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A 6-DOF 6-CPS orthogonal parallel manipulator was presented, based on the theory of the structure synthesis for parallel robot mechanisms. The forward kinematics of the manipulator was studied to analyze the positive positional solutions. With the restrict condition of the lengths of six actuated limbs, the constrained equations were established, and the nonlinear equations (NLEs) were obtained to solve forward positional problems of the mechanism. Based on the Chen-Lai algorithm, by means of which chaos was generated or modified, a discrete-time dynamical system was feedback controlled to be transferred to a chaotic system that has the expected Lyapunov exponent and good ergodicity. The chaotification-based Newton iterative algorithm (CBNIA) was utilized to solve NLEs for forward displacement analysis of 6-CPS orthogonal parallel manipulators. The results of numerical verification show that through the CBNIA, all forward solutions can be quickly found that are very coincident with the inverse ones.