Rational Kinematics

A rational study of kinematics is a treatment of the subject based on invariants, i.e., quantities that remain essentially unchanged under a change of observer. An observer is understood to be a reference frame supplied with a clock (Truesdell 1966). This study will therefore include an introduction to invariants. The language of these is tensor analysis and multilinear algebra, both of which share many isomorphic relations, These subjects are treated in full detail in Ericksen (1960) and Bowen and Wang (1976), and hence will not be included here. Only a short account of notation and definitions will be presented. Moreover, definitions and basic concepts pertaining to the kinematics of rigid bodies will be also included.

Results concerning the motion of a rigid body between two distinct configurations are discussed in this chapter. These configurations are assumed to be finitely separated, i.e., the displacements undergone by the points of a bounded subset of the body are assumed to be finite. Infinitesimally separated configurations of a rigid body are dealt with in Chapters 3 and 4. The main results of this chapter are Euler’s Theorem, Chasles’ Theorem,the characterization of a rigid-body motion through its screw parameters, and the Aronhold-Kennedy Theorem. The concepts of screw and pose of a body are introduced, and some results concerning the displacement field of a rigid-body motion, as well as the compatibility conditions that this field verifies, are derived. Contrary to the common practice, quaternions are deliberately avoided here, the reason for such avoidance being, as explained in Chapter 1, that these require a very special algebra. One aim of this chapter is to show that rotations can be fully studied with linear algebra. The reader interested in quaternions is referred to the original works of Kelland and Tait (1882) and Hamilton (1899). A comprehensive review of the subject is given in (Spring 1986).

In this chapter the angular velocity of a rigid-body motion is introduced as a skew-symmetric tensor, its linear vector invariant being defined as the angular-velocity vector of the given motion. The linear relations between the angular-velocity vector and the time rates of change of the natural, the linear, and the quadratic invariants of the rotation tensor are derived. The relation between the angular-velocity vector and the time-rate of change of the quadratic invariants—Euler’s parameters—of the rotation tensor have been reported previously, e.g., in Wittenburg (1977) and Kane, Likins, and Levinson (1983). A comprehensive study of the relations between the first and second time derivatives of the Euler parameters and the angular-velocity and angular-acceleration vectors was reported by Nikravesh, Wehage. and Kwon (1985). Apart from these, the other relations are derived for the first time in invariant form. A preliminary derivation of the relation between the angular-velocity vector and the time rate of change of the linear invariants was first introduced in Angeles (1985). Spring (1986) includes a table showing some of the results that are derived here. Furthermore, a theorem related to the velocity distribution in a rigid body, paralleling that of Chasles’ of Chapter 2, is proven. Next, thė Theorem of Aronhold-Kennedy, pertaining to the relative motion of three rigid bodies, is proven. Additional theorems related to the velocity distribution throughout a moving rigid body are presented and proven, and the concept of twist of a rigid body is introduced. Finally, the problem of determining the angular velocity of a rigid-body motion from point-velocity data is discussed, and compatibility equations which the given data should verify, are derived.

The concept of angular acceleration of a rigid-body motion is introduced in this chapter. Moreover, the relationships between the angular acceleration of the rigid body and the time derivatives of the associated rotational invariants are derived, while the acceleration field of the body is derived in terms of the angular-acceleration tensor. Unlike the displacement and the velocity fields, the acceleration field, in general, contains one point of zero acceleration that is unique. Indeed, as shown in this chapter, the angular-acceleration tensor is, in general, nonsingular, and becomes singular only in the particular case in which the body instantaneously undergoes a rotation about a stationary axis. Thus, in the case of acceleration fields, no theorem similar to that of Chasles’ for displacement fields, or its counterpart for velocity fields, exists.

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.