The inertia matrix of any rigid body is the same as the inertia matrix of some system of four pointmasses. In this work, the possible disposition of these point-masses is investigated. It is found that every system of possible point-masses with the same inertia matrix can be parameterised by the elements of the orthogonal group in four-dimensional modulo-permutation of the points. It is shown that given a fixed inertia matrix, it is possible to find a system of point-masses with the same inertia matrix but where one of the points is located at some arbitrary point. It is also possible to place two point-masses on an arbitrary line or three of the points on an arbitrary plane. The possibility of placing some of the point-masses at infinity is also investigated. Applications of these ideas to rigid body dynamics are considered. The equation of motion for a rigid body is derived in terms of a system of four point-masses. These turn out to be very simple when written in a 6-vector notation.

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Planar Revolute-Coupled Kinematic Chains in Critical Configurations and their DualsTrevor H. Davies * Luís Paulo Laus † Abstract:The motivation for studies of dual coupling networks is explained and a definition is suggested for duality. Dual couplings are defined and several couplings dual with the revolute coupling are described. Six linkages, instantaneously in a critical configuration, provide examples. The consequences of these critical configurations are explored for both the linkages and the coupling networks that are dual with them.

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