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´ ∗ ∗Laboratório de Automa¸cãoe Controle - LAC/PTC, Escola Politécnica da USP Av. Prof. Luciano Gualberto, trav 3, 158 Cidade Universitária, São 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¸cõesexistentes entre expressõese formula¸cões em Robótica (em particular na Teoria das Helicóides) com Geometria Diferencial do ponto de vista de Cartan. A idéiade partida foi inspirada em livros de Geometria Diferential e suas aplica¸cões.A conexão de Cartan éo ponto de partida para a interpreta¸cãoposterior, assim como a derivada covariante. Apresenta-se como a cinemática e a dinâmica de sistemas de corpos r´ıgidos, com especial interesse para a área de modelagem de sistemas robóticos, sãodeduzidas, inicialmente no caso de rota¸cões puras, e depois para o caso de movimento geral no espa¸coEuclidiano. Os conceitos são generalizados para variedades Riemannianas, e conclusõesimportantes sobre Robótica são tiradas para estes espa¸cos (no que se refere a cadeias cinemáticas). Finalmente, o uso de conexõesde Cartan para sistemas com muitos frames de referências (pelo menos um inercial) éapresentado.

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 ω × r0, where there is a isomorphism between dynamic in SE(3), which describes the complete vector ω 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 ∈ 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. R(t) could velocity) presented in the last section (all groups be viewed as a trajectory in the Lie group SO(3). share this property). The positions of an arbitrary point in E3 are related by: I x S1 SO(2) 0 ~ r(t) = R(t)r (t) (1) 1 G . If one applies the time derivative in equation G d (1), one has d G q a) b) r˙ = R˙ r0 + Rr˙0 = R RT R˙ r0 + r˙ 0 = R Ωr0 + r˙ 0 (2) Figure 1: Transformation Between Frames where Ω = RT R˙ is an anti-symmetric (time variant) matrix called body angular velocity, that is a As Γ(t) is invertible, the above formula can trajectory in the Lie algebra so(3), that could be be written as Γ−1Γ˙ = ξ and the (left) Maurer- called hodograph. Both time variant matrices R(t) Cartan form is defined by Ξ = Γ−1 dΓ, where Γ and Ω(t) must be seen as time variant operators in is a generic matrix of the group (Sattinger and

