a matter of debate over the years, it has been shown that the Lagrange-d'Alembert principle can be derived from balance of forces, Newton's second law, together with balance of torques ( [3]). Further more, Korteweg showed conclusively that the general equations of nonholonomic mechanics are not variational ( [4]). Greater detail and summarized histories of the eld can be found in ( [2], [5]). Of particular physical interest are mechanical systems with Lie group symmetries. In this, nonholonomic constraints give rise to new and interesting dynamics that are not possible in holonomic systems. Their motion does not necessarily obey Noethers theorem for holonomic mechanics, which says that every generator of a symmetry group action on the Lagrangian leads to a conserved conjugate momentum ( [6], [7]). However, horizontal symmetries can lead to conserved momenta ( [8]), and, more generally, "it is possible to nd vector elds that are not the generators of the group action but nevertheless identify the directions in which the momentum is preserved" ( [9]). This allows nonholonomic systems to conserve energy while exhibiting behavior normally associated with dissipative systems, such as neutrally stable and asymptotically stable relative equilibria ( [10]). In this dissertation we will examine a nonholonomic system with Lie group symmetry: the Chaplygin sleigh coupled to an oscillator moving through a potential uid in two dimensions. This example is chosen to illustrate several general features. The sleigh system in the plane has SE(2) symmetry. This group symmetry will be used to separate the dynamics of the system into those along the group directions and those not. The oscillator motion is not along the group and so acts as an additional conguration space coordinate that plays the role of internal shape. The potential uid serves as an interactive environment for the sleigh. 2

19 The interaction between the uid and sleigh depends not only on the sleigh body shape and size but also on its motion. The motion of the sleigh causes motion in the surrounding uid and vice-versa. Since the sleigh body is coupled to the oscillator, the oscillator will have indirect interaction with the uid. This oscillator serves as internal shape and interacts with the external environment of the sleigh through its coupling to the sleigh body and the nonholonomic constraint; it will be shown that this interaction can produce a variety of types of motion depending on the sleigh geometry. In particular, when the internal shape of the system is actively controlled, it will be proven that the sleigh can be steered through the plane towards any desired position. In this way the sleigh-uid-oscillator system will demonstrate how a rigid body can be steered through an interactive environment by controlling things wholly within the body itself and without use of external thrust. The material covered in this dissertation is separated broadly as follows. Chapter 2 will present the reader with necessary background information. Section 2.1 will provide mathematical denitions and results concerning, among other things, nonholonomic constraints, Ehresmann connections, Lie groups, and the equations of motion governing nonholonomic systems. Then section 2.2 will provide an example of all these things in the classic nonholonomic system of the Chaplygin sleigh modied to include an oscillator. Afterwards, the interactions of a rigid body with an ideal Kircho uid are derived in section 2.3. All of the material in chapter 2 will be combined in chapter 3 when discussing the system of the Chaplygin sleigh with oscillator moving through a Kircho uid. General expressions for the Lagrangian, connection, momenta, and equations of motion are calculated in section 3.1 before being made explicit for the choice of an elliptical sleigh in section 3.2. These expressions are analyzed and, where possible, solved for several increasingly complex cases in section 3.3 with simulations of the motion of example systems provided in section

20 The question of controllability of the sleigh is taken up in chapter 4 and broken into two cases: the sleigh without rotational blade oset in section 4.1 and the sleigh with rotational blade oset in 4.3. Simulations of both cases are given in sections 4.2 and 4.4 respectively. Analytic solutions of the sleigh momenta equations of motion for xed choice of the adjustable mass position are provided in section 4.5. Finally, results are summarized in chapter 5 with possible future work suggested. 4

21 CHAPTER II Background This chapter aims to acquaint the reader with necessary background and earlier works regarding nonholonomic mechanics, Kircho uids, and the classic Chaplygin sleigh. Nonholonomic mechanical systems are mechanical systems which obey a certain type of nonintegrable constraints. These constraints often lead to unintuitive behavior and require a dierent analysis than that of holonomic systems. Mathematical background information regarding nonholonomic constraints and equations of motion for nonholonomic systems, as well as systems exhibiting Lie group symmetries are provided in section 2.1. A well known specic example of the Chaplygin Sleigh with an added complication of an oscillating mass is examined in section 2.2, and nally, the interactions between a rigid body and a surround Kircho Fluid are detailed in section

