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Differential-Geometric Modelling and Dynamic Simulation of Multibody Systems

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Differential-Geometric Modelling and Dynamic Simulation of Multibody Systems Strojarstvo 51 (6) 597-612 (2009) A. MÜLLER et. al., ��������������������������������������������������Differential-Geometric Modeling and Dynamic... ����597 CODEN STJSAO ISSN 0562-1887 ZX470/1418 UDK 531.392:531.314:004.94:512.81 Differential-Geometric Modelling and Dynamic Simulation of Multibody Systems Andreas MÜLLEr1) and Zdravko TErZE2) Original scientific paper A formulation for the kinematics of multibody systems is presented, 1) University Duisburg-Essen which uses Lie group concepts. With line coordinates the kinematics is Chair of Mechanics and Robotics parameterized in terms of the screw coordinates of the joints. Thereupon, the Lagrangian motion equations are derived, and explicit expressions are Lotharstrae 1, Germany given for the objects therein. It is shown how the kinematics and thus the 2) Fakultet strojarstva i brodogradnje Sveučilišta motion equations can be expressed without the introduction of body-fixed u Zagrebu (Faculty of Mechanical Eng. & reference frames. This admits the processing of CAD data, which refers to Naval Arch, University of Zagreb), a single (world) frame. For constrained multibody systems, the Lagrangian Ivana Lučića 5, HR-10000 Zagreb motion equations are projected to the constraint manifold, which yields the Republic of Croatia equations of Woronetz. The mathematical models for numerical integration routines of MBS are surveyed and constraint gradient projective method [email protected] for stabilization of constraint violation is presented. Diferencijalno-geometrijsko modeliranje i dinamička Keywords simulacija diskretnih mehaničkih sustava s kinematičkim CAD vezama Constraint violation Lagrange equations Izvornoznanstveni članak Lie groups U radu je prikazano matematičko modeliranje kinematike diskretnih Multibody systems mehaničkih sustava s kinematičkim vezama pomoću Lievih grupa. Numerical integration Kinematika sustava parametarizirana je koristeći vijčane koordinate Woronetz equations zglobova kinematičkog lanca. Nastavljajući se na takav kinematički model, Lagrangeove dinamičke jednadžbe gibanja sustava izvedene su u nastavku Ključne riječi rada. Koristeći takav pristup, pokazano je kako se kinematički model, a CAD također i dinamičke jednadžbe gibanja mehaničkog sustava, mogu izvesti Diskretni mehanički sustavi s kinematičkim bez upotrebe lokalnih koordinatnih sustava vezanih za pojedina tijela vezama kinematičkog lanca. Takvo matematičko modeliranje omogućava izravnu Jednadžbe Woronetza upotrebu CAD podataka koji se, u pravilu, izražavaju u jedinstvenom Lagrangeove jednadžbe koordinantnom sustavu. U slučaju dodatnih kinematičkih ograničenja Lieve grupe narinutih na sustav, jednadžbe gibanja izvedene su projiciranjem Lagrangeovih jednadžbi na višestrukost ograničenja, čime se model Numerička integracija izražava u obliku jednadžbi Woronetza. U radu su, nadalje, prikazane Numeričke povrede kinematičkih ograničenja formulacije matematičkih modela koji se koriste kao podloga numeričkih algoritama za vremensko integriranje jednadžbi dinamike, a također je, Received (primljeno): 2009-04-30 uz izrađeni numerički primjer, opisana i metoda stabilizacije numeričkih Accepted (prihvaćeno): 2009-10-30 rješenja na višestrukosti ograničenja. 1. Introduction concepts, is the field of geometric integration [10]. The subject here is the integration of dynamic systems on Geometric methods in dynamics and control have manifolds, of which systems on Lie groups are special become widely established and valued approaches. The cases [12, 27]. actual meaning of ’geometric’ varies with the context In mechanism theory, one is concerned with the in which it is used. It generally refers to the differential geometry of finite motions, and on velocity level with geometry of the mathematical structures underlying their differential geometry. As it turns out in the case of a dynamic system. As an example, non-linear control rigid multibody body systems (MBSs), the differential- theory for continuous systems is solely built upon the geometric objects correspond to with the line geometry differential-geometry of the control system [5, 21-23]. In of the joints and bodies. The underlying reason is the other words, generally, one deals with control systems isomorphism of the algebra of screws and the algebra of the on manifolds. Another area that makes use of geometric Lie-group of rigid body motions [20, 34]. The latter is the 598 A. MÜLLER et. al., Differential-Geometric Modeling and Dynamic... Strojarstvo 51 (6) 597-612 (2009) group of isometric orientation preserving transformations MBS. It is assumed hereafter that Γ has one connected of the three-dimensional Euclidean space. In essence, the component. In this case Γ has γ = m − N fundamental parameterization of screws with line coordinates gives loops (FLs), where m:= |J| is the number of joints and N:= rise to geometric treatment of