937 Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014

0 Weaver, 1986). It is clear that those formulas are In fact, if one applies Dt to r , it results in related. Although it is tempting to divide both Ωr0 + r˙ 0, which is the velocity of point r0 as seen sides of the last formula by dt, it is not mathemat- in the inertial frame, but in the coordinates of the ically rigorous. In the following, it is illustrated non-inertial frame S0. the determination of a Maurer-Cartan form that Another time derivative applied in equation is mathematically rigorous. (1) would produce the transformation formula for accelerations, that involves several additional in- Examples: [Group SO(2)] This one-dimensional ertial accelerations, that corrects the Newton’s law group is the planar rotation and corresponds to for non-inertial systems. The result is very simple (or is a representation of) the unitary circle S1, to obtain, but the same could also be obtained by: as depicted in figure 1 − b). To any point in S1 (except one) there is a function θ : S1 → R that Definition 2.2 (Acceleration of Connection) associates an number in the interval [0, 2π) (that The acceleration for a given connection Ω, that is, an angle) to that point. If S1 is the unitary cir- applies to a particle with mass m is given by: cle in the complex plane, any point in S1 (except jθ ∂ for one) is given by R(θ) = e , or: a0 = D2r0 = + Ω (Ωr0 + r˙ 0) = (5) t ∂t " # cos θ − sin θ 0 ˙ 0 0 2 0 R(θ) = (3) ¨r + Ωr + 2Ωr˙ + Ω r sin θ cos θ where ¨r0 is the relative acceleration, Ω2r0 is the In figure 1 − b), it is represented a tangent centrifugal acceleration, is 2Ωr˙ 0 the Coriolis ac- (velocity) vector field θ˙(∂/∂θ), where ∂/∂θ is the celeration. This is the acceleration of point r0 as natural vector field such that dθ(∂/∂θ) = 1. A seen in the inertial frame S but expressed in the curve R(t) in this group is equivalent to a curve of non-inertial frame S0. the form R(t) = ejθ(t), where θ(t) is the angular time function. The Maurer-Cartan form in this case is deduced in the following way: given the 2.1 Dynamics of points in SO(3) matrix R in equation (3), the differential is given In order to describe the dynamic of a point with by " # mass m, the Newton’s law is reformulated as − sin θ − cos θ dR = dθ 2 0 0 cos θ − sin θ mDt r = f (6) Then, the Maurer-Cartan form is given by where f 0 represents the resulting force in the non- inertial frame S0. This could also be written in " # terms of the momentum of the particle, calculated −1 cos θ − sin θ 0 0 Ξ = R dR = · (4) as P = mDtr , and the Newton’s law would be sin θ cos θ 0 0 reformulated as DtP = f . It is easy to see that " # " # − sin θ − cos θ 0 − dθ all the inertial forces will appear. dθ = cos θ − sin θ dθ 0 2.2 Dynamics of rigid body in SO(3) If one applies the Maurer-Cartan form to the vector field θ˙(∂/∂θ), one has In order to obtain the dynamics of the rigid body motion, it is necessary to work in another repre- " # " # sentation of the Lie group SO(3) and Lie algebra 0 − dθ(θ˙ ∂ ) 0 −θ˙ so(3), that is the adjoint representation in the very ∂θ = ˙ ∂ ˙ Lie algebra so(3). In this new representation, the dθ(θ ∂θ ) 0 θ 0 covariant derivative must be defined by: which is the definition of Ω(t). In a certain sense, one could say that the (an- Definition 2.3 (Covariant Derivative) ˙ gular) velocity field θ(∂/∂θ) is transformed in a Given a connection Ω, the covariant derivative in hodograph by the Maurer-Cartan form (note that the adjoint representation is an operator that act it is unidimensional). Returning to the general in elements of so(3) given by: case of SO(3), it is possible to define: ∂ D = + [Ω, ·] (7) Definition 2.1 (Covariant Derivative) t ∂t Given a connection Ω (that in the Euclidean case is a Maurer-Cartan form), the covariant where [A, B] = AB − BA is the commutator. derivative (or total derivative) is an operator that There are two copies of so(3) that must be act in vectors and is given by: considered: 1) the copy where lives angular ve- ∂ locity and the quantities related, and 2) the copy D = + Ω t ∂t where lives vector positions, point velocities and

