Model-based controllers potentially offer a higher positioning accuracy for robotic systems. The direct kinematics solution is an essential part of these controllers. However, the direct kinematics problem for parallel manipulators is usually very complicated and in general does not have a closed-form solution. This problem usually leads to multiple solutions. When a control application or dynamics simulation is considered, only one of the solutions is acceptable. Although there are some numerical methods that obtain all possible solutions, specifying the one acceptable solution among them is still a challenging problem. It is shown that the tool space of a parallel manipulator with one inverse kinematic solution can be categorized into special regions called basic regions. In this paper, a new concept for basic regions is proposed that extends the concept to non-cuspidal fully parallel manipulators with multiple inverse and multiple direct kinematics solutions. Then, for general non-cuspidal fully parallel manipulators, a numerical algorithm is proposed that determines the basic region domains in the tool space. Finally, a novel method is proposed which utilizes the basic regions theory to determine which direct kinematics solution is acceptable. The proposed method presents a general solution to the direct kinematics problem of non-cuspidal parallel manipulators in trajectory following. The provided solutions are reliable and can be refined up to an arbitrary accuracy. The proposed method is illustrated using a 3-RRR planar parallel manipulator.

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