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Article(id=1312006261491725263, tenantId=1146029695717560320, journalId=1311385739772133495, issueId=1311999647825101743, articleNumber=null, orderNo=null, doi=10.6041/j.issn.1000-1298.2025.12.075, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1724515200000, receivedDateStr=2024-08-25, revisedDate=1728748800000, revisedDateStr=2024-10-13, acceptedDate=null, acceptedDateStr=null, onlineDate=1790736268637, onlineDateStr=2026-09-30, pubDate=1766592000000, pubDateStr=2025-12-25, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1790736268637, onlineIssueDateStr=2026-09-30, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1790736268637, creator=13701087609, updateTime=1790736268637, updator=13701087609, issue=Issue{id=1311999647825101743, tenantId=1146029695717560320, journalId=1311385739772133495, year='2025', volume='56', issue='12', pageStart='1', pageEnd='818', issueExtLink='null', onlineDate='null', pubDate='1766592000000', pubDateStr='2025-12-25', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=1, specialIssue=null, createTime=1790734691815, creator='13701087609', updateTime=1790738411040, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext={EN=IssueExt(id=1312015247465083866, tenantId=1146029695717560320, journalId=1311385739772133495, issueId=1311999647825101743, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1312015247465083867, tenantId=1146029695717560320, journalId=1311385739772133495, issueId=1311999647825101743, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null, downloadFileDto=null}, startPage=811, endPage=818, ext={EN=ArticleExt(id=1312006261743383505, articleId=1312006261491725263, tenantId=1146029695717560320, journalId=1311385739772133495, language=EN, title=Kinematics and Dynamics Analysis of 2-RRC/1-RRS Parallel Manipulator, columnId=1312006261667886032, journalTitle=Transactions of the Chinese Society for Agricultural Machinery, columnName=Mechanical Design & Manufacturing and Automation, runingTitle=null, highlight=null, articleAbstract=

A novel structure model of 2-RRC/1-RRS parallel manipulator mechanism was proposed for the screening of agricultural materials, and its kinematics and dynamic characteristics were studied. Firstly, based on the motion characteristics of each branch chain in the 2-RRC/1-RRS parallel manipulator mechanism, the manipulator structure characteristics and the coupling relationship between the motion variables were analyzed. Secondly, the kinematics of the 2-RRC/1-RRS parallel mechanism was deduced, and the analytical expression of the inverse kinematics of the robot was given. Then based on the discussion of kinetic energy and potential energy of the 2-RRC/1-RRS parallel manipulator, the dynamic equation of the manipulator was derived by Lagrange equation method of classical mechanics. Finally, the dynamic characteristics of the 2-RRC/1-RRS parallel manipulator mechanism, such as motion parameters, driving torque and energy consumption, were discussed by an example. The results showed that the parameters of moment of inertia in the dynamics equation of the 2-RRC/1-RRS parallel manipulator system were closely related to the structure size of the manipulator, the mass size and mass distribution of each component, and the motion configuration of the manipulator, so as to determine the kinematics and dynamics characteristics of the 2-RRC/1-RRS parallel manipulator. The research content was of great significance to deeply understand the kinematics and dynamic characteristics of 2-RRC/1-RRS parallel manipulator, as well as the structural optimization design and the high-precision control of the manipulator, which can provide an idea for the innovative design of new equipment for agricultural material screening.

, authors=Shanzeng LIU1, 2, Zhujun ZHANG1, Dechao MENG1, authorsList=Shanzeng LIU, Zhujun ZHANG, Dechao MENG, authorCompany=null, correspAuthors=null, authorNote=null, correspAuthorsNote=null, copyrightStatement=null, copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, fund=null), CN=ArticleExt(id=1312006278713536602, articleId=1312006261491725263, tenantId=1146029695717560320, journalId=1311385739772133495, language=CN, title=2-RRC/1-RRS并联机器人运动学与动力学分析, columnId=1312006261831463890, journalTitle=农业机械学报, columnName=机械设计制造及其自动化, runingTitle=null, highlight=null, articleAbstract=

针对农业物料筛分问题，本文提出了一种2-RRC/1-RRS并联机器人机构模型，对其运动学和动态特性进行了研究。基于2-RRC/1-RRS并联机器人机构各支链运动特点，开展了机器人结构特征与运动变量耦合关系分析。进行了2-RRC/1-RRS并联机器人机构运动学分析，给出了机器人运动学反解解析表达式。基于2-RRC/1-RRS并联机器人系统动能和势能分析，利用经典力学Lagrange方程法，推导了机器人动力学方程。通过算例探讨了2-RRC/1-RRS并联机器人机构运动参量、驱动力和能耗变化规律。研究结果对认知2-RRC/1-RRS并联机器人运动学特征、动态特性以及结构优化设计和系统的高精度控制等具有重要意义，为农业物料筛分装备设计提供了新思路。

, authors=刘善增1, 2, 张著军1, 孟德超1, authorsList=刘善增, 张著军, 孟德超, authorCompany=null, correspAuthors=null, authorNote=

刘善增（1977—），男，副教授，博士，主要从事机构学与机器人技术研究，E-mail：

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2.伦敦国王学院，伦敦　WC2R 2LS, bio={"content":"

刘善增（1977—），男，副教授，博士，主要从事机构学与机器人技术研究，E-mail：

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刘善增（1977—），男，副教授，博士，主要从事机构学与机器人技术研究，E-mail：

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journalId=1311385739772133495, articleId=1312006261491725263, language=CN, label=图6, caption=转动惯量变化曲线, figureFileSmall=06QxigegqYmad8uTVH8+iA==, figureFileBig=SlOVcBThT2Vwb4SXEaHKtg==, tableContent=null), ArticleFig(id=1312006283205636232, tenantId=1146029695717560320, journalId=1311385739772133495, articleId=1312006261491725263, language=EN, label=Tab. 1, caption=

Driving torque

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项目|τ1||τ2||τ3|
最大值0.565 918.960 515.174 1
最小值0.166 711.272 412.885 0
平均值0.375 514.901 514.129 0
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驱动转矩

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项目|τ1||τ2||τ3|
最大值0.565 918.960 515.174 1
最小值0.166 711.272 412.885 0
平均值0.375 514.901 514.129 0
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Moment of inertia

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项目
最大值0.4530.7520.5040.1490.1700.179
最小值0.2680.2350.3060.1070.0850.047
平均值0.3750.4320.3880.1260.1250.140
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转动惯量

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项目
最大值0.4530.7520.5040.1490.1700.179
最小值0.2680.2350.3060.1070.0850.047
平均值0.3750.4320.3880.1260.1250.140
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2-RRC/1-RRS并联机器人运动学与动力学分析
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刘善增 1, 2 , 张著军 1 , 孟德超 1
农业机械学报 | 机械设计制造及其自动化 2025,56(12): 811-818
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农业机械学