Reduced workspace is the main parallel robot disadvantage. It is generally due to the robot configuration, mainly the platform orientation constraint, the present work intends to find the maximum sphere within the orientation workspace, i.e., the singularity-free orientation regions. These regions are related to the platform orientation through Roll-Pitch-Yaw angles. Therefore, an optimization genetic algorithm is used to determine the initial platform orientation corresponding to the largest sphere volume. In this algorithm, the geometrical parameters and the direct and inverse singularities are the optimization constraints. The geometrical constraints may be studied using vectorial analysis. The reciprocity property from screw theory is implemented to analyze the direct and inverse kinematic. In this work it is used a methodology to verify the singularity closeness measure associated with direct kinematic. This measure is related to the rate of work done by each leg upon the platform twist. To determine how close is the parallel robot to a direct singularity a index value is proposed. It is considered that the passive joints reachable regions may be limited by a cone, whereby the cone symmetric axis is the same than the passive joint axis. In the optimization problem, the sphere volume, i.e., the maximal angular displacement of the moving platform around any axis is the objective function. Thus, the genetic algorithm individuals explore all feasible regions looking for an optimal solution.