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Multibody Dynamics Notation Multibody dynamics notation Citation for published version (APA): Traversaro, S., & Saccon, A. (2016). Multibody dynamics notation. Technische Universiteit Eindhoven. Document license: Unspecified Document status and date: Published: 04/07/2016 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 26. Sep. 2021 Multibody Dynamics Notation by Silvio Traversaro and Alessandro Saccon Report Title Multibody Dynamics Notation Authors Silvio Traversaro and Alessandro Saccon Date 4 July 2016 Research Group Dynamics and Control TU/e Department Department of Mechanical Engineering, Eindhoven University of Technology Report locator DC 2016.064 1 Introduction This document introduces a notation to be used for describing kinematics and dynamics quantities, as well as their coordinate transformations, of a mechanical system composed by rigid bodies. The notation strives to be compact, not ambiguous, and in harmony with Lie Group formalism. The notation borrows from the well known notation introduced in [7], which is also used, with slight modifications, in [15]. The notation used in [7], is, unfortunately, not fully in accordance with Lie group formalism used in, e.g., [13, 14, 10], that is, however, less compact than [7], leading to long expressions when several rigid bodys are present. This report is an attempt to get the best from these two worlds, also focusing on an unambiguous notation to describe the Jacobians relating the generalized coordinates to Cartesian linear and angular velocities. 2 A quick overview on the developed notation Quick reference list for the symbols used in this document. Precise definition is given in the text below. A,B coordinate frames p an arbitrary point oB origin of B [A] orientation frame associated to A B[A] frame with origin oB and orientation [A] Ap coordinates of p w.r.t. to A A oB coordinates of oB w.r.t. to A A HB homogeneous transformation from B to A A XB velocity transformation from B to A C vA;B twist expressing the velocity of B wrt to A written in C C ^ C vA;B 4 × 4 matrix representation of vA;B C vA;B× 6 × 6 matrix representation of the twist cross product C ∗ vA;Bׯ 6 × 6 matrix representation of the dual cross product Bf coordinates of the wrench f w.r.t. B B AX wrench transformation from B to A B Bf; vA;B pairing between wrench and velocity B(ML)B 6 × 6 inertia tensor of link (=rigid body) L expressed with respect to frame B C JA;B Jacobian relating the velocity of B with respect to A expressed in C C JA;B=F Jacobian relating the velocity of B with respect to A expressed in C, where the floating base velocity is expressed in F 1 3 Math preliminaries 3.1 Notation The following notation is used throughout the document. • The set of real numbers is denoted by R. Let u and v be two n-dimensional column vectors of real numbers, i.e. u; v 2 Rn, then their inner product is denoted as uT v, with \T " the transpose operator. n×n • The identity matrix of dimension n is denoted In 2 R ; the zero column n vector of dimension n is denoted 0n 2 R ; the zero matrix of dimension n× n×m m is denoted 0n×m 2 R . • The set SO(3) is the set of R3×3 orthogonal matrices with determinant equal to one, namely 3×3 T SO(3) := f R 2 R j R R = I3; det(R) = 1 g: (1) When endowed with matrix multiplication, SO(3) becomes a Lie group, the Special Orthogonal group of dimension three. • The set so(3), read little so(3), is the set of 3×3 skew-symmetric matrices, 3×3 T so(3) := f S 2 R j S = −S g: (2) When endowed with the matrix commutator as operation, the set becomes a Lie algebra. • The set SE(3) is defined as n R p 4×4 3o SE(3) := 2 R j R 2 SO(3); p 2 R : (3) 01×3 1 When endowed with matrix multiplication, it becames the Special Eu- clidean group of dimension three, a Lie group that can be used to represent rigid transformations and their composition in the 3D space. • The set se(3) is defined as n Ω v 4×4 3o se(3) := 2 R j Ω 2 so(3); v 2 R : (4) 01×3 0 When endowed with the matrix commutator as operation, se(3) becomes the Lie algebra of the Lie group SE(3). • Given the vector w = (x; y; z) 2 R3, we define w^ (read w hat) as the 3×3 skew-symmetric matrix ^ 2x3 2 0 −z y3 ^ w = 4y5 := 4 z 0 −x5 2 so(3): (5) z −y x 0 2 Given the skew-symmetric matrix W = w^, we define W _ 2 R3 (read W vee) as _ 2 0 −z y3 2x3 _ 3 W = 4 z 0 −x5 := 4y5 2 R : (6) −y x 0 z Clearly, the vee operator is the inverse of the hat operator. • Given a vector v = (v; !) 2 R6, v and ! 2 R3, we define ^ v !^ v v^ = := 2 se(3): (7) ! 01×3 0 • Similarly to what done for vectors in R3 few lines above, we define the vee operator as the inverse of the hat operator such that ^ _ ! v v 6 := = v 2 R : (8) 01×3 0 ! • Given two normed vector spaces E and F and a function between them f : E 7! F we define as the derivative of f in x0 2 E as the linear function Df(x0): E 7! F such that: jjf(x) − f(x ) − Df(x ) · (x − x )jj lim 0 0 0 = 0 (9) x!x0 jjx − x0jj If E = R and F = R then Df(x0) is the linear function tangent to f(x), that is usually indicated as: df Df(x0) · x = (x − x0) + f(x0): dx x=x0 n m If E = R and F = R then Df(x0) is: 2 @f @f @f 3 1 1 ::: 1 @x1 @x2 @xn 6 @f2 @f2 ::: @f2 7 6 @x1 @x2 @xn 7 Df(x0) · x = 6 . 7 (x − x0) + f(x0): 6 . .. 7 4 . 5 @fm @fm ::: @fm @x1 @x2 @xn x=x0 Given a function f of p normed vector spaces f : E1 × E2 :::Ep 7! F we define as D1f the derivative of f with respect to E1, with D2f the derivative of f with respect to E2 and so on so forth. We refer to [17, Chapter 2] and to [1, Section 2.3] for further details on the derivative notation (D, D1, D2, ::: ) used in this document. 3 4 Points and coordinate frames A frame is defined as the combination of a point (called origin) and an orien- tation frame in the 3D space [4, 18]. We typically employ a capital letter to indicate a frame. Given a frame A, we will indicate with oA its origin and with [A] its orientation frame. Formally, we write this as A = (oA; [A]). Frames can be time varying. They can be used, e.g., to describe the position and orientation in space of a rigid body as time evolves. They are also used to expressed a coordinate system for a wrench exchanged by two bodies or used to define a coordinate system to describe a robot task (like a frame attached to the center of mass and oriented as the inertial frame). Newton's mechanics requires the definition of an inertial frame. In this document, we usually indicate this frame simply with A (the Absolute frame). As common practice, for robots operating near the Earth surface, we will assume the frame A to be fixed to the world's surface, disregarding non-inertial effects due to the Earth's motion. 4.1 Coordinate vector of a point Given a point p, its coordinates with respect to a frame A = (oA; [A]) are collected in the coordinate vector Ap.
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