AbstractThe main goal of the present work is to provide an add‐on scheme for the formulation of multibody dynamics, based on natural coordinates, in regard to ideally balanced rigid bodies with high rotational spin, e.g. gyroscopes. The underlying aim of this approach is to achieve higher numerical accuracy whenever the preferred axis of rotation coincides with the balanced main axis of the body. This will be achieved by seperating the spin of the balanced rigid body along the denoted axis as an additional angular coordinate, whereas the other rotations will be covered by a carried frame, parameterized via natural coordinates. At the same time the carried frame provides a link to the existing modelling framework in terms of natural coordinates, enabling a straightforward implementation into existing multibody systems (e.g. rotary crane [2]). (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

AbstractIn the past, a lot of effort has gone into the development of structure‐preserving time‐stepping schemes for forward dynamic problems. This is due to the superior numerical stability of these integrators. Guided by previous developments in the design of energy‐momentum integrators for forward dynamic problems, a Hamiltonian conserving indirect optimal control method will be introduced. For the state equations, a consistent variant of the midpoint evaluation introduced in [1] will be applied. Based on this specific discretization of the state equations, a discretization of the costate equations will be introduced, which is based on the notion of a discrete derivative and which leads to the algorithmic conservation of the discrete Hamiltonian. The newly developed method will be compared with a direct transcription method. We will test the newly proposed method within two numerical examples, which are the optimal control of a particle in a gravitational field and a 3‐link manipulator. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The present work deals with optimal control problems governed by differential‐algebraic equations (DAEs). In particular, the control effort, which is necessary for moving a multibody system from one configuration to another, will be minimized. The orientation of the rigid bodies will be described using directors, which facilitates the integration of the equations of motion with an energy‐momentum consistent time‐stepping scheme [1]. This type of structure‐preserving integrators offer outstanding numerical stability and robustness properties in comparison to the often applied generalized coordinates formulation. In the context of optimal control, other kinds of consistent integrators have been applied previously in [2] and [3]. We will test the different formulations with two numerical examples, a 3‐link manipulator and a satellite. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The formulation of multibody dynamics in terms of natural coordinates (NCs) leads to equations of motion in the form of differential-algebraic equations (DAEs). A characteristic feature of the natural coordinates approach is a constant mass matrix. The DAEs make possible (i) the systematic assembly of open-loop and closed-loop multibody systems, (ii) the design of state-of-the-art structure-preserving integrators such as energy-momentum or symplectic-momentum schemes, and (iii) the direct link to nonlinear finite element methods. However, the use of NCs in the optimal control of multibody systems presents two major challenges. First, the consistent application of actuating joint-forces becomes an issue since conjugate joint-coordinates are not directly available. Second, numerical methods for optimal control with index-3 DAEs are still in their infancy. The talk will address the two aforementioned issues. In particular, a new energy-momentum consistent method for the optimal control of multibody systems in terms of NCs will be presented.

AbstractThe present work deals with inverse multibody dynamics problems, more precisely with optimal control problems governed by differential‐algebraic equations. In our case, the control effort, which is necessary for moving a body from one position to another, will be minimized.Beside the commonly used generalized coordinates formulation of the equations of motion, the application of redundant coordinates will permit the derivation of conserving integrators. In addition to existing structure preserving integrators [2], the equations of motion, which serve as constraints within the optimal control scheme, may also be discretized with an energy‐momentum conserving integrator. Schemes with this property typically exhibit superior numerical stability properties.Two different possibilities exist, when applying an energy‐momentum method. The system may be reduced by applying the discrete nullspace method, which yields the elimination of the algebraic constraints [3]. This leads to a formulation with a higher degree of nonlinearity, which complicates the calculation of the gradients, that are necessary for the applied optimal control method. Alternatively, the index 3‐DAEs may be used directly. Previously, this has been done in [1] within an energy preserving direct transcription method. In that case, the simple structure of the discrete equations of motion facilitates the calculation of the gradients, which may be used for calculating the discrete necessary conditions of optimality. These equations can be solved by Newton's method.We test the different formulations with the help of an underactuated overhead crane as numerical example. This example is taken from [4], where it was used in the context of trajectory tracking. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

AbstractThe present work deals with inverse multibody dynamics problems, more precisely with optimal control problems for dynamical systems governed by differential‐algebraic equations. The used numerical integration method relies on the Hamilton formulation of the equations of motion. We will apply an energy and momentum conserving time discretization which typically exhibits superior numerical stability properties. The rigid body equations of motion are basically formulated using the director formulation, which requires the implementation of internal constraints. To eliminate the constraints the system will be reduced to its minimal dimension by applying the discrete nullspace method [3]. For the derivation of the necessary conditions of optimality the calculus of variation is used [4]. This approach leads to an indirect transcription method. The arising algorithm yields a sequence of discrete configurations together with a sequence of actuating forces. We test the formulation with the help of representative numerical example problems. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

In the present paper unit quaternions are used to describe the rotational motion of a rigid body. The unit‐length constraint is enforced explicitly by means of an algebraic constraint. Correspondingly, the equations of motion assume the form of differential‐algebraic equations (DAEs). A new route to the derivation of the mass matrix associated with the quaternion formulation is presented. In contrast to previous works, the newly proposed approach yields a non‐singular mass matrix. Consequently, the passage to the Hamiltonian framework is made possible without the need to introduce undetermined inertia terms. The Hamiltonian form of the DAEs along with the notion of a discrete derivative make possible the design of a new quaternion‐based energy–momentum scheme. Two numerical examples demonstrate the performance of the newly developed method. In this connection, comparison is made with a quaternion‐based variational integrator, a director‐based energy–momentum scheme, and a momentum conserving scheme relying on the discretization of the classical Euler's equations. Copyright © 2009 John Wiley & Sons, Ltd.

AbstractUnit–quaternions (or Euler parameter) are known to be well–suited for the singularity–free parametrization of finite rotations. Despite of this advantage, unit quaternions were rarely used to formulate the equations of motion (exceptions are the works by Nikravesh [1] and Haug [2]). This might be related to the fact, that the unit–quaternions are redundant, which requires the use of algebraic constraints in the equations of motion. Nowadays robust energy consistent integrators are available for the numerical solution of these differential–algebraic equations (DAEs). In the present work a mechanical integrator for the quaternions will be derived. This will be done by a size–reduction from the director formulation of the equations of motion, which also has the form of DAEs. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)