differential-geometric |B| − 1. Joints and bodies are respectively denoted by Jα, entities. Screw theoretic considerations also gave rise to α = 1, . ,m and Bk, k = 0, . ,N. Body B0 stands for the the development of advanced methods for the design of ground. An edge Jα∈Γ is an unordered pair Jα≡ (Bk, Bl) mechanisms [14, 25, 36]. Besides these geometric and (for brevity, we write α∈Γ or (k, l) ∈Γ if Jα ≡ (k, l)). kinematic applications, explicit use of the Lie group A spanning tree of Γ is denoted by G = (B, JG), and the structure admits to expressed motion equations of corresponding cotree by H (Γ,G) = (B, JH), or simply with mechanisms in a very compact and elegant way [4, 11, G H. Vertex B0 is chosen as the root of the tree. J is the set 17, 24, 26, 28] and also admits application of integration of tree-joints, and JH := J \ JG is the set of cut-joints. G schemes on Lie groups [25]. The difference of classical comprises N tree joints. The term cut-joint indicates that, matrix formulations and recursive formulations [1, if we ’cut’ these joints, we get a tree topology. 7-8, 32-33] is that these formulations are entirely formulated in terms of screw quantities, corresponding J to the mechanism’s joint geometry (by virtue of the 6 B parameterization with line coordinates). Furthermore, 5 J B 7 the coordinate invariance allows for a formulation 2 without body-fixed reference frames and hence for the B7 B6 J8 J9 incorporation of CAD data. This fact will be emphasized J2 throughout the paper. J5 The paper is organized as follows. Section 2 deals B with the geometry and summarizes the representation of 4 configurations of rigid MBSs. After recalling the graph J4 representation of the MBS topology, configurations B1 B are expressed as elements of the 3-dimensional special 3 Euclidean group. The configuration space is then J parameterized using canonical (axis/angle) and non- J 3 canonical parameters. A formulation is given that does not 1 need the definition of body-fixed frames. The kinematics is addressed in section 3. The closed form expression for rigid body velocities in terms of the instantaneous joint screws and the time derivatives of the parameters is given. This is a recursive relation that can also be expressed in matrix form. In section 4 the dynamics is addressed and the Lagrangian motion equations are given. Their differential-geometric meaning is briefly outlined, making use of the Riemannian metric defined by the generalized mass matrix. The motion equations are also given in matrix form. For completeness, the equations of Woronetz are given, i.e. a formulation of the motion equations of constrained MBSs, which does not involve Lagrange multipliers. In section 5 the mathematical models for numerical integration routines of MBS are surveyed. Furthermore, constraint gradient projective method for stabilization of constraint violation is addressed and a numerical example is presented. Figure 1. An example for the graph description of a MBS. Slika 1. Primjer graf-opisa diskretnog mehaničkog sustava s kinematičkim vezama 2. MBS geometry d 2.1. The topological graph of an MBS Let G be the directed tree w.r.t. to the root B0, which is the directed subgraph of G such that to every vertex The MBS topology is represented as an oriented graph there is a path starting from the root of G (Figure 1). The Γ = (B, J), where the set of vertices B represents the directed tree induces the following ordering relation of bodies, and the set of edges J represents the joints of the joints and bodies: Strojarstvo 51 (6) 597-612 (2009) A. MÜLLER et. al., ��������������������������������������������������Differential-Geometric Modeling and Dynamic... ����599 • B is the direct predecessor of body B , denoted by B N l k l Cabs := {c ∈ SE (3) | Rα ∈ Gα}. (2) 1 d = Bk − 1, iff (l, k) ∈ G (for short l = k − 1) This is called the configuration space of a holonomic • Jβ is the direct predecessor of joint Jα, denoted by d d MBS in absolute representation. Jβ = Jα − 1, iff Jβ = (·, k) ∈ G Λ Jα= (k, *) ∈ G (for short β = α − 1) Cabs is a rather abstract notion that covers MBSs with arbitrary topology. If we were to use this representation • Jβ is a predecessor of joint Jα, denoted by Jβ ≤ Jα, iff there is a finite m, such that β = α − 1 − 1 · · · − 1 (m for our actual computations, we would need to introduce times −1). constraints that restrict the relative motion of adjacent bodies to the motion spaces. In fact, this leads to absolute Note that possibly k − 1 = l − 1 for k ≠ l. E.g. in Figure coordinate formulations. What we will do instead is to 1., J = J − 1, J = J − 1, J = J − 1, so that J < J . For 1 2 2 6 6 7 1 7 eliminate the constraints by resolving for the relative each body B , there is a unique tree joint J connecting B k α k configuration of tree-joints, and introduce constraints to its predecessor, i.e. J = (·, k) ∈ Gd. This fact is denoted α only for the cut-joints. by α = J (k). In other words, body Bk is connected to Bk−1 via Jα. Finally, Jroot (k) denotes the smallest α ≤ J (k). In figure 1, J (6) = 7, J (5) = 6 and Jroot (k) = 1 for all k.
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