Kinematic Chains

The coupling of rigid bodies by means of mechanical constraints constitutes a kinematic chain. This coupling takes place pairwise, and hence, it is given the name kinematic pair. In this chapter the basic classification of kinematic pairs, namely, lower and upper kinematic pairs, is introduced and the discussion will be mainly devoted to a study of the former. The latter are discussed briefly in Section 5.8. Furthermore, kinematic chains coupled by lower kinematic pairs are classified into simple and complex. The former, in turn, can be either open or closed. In any case, the degree of freedom of the chain is determined resorting either to a Chebyshev-Grübler-Kutzbach formula or to the Jacobian matrix of the chain under study. It is shown that the said type of formulae, based solely on the topology of the chain, has limited applicability regarding the determination of the chain’s degree of freedom. On the other hand, the Jacobian of the chain provides a widely applicable means of determining the degree of freedom of not only simple, but also complex kinematic chains. Regarding the latter, two particular types of kinematic structures are distinguished, namely, tree structures and chains with multiple closed loops. The former are discussed briefly, for they are not essential in this context; the latter are studied in detail regarding the determination of their degree of freedom. Next, an item that is of the utmost relevance in dynamics is introduced, namely, the kinematic constraint equations of a general mechanical system.