CAJ | 학술논문

为获得stewart平台机器人位置正解的全部封闭解,使用分次字典序Groebner基和Sylvester结式相结合的代数方法进行研究.使用旋转矩阵建立机器人位置正解封闭数学模型;利用计算代数中的分次字典序Groebner基算法,计算封闭数学模型的分次字典序Groebner基,从获得的41个基中选取10个基,构造10阶Sylvester结式,从而推导出该机器人的一元二十次输入输出方程.从理论上阐明了存在多个不同的结式可以获得该机器人的位置正解,使用同伦连续法对同一个数字算例进行计算验证,两种方法得到结果一致.结果表明,该算法避免了复杂数学公式推导,构造的结式尺寸较小,方便计算机程序实现.
위획득stewart평태궤기인위치정해적전부봉폐해,사용분차자전서Groebner기화Sylvester결식상결합적대수방법진행연구.사용선전구진건립궤기인위치정해봉폐수학모형;이용계산대수중적분차자전서Groebner기산법,계산봉폐수학모형적분차자전서Groebner기,종획득적41개기중선취10개기,구조10계Sylvester결식,종이추도출해궤기인적일원이십차수입수출방정.종이론상천명료존재다개불동적결식가이획득해궤기인적위치정해,사용동륜련속법대동일개수자산례진행계산험증,량충방법득도결과일치.결과표명,해산법피면료복잡수학공식추도,구조적결식척촌교소,방편계산궤정서실현.
To obtain all closed-form solutions of the forward displacement of Stewart platform robot,the Groebner-Sylevester hybrid approach was presented. First of all,the orientation matrix was used to construct a closed-form mathematical model for the robot. Then,the reduced Groebner basis under degree lexicographic ordering for the closed-form equations was obtained by using computer algebra. Selecting 10 Groebner basis from 41 ones,a 10 × 10 Sylvester' s matrix can be constructed,which is relatively small in size. A 20th degree univariate equation was obtained from the determinate of the matrix. The process shows that there are many different resultants,which lead to the same univariate equation. The same results could be obtained by using the continuation method. The results show that the proposed algorithm,which is simple and requires less computation time,makes it easy for compaters to realize the programs.