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CONNECTIONS BETWEEN SCREW THEORY and CARTAN's CONNECTIONS Diego Colón ∗Laboratório De Automaç˜Ao E Controle

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CONNECTIONS BETWEEN SCREW THEORY and CARTAN's CONNECTIONS Diego Colón ∗Laboratório De Automaç˜Ao E Controle Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014 CONNECTIONS BETWEEN SCREW THEORY AND CARTAN'S CONNECTIONS Diego Colon´ ∗ ∗Laborat´orio de Automa¸c~aoe Controle - LAC/PTC, Escola Polit´ecnica da USP Av. Prof. Luciano Gualberto, trav 3, 158 Cidade Universit´aria, S~ao Paulo, BRAZIL Email: [email protected] Abstract| In this paper some relations that exist between expressions in Screw Theory of Robotics and Differential Geometry, from the point of view of Cartan, are presented. The ideas were inspired in books of Differential Geometry, Lie groups and Lie algebras and its applications. The Cartan's connection is the initial concept, followed by the covariant derivative. It is presented how kinematics and dynamics are obtained, in case of pure rotations and general movement in Euclidean space. The generalization of these concepts to Riemannian manifolds are then presented, and some conclusions are drawn concerning the possibility of kinematic chain calculations in such space (propagation of velocities). Finally, the idea of using connections and covariant derivatives are extended to multi-frame cases (as is common in Robotics) in Euclidean space. Keywords| Screw Theory, Riemannian geometry, Cartan's Connection. Resumo| Este artigo apresenta algumas rela¸c~oesexistentes entre express~oese formula¸c~oes em Rob´otica (em particular na Teoria das Helic´oides) com Geometria Diferencial do ponto de vista de Cartan. A id´eiade partida foi inspirada em livros de Geometria Diferential e suas aplica¸c~oes.A conex~ao de Cartan ´eo ponto de partida para a interpreta¸c~aoposterior, assim como a derivada covariante. Apresenta-se como a cinem´atica e a din^amica de sistemas de corpos r´ıgidos, com especial interesse para a ´area de modelagem de sistemas rob´oticos, s~aodeduzidas, inicialmente no caso de rota¸c~oes puras, e depois para o caso de movimento geral no espa¸coEuclidiano. Os conceitos s~ao generalizados para variedades Riemannianas, e conclus~oesimportantes sobre Rob´otica s~ao tiradas para estes espa¸cos (no que se refere a cadeias cinem´aticas). Finalmente, o uso de conex~oesde Cartan para sistemas com muitos frames de refer^encias (pelo menos um inercial) ´eapresentado. Palavras-chave| Teoria das Helic´oides, Geometria Riemanniana, Conex~ao de Cartan. 1 Introduction the space (Sharpe, 1997). Cartan's connection is a central concept in modern Particle Physics, as Concepts of Differential Geometry and Lie Groups it can represent eletromagnetic potential, gluons and Algebras have been used in Robotics for a (particles that carry the strong force) and many long time, with the first appearing in classical ref- others (Schwarz, 1996). In the case of Screw The- erences as (Craig, 1989), where transformation be- ory, moving frames are fields of screws, and Car- tween two reference frames are matrices in the ma- tan's connections are fields of twists. trix group SE(3) (or SE(2), if the manipulator's Connections are essential in the definition of movement is planar). More recently, Screw The- time derivatives in non-inertial moving frames (co- ory, that was developed by (Ball, 1876), started to variant derivative or total derivative), and by us- be applied, but the modern definitions of screws, ing covariant derivative, it is possible to rewrite twists and wrenches as elements of SE(3), se(3) Newton's law and Euler's equation in a form that and se∗(3), (respectively, a Lie group, its Lie al- is invariant in any non-inertial frame. This frame- gebra and Lie co-algebra), are far more recent work even predict that serious troubles must be (Sattinger and Weaver, 1986). Today, there are tackled in a possible generalization of Robotics to very complete references on the subject (Murray the Riemannian case, as it is easy to prove (and et al., 1994). will be done in this work) that no reference frame Cartan's connections are geometrical objects exists that can be used as base of a kinematic that are fundamental to modern Geometry and chain (propagation of velocities and accelerations) Physics, besides the concept of moving frames like is done in the Euclidean space. (Schwarz, 1996), (Choquet-Bruhat et al., 1989). The objective of this work is manifold, but In simple terms, given a manifold M (that could could be roughly divided in the following objec- be Riemannian), a moving frame associates to tives: 1) convince the reader that angular veloc- each point in M a reference frame (in general, non- ities and twists are generalized by the concept of inertial), and a Cartan's connection is a derivative Cartan's connection; 2) convince the