938 Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014 etc. An essential concept in the dynamic formu- the frame in relation to an inertial frame S (the lation is the angular momentum, that is related space frame). Again, the three vectors of S0 are ar- to the angular velocity by the a linear operator I, ranged as column vectors in an orthogonal matrix that is called the inertial operator. In the book R, that belongs to the Lie group SO(3) (in planar (Sattinger and Weaver, 1986), it is shown how movement, the group would be SO(2)). The ori- to calculate it, by working in the second copy of gin of S0 in relation to S is represented by a vector so(3), that we avoid here. The angular momen- γ ∈ E3, that are arranged in a larger 4 × 4 matrix tum is given by of the form " # R γ Γ = Λ = I(Ω) 0 1 and the Newton’s law (or Euler’s) is given by: which belongs to the Lie group SE(3). Another point of view is important here, that was not used DtΛ = T yet. To each γ, that indicates a point in E3, it is where T is the torque matrix acting in the rigid associated a copy of the Lie group SO(3) (let us body. After some substitution, we have call it SO(3)γ ). The Lie group SE(3) can then be identified with the space E3 × SO(3), which is of course, another Lie group (every vector space is ∂ DtΛ = DtI(Ω) = I(Ω) + [Ω, I(Ω)] = a particular kind of Lie group, known as Abelian ∂t Lie group). SE(3) ∼ E3 × SO(3) is also a special ˙ I(Ω) + [Ω, I(Ω)] = T kind of differentiable manifold known as (principal) fiber bundle.A section of a (principal) fiber After choosing a basis for so(3) given by bundle can be imagined as a field that associates to each point in E3 an element of SO(3), which h 0 0 0 i h 0 0 1 i h 0 −1 0 i is a frame. Any smooth field of frames represents E1 = 0 0 −1 ,E2 = 0 0 0 ,E3 = 1 0 0 0 1 0 −1 0 0 0 0 0 possible rigid body motions in all directions. A and decomposing Ω = Ω E + Ω E + Ω E , it is curve in SE(3), that is Γ(t), describes the motion 1 1 2 2 3 3 of a rigid body (translational and rotational, see, possible to show that I(Ei) = IiEi, where Ii are eigenvalues of the operator (until here, the Ein- for example, (Ivey and Landsberg, 2003)). stein convention is not used. Begin using here!). As is done in Screw Theory (see for exam- As Ω = Ω E , then ple (Selig, 2005)), the group action/representation i i of SE(3) in operators over E3 represents, in the active point of view, a translation and rotation ˙ (screw movement) of a vector in E3 and, in the ΩiI(Ei) + [ΩjEj, ΩkI(Ek)] = ˙ passive point of view, a coordinate transformation ΩiIiEi + [ΩjEj, ΩkIkEk] = from S0 to S. If Lagrangean dynamic formulation (Ω˙ mIm + IkΩjΩkikm)Em = TmEm would be applied, SE(3) would be the configuration space for the rigid body motion. The space where ikm is the totally anti-symmetric tensor. E3 is homogeneous in the sense that no point is This results in (the Euler’s equation): different of all the others (see (Kobayashi and No- mizu, 1969)). Every homogeneous space is the module of a Lie group and one of its sub-groups. I Ω˙ = (I − I )Ω Ω + T 1 1 2 3 2 3 1 In particular, E3 = SE(3)/SO(3). The motion of ˙ I2Ω2 = (I3 − I1)Ω3Ω1 + T2 points are simply curves in E3.

I3Ω˙ 3 = (I1 − I2)Ω1Ω2 + T3 If equation (1) is modiﬁed to include frame translation, then equations (2) could be written 0 In the next section, the calculations presented as r˙ = RDtr +γ ˙ . In the group SE(3), this ex- here (that came from (Sattinger and Weaver, pression is written as 1986)) will be generalized to SE(3), that is, for screws, twists and wrenches). Although the for- " # " # " 0 # mulas are well known in Screw Theory, the author r˙ R γ r = {Dt} could not ﬁnd a similar formulation in the litera- 0 0 1 1 ture (using covariant derivative and connections). where 3 Kinematics and Dynamics in SE(3) " # " # I 0 ∂ Ω RT γ˙ 3.1 Basics of SE(3) Kinematics D = + t 0 1 ∂t 0 0 Again, we attach a reference frame S0, that is non inertial, to a rigid body (that is, the body is the covariant derivative Dt to this new group frame), and also a vector to describe the origin of and connection that will be analyzed in the sequel

