Cartan connection applied to dynamic calculation in robotics Diego Colón, 1 Phone +55-11-3091-5650 Email [email protected] 1 Laboratório de Automação e Controle - LAC/PTC, Escola Politécnica da Universidade de São Paulo, São Paulo, Brazil Received: 31 January 2017 / Accepted: 6 June 2018 Abstract A Cartan connection is an important mathematical object in differential geometry that generalizes, to an arbitrary Riemannian space, the concept of angular velocity (and twists) of a non-inertial reference frame. In fact, this is an easy way to calculate the angular velocities (and twists) in systems with multiple reference frames, like a robot or a complex mechanism. The concept can also be used in different representations of rotations and twists, like in the Lie groups of unit quaternions and unit dual quaternions. In this work, the Cartan connection is applied to the kinematic and dynamic calculations of a systems of with multiple reference frames that could represent a robotic serial chain or mechanism. Differently from previous works, the transformation between frames is represented as unit quaternions, which are known to be better mathematical representations from the numerical point of view. It is also generalized to this new quaternion representation, previous results like the extended Newton’s equation, the covariant derivative and the Cartan connection. An example of application is provided. AQ1 Keywords Quaternions Robotics Lie groups Screw theory Technical Editor: Victor Juliano De Negri. 1. Introduction Concepts of differential geometry and Lie groups and Lie algebras have been used in many areas of engineering for a long time. More recently, a renewed interest could be verified in mechanism and robotics [1, 2, 10, 16, 23], where dual quaternions were used, and very complete reference books, like [3], appeared. Classical references in robotics, like [9, 22], used only vector analysis, but some Lie group’s concepts (like orthogonal matrices and the Lie group SE(3)) were already in use. A systematic use of differential geometry, on the other hand, is far more recent [14, 19]. In control theory, differential geometry has also been explored [8, 15]. A Cartan connection is a concept that is fundamental to modern physics and differential geometry, besides the concept of moving frames [4, 18, 20]. In simple terms, given a manifold M (that could be Riemannian), and a field S of reference frames in M, which associates with each point in p ∈ M a reference frame Sp (generally non-inertial), a Cartan connection Ξ is a derivative field of S such that Ξp contains the information of how the frame Sp changes in each direction. This is the generalized gradient of S, as it also depends on the curvature of M (see Fig. 1(a) for an illustration of a field of frames). Computationally, S is a field of matrices, as well as Ξ , but of distinct types. In practical terms, a Cartan connection is the angular velocity field in case of pure rotational motion (motion in SO(3)), and a twist (differential screw) in general rigid motion in SE(3) [14]. The inspiration to make this work came from [17] and started in the paper [6], where this author continues the approach in [17] and applies the Cartan connection to obtain some formulas for kinematic chain calculations in robotics, doing them in the Lie groups SO(3), SE(3) and its Lie algebras. Another essential concept, associated with the Cartan connections, is the covariant derivative, and by using it, it is possible to write the Newton’s law and the Euler’s equations in forms that are invariant in any non-inertial frame. This allows the dynamic calculations to be written in an arbitrary non-inertial reference frame. Another advantage of the Cartan connection is that it can be applied to the description of motions in different representations, as in the quaternions, dual quaternions and general Clifford algebras representations [19]. In fact, a pure rotational motion is described, in terms of the unit quaternions, in the Lie group SU(2) (special unitary complex matrices) and the general rigid motion in the Lie group of unit dual quaternions (see preliminary results in [7, 24]). Fig. 1 Frame field and principal fiber bundle. In this work, it is shown how the dynamic calculations can be done in the framework of Cartan connections, continuing the work presented in [6, 7, 24]. In Sect. 2, the kinematics of pure rotational motion (with several reference frames) and rotation plus translation, using the Lie groups SO(3) and SE(3), is are shortly described. These are only reproduced results already obtained by this author in [5, 6]. The precise notion of Cartan connections and covariant derivative are presented, which is are