22 2.1 Nonholonomic Mechanics Lagrangians on Manifolds We start by presenting mathematical framework with which to describe mechanical systems. Let Q be the conguration space describing a mechanical system, a smooth manifold for which any point q Q, species a possible state of the system. Let L : T Q R be the Lagrangian for this system. For such a system, Hamilton's principle states that the time evolution of the system will be a critical point of the action. Then the path taken between two system's states q a and q b corresponding to the time moments a and b satises δ b a L(q, q) dt = 0. (2.1.1) This is equivalent to the well known EulerLagrange equations which, in local coordinates q = (q 1,..., q n ), take the form of a system of second-order dierential equations d L dt q = L, i = 1,..., n. (2.1.2) i qi The Ehresmann Connection We introduce now the notation A to represent an Ehresmann connection, the curvature of which is denoted B. A more general discussion of Ehresmann connections can be found in ( [2]). Here we will just give a denition and discuss certain special types of connection. First consider a bundle Q with projection mapping π : Q R where R is the base for Q. Let T q Q denote the tangent mapping at any point q Q. For each q, call the kernel of T q Q the 6

23 vertical space, which is denoted V q. Then an Ehresmann connection, A, is a vector-valued one-form on Q which satises: 1. A q : T q Q V q is a linear map for each q Q, 2. A is a projection operator: A q (v q ) = v q for each v q V q. The important characteristic of an Ehresmann connection is that it splits the tangent space at each point in Q into two parts. That is, for each q Q the vector space T q Q can be written as a direct sum of V q and H q, where H q = ker(a q ). These are called the vertical and horizontal spaces. The connection can be used to project any tangent vector at a point q onto it's vertical component. The curvature B of a given connection A is the vertical vector valued two form on Q dened by B(X, Y ) = A([hor X, hor Y ]). The bracket here is the JacobiLie bracket of vector elds obtained by extending the vectors to vector elds. The curvature has geometric meaning, and further information can be found in [2], but this will not be used here Nonholonomic Constraints Mechanical systems sometimes move under external constraints. Examples of this include pendulums that oscillate at a xed distance from a pivot point and wheels that roll across a surface without slipping. These constraints come in two varieties, holonomic and nonholonomic. The xed length pendulum is an example of a system moving under a holonomic constraint. While the pendulum bob is nominally allowed to move in three spacial dimensions with associated momenta, resulting in a six dimensional conguration space, the xed length of the pendulum means that all motion is limited to the surface of a sphere. Any 7

24 momenta perpendicular to the surface must be zero. This means that the system can be limited to a four dimensional conguration space. This is the dening property of holonomic constraints, that they allow the system to be reduced in such a way as to lower the dimension of the conguration space. In other words, holonomic constraints are integrable. Nonholonomic constraints, of which rolling without slipping is an example, are not integrable in this way. In this paper we will be interested exclusively in nonholonomic constraints that are linear and homogeneous in the velocity. Earlier we had for any Lagrangian system: δ b a L(q, q) dt = b a ( L q d ) L δq i dt = 0. (2.1.3) i dt q i Nonholonomic constraints place limits on the type of motion of the system and so place limits on the δq i 's. At each q Q there is a subspace D q T q Q such that only variations δq which are in D q are allowed. Dene D = D q. Only curves on Q for which the velocities are entirely within the distribution D are allowed. Since nonholonomic constraints are, by denition, nonintegrable, D cannot be a subspace of T q Q, and so the conguration space cannot be reduced to a lower dimensional subspace. In local coordinates (q 1,..., q n ) these constraints take the form below, where j ranges over the number of constraints imposed (which necessarily must be less than the dimension of the conguration space). n A j k (q) qk = 0 (2.1.4) k=1 Non-integrability of these constraints means that they cannot be expressed as the time 8

25 derivatives of any functions of the coordinates alone. That is, there cannot exist any functions b j (q) such that the system of equations (2.1.5) are equivalent to those in system (2.1.4). n k=1 B j q k qk = 0 (2.1.5) Nonholonomic constraints allow for a certain natural choice for Ehresmann connection called the nonholonomic connection. The nonholonomic connection is the connection that, at each point, acts as a projection operator on all tangent vectors which obey the nonholonomic constraints. The horizontal space is then the subspace of T q Q orthogonal to D q. The nonholonomic connection can be used to write equations of motion governing a system subject to nonholonomic constraints. As before, let Q be the conguration space for a mechanical system with Lagrangian, and let (q 1,..., q n ) be local coordinates on Q. Following the method in [11] we choose to work with the nonholonomic constraints expressed through forms. Assume this mechanical system is subject to constraints on the allowed velocities such that only velocities in the kernel of the system of one forms {ω a } p a=1 are allowed. There are p constraints with 1 p n. We can also choose to write the coordinates as q = (q 1,...q n ) = (r α, s a ) with 1 α n p and 1 a p so that the forms can be expressed as ω a (q) = ds a + A a α(r, s)dr α These forms limit the allowed velocities of this system to those in the kernel of all of the ω a. Let (v α, v a ) be a velocity in T q Q that satises the nonholonomic constraints. Then we know 9

