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For the elastodynamic simulation of a spatially discretized beam, asynchronous variational integrators (AVI) offer the possibility to use different time steps for every element [1]. They are symplectic and conserve discrete momentum maps and since the presented integrator for geometrically exact beam dynamics [2] is derived in the Lie group setting (SO (3) for the representation of rotational degrees of freedom), it intrinsically preserves the group structure without the need for constraints [3]. A decrease of computational cost is to be expected in situations, where the time steps have to be very low in certain parts of the beam but not everywhere, e.g. if some regions of the beam are moving faster than others. The implementation allows synchronous as well as asynchronous time stepping and shows very good energy behaviour, i.e. there is no drift of the total energy for conservative systems. Discrete variational mechanicsThe theory of discrete variational mechanics has its roots in the optimal control literature of the 1960's. The past ten years have seen a major development of discrete variational mechanics and corresponding numerical integrators, largely due to pioneering work by Jerrold Marsden and his collaborators, e.g. in [4]. The discrete Lagrangian L d approximates the action in a time interval [t j−1 , t j ]. For linear vector spaces, i.e. q ∈ R n , this leads to the discrete action sumand the discrete variational principle δS d = 0 results in the discrete Euler-Lagrange equationswhich are symplectic and conserve discrete momentum maps due to the variational derivation [4]. Asynchronous variational integrator for the beamAsynchronous variational integrators (AVI), as described by Lew et. al. [1], offer the possibility to use different time steps for every element of the spatial discretization. The use of AVI promises less computational costs by increasing the time step sizes for slowly moving parts of the beam. We review the kinematic description of the geometrically exact beam model in the ambient space R 3 presented in [2]. The configuration of a beam is defined by specifying the position of its line of centroids by means of a map φ : [0, L] → R 3 , and the orientation of cross-sections at points φ(S) by means of a moving basis {d 1 (S), d 2 (S), d 3 (S)} attached to the cross section. The orientation of the moving basis is described with the help of an orthogonal transformation Λ : [0, L] → SO(3) such that d I (S) = Λ(S)E I , I = 1, 2, 3where {E 1 , E 2 , E 3 } is a fixed basis referred to as the material frame. The configuration of the beam is thus completely determined by the maps φ and Λ in the configuration spacewith the Lie group SE (3) being the Euclidean group. If we take into account that the thickness of the rod is small compared to its length and that the material is homogenous and isotropic, we can consider the stored energy to be given by a quadratic

In underactuated dynamical systems, the number of control inputs nu is smaller than the number of degrees of freedom nq.Real world examples include e. g. flexible robot arms or cranes. In these two exmples the goal is to prescribe the trajectory of an end effector and find the necessary control variables. One approach to model these problems is to introduce servo constraints in the equations of motion that enforce a given trajectory for some part of the system [1].