939 Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014

(in this case, the Cartan’s connection continues where the last term in the sum is the relative to be a Maurer-Cartan form, but for the group velocity, the matrix in (8) appears multiplying a SE(3)). position, which means that it has the role of an angular velocity. 2 0 Theorem 1 Given two reference frames S and S Evidently, to describe acceleration of a point, 3 in an Euclidean space E , such that the first one is as well as its dynamics, the equations (5), (6) and inertial, the frame curve Γ(t) is a curve in SE(3) (7) should be reformulated to the new definition ˙ and its velocity curve Γ has an hodograph in se(3), of connection, as presented in equation (8). which is also a curve of twists: " # Ω RT γ˙ 3.2 Dynamics of Rigid Bodies ξ(t) = (8) 0 0 A twist ξ ∈ se(3) is related to a screw by the exponential formula eξ (Murray et al., 1994). where Ω(t) is the rotation matrix curve in so(3). Chasles’ theorem states that any rigid movement is a screw. In a sufficiently small neighborhood of Proof: In order to deduce the formulas for trans- I ∈ SE(3), there is a one-to-one correspondence formation of velocities, the action of SE(3) in the between twists and screws, that is established space E3 rotates and translates the vectors: by the exponential functions of Lie groups (see " # " #" # (Sattinger and Weaver, 1986)). If g(t) ∈ SE(3) is r R γ r0 = (9) the motion of a rigid body in relation to an inertial 1 0 1 1 frame, Γ−1(t)Γ(˙ t) ∈ se(3) is called the body veloc- −1 ˙ that relates the coordinates of an arbitrary point ity and Γ (t)Γ(t) ∈ se(3) the spatial velocity, and in S0 and S. By differentiating both sides of (9), they are related by the adjoint action of SE(3) in one has: se(3). The same ideas presented in subsection 2.2 apply in this case, which results in the dynamic equations for rigid bodies. As generalized forces " # " #" # " #" # r˙ R˙ γ˙ r0 R γ r˙0 = + are 1-forms in the configuration space (Bullo and 0 0 0 1 0 1 0 Lewis, 2005), cotangent vectors in SE(3) are gen- (10) eralized forces in a rigid body. Due to the Maurer- Cartan form (a particular type of connection), all where the second formula at right is the action the generalized forces can be mapped, by right and 3 of SE(3) in tangent spaces Tr0 E , and the first left translations, to the Lie co-algebra se(3)∗, and will be analyzed in que sequel. By applying the those elements are called wrenches. A wrench F(t) ˙ formula R = RΩ, all the tangent vectors along acting (as a co-vector) on the rigid body with twist the curve Γ(t) can be transformed to elements in ξ(t) (the vector) is the number hF, ξi(t), which is se(3). This curve ξ(t) is called hodograph. The the instantaneous work done by the wrench. In Lie algebra se(3) has a semi-direct decomposi- section 5, a continuation o this section will be pre- tion t(3) n so(3), where t(3) is an abelian alge- sented. bra (representing translational velocities) and the second, not abelian, representing rotations. It is also the Levi decomposition of se(3) (Sattinger 4 Kinematics in Riemannian Space and Weaver, 1986). The formula in equation (10) In this section, the Cartan’s connection will be can be simply written as: presented in its full generality (that is, it will not be a Maurer-Cartan form anymore) as the space " # " #" # r˙ RΩγ ˙ r0 where the bodies move, that is M, is Riemannian. = + Also, an arbitrary number of moving frames (as 0 0 0 1 necessary in Robotics) will be assumed. In prin- (11) ciple, none of them will be inertial, and it will be " #" # " #" # R γ r˙0 R γ Ω RT γ˙ shown that it represents a big difficulty for the = · generalization of kinematic chains, that needs an 0 1 0 0 1 0 0 inertial frame of reference. " # " #" # r0 R γ r˙0 In fact, M is not a homogeneous space any- + 1 0 1 0 more, and cannot be given by module of two groups. The space of all frames in all points is That could even be written as: no longer a group, but a principal fiber bundle P with base M and canonical projection π : P → M, " # " # (" #" # " #) with fibers isomorphic to SO(3), which is the sym- r˙ R γ Ω RT γ˙ r0 r˙0 = + metry group (Kobayashi and Nomizu, 1969), (Ivey 0 0 1 0 0 1 0 and Landsberg, 2003). The action of SO(3) in P (12) is fiber preserving, that is, only rotates a frame,

