Complete Real Solution of the Five-orientation Motion Generation Problem for a Spherical Four-bar Linkage

For a spherical four-bar linkage,the maximum number of the spherical RR dyad(R:revolute joint)of five-orientation motion generation can be at most 6.However,complete real solution of this problem has seldom been studied.In order to obtain six real RR dyads,based on Strum's theorem,the relationships between the design parameters are derived from a 6th-degree univariate polynomial equation that is deduced from the constraint equations of the spherical RR dyad by using Dixon resultant method.Moreover,the Grashof condition and the circuit defect condition are taken into account.Given the relationships between the design parameters and the aforementioned two conditions,two objective functions are constructed and optimized by the adaptive genetic algorithm(AGA).Two examples with six real spherical RR dyads are obtained by optimization,and the results verify the feasibility of the proposed method.The paper provides a method to synthesize the complete real solution of the five-orientation motion generation,which is also applicable to the problem that deduces to a univariate polynomial equation and requires the generation of as many as real roots.