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A new class of six-degree-of-freedom (DOFs) spatial parallel platform mechanism is considered in this paper. The architecture consists of a mobile platform connected to the base by three identical kinematic chains using five-bar linkages. Recent investigations showed that parallel mechanisms with such a topology for the legs can be efficiently statically balanced using only light elastic elements. This paper follows up with a workspace analysis and optimization of the design of that parallel mechanism. More specifically, considering a possible industrial application of the architecture as a positioning and orienting device of heavy loads, an optimization procedure for the maximization of the volume of the three-dimensional (3-D) constant-orientation workspace of the mechanism is first presented. As the mechanism could also have great potential as a motion base for flight simulators, we develop here a discretization method for the computation and graphical representation of a new workspace with coupled translational and rotational DOFs. This workspace can be defined as the 3-D space which can be obtained when generalized coordinates x,y and torsion angle ψ in the tilt-and-torsion angles parametrization are constant. A second procedure is then presented for the maximization of the volume of this second subset of the complete workspace. For both approaches, our purpose is to attempt an optimal design of the mechanism by maximizing the volume of the associated 3-D Cartesian region that is free of critical singularity loci.