940 Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014 and not change its origin. The motion of a point measures how much the structural equation is not is a curve r(t) in M. A rigid body motion is a satisfied. For the rigid body motion, the angu- curve Γ(t) ∈ P , and its projection γ(t) = π(Γ(t)) lar velocity curve Ω(t), that is calculated from the is the translational motion of the body. The prin- connection similarly as done in the example, is cipal fiber bundle has a symbolic notation, that is no longer a curve in g (in this case so(3)), but (P, M, π, SO(3)) (Kobayashi and Nomizu, 1969). has its values distributed along copies of so(3) A moving frame in P is a section (field) that in each point of M. Hodographs are not de- attributes to each point x ∈ M a reference frame fined in this cases (Kobayashi and Nomizu, 1969), sx. This could be seen as application s : M → P (Sharpe, 1997). Now, the transformation formu- that satisfy π ◦ s = IdM (the identity application las between frames and Cartan’s connections are in M). A moving frame could also be seen as deduced (see figure 2). a set of tangent vector fields {xi}, that are lin- early independent in each point (the number of Theorem 3 (Frame Transformation) Given vectors is the dimension of M). Any other mov- two arbitrary reference (moving) frames S, S0 0 ing frame could then be written in terms of those (neither necessarily inertial), and X, X its 0 vectors, which results in a field of SO(3) matrices matrices of frames, X = XA the relation between 0 0 0 X : M → P . them, and ∇X = X Ξ , ∇X = XΞ its directional The concept of Cartan’s connection, in its full derivatives, the relation between the fields are generality, appears in the following way: to a given given by: moving frame s, it is associated a 1-form field Ω 0 with values in so(3), that represents the frame an- Ξ = A−1 dA + A−1ΞA (13) gular velocity. It is formally defined as the Dar- and is called a frame (gauge) transformation, and boux’s derivative of the moving frame, and it is Ξ, Ξ0 are two connections. calculated as s∗(ψ) (the pullback of ψ by s), where ψ is the Maurer-Cartan form of SO(3) (Ivey and Proof: The formula in (13) can be easily demon- Landsberg, 2003). In general: strated (Spivak, 2005):

Definition 4.1 (Cartan’s Connection) Given 0 0 0 a Riemannian space M, a Cartan’s connection ∇X = X Ξ = ∇(XA) = (14) Ξ is a field of 1-forms, with values in some Lie X dA + ∇XA = X dA + XΞA algebra g of some Lie group G, which contains 0 −1 −1 the geometric information of how frames change which results in Ξ = A dA + A ΞA. in any direction. 2 Then, the formula in the theorem is a trans- The Cartan’s connection is no longer a formation between connections in different frames. Maurer-Cartan form of a group (as was in the SE(3) case) but is related to such a form (in fact, Definition 4.2 (Covariant Derivative) it appears in connections’ definition). In fact, Given a (Riemannian) space M and a connection when a principal fiber bundle reduces to a Lie Ξ with values in g, the associated covariant group, a Cartan’s connection reduces to a Maurer- derivative is given by: Cartan form (Sharpe, 1997). A natural question to ask is: what are the conditions for a smooth ∂ Dt = I + Ξ (15) field s : M → P be a field of frames ? The an- ∂t swer: where I is the identity in g.

Theorem 2 (Cartan’s Theorem) Let G a ma- If three moving frames are involved (see figure trix Lie group and its Lie algebra g with Maurer- 2), which is necessary in Robotics, the following Cartan form ψG. Let also be a manifold M and a results are important. Suppose there are three 1-form Ω with values in g such as dΩ + Ω ∧ Ω = 0 moving frames S, S0 and S00, with matrices X, 0 00 (∧ is the exterior product). Then, in each point X and X , and the transformation matrices from 0 00 0 0 x ∈ M, there is a neighborhood U and an applica- one frame to another X = XA and X = X A . tion s : U ⊂ M → G such as s∗(ψ ) = Ω. Also, Consider also its directional derivatives given by G 0 0 0 00 00 00 given two such applications s1, s2, they must sat- ∇X = XΞ, ∇X = X Ξ and ∇X = X Ξ , 0 00 isfy s1 = as2 for some a ∈ G. where the matrices Ξ, Ξ and Ξ are the Cartan’s connection. Then: The equation dΩ + Ω ∧ Ω = 0 is called the structural equation, and is satisfied by only in eu- Proposition 4.1 (Transformation Formulas) clidian spaces (by Maurer-Cartan forms). Any Given the moving frames S, S0 and S00, then the other connection will not fit, which means that transformation formula for the connections are: there exists a 2-form Θ with values in g called 00 0 −1 0 0 −1 0 curvature which is equal to Θ = dΩ + Ω ∧ Ω and Ξ = (AA ) d(AA ) + (AA ) Ξ(AA ) (16)