26 that v a = A a α(r, s)v α (2.1.6) Using this, we can dene a constrained Lagrangian by taking the general Lagrangian and explicitly dening the velocities as those obeying the constraints. L(q, q) = L(r α, s a, ṙ α, ṡ a ) = L(r α, s a, ṙ α, A a α(r, s)δr α ) = L c (r α, s a, ṙ α ) The equations of motion, in terms of this constrained Lagrangian are then calculated to be d L c dt ṙ L c α r + L c α Aa α B b αβ = Ab α r β Ab β r α + Aa α s = L c a ṡ b Bb αβṙ β (2.1.7) A b β s A b a Aa α β s a (2.1.8) The A a α are coordinate expressions of the nonholonomic connection, and the B B αβ are coordinate expressions for the corresponding curvature Symmetry Often, physical systems of interest will exhibit symmetries. We will be particularly interested in those symmetries which can be described by a Lie group. A Lie group is a smooth manifold with group structure for which group multiplication and multiplicative inversion are continuous functions. Let G be a Lie group that acts on Q. We dene the orbit of a point q Q by 10

27 Orb(q) = {gq g G} (2.1.9) Assume that this action is free and proper. The group action is considered free i there is no q Q such that Orb(q) = q, and it is proper if the mapping (q, g) gq is proper (inverse images of compact sets are compact) q Q, g G. Under these assumptions, we refer to the quotient space Q/G as the shape space. Each point in the shape space of Q/G is an orbit as dened above. This shape space is a smooth manifold as long as the group action is free and proper. Thus, the conguration space has the structure of a principle ber bundle with the shape space as the base. This allows the local coordinates on Q to be chosen as q = (r, g) with r being local coordinates on the shape space Q/G and g being coordinates on the group G. In these coordinates, the action of the group on Q is given by left multiplication on the group coordinate g with no eect on the shape space coordinates. The group G is said to be a symmetry group for the system if the Lagrangian is invariant under the induced action of G on T Q. Assume now that the Lagrangian is the dierence between the kinetic and potential energies, with the kinetic energy given by a Riemannian metric on Q. The presence of a group symmetry allows us to introduce another example of an Ehresmann connection, the mechanical connection. The horizontal space of the mechanical connection at each point q is the subspace of T q Q that is orthogonal (relative to the kinetic energy metric) to the tangent space of the group orbit through q, see [7] for details. That is, the horizontal component of velocity is hor q q = ṙ. The vertical space at q is the tangent space to the group orbit through q. Doing this leads to the Lagrange-d'Alembert equations of motion for a G-invariant La- 11

28 grangian, a derivation of which can be found in [12]. d l dt ṙ l l r = Ω, i ṙb + iṙγ, A γ, iṙa + E, Ω, (2.1.10) d l dt Ω = l l ad Ω Ω + Ω, i ṙe, ġ = g(ω iṙa). In the above equations γ is dened by de = γ(r), e, where e(r) = (e 1 (r),..., e k (r)) is a basis of the Lie algebra of the group G, A is a connection on Q/G, B is its curvature, ad is the adjoint operator on a Lie algebra, ad is its dual, E is a form dened by E = γ ad A, and Ω = Aṙ + ξ. The quantity Ω is sometimes referred to as the body angular velocity, see [8] and the next section for more details, including the benets of this choice of variables Nonholonomic Systems with Symmetry Let us now work with a mechanical system with conguration space Q, Lagrangian L, and nonholonomic constraints characterized by a constraint distribution D. Assume that the Lagrangian is G-invariant as above. Let us further assume that the constraint distribution is also G-invariant in the sense that for any g G and q Q the induced action of G on T Q maps D q to D gq. Then there is an induced Lagrangian on the reduced velocity phase space T Q/G with constraint distribution given by D/G. Finally, we assume that the Lagrangian is given by the dierence between the kinetic and potential energies. The Lagrange-d'Alembert principle outputs reduced equations called the Lagrange-d'Alembert- Poincaré equations ( [11]). Let (r, g) be bundle coordinates on the conguration space, where as before, local coordinates 12