a classical differential geometric concepts, but essential to the approach with quaternions. In Sect. 3, some concepts of differential geometry and of the Lie group SU(2) that are not standard are presented. In particular, the principal fiber bundle with group SU(2) is presented with details, as it is essential to kinematics and dynamics with unit quaternions. The Cartan connection and the covariant derivative for this case is explored. In Sect. 4, the kinematics for a serial robot is presented and the recursive formula for the Cartan connection is generalized to the case of the group SU(2), which is the main formula for kinematics and is analogous to the recursive formula for the angular velocity calculation. This result will be necessary to the next section. In Sect. 5, the recursive formulas for the dynamic calculations are presented for translational and rotational motions, but in the case of the Lie group SU(2) of unit quaternions. In Sect. 6, the method is applied to a classical example in the literature. In Sect. 7, conclusions and suggestions for future work are presented. The point of view of applying the Cartan connection to kinematic and dynamic calculation in robotics is original of this author, to the best of his knowledge. The new results are presented in Sects. 4, 5 and 6. 2. Kinematics in SO(3) and in SE(3): Cartan connection’s point of view In the following two subsections, a concise review of the basic concepts presented in the literature and some previous results obtained in other publications of this author, particularly [5, 6, 7], are presented. 2.1. Kinematics in SO(3) and SE(3) 3 Following the steps of [5, 17], any point in the Euclidean space R is represented by a 3-vector r′ in the coordinates of a non-inertial frame S′ , and by a 3-vector r in the coordinates of the inertial frame S. Suppose that both frames have the same origin. The transformation matrix from the coordinates of S′ to the coordinates of S is R = R(t) ∈ SO(3) , whose columns are the 3-vectors of the basis of S′ expressed in the coordinates of S. Those 3-vectors are related by r(t) = R(t)r′(t) and after the application of the time derivative, one has ˙r = R (Ωr′ + ˙r′) , where Ω = RT R˙ is an anti-symmetric (time variant) matrix called body angular velocity, that is a trajectory in the Lie algebra so(3) , and represents the angular velocity of S′ in relation to S but 3 in the coordinates of S′ . The matrices R(t) and Ω(t) are operators in R that are representations of the Lie group SO(3) and the Lie algebra so(3) , respectively, in the 3 3-vector space R [19]. The frame S′ could also be attached to a rigid body, and R(t) could be viewed as a trajectory in SO(3) describing the rotational motion of the rigid body. The simplest form of a Cartan connection is as a field of angular velocity matrices Ω , when the motion of the reference frame (or rigid body) is purely rotational. It can be visualized as a derivative (gradient) field of the frame field represented in Fig. 1(a). As presented in [17], the covariant derivative is an operator that acts in the 3-vectors 3 of R and is given by Dt = ∂/∂t + Ω . If one wants to calculate the time derivative of any 3-vector in a non-inertial reference frame S′ , this operation must be used. In ′ ′ ′ fact, if one applies Dt to r , it results in Ωr + ˙r , which is the velocity of the point r′ as seen in the inertial frame S, but in the coordinates of the non-inertial reference ′ ′ 2 ′ frame S . Another application of the time derivative results in a = Dt r , which is the definition of the acceleration in non-inertial reference frames, and given by ′ 2 ′ ′ ˙ ′ ′ 2 ′ ′ 2 ′ a = Dt r = ¨r + Ωr + 2Ω˙r + Ω r , where ¨r is the relative acceleration, Ω r is the centrifugal acceleration, and 2Ω˙r′ is the Coriolis acceleration. This is the acceleration of point r′ as seen in the inertial frame S but expressed in the coordinates of the non-inertial frame S′ . In order to describe the motion of an arbitrary point of a rigid body, one has to assume that ˙r′ = ¨r′ = 0 . It is easy to deduce that, in the case of a translational + rotational motion of S′ in relation to S, ′ 2 ′ −1 the acceleration is given by a = Dt r + R o¨ , where o is the position of the origin of S′ in relation to S and in the coordinates of S. 3 There is a Lie algebra isomorphism between (so(3), [, ]) and (R , ×) , where [A, B] = AB − BA is the Lie bracket and × is the vector product. This means that 3 there is a correspondence between Then, to the the 3-vectorss r, r′ ∈ R corresponds to and respectively the matrices X, X′ ∈ so(3) describing the motion of a point such that the Lie bracket of those matrices correspond to the vector product of the two vectors .