Robot and Multibody Dynamics: Analysis and Algorithms

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Spatial Vectors Abstract. This chapter establishes notation and several foundational concepts required for the study of multibody kinematics and dynamics.

Beginning with the notion of coordinate-free representations, we introduce homogeneous transformations, spatial vectors and their properties. Spatial vectors are used to define spatial velocities, accelerations and forces for coordinate frames and bodies. Rigid body transformation matrices that transform spatial velocities and forces across frames on a rigid body are also introduced. This chapter also studies time derivatives of spatial quantities with respect to different frames.

This chapter develops the dynamical equations of motion for a single rigid body in multiple different ways: about the body center of mass, about an arbitrary point on the body, using body and inertial frame derivatives, and finally using an inertially fixed velocity reference point. The properties of each of these formulations are explored. These alternative formulations also help illustrate the analytical and transformation properties of spatial quantities.

This chapter develops kinematics models for serial-chain rigid multibody systems. Although serial-chains are perhaps the simplest examples of multibody systems, many of the concepts and techniques developed for these systems apply, with only modest extensions, to general multibody systems. While the mass and rotational inertia of a body are the elemental quantities for studying single rigid body dynamics, the mass matrix plays the corresponding role for multibody systems.

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One crucial difference is that the mass matrix varies with the configuration of the system. In this chapter, we derive expressions for the mass matrix of a serial-chain multibody system, study its properties, and develop related computational algorithms. We now study the dynamics of serial-chain multibody systems and derive their equations of motion.


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The approach builds upon the single rigid body dynamics and serial-chain kinematics developments in Chaps. The component link-level equations of motion are assembled into compact system-level operator versions of the equations of motion using the serial-chain mass matrix. Algorithms for the system inverse dynamics are also discussed.


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  5. This chapter introduces articulated body models for the component links in a serial-chain system. This model is an alternative to the terminal and composite body models discussed earlier. While the composite body model is appropriate for the inverse dynamics problem, the articulated body model is better suited for the forward dynamics problem.

    While the composite body model is appropriate for the inverse dynamics problem, the articulated body model is better suited for the for- ward dynamics problem. This chapter uses the articulated body model to derive a new Innovations Operator Factorization of the serial—chain mass matrix, M. The Innovations factorization is subsequently used to obtain an explicit analytical operator expression for the inverse of the mass matrix. These factoriza— tions have many important uses, which will be explored in subsequent chapters.

    For canonical serial-chain systems, spatial operator techniques have been used to establish important analytical results, such as decompositions and factorizations of the mass matrix, and an operator expression for its inverse. This analytical ground- work led to low-order computational algorithms, such as for inverse dynamics, forward dynamics, and for computing the mass matrix.

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