Efficient embedding of a geometrically exact Cosserat rod in multibody dynamics using kinematic coupling constraints

In vehicle design and production, one typically has to deal with flexible components. Simulation of these flexible components as part of a multibody system is especially challenging, if fast simulations up to realtime are demanded. This is the case, e.g., in interactive assembly simulations of cables and hoses. In our work, we use the discrete Cosserat rod model as presented in [1], which is based on finite differences. It uses the special structure of one-dimensional components and, thus, is very efficient, but accurate at the same time. Figure 1: Sketch of the coupling example. To couple this flexible rod model (F) with the surrounding rigid multibody system (R), we propose a coupling approach that uses a kinematic coupling constraint 0 = gco(q R ,q co F ), where only the coupling states qco R ,q co F appear, and no inner states q in R ,q in F . Starting formally with a monolithic description of the coupled problem, a force-displacement co-simulation is developed. Here, the multibody system is driven by constraint-forces Fcoupl =−co ∂qR T Λ, (1) while the coupling states of the flexible structure, i.e. the end-points of the cable, are prescribed. The prescribed states q F , q̇ pre F can be computed via the kinematic coupling from the multibody states q co R , q̇ co R , whereas the constraint-forces, i.e. the Lagrange multiplier Λ, could be derived by solving the monolithic index-1-system. However, for efficiency reasons the forces are approximated with a size-reduced monolithic system, where we exploit the zero-blocks ∂gco ∂qin R = 0 and ∂gco ∂qin F = 0 in the constraint gradient. Only equations of motion that correspond to coupling states appear in this smaller system, whose size is related to the number of coupling states (details will be in [2]). Depending on the mass ratio of the subsystems, the approximation can cause an instability. This observation is similar to the one in [3, 4]. Our coupling approach does not introduce a bushing-element for the kinematic coupling and, thus, no artificial stiffness. Moreover, it is easy to formulate more complex coupling joints, which is non-trivial with bushing-elements, cf. [5]. For time integration we use a parallel co-simulation, since it is more efficient than a sequential one. Therefore, both coupling terms, forces and displacements, have to be extrapolated for the integration up to the next macro-point, cf. [6]. On the one hand, the multibody system can be integrated with any standard MBS-solver. On the other hand, for realtime-capability, a linear-implicit method is used to simulate the flexible component, cf. [7]. In a first coupling example, the cable model is embedded in a small multibody system, as sketched in figure 1. One end of the cable is fixed at the wall, the other one is coupled to the body with mass m3. In this example, the cable is fully clamped to the rigid body, but also changes of the coupling joint are possible with little effort. A challenging aspect is the staggered discretization of the rod model, as depicted in figure 2. Rotational degrees of freedom – parameterized with quaternions p 1 2 , . . . , pN− 1 2 – are located on the edges and not in the nodes, such that there are no discrete rotational variables at the cable end points. To handle clamped boundary rotations p0, a virtual ghost quaternion p− 2 is introduced beyond the cable end point, and is defined as the spherical linear extrapolation of p 1 2 via p0. Thus, the virtual ghost quaternion appears in the rotational coupling constraint 0 = grot co ( φ , p0(p− 2 , p 2 ) ) , (2) where φ represents the rotation of the coupling body of the multibody system. To derive the coupling forces (exactly or approximately), constraint gradients and corresponding equations of motion are necessary. Therefore, we have to set up virtual equations of motion for the ghost quaternion p− 2 , which is, however, computationally cheap, since all involved terms are evaluated already. Figure 2: Staggered discretization of the cable model. References[1] H. Lang, J. Linn, M. Arnold. Multibody dynamics simulation of geometrically exact Cosseratrods. Multibody System Dynamics, Vol. 25, No. 3, pp. 285–312, 2011. [2] F. Schneider, M. Burger, M. Arnold, B. Simeon. A new approach for force-displacement co-simulation using kinematic coupling constraints. In preparation for submission to Z. Angew.Math. Mech., 2014. [3] R. Kübler, W. Schiehlen. Two Methods of Simulator Coupling. Mathematical and ComputerModelling of Dynamical Systems, Vol. 6, No. 2, pp. 93–113, 2000. [4] M. Arnold, M. Günther. Preconditioned dynamic iteration for coupled differential-algebraic sys-tems. BIT Numerical Mathematics, Vol. 41, No. 1, pp. 001–025, 2001. [5] B. Esterl, T. Butz, B. Simeon, B. Burgermeister. Real-time capable vehicle–trailer coupling by al-gorithms for differential-algebraic equations. Vehicle System Dynamics, Vol. 45, No. 9, pp. 819–834, 2007. [6] M. Busch. Zur effizienten Kopplung von Simulationsprogrammen. kassel university press GmbH,Kassel, 2012. [7] M. Arnold, B. Burgermeister, A. Eichberger. Linearly implicit time integration methods in real-time applications: DAEs and stiff ODEs. Multibody System Dynamics, Vol. 17, No. 2-3, pp. 99–117, 2007.