A six-DOF mechanism of a Steward (or Gough–Steward) type of platform is isolated among parallel-structure mechanisms. Steward [1] and Gough and Whitehall [2] propose a specific application of platforms in special test benches. The general theory of multiple-DOF platforms was developed in the 1980s–1990s [3–9]; however, since then serious investigations have been performed on this subject [10–21]. Analytical methods of solving analytical problems of the statics, kinematics, and dynamics of six-DOF platforms make it possible to establish qualitative properties. Computer programs are currently employed for practical solution of mechanics problems of these platforms. The direct geometry problem for the parallel-structure mechanisms under consideration is formulated in the following manner: the position of the platform (a free solid body with six degrees of freedom in space) is assigned by six geometric parameters, and the platform is hinge-connected to a stationary base with rigid driving limbs equipped with linear drive for varying their lengths (within certain limits). In classical studies involving the mechanics of six-DOF platforms, it is assumed that all degrees of freedom on the stationary base are situated in the same plane, and the hinges on the moving platform are arranged in a single moving plane. In certain structures (for example, [18]), these restrictions are removed, and the hinges on the stationary base and moving platform may be arranged at the whim of the investigator. Problems involving the selection of the position of the spherical hinges of the driving limbs on the stationary base and platform are solved separately on the basis of different requirements (for example, provision for maximum stiffness of the structure on the whole, leaving zones free, etc.); it is then considered that arrangement of the hinges is assigned. Basic Geometric Relationships. It is necessary and sufficient that there be six driving limbs for single-valued assignment of the position of the moving platform on the one hand, and for static determinacy on the other. In the direct geometry (or kinematics) problem, values of the lengths li (distance between centers of spherical hinges) of the six limbs of variChemical and Petroleum Engineering, Vol. 50, Nos. 9–10, January, 2015 (Russian Original Nos. 9–10, Sept.–Oct., 2014)