Dual-number quaternion algebra has been used to obtain explicit, closed-form algebraic expressions for the displacements, velocities, velocity ratios, forces, torques, mechanical advantages, transmission angle, and locking positions in spatial four-link mechanisms having one turning pair and three turn-slides. The results have been programmed for automatic digital computation.

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This paper presents the application of dual-number matrices to the formulation of displacement equations of robot manipulators with completely general geometry. Dual-number matrices make possible a concise representation of link proportions and joint parameters; together with the orthogonality properties of the matrices we are able to derive, in a systematic manner, closed-form solutions for the joint displacements of robot manipulators with special geometry as illustrated by three examples. It is hoped that the method presented here will provide a meaningful alternative to existing methods for formulating the inverse kinematics problem of robot manipulators.

The objective of this paper is to make instantaneous invariants a more accessible tool for problem solving in the field of kinematics. We present a systematic procedure to determine the instantaneous invariants of a rigid body under geometric constraint, develop a process by which kinematic properties of rigid body motion can be expressed in terms of instantaneous invariants, and apply instantaneous invariants to solve typical kinematic synthesis problems. Four examples are given in detail for illustrative purposes.

The kinematics, statics, and inertia-force analysis of epicyclic bevel-gear trains have been described. Beginning with the general case of nonparallel, nonintersecting axes, the results are specialized to planetary bevel-gear trains. Dual (3 × 3) matrices have been found useful in the analysis, which is an outgrowth of earlier work on spur-gear trains.