941 Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014

z’ Lemma 5.1 Given the matrix (RR0)−1 d(RR0), 0 T 0 0 0 z’’ it reduces to (R ) ΩR + Ω , and Ω + Ω for the z S’ y’ S’’ case of SO(2). y’’ The covariant derivative in this case is obvi- S ously deﬁned, and the following theorem shows that the Newton’s law can be correctly given by a x y x’’ generalization of equation (6). 00 Theorem 4 The Newton’s law in the frame S Figure 2: Transformation Between Reference 00 0 2 00 has the expression f = m(Dt) r with covariant Frames derivative given by:

Proof: By applying formula (13) two times, one ∂ 0 T 0 0 Dt = + (R ) ΩR + Ω (18) has: ∂t Proof: By applying two times the time derivative 0 00 00 0 0 0 0 in the expression r = RR r , one has Ξ = (A )−1 dA + (A )−1 A−1 dA + A−1ΞA A 00 0 00 0 T 0 00 0 0 0 0 0 r¨ + 2Ω r˙ + 2(R ) ΩR r˙ + = (A )−1 dA + (A )−1A−1 dAA = (A )−1 h 0 0 T 0 i 00 h 0 2 0 T 2 0 i 00 h 0 0 i 0 0 Ω˙ + (R ) Ω˙ R r + (Ω ) + (R ) Ω R r + dA + A−1 dAA + (AA )−1Ξ(AA ) = 0 T 0 0 00 0 0 0 0 2(R ) Ω(R )Ω r (AA )−1 d(AA ) + (AA )−1Ξ(AA ) Applying now the formula (18) two times, one has and that concludes the proof. 2 2 0 00 ∂ 0 00 According to (Spivak, 2005), if the space M (D )2r = + (R0)T ΩR + Ω0 r = t ∂t is flat (that is, the curvature 2-form Θ is null), ∂ 0 00 0 00 00 the principal fiber bundle can be reduced to a Lie + (R0)T ΩR + Ω0 r˙ + (R0)T ΩR r + Ω0r group, and it is possible to find a moving frame in ∂t which the connection reduces to a Maurer-Cartan After some manipulation, it is easy to show that form. In other words, its is possible to find a both formulas coincide, which concludes the proof. frame, say S, in which the corresponding connec- 2 tion is Ξ = 0. If such a connection exists, then its is possible to find a frame where no inertial forces Corollary 5.1 In case of SO(2) (planar move- are present, in other words, it is possible to find ment), one has an inertial reference frame. In other words, it says 00 0 00 0 00 0 2 00 r¨ + 2 Ω + Ω r˙ + Ω˙ + Ω˙ r + Ω + Ω r that Robotics calculations (for example, the determination of angular velocities of links) would be Proof: By applying the commutative property of in trouble, as it is strongly based in the existence Ω and Ω0 with other matrices, it is easy to com- of an spatial (inertial) reference frame. The study plete the proof. 2 of how to do such calculations will be subject of In case of SE(2) (that is, planar movement), future research. one has:

5 Returning to Kinematics in SE(2) Theorem 5 The Maurer-Cartan form for SE(2) 00 in frame S has expression In this ﬁnal section, it will be shown, in the con- " 0 0 T T 0 # 00 Ω + Ω (R ) Ωγ + R dγ + dγ text of Screw Theory in Euclidean planar space, ξ = (19) how the various formulas in kinematic and dy- 0 0 namic chains can be deduced in the context of Proof: By calculating the inverses of a gen- Cartan’s connections (in this case, Maurer-Cartan eral matrix in SE(2) and its exterior derivatives forms). To the best of the author knowledge, the dA, dA0 and substituting these formulas in (19), theorems and lemmas in this section, in the con- one has: text of Cartan’s, are not in the literature. In case of Euclidean space, one has: " #" # (R0)T −(R0)T γ0 RT −RT γ ξ00 = · 0 1 0 1 00 0 −1 0 0 −1 0 Ξ = (AA ) d(AA ) + (AA ) 0(AA ) (17) (" #" # dR0 dγ0 R0 γ0 0 −1 0 · = (AA ) d(AA ) 0 0 0 1 " #" #) In the particular case of frames with the same R γ dR dγ origin (that is, geometry of SO(3), it is easy to + 0 1 0 0 prove the following lemma.