reader that field of a moving frame that generalizes the con- the Newton/Euler's equations could be formu- cept of gradient. A common physical interpre- lated by using the concept of covariant derivative; tation (in Mechanics) is that a Cartan's connec- 3) To show that this way of formulating Screw tion generalizes the angular velocity of the mov- Theory is the correct way to generalize for the ing frame (or of a moving rigid body) to a Rie- case of Riemannian geometry, as well as show the mannian manifold, and it contains geometric in- problems that must be surpassed) and 4) Present formation, which is related to the curvature of some formulas (and its proofs) that this author 936 Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014 could not find in the specialized literature. Con- space E3 that are representations of group SO(3) cepts and calculations will be presented to support and Lie algebra so(3) in the vector space E3. In itens 1), 2) and 3), as well as references. fact, kinematics and dynamics of points use repre- The point of departure of this work was the in- sentations in E3, but kinematics and dynamics of variant Newton-Euler's formulation of rigid body rigid bodies uses the adjoint representation, that dynamics presented by (Sattinger and Weaver, are Lie groups and algebras representations in the 1986), section 14, pages 44-48, that is adapted very Lie algebra (in this case so(3)), which is also and presented in section 2. It treats the kine- a vector space. Another important fact is the Lie matics and dynamics of a non-inertial frame (that algebra isomorphism between so(3) and R3, with could represent a rigid body) that only performs the Lie bracket [A; B] = AB − BA corresponding rotations around a fixed point. The precise no- to the vector product × of Analytic Geometry. tion of connections and covariant derivative is pre- The action of so(3) in E3 given by applying Ω in sented, as well as its dynamics. In section 3, it r0, (that is Ωr0), is equivalent to the vector prod- is presented the basic idea of the kinematic and uct ! × r0, where there is a isomorphism between dynamic in SE(3), which describes the complete vector ! and matrix Ω. The vector r_ 0 is called rel- movement of points and rigid bodies in the Eu- ative velocity, and equation (2) shows the trans- clidean three-dimensional space E3. The reader formation formula from the non-inertial frame to will easily recognize Screw Theory concepts, but the inertial one for the velocities. it is a preparation for the next sections. The matrix Ω(t) contains the information of In section 4, it is presented important con- how the (moving) frame S0 moves, and in the book cepts of Riemannian spaces, that are useful to (Sattinger and Weaver, 1986), it is said that it generalize Robotics to Riemannian space, and it is is a (Cartan's) connection, although the point of shown that the kinematics calculation are in trou- view to be presented here are not developed there ble in this case, due to the space curvature (there (and, as said before, the body is fixed by a point). is no inertial reference frame). In section 5, after Also, as the space is Euclidean, Cartan's connec- the results obtained in the previous sections, it is tion reduces to a Maurer-Cartan form restricted shown how to determine kinematics and dynam- to a curve R(t), that is a special kind of connection ics of manipulators (several frames) in Euclidean (from the didactic point of view, the understand- space (in the Cartan's connections context), and ing of Maurer-Cartan is a preparation for under- some formulas are proved (that are believed not to standing connections). be in the literature). Finally, in section 6, conclu- In order to further discuss Maurer-Cartan sions and directions of future works are presented. (see, for example, (Ivey and Landsberg, 2003)), it is important to remember the following: given a 2 Kinematics and Dynamics in SO(3) Lie group G, its Lie algebra g is the tangent space in the identity element I 2 G, like presented in fig- Following the steps of the book (Sattinger and ure 1 − a). Any tangent vector in G, represented Weaver, 1986), any point in the Euclidean space by a velocity Γ_ of a curve Γ(t), can be translated E3 is represented by a three dimensional vector r0 to the identity element I by a left multiplication in a non-inertial frame S0, and by a vector r in by Γ. This mapping is called left translation, and the inertial frame S. Both frames have the same the formula Γ_ = Γξ allows to represent any ve- origin. The transformation matrix between the locity Γ_ as elements of the Lie algebra g. It is two frames, that depends on time, is represented clear that it is the generalization, for an arbitrary by R = R(t), whose columns are the vectors of group, of the formula R_ = RΩ in SO(3) (body's S0 expressed in the coordinates of S.
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