Based on the optimization theory and the direct elimination technique, we present a systematic approach to explore the single reciprocal screw axis of Bricard six-revolute mechanisms numerically and to confirm its geometric properties graphically. To overcome the difficulty of deriving the analytical closed-form solutions and the shortcomings of needing an accurate initial estimate and having a highly computational sensitivity in the popular Newton-type procedure, the nonlinear programming algorithm establishes a general computational solution to the matrix displacement equation. Using the transformation of line and screw coordinates and the fundamentals of reciprocity of screws, we develop a simple numerical computational approach to obtain the reciprocal screw axis of mechanisms through the linear dependence or independence of linear algebra. Geometric explanations with 3-D computer graphics are also offered to comprehensively understand their algebraic surface of the reciprocal screw axes and to confirm the correctness and validity of the derived algorithm.

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