942 Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014

References " # (R0)T RT −(R0)T RT γ − (R0)T γ0 Ball, R. S. (1876). The theory of screws: A study ξ00 = · 0 1 in the dynamics of a rigid body, Mathematis- che Annalen 9(4): 541–553. " # ( dR)R0 + R dR0 − dRγ0 + dγ + R dγ0 Bullo, F. and Lewis, A. D. (2005). Geometric Con- 0 0 trol of Mechanical Systems: Modeling, Anal- ysis, and Design for Simple Mechanical Con- By evoking lemma 5.1, the result follows. trol Systems, number 49 in Texts In Applied 2 Mathematics, Springer. The connection can be calculated for each Choquet-Bruhat, Y., DeWitt-Morette, C. and frame of sequence (kinematic chain) both in the 3 Dillard-Bleick, M. (1989). Analysis, Mani- representation of SE(3) in E and in the adjoint folds and Physics: Part II: Applications, re- representation (that is, the covariant derivative vised edn, North-Holland. similar to the one in equation (7)). It is then possible to formulate rigid body dynamics in this Craig, J. J. (1989). Introduction to Robotics: Me- representation. chanics and Control, Addison-Wesley Long- man.

6 Conclusions Ivey, T. A. and Landsberg, J. M. (2003). Car- tan for Beginners: Differential Geometry It was presented some concepts and results re- via Moving Frames and Exterior Differen- lating the Screw Theory (used in Robotics) to tial Systems, Vol. 61 of Graduate Studies in concepts in Differential Geometry and Lie groups Mathematics, American Mathematical Soci- and Algebras. In particular, it was shown that ety. Cartan’s connections (a differential geometric concept) is the natural generalization for angular ve- Kobayashi, S. and Nomizu, K. (1969). Founda- locities and twists for general Riemannian spaces. tions of Differential Geometry, Vol. one, In- It was also shown that Newton-Euler formulation terscience Publishers. of Robotics are written in this context in an invari- Murray, R. M., Li, Z. and Sastry, S. (1994). A ant form (by using connections) and more research Mathematical Introduction to Robotic Manip- has to be done in order to generalize Robotics to ulation, CRC Press, Corporate Blvd., Boca the Riemannian case (necessity to redefine kine- Raton, Fl 33431. matic and dynamic chains, as no inertial reference frame is possible in general). To the best of the Sattinger, D. H. and Weaver, O. L. (1986). author’s knowledge, such an exposition is lacking Lie Groups and Algebras with Applications in the Robotic’s Literature, as well as some of the to Physics, Geometry and Mechanics, num- formulas and theorem’s proofs. ber 61 in Applied Mathematical Sciences, In future works, the question of how to cal- Springer-Verlag. culate angular velocities of robotic chains in Rie- Schwarz, A. S. (1996). Topology for Physi- mannian spaces, as well as questions related to the cists, Grundlehren der Mathematischen Wis- dynamics of rigid bodies, will be investigated, as senschaften - A Series of Comprehensive well as possible practical applications of the re- Studies im Mathematics, Springer. sults presented. Also, several other formulas and theorems had to be excluded from this paper, due Selig, J. M. (2005). Geometric Fundamentals of to the lack of space in a conference paper, as well Robotics, second edn, Springer. as important practical matters like singular con- figurations. In a (near) future extended version of Sharpe, R. W. (1997). Differential Geome- this work, to be submitted to a journal, all those try: Cartan Generalization of Klein Erlan- questions will have their deserved places. gen Program, number 166 in Graduate Texts in Mathematics, Springer.

Acknowledgments Spivak, M. (2005). Comprehensive Introduction to Diﬀerential Geometry, Vol. Two, 3rd edn, The author thanks to FAPESP for the grant to Publish or Perish, INC. participate in this reunion, to Prof. Mauricio Be- cerra Vargas for suggesting valuable bibliograph- ical references, and to the anonymous reviewers, that kindly oﬀered suggestions for the improve- ment of this work.

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