张铁等：基于BP神经网络的机器人波动摩擦力矩修正方法 ·1091· 精度. (刘俊.BP神经网路在多维非线性函数拟合中的应用.商洛 (3)神经网络验证实验表明：使用辨识得到的 学院学报，2014,28(6)：19) 神经网络函数对原来的计算力矩进行补偿，可使修 [9]Lin Y F,Deng H M,Shi X Y.Application of BP neural network based on newly improved particle swarm optimization algorithm in 正后的力矩误差在机器人平稳运动时基本保持在 fitting nonlinear function.Comput Sci,2017,44(11A):51 [-1,1]Nm的范围内，且有78.82%的力矩误差 (林宇锋，邓洪敏，史兴宇.基于新的改进粒子群算法的B 可保持在[-0.5,0.5]Nm的范围之内.相比于修 神经网络在拟合非线性函数中的应用.计算机科学，2017,44 正前的关节力矩误差，修正后的力矩误差的方差为 (11A):51) 0.1659N2m2,是修正前的24.23%. [10]Wu W X,Zhu S Q,Jin X L.Dynamic identification for robot manipulators based on modified Fourier series.J Zhejiang Univ Eng Sci,2013,47(2):231 参考文献 (吴文祥，朱世强，靳兴来.基于改进傅里叶级数的机器人 动力学参数辨识.浙江大学学报：工学版，2013,47(2)： [1]Iwatani M,Kikuuwe R.Some improvements in elastoplastic fric- 231) tion compensator.SICEJ Control Meas Syst Integration,2017,10 [11]Khalil W,Gautier M,Lemoine P.Identification of the payload (3):141 inertial parameters of industrial manipulators /Proceedings 2007 [2]De Luca A,Mattone R.Sensorless robot collision detection and IEEE International Conference on Robotics and Automation.Ro hybrid force/motion control /Proceedings of the 2005 IEEE Inter- ma,2007:4943 national Conference on Robotics and Automation.Barcelona, [12]Sousa C D,Cortesao R.Physical feasibility of robot base inertial 2005:999 parameter identification:a linear matrix inequality approach.Int [3]Haddadin S,De Luca A,Albu-Schaffer A.Robot collisions:A J Rob Res,2014,33(6):931 survey on detection,isolation,and identification.IEEE Trans [13]Xu F Y.Design method of filter for audio noise reduction based Rob,2017,33(6):1292 on application of Matlab.Audio Eng,2017,41(2):28 [4] Gandhi P S,Ghorbel F H,Dabney J.Modeling,identification, (徐帆云.基于Matlab的音频降噪滤波器设计.电声技术， and compensation of friction in hamonie drives//Proceedings of 2017,41(2):28) the 41st IEEE Conference on Decision Control.Las Vegas, [14]Yuan F.Comparison of several filtering methods based on MAT- 2002:160 LAB.Poncer Supply Technol Appl,2013(10):50 [5]Liao H B,Fan S X.Fan D P.Friction compensation of harmonic (远飞.基于MATLAB的几种滤波方法比较.电源技术应用， gear based on location relationship.Proc Inst Mech Eng Part IJ 2013(10):50) Syst Control Eng,2016,230(8):695 [15]Taghirad H D.Belanger PR.Modeling and parameter identifica- [6]Wu W X.Joint Friction Analysis and Low-speed Hight-precision tion of harmonic drive systems.J Dyn Syst Meas Control,1998 Motion Control of Multi-DOF Serial Robots Dissertation].Hang- 120(4):439 zhou:Zhejiang University,2013 [16]Gandhi P S.Modeling and Control of Nonlinear Transmission At- (吴文祥.多自由度串联机器人关节摩擦分析与低速高精度 tributes in Harmonic Drive Systems Dissertation].Houston:Rice 运动控制[学位论文].杭州：浙江大学，2013) University,2001 [7]Zhu SQ,Wu W X,Wang X Y,et al.Slow motion control of seri- [17]Ye H W,Yang L F,Liu X Z.Optimizing weight and threshold of al robots with harmonic drives considering friction compensation. BP neural network using SFLA:applications to nonlinear function Trans Chin Soc Agric Mach,2013,44(11):293 fitting /Fourth International Conference on Emerging Intelligent (朱世强，吴文祥，王宣银，等.考虑摩擦的谐波驱动机器人 Data and Web Technologies.Xi'an,2013:211 低速运动控制方法.农业机械学报.2013,44(11)：293) [18]Liu S,Gang T Q.Adaptive back-stepping control of the harmonic [8]Liu J.The application of BP neural network in multidimensional drive system with LuGre model-based friction compensation.A/P nonlinear function.J Shangluo Unig,2014,28(6):19 Conf Proc,.2018,1944:020027-1张 铁等: 基于 BP 神经网络的机器人波动摩擦力矩修正方法 精度. (3)神经网络验证实验表明:使用辨识得到的 神经网络函数对原来的计算力矩进行补偿,可使修 正后的力矩误差在机器人平稳运动时基本保持在 [ - 1,1] N·m 的范围内,且有 78郾 82% 的力矩误差 可保持在[ - 0郾 5,0郾 5] N·m 的范围之内. 