Abstract：In order to improve the efficiency and universality of the forward kinematics solution of six-degree-of-freedom parallel mechanisms, a forward kinematics solution algorithm of 6-UPS (universal-prismatic-spherical) mechanism is proposed. Firstly, two angle sensors are added to the Hook hinge of any branch of the 6-UPS mechanism, and the rotation angles in the two directions of the Hook hinge are measured. Then 12 equations are constructed based on the rotation matrix, and a simple one-variant quadratic equation system is obtained by reducing the order of equation through algebraic elimination. Finally, the forward kinematics solution of the 6-UPS mechanism is obtained. The proposed method is verified by the numerical examples, and the definite forward kinematics solution is obtained. The method not only reduces the difficulty of data processing, but also solves a unique forward solution, and therefore the problem is avoided that there are multiple forward solutions in parallel mechanism.
 Stewart D. A platform with six degrees of freedom[J]. Proceedings of Institution of Mechanical Engineers, Part C-Journal of Mechanical Engineering Science, 1965, 223(1):279-281.  颜旗权.6自由度飞行模拟器的研究[D].哈尔滨:哈尔滨工程大学,2008. Yan Q Q. Research on 6-DOF flight simulator[D]. Harbin:Harbin Engineering University, 2008.  方喜峰,赵若愚,吴洪涛,等.6-UPS交叉杆型并联机床的运动学正解算法[J].机械工程学报,2012,48(13):56-60. Fang X F, Zhao R Y, Wu H T, et al. Forward kinematics analysis of 6-UPS parallel machine tool with cross rod[J]. Journal of Mechanical Engineering, 2012, 48(13):56-60.  刘艳梨,吴洪涛,李耀,等.6-UPS并联机器人快速正向运动学研究[J].农业机械学报,2019,50(4):374-381,400. Liu Y L, Wu H T, Li Y, et al. Fast forward kinematics of 6-UPS parallel robot with representative points[J]. Transactions of the Chinese Society for Agricultural Machinery, 2019, 50(4):374-381,400.  韩方元,赵丁选,李天宇.3-RPS并联机构正解快速数值算法[J].农业机械学报,2011,42(4):229-233. Han F Y, Zhao D X, Li T Y. A fast forward algorithm for 3-RPS parallel mechanism[J]. Transactions of the Chinese Society for Agricultural Machinery, 2011, 42(4):229-233.  Abbasnejad G, Zark S, Imani M. Forward kinematics analysis of a 3-PRS parallel manipulator[C]//World Academy of Science, Engineering and Technology. Berlin, Germany:Springer, 2010:329-335.  王超群,吴洪涛.6-PUS并联机构的运动学正解分析[J].中国机械工程,2013,24(7):853-856. Wang C Q, Wu H T. Kinematics forward analysis of 6-PUS parallel mechanism[J]. China Mechanical Engineering, 2013, 24(7):853-856.  郭宗和,杜晴晴,杨启志.新型3-PRRS并联机构的位置正反解分析[J].中国机械工程,2015,26(4):451-455, 463. Guo Z H, Du Q Q, Yang Q Z. Forward and inverse position kinematics analysis of a new 3-PRRS parallel mechanism[J]. China Mechanical Engineering, 2015, 26(4):451-455,463.  黄真,赵永生,赵铁石.高等空间机构学[M].北京:高等教育出版社,2014. Huang Z, Zhao Y S, Zhao T S. Advanced spatial mechanism[M]. Beijing:Higher Education Press, 2014.  Morell A, Tarokh M, Acosta L. Solving the forward kinematics problem in parallel robots using support vector regression[J]. Engineering Applications of Artificial Intelligence, 2013, 26(7):1698-1706.  任子武,王振华,孙立宁.基于改进人工蜂群算法的并联机器人正运动学解[J].机械工程学报,2013,49(13):48-55. Ren Z W, Wang Z H, Sun L N. Forward kinematics solution of parallel robot based on improved artificial bee colony algorithm[J]. Journal of Mechanical Engineering, 2013, 49(13):48-55.  薛剑,唐志勇,裴忠才.6-3 Stewart平台位置正解的机构简化数值方法[J].北京航空航天大学学报,2014,40(7):921-926. Xue J, Tang Z Y, Pei Z C. Numerical method of forward position solution for 6-3 Stewart platform based on mechanism simplification[J]. Journal of Beijing University of Aeronautics and Astronautics, 2014, 40(7):921-926.  Merlet J P. Closed-form resolution of the direct kinematics of parallel manipulators using extra sensors data[C]//IEEE International Conference on Robotics and Automation. Piscataway, USA:IEEE, 1993:200-204.  Baron L, Angeles J. The decoupling of the direct kinematics of parallel manipulators using redundant sensors[C]//IEEE International Conference on Robotics and Automation. Piscataway, USA:IEEE, 1994:974-979.  吕崇耀.6-6 SPS 6自由度Stewart并行机构位姿正解单解存在条件[J].机器人,2000, 22(2):148-153. Lü C Y. Exact closed-form forward kinematics of 6-66DoF Stewart in parallel mechanisms[J]. Robot, 2000, 22(2):148-153.  刘艳梨,程世利,蒋素荣,等.带位移传感器的6-UPS并联机构运动学正解[J].机械工程学报,2018,54(5):1-7. Liu Y L, Cheng S L, Jiang S R, et al. Forward kinematics of 6-UPS parallel manipulators with one displacement sensor[J]. Journal of Mechanical Engineering, 2018, 54(5):1-7.