Dynamics of Underactuated Multibody Systems

Multibody system dynamics deals with the modeling, analysis, and simulation of mechanical and mechatronic systems undergoing large nonlinear motion. Underactuated multibody systems are an interesting type of controlled multibody systems, which are characterized by possessing less actuators than degrees of freedom. The treatment of such multibody systems is a theoretically challenging task with increasing relevance for industry and requires advanced modern methods in modeling, control design and system optimization. These are the centerpiece of this book, and this chapter gives a short introduction to this book.

In this chapter, the fundamentals of modeling and generation of the equation of motion using the Newton-Euler formalism is presented for rigid multibody systems. Its extension to flexible multibody systems using the floating frame of reference approach is also discussed. Thereby, the given presentation concentrates on a descriptive approach in order to derive the equation of motion in minimal form. This approach provides the equation of motion in symbolic or semi-symbolic form which is especially helpful for real-time simulations, optimizations and nonlinear control design.

Nearly all real world engineering systems comprise some type of nonlinearity. For output trajectory tracking of systems with continuous nonlinearities feedback linearization and feedforward control based on exact model inversion are powerful modern control techniques. This chapter provides an introductional representation of feedback linearization and model inversion. The goal is to present this topic in sufficient depth to understand the application of these concepts to underactuated multibody systems in the later chapters. In the first part of this chapter some important analysis and basic principles for nonlinear control of single-input single-output systems are developed. In the second part, output trajectory tracking using inversion-based feedforward control is presented. The extension to multiple-input multiple-output systems is predominantly straightforward and is summarized afterwards.

The methods of feedback linearization and feedforward control based on exact model inversion are powerful tools for output trajectory tracking of nonlinear systems. Using these methods, one key step is the determination of the input–output normal form. In general, this step depends heavily on symbolic calculations of Lie derivatives of the system outputs in state space. However, establishing a symbolical state space description of a multibody system requires the symbolical inversion of the mass matrix. This yields, even in small mechanical systems, very complex state space descriptions. Therefore, the straightforward application of these nonlinear control methods to multibody systems is, in general, limited to systems with very few degrees of freedom. In this chapter, it is shown that it is often possible to determine the input–output normal form and the resulting control laws by direct symbolic manipulations on the second order differential equations of motion of a multibody system. Thereby, no explicit utilization of their state space description is necessary, since the special structure of the equations of motion of multibody systems is utilized. However, in order to get a full understanding of this procedure and the underlying differential geometric control techniques, it is very important to relate to the corresponding theory in state space, which is presented in the previous chapter.

The control strategies presented in the previous chapter for trajectory tracking of multibody systems originate in differential-geometric concepts developed for general nonlinear systems. These are based on an explicit coordinate transformation into the nonlinear input?output normal form. An alternative control design approach for multibody systems is the use of so-called servo-constraints. The basic idea is the enforcement of output trajectory tracking by introducing constraint equations, yielding a set of differential-algebraic equations. Thus, this approach is closely related to classical geometric constraints in multibody systems. While this control design approach has a fundamentally different philosophy, it also shows many similarities to the coordinate transformation-based procedures developed in the previous chapters.

Very important cases of underactuated multibody systems are flexible multibody systems. The actuation of a multibody system occurs in many cases at its joints, and thus, the elastic degrees of freedom have no associated control input. Thus, elasticity naturally causes underactuation. Due to its high technical relevance in modern light-weight machine designs the control of flexible multibody systems is an ongoing field of research. Many of the results derived in previous sections can be easily adopted to flexible multibody systems. Therefore, the main focus in this chapter is the derivation and discussion of some additional results and insights which are useful for trajectory tracking control of flexible multibody systems.

The analysis and discussions in the previous chapters show that a minimum phase system allows a much easier control design than a non-minimum phase system. For minimum phase systems feedback linearization is possible and the design of feedforward control is significantly simplified. However, the initial design of an underactuated multibody system might be non-minimum phase and requires the demanding computation of feedforward control by stable inversion. In this chapter, an optimization based structural and control design methodology is proposed in order to obtain minimum phase system designs. The proposed design methodology already considers structural design and control design concurrently in the early stage of the design process. The proposed integrated design approach is based on an optimization procedure for either, the system output, the structural design, or both.

Due to the many theoretical and practical challenges which underactuated multibody systems pose, they are a fascinating research field with increasing industrial relevance in modern machine design. Especially the appealing use of light-weight design techniques, which often include body elasticity or passive joints, require a thorough understanding of the dynamics of those underactuated multibody systems. The efficient treatment of underactuated multibody systems requires a sound basis in modeling and nonlinear control design. These must be combined with advanced computational strategies and a utilization of the typical structure and properties of underactuated multibody systems. The presented research work covers these topics in a self contained way, and represents the state of the art with many newly derived results. These are subsequently combined in an integrated optimal system design approach.