The present paper illustrates the potential of the adjoint method for a wide range of optimization problems in multibody dynamics such as inverse dynamics and parameter identification. Although the equations and matrices included show a complicated structure, the additional effort when combining the standard forward solver to the adjoint backward solver is kept in limits. Therefore, the adjoint method shows an efficient way to incorporate inverse dynamics to engineering multibody applications, e.g., trajectory tracking or parameter identification in the field of robotics. The present paper studies examples for both, parameter identification and optimal control, and shows the potential of the adjoint method in solving classical optimization problems in multibody dynamics.

In the case of complex multibody systems, an efficient and time-saving computation of the equations of motion is essential; in particular, concerning the inertia forces. When using the floating frame of reference formulation for modeling a multibody system, the inertia forces, which include velocity-dependent forces, depend nonlinearly on the system state and, therefore, have to be updated in each time step of the dynamic simulation. Since the emphasis of the present investigation is on the efficient computation of the velocity-dependent inertia forces as along with a fast simulation of multibody systems, a detailed derivation of the latter forces for the case of a general rotational parameterization is given. It has to be emphasized that the present investigations revealed a simpler representation of the velocity-dependent inertia forces compared to results presented in the literature. In contrast to the formulas presented in the literature, the presented formulas do not depend on the type of utilized rotational parameterization or on any associated assumptions.

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Many models of three-dimensional rigid body dynamics employ Euler parameters as rotational coordinates. Since the four Euler parameters are not independent, one has to consider the quaternion constraint in the equations of motion. This is usually done by the Lagrange multiplier technique. In the present paper, various forms of the rotational equations of motion will be derived, and it will be shown that they can be transformed into each other. Special attention is hereby given to the value of the Lagrange multiplier and the complexity of terms representing the inertia forces. Particular attention is also paid to the rotational generalized external force vector, which is not unique when using Euler parameters as rotational coordinates.

In multibody dynamics, the flexibility effects of each body are captured by using a linear combination of elastic mode shapes. If a co-rotational and co-translating frame of reference is used together with eigenvectors of the unconstraint body, which are free-surface modes, some spatial integrals in the floating frame of reference configuration do vanish. The corresponding coordinate system is the so-called Tisserand (or Buckens) reference frame. In the present contribution, a technique is developed for separating an arbitrary elastic mode shape into a pseudo-free-surface mode and rigid body modes. The generated pseudo-free-surface mode has most of the advantageous characteristics of a free-surface mode, and spans together with the rigid body modes the same solution space as it is spanned by the original mode shape. Due to the fact that, in the floating frame of reference configuration, the rigid body motions are already described by special generalized coordinates, only the resulting pseudo-free-surface modes are finally used to capture the flexibility effects of each body. A result of the generated pseudo-free-surface modes is that some of the spatial integrals do vanish and, thus, the equations of motion are significantly simplified. Two examples are presented in order to illustrate and to demonstrate the potential of the proposed method.