相比于修 正前的关节力矩误差,修正后的力矩误差的方差为 0郾 1659 N 2·m 2 ,是修正前的 24郾 23% . 参 考 文 献 [1] Iwatani M, Kikuuwe R. Some improvements in elastoplastic fric鄄 tion compensator. SICE J Control Meas Syst Integration, 2017, 10 (3): 141 [2] De Luca A, Mattone R. Sensorless robot collision detection and hybrid force / motion control / / Proceedings of the 2005 IEEE Inter鄄 national Conference on Robotics and Automation. Barcelona, 2005: 999 [3] Haddadin S, De Luca A, Albu鄄Schaffer A. Robot collisions: A survey on detection, isolation, and identification. IEEE Trans Rob, 2017, 33(6): 1292 [4] Gandhi P S, Ghorbel F H, Dabney J. Modeling, identification, and compensation of friction in harmonic drives / / Proceedings of the 41st IEEE Conference on Decision & Control. Las Vegas, 2002: 160 [5] Liao H B, Fan S X, Fan D P. Friction compensation of harmonic gear based on location relationship. Proc Inst Mech Eng Part I J Syst Control Eng, 2016, 230(8): 695 [6] Wu W X. Joint Friction Analysis and Low鄄speed Hight鄄precision Motion Control of Multi鄄DOF Serial Robots [Dissertation]. Hang鄄 zhou: Zhejiang University, 2013 (吴文祥. 多自由度串联机器人关节摩擦分析与低速高精度 运动控制[学位论文]. 杭州: 浙江大学, 2013) [7] Zhu S Q, Wu W X, Wang X Y, et al. Slow motion control of seri鄄 al robots with harmonic drives considering friction compensation. Trans Chin Soc Agric Mach, 2013, 44(11): 293 (朱世强, 吴文祥, 王宣银, 等. 考虑摩擦的谐波驱动机器人 低速运动控制方法. 农业机械学报, 2013, 44(11): 293) [8] Liu J. The application of BP neural network in multidimensional nonlinear function. J Shangluo Univ, 2014, 28(6): 19 (刘俊. BP 神经网络在多维非线性函数拟合中的应用. 商洛 学院学报, 2014, 28(6): 19) [9] Lin Y F, Deng H M, Shi X Y. Application of BP neural network based on newly improved particle swarm optimization algorithm in fitting nonlinear function. Comput Sci, 2017, 44(11A): 51 (林宇锋, 邓洪敏, 史兴宇. 基于新的改进粒子群算法的 BP 神经网络在拟合非线性函数中的应用. 计算机科学, 2017, 44 (11A): 51) [10] Wu W X, Zhu S Q, Jin X L. Dynamic identification for robot manipulators based on modified Fourier series. J Zhejiang Univ Eng Sci, 2013, 47(2): 231 (吴文祥, 朱世强, 靳兴来. 基于改进傅里叶级数的机器人 动力学参数辨识. 浙江大学学报:工学版, 2013, 47 ( 2 ): 231) [11] Khalil W, Gautier M, Lemoine P. Identification of the payload inertial parameters of industrial manipulators / / Proceedings 2007 IEEE International Conference on Robotics and Automation. Ro鄄 ma, 2007: 4943 [12] Sousa C D, Cortes觔o R. Physical feasibility of robot base inertial parameter identification: a linear matrix inequality approach. Int J Rob Res, 2014, 33(6): 931 [13] Xu F Y. Design method of filter for audio noise reduction based on application of Matlab. Audio Eng, 2017, 41(2): 28 (徐帆云. 基于 Matlab 的音频降噪滤波器设计. 电声技术, 2017, 41(2): 28) [14] Yuan F. Comparison of several filtering methods based on MAT鄄 LAB. Power Supply Technol Appl, 2013(10): 50 (远飞. 基于 MATLAB 的几种滤波方法比较. 电源技术应用, 2013(10): 50) [15] Taghirad H D, Belanger P R. Modeling and parameter identifica鄄 tion of harmonic drive systems. J Dyn Syst Meas Control, 1998, 120(4): 439 [16] Gandhi P S. Modeling and Control of Nonlinear Transmission At鄄 tributes in Harmonic Drive Systems [Dissertation]. Houston: Rice University, 2001 [17] Ye H W, Yang L F, Liu X Z. Optimizing weight and threshold of BP neural network using SFLA: applications to nonlinear function fitting / / Fourth International Conference on Emerging Intelligent Data and Web Technologies. Xi蒺an, 2013: 211 [18] Liu S, Gang T Q. Adaptive back鄄stepping control of the harmonic drive system with LuGre model鄄based friction compensation. AIP Conf Proc, 2018, 1944: 020027鄄1 ·1091·