29 on the shape space are denoted r and group coordinates are denoted g. Let g be the Lie algebra of the group G. These coordinates induce the coordinates (r, ṙ, ξ) on T Q/G, where ξ = g 1 ġ, see [8]. Now, assuming that the constraint distribution and group orbits span the entire tangent space T q Q at each point q Q, we dene a new Lie algebra variable Ω via the nonholonomic connection by the formula Ω = Aṙ + ξ where A is the nonholonomic connection. This new variable is called the body angular velocity and is used to replace velocities along the symmetry group directions in the reduced phase space. The benet of this change of variables is that in these coordinates the constraints become particularly simple and read Ω a = 0 for all indices a ranging over the coordinates in the symmetry directions which lie outside the constraint space. Then, following [8], the Lagrangian becomes l(r α, ṙ α, Ω A ) = 1 2 g αβṙ α ṙ β I ABΩ A Ω B + λ a αṙ α Ω a U(r), (2.1.11) where repeated indices are summed over. In the above, g αβ is the induced kinetic energy metric on Q/G and I AB is the locked inertia tensor. The coecients λ a α create an interconnection between the reduced space Q/G and the shape space. These are given by: λ a α = 2 l ξ a ṙ 2 l α ξ a ξ B AB α (2.1.12) As shown in [11], only when the mechanical and nonholonomic connections are identical, the terms λ a α are found to vanish. In this particular case, the constrained reduced Lagrangian 13

30 becomes block diagonal and reads l c = 1 2 g αβṙ α ṙ β I abω a Ω b U(r). (2.1.13) Then equations can be expressed in coordinates. They become the reduced nonholonomic Lagrange-d'Alembert equations ( [2]). d l c dt r l c = Icd α r α r p cp α d DbαI c bd p c p d Bαβp c c ṙ β (2.1.14) D βαb I bc p c ṙ β K αβγ ṙ β ṙ γ d dt p b = C c abi ad p c p d + D c bαp c ṙ α + D αβb ṙ α ṙ β In expression above l c (r α, ṙ α, p a ) is the constrained Lagrangian with r α coordinates in shape space and p a = lc Ω, e a(r) components of the momentum map in the body representation. I ad are components of the inverse locked inertia tensor; Bαβ a are the local coordinates of the curvature associated with the nonholonomic connection A. The coecients D c bα, D αβb, and K αβγ are given by D c bα = D αβb = K αβγ = k a =1 k a =m+1 k a =m+1 C c a ba a α + γ c bα + λ a α(γ a bβ λ a γb a αβ k b =1 k a =m+1 C a b ba b β ) λ a αc a abi ac 14

31 where k is the dimension of the symmetry group, and m is the dimension of the constraint space. The curvature is calculated in coordinates by αβ = Aa α r β B a Aa β r α Ca b c αa Ab c β + γb a βa b α γb a αa b β where primed indices run over the group symmetry directions, and the γ expressed in coordinates are calculated according to e b r α = k c =1 γ c bαe c. These reduced equations are also called the Lagrange-d'Alembert-Poincaré equations. 2.2 The Classic Chaplygin Sleigh with Oscillating Mass We rst present the classic, two dimensional Chaplygin sleigh, which is a rigid body of mass M moving through the plane supported by three points. Two of these points move frictionlessly, while the third point is a blade that prevents motion perpendicular to the blade orientation. The conguration of this sleigh is fully described by the position of the blade contact point in a xed coordinate frame as well as the angle θ between the blade and the x axis. We start with the Lagrangian for a free rigid body: L(θ, x, y) = 1 2 M(ẋ2 + ẏ 2 ) I θ 2 (2.2.1) The parameter I is the moment of inertia of the sleigh about the blade contact point. To this we add a point mass, m, on a spring. The spring equilibrium position is directly above 15

32 the blade contact point, and the mass is allowed to oscillate along the blade direction. This system is examined in detail in [12]. Then the sleigh-oscillator Lagrangian becomes L = 1 2 I θ M(ẋ2 + ẏ 2 ) + 1 m[(ẋ, ẏ) + ṙ(cos(θ), sin(θ)) + 2 r θ( sin(θ), cos(θ))] 2 U(r) = 1 2 I θ M(ẋ2 + ẏ 2 ) m[ẋ2 + ẏ 2 + ṙ 2 + r 2 θ 2 + 2ṙ(ẋcos(θ) + ẏsin(θ)) +2r θ(ẏcos(θ) ẋsin(θ))] U(r). In the above expression, < x, y > denotes the position of the blade contact point, and the position of the oscillatory mass is expressed as the vector sum of the blade contact point and the time dependent oscillator length, r. U(r) is the oscillator potential energy. In order to more naturally handle the blade constraint we utilize a body reference frame for the sleigh. The body frame consists of three vectors: a vector directed perpendicular to the sleigh's plane to measure angular position, a vector along the blade direction, and another vector perpendicular to the blade in the sleigh's plane. Denote the velocity components along these directions by (ξ 1, ξ 2, ξ 3 ) where ξ 1 = θ is the angular velocity and (ξ 2, ξ 3 ) = (ẋsin(θ) + ẏcos(θ), ẋcos(θ) + ẏsin(θ) are the linear velocity components along the blade and perpendicular to the blade, respectively. The Lagrangian for the rigid body with oscillating mass is written using these velocities in the body frame as: 16

33 l(r, ṙ, ξ) = 1 2 (I + mr2 )ξ (M + m)(ξ2 2 + ξ 2 3) mṙ2 + mṙξ 2 + mrξ 1 ξ 3 U(r) (2.2.2) This sleigh-oscillator system has SE(2) symmetry and is a simple example of all those mathematical constructs detailed in section 2.1. The Lagrangian contains terms that interconnect the shape and group dynamics, specically the term mṙξ 2. Following the procedure from section 2.1.5, the nonholonomic connection for this example is computed to be 0 A = m m+m. 0 Recall that this connection eliminates this interconnecting term in the constrained Lagrangian and simultaneously respects the blade constraint. Indeed, dene new velocity by the formula Ω = ξ + Aṙ. The blade constraint is then simply Ω 3 = 0. After imposing this constraint, the constrained Lagrangian for the Chaplygin sleigh becomes l c (r, ṙ, Ω) = 1 Mm mṙ2 + 1 ( ) (I + mr 2 )Ω (M + m)ω 2 2 U(r). (2.2.3) 2 M

34 The equations of motion are Mm M + m r = Mmr (M + m)(i + mr 2 ) 2 p2 1 U r, (2.2.4) mr ṗ 1 = (M + m)(i + mr 2 ) p m 2 r 1p 2 + (M + m)(i + mr 2 ) p 1ṙ, mr ṗ 2 = (I + mr 2 ) 2 p2 1 where the momenta are given by the usual canonical denition, p i = l Ω i given in equation with the Lagrangian p 1 p 2 = I + mr2 0 0 M + m Ω 1 Ω 2 (2.2.5) It will often be necessary to convert between momenta and velocities, so inverting the matrix above gives Ω 1 Ω 2 λ 1 (I + mr 2 )(M + m) = λ M + m 0 0 I + mr 2 p 1 p 2. (2.2.6) Additionally, we can recover the blade contact point from the momenta by remembering how the original velocities were dened with respect to a xed coordinate system. The resulting equations are referred to as the reconstruction equations 18

35 θ = ξ 1 = Ω 1 A 1 ṙ (2.2.7) ẋ = ξ 2 Cos(θ) = Cos(θ)(Ω 2 A 2 ṙ) ẏ = ξ 2 Sin(θ) = Sin(θ)(Ω 2 A 2 ṙ) The equations of motion, along with the reconstruction equations relating the position of the blade contact point to the momenta, completely determine the motion of the sleigh and oscillating mass. 2.3 A Rigid Body in a Kircho Fluid Immersing an object in a uid changes its motion drastically, principally by coupling motion along dierent axes. In general, this interaction can be very complicated, but under certain idealized assumptions it can be shown that the interaction between a uid and a rigid body depends only on the body. Then the eect of adding the uid to the system can be represented by a uid inertia tensor added to that of the body. Consider a rigid body moving in a potential uid. Assume the uid is irrotational, inviscid, and remains at rest at the uid boundaries (generally assumed to be at innity). Further assume that the normal components of the velocities of the solid and uid particles are equal at the boundary between them; thus no cavities are formed in the uid. Then Kirchho's theory shows that the dynamics of the solid-uid system can be described using only the solid variables. We follow ( [13]) to demonstrate this. The kinetic energy of the uid is given by 19

36 T F = 1 2 F ρ F u 2 dv. (2.3.1) Here, u is the velocity eld of the uid and ρ F is the uid density. Since this is a potential uid ow, the eld u can be written as a gradient: u = φ, (2.3.2) φ = 0. (2.3.3) The latter equation follows from the assumption that the uid is irrotational. In principle, u as well as φ are functions of time, and it is possible that equation may be true only at the initial time. However, the uid particles are identical, and so there is a relabeling symmetry that any reasonable uid Lagrangian will respect. Then the equations of motion will preserve circulation, so that equation 2.3.3, if true initially, will hold for all times. The kinetic energy of the rigid body reads T B = (mvi 2 + i=1 3 j=1 I ij B ω iω j + 2K ij B v iω j ), (2.3.4) where m is the mass, and the constants I ij B and Kij B are determined by the shape and mass distribution of the rigid body. In more compact notation, the kinetic energy of the body becomes T B = 1 2 ξt I B ξ where ξ and I B are dened as the tangent vector to conguration space and body inertia 20

37 tensor respectively to reproduce the previous equation. The goal then is to write the uid energy in a similar manner to the above expression for the body kinetic energy. For simplicity, assume that the uid density ρ F is equal to that of the submerged body. Then there is no constant buoyant force acting on the body. Using equation as well as the identity div(φ φ) = φ φ + φ φ and Green's Theorem, expression for the kinetic energy of the uid becomes T F = 1 ρ F φ φ ndv. 2 B Only the boundary between the solid and uid, B, is involved because the boundary term arising at innity is zero due to the assumption that the uid is at rest at innity. The velocity potential φ is a solution to Laplace's equation with Neumann boundary conditions, namely that the component of the velocity of the body normal to its surface is equal to that of the uid at the interface. Thus, φ can be expressed as a linear superposition of solutions to Laplace's equation that satises equality of the components of the uid and sleigh body velocities at the interface as expressed in a basis to be chosen ( [13]). In this way, the kinetic energy of the uid becomes expressed entirely in terms of the submerged body: T F = 1 2 ξt I F ξ. 21

38 Thus, the Lagrangian for the system is L = 1 2 ξt (I B + I F )ξ. 22

39 CHAPTER III The Chaplygin Sleigh with Oscillator in a Kircho Fluid This chapter will present a system that combines all of the features previously discussed. Specically, we will consider a Chaplygin Sleigh in a Kircho Fluid with an oscillating mass. This is a nonholonomic system with Lie group symmetry, specically the special euclidean group, SE(2). The dynamics will be split into those along this group and a shape space consisting of the oscillator. The general case of a sleigh of arbitrary shape is presented rst in section 3.1. However, the explicit form of the uid interaction terms depends on the sleigh shape, and so section 3.2 focuses on deriving the equations of motion for an elliptical sleigh body. These equations are analyzed on a case by case basis, starting with the simplest case of a circular, balanced sleigh and growing increasingly more complex in section 3.3. They are then used to create simulated motion of example sleigh-uid systems for each case, including with and without oscillator as well as with various combinations of parameters of the sleigh body chosen to be zero or nonzero in section

40 3.1 The Chaplygin Sleigh in a Kircho Fluid with Oscillating Mass Consider again the Chaplygin sleigh with a mass m allowed to oscillate along the blade direction. At the blade contact point the sleigh is prohibited from moving along the direction normal to the blade. The Lagrangian for this system is given by equation as well as the condition that ξ 3 = Ω 3 = 0. Now consider the sleigh-oscillator system moving through a potential uid that has no interaction with the oscillatory mass. Assume the uid is ideal in the sense that it is incompressible, has no vorticity, and no viscosity. Also assume that the velocity potential and its derivatives are everywhere nite and vanish at innity. The energy of this system is the sum of the energy of the sleigh-oscillator (2.2.2), and the kinetic energy of the uid. As shown in section 2.3, for an ideal uid this kinetic energy is expressed entirely in terms of the sleigh motion. The inuence of the uid on the sleigh's motion is expressed via a uid inertia tensor that is added to that of the sleigh body ( [13]). I F K1 F K2 F I F = K1 F M11 F M12 F. K2 F M21 F M22 F The cumulative Lagrangian is then that of the sleigh-oscillator system alone plus an additional kinetic energy of the sleigh that comes from this uid inertia tensor. 24

41 L = 1 2 (I + mr2 )ξ (M + m)(ξ2 2 + ξ 2 3) mṙ2 + mṙξ 2 + mrξ 1 ξ kr [IF ξ K F 1 ξ 1 ξ 2 + 2K F 2 ξ 1 ξ 3 + M F 11ξ M F 12ξ 2 ξ 3 + M F 22ξ 2 3] (3.1.1) We can now utilize a connection, chosen to be both mechanical and nonholonomic, to change coordinate systems into one which block diagonalizes the Lagrangian. In order to nd the required connection, we rst dene the new velocities ξ a = Ω a A a ṙ. (3.1.2) The third component of the connection is assumed to be zero in accordance with the nonholonomic constraint. Then the Lagrangian in terms of the new velocities is L = 1 2 (I + mr2 )(Ω 1 A 1 ṙ) (M + m)((ω 2 A 2 ṙ) 2 + Ω 2 3) mṙ2 + mṙ(ω 2 A 2 ṙ) + mr(ω 1 A 1 ṙ)ω kr [IF (Ω 1 A 1 ṙ) 2 + 2K F 1 (Ω 1 A 1 ṙ)(ω 2 A 2 ṙ) +2K F 2 (Ω 1 A 1 ṙ)ω 3 + M F 11(Ω 2 A 2 ṙ) 2 + 2M F 12(Ω 2 A 2 ṙ)ω 3 + M F 22Ω 2 3] = 1 2 (I + mr2 )(Ω A 2 1ṙ 2 2Ω 1 A 1 ṙ) (M + m)(ω2 2 + A 2 2ṙ 2 2Ω 2 A 2 ṙ + Ω 2 3) mṙ2 + mṙω 2 ma 2 ṙ 2 + mr(ω 1 Ω 3 Ω 3 A 1 ṙ) 1 2 kr [IF (Ω A 2 1ṙ 2 2Ω 1 A 1 ṙ) +2K F 1 (Ω 1 Ω 2 Ω 2 A 1 ṙ Ω 1 A 2 ṙ + A 1 A 2 ṙ 2 ) + 2K F 2 (Ω 1 Ω 3 Ω 3 A 1 ṙ) 25

42 +M F 11(Ω A 2 2ṙ 2 2Ω 2 A 2 ṙ) + 2M F 12(Ω 2 Ω 3 Ω 3 A 2 ṙ) + M F 22Ω 2 3] = 1 2ṙ2 [(I + mr 2 + I F )A (M + m + M F 11)A K F 1 A 1 A 2 2mA 2 + m] 1 2ṙ[(I + mr2 + I F )2Ω 1 A 1 + (M + m + M F 11)2Ω 2 A 2 2mΩ 2 + 2mrΩ 3 A 1 +2K F 1 Ω 1 A 2 + 2K F 1 Ω 2 A 1 + 2K F 2 Ω 3 A 1 + 2M F 12Ω 3 A 2 ] (I + mr2 + I F )Ω (M + m + M F 11)Ω (M + m + M F 22)Ω (mr + K F 2 )Ω 1 Ω 3 + K F 1 Ω 1 Ω 2 + M F 12Ω 2 Ω 3. (3.1.3) Looking at the ṙ term, all coecients of Ω a should go to zero when the connection is chosen correctly. This means: Ω 1 [(I + mr 2 + I F )A 1 + K F 1 A 2 ] = 0 Ω 2 [(M + m + M F 11)A 2 m + K F 1 A 1 ] = 0 Solving this system for the components of the connection gives the following A 1 = A 2 = mk1 F (K1 F ) 2 (I + mr 2 + I F )(M + m + M11) F (3.1.4) m(i + mr 2 + I F ) (I + mr 2 + I F )(M + m + M11) F (K1 F ) 2 (3.1.5) This choice of mechanical connection, which also serves as a nonholonomic connection because it respects the nonholonomic constraint as in section 2.1.5, can then be plugged back in to the denition of the new velocities, Ω a. In terms of this connection and velocities the 26

43 resulting Lagrangian is L = 1 2ṙ2 [(I + mr 2 + I F )A (M + m + M F 11)A K F 1 A 1 A 2 (3.1.6) 2mA 2 + m] ṙω 3 [mra 1 + K F 2 A 1 + M F 12A 2 ] (I + mr2 + I F )Ω (M + m + M F 11)Ω (M + m + M F 22)Ω (mr + K F 2 )Ω 1 Ω 3 + K F 1 Ω 1 Ω 2 + M F 12Ω 2 Ω 3 U(r). Using the canonical denition the momenta p i = l Ω i are found to be p 1 = (I + mr 2 + I F )Ω 1 + (mr + K F 2 )Ω 3 + K F 1 Ω 2, (3.1.7) p 2 = (M + m + M F 11)Ω 2 + K F 1 Ω 1 + M F 12Ω 3, p 3 = ṙ[mra 1 + K F 2 A 1 + M F 12A 2 ] + (M + m + M F 22)Ω 3 = + (mr + K F 2 )Ω 1 + M F 12Ω 2 1 (I + mr 2 + I F )(M + m + M F 11) (K F 1 ) 2 [[(mr + K F 2 )(M + m + M F 11) M F 12K F 1 ]p 1 + [M F 12(I + mr 2 + I F ) (mr + K F 2 )K F 1 ]p 2 ] ṙ[mra 1 + K F 2 A 1 + M F 12A 2 ]. Inverting the rst equations leads to expressions for the velocities in terms of the momenta 27

44 Ω 1 Ω 2 λ 1 (I + mr 2 + I F )(M + m + M11) F (K1 F ) 2 = λ M + m + M 11 F K1 F p 1 K1 F I + mr 2 + I F p 2 (3.1.8) Using these relations p 3 can be expressed in terms of the other momenta p 3 = λ[[(mr + K F 2 )(M + m + M F 11) M F 12K F 1 ]p 1 +[M F 12(I + mr 2 + I F ) (mr + K F 2 )K F 1 ]p 2 ] ṙ[mra 1 + K F 2 A 1 + M F 12A 2 ] (3.1.9) These expressions can be used in equations to calculate how r(t) and the momenta evolve in time. The oscillator equation of motion is expressed in this case as d l c dt ṙ l c r = A 1p 3 Ω 2 A 2 p 3 Ω 1 (3.1.10) Provided below are explicit calculations of partial derivatives with respect to r and full time derivatives of the Lagrangian as well as the components of the mechanical connection. l c ṙ = ṙ[(i + mr 2 + I F )A (M + m + M F 11)A K F 1 A 1 A 2 2mA 2 + m] 28

45 d l c dt ṙ = d dt [ṙ[(i + mr2 + I F )A (M + m + M F 11)A K F 1 A 1 A 2 2mA 2 + m]] = r[(i + mr 2 + I F )A (M + m + M F 11)A K F 1 A 1 A 2 2mA 2 + m] +ṙ[2mrṙa (I + mr 2 + I F )A 1 A 1 + 2(M + m + M11)A F 2 A 2 + 2K1 F (A 1 A 2 + A 2 A 1 ) 2mA 2 ] A 1 = λ 2 2m 2 K F 1 (M + m + M F 11)rṙ A 2 = [2m 2 r 2(I + mr 2 + I F )(M + m + M F 11)m 2 rλ]λṙ = 2m 2 r(k F 1 ) 2 λ 2 ṙ l c r = ṙ 2 [mra (I + mr 2 + I F )A 1 A 1r + (M + m + M F 11)A 2 A 2r + K1 F (A 1 A 2r + A 2 A 1r ) 2mA 2r ] + mrω 2 1 A 1r = 2m 2 K1 F (M + m + M11)rλ F 2 A 2r = 2m 2 rλ 2m 2 (I + mr 2 + I F )(M + m + M11)rλ F 2 = 2m 2 r(k1 F ) 2 λ 2 Then the momenta are found to evolve in time according to ṗ 1 = λ[[(mr + K F 2 )(M + m + M F 11) M F 12K F 1 ]p 1 (3.1.11) 29

46 + [M F 12(I + mr 2 + I F ) (mr + K F 2 )K F 1 ]p 2 ] ṙ[mra 1 + K F 2 A 1 + M F 12A 2 ][λ (K F 1 p 1 (I + mr 2 + I F )p 2 ) + ṙa 2 ] ṗ 2 = λ[[(mr + K2 F )(M + m + M11) F M12K F 1 F ]p 1 + [M12(I F + mr 2 + I F ) (mr + K2 F )K1 F ]p 2 ] ṙ[mra 1 + K2 F A 1 + M12A F 2 ] [λ((m + m + M11)p F 1 K1 F p 2 ) A 1 ṙ]. In all of these equations the shape of the sleigh as well as orientation of the blade with respect to the sleigh are left arbitrary. Additionally, the uid inertia terms are written as independent parameters. This isn't technically correct as section 2.3 shows that the uid inertia terms depend on the sleigh shape. A more rened treatment of this system, in which the uid inertia tensor is expressed in terms of sleigh body parameters, is provided in section The Elliptical Sleigh in a Kircho Fluid with Oscillating Mass The terms in the uid inertia tensor depend on the shape of the sleigh as shown in ( [13]). Specically, they are calculated by taking a certain integral over the boundary of the sleigh. In ( [14]) the uid inertia tensor for the specic case of an elliptical sleigh is shown to be I F = M AB 30

47 (A 2 B 2 ) 2 + a 2 A 2 + b 2 B 2 bb 2 cos α aa 2 sin α aa 2 cos α bb 2 sin α 4 bb 2 cos α aa 2 sin α B 2 cos 2 α + A 2 sin 2 A α 2 B 2 sin(2α) 2. aa 2 cos α bb 2 A sin α 2 B 2 sin(2α) A 2 cos 2 α + B 2 sin 2 α 2 Here A and B are the lengths of the semimajor and semiminor axes of the ellipse, a and b give the oset of the blade contact point from the sleigh center of mass, and α measures the angle between the ellipse axes and the sleigh blade. The oset of the blade adds o diagonal terms in the body inertia tensor as well. The general body inertia tensor is written as follows: M( A2 +B 2 + a 2 + b 2 ) + mr 2 Mb Ma + mr 4 I B = Mb M + m 0. Ma + mr 0 M + m Combining this with the uid inertia tensor, i.e., setting I = I B + I F results in a total inertia tensor. The entries in the total inertia tensor can be used to simplify the equations of motion as well as the connection presented in the previous section. Expression for the connection can be written in terms of elements of the total inertia tensor as A = mi 12 I12 2 I 11I 22 mi 11 I 11 I 22 I (3.2.1) and the momenta equations, in the previous section, become ṗ 1 = I 11p 2 I 12 p 1 p I 11 I 22 I A 2 p 3 ṙ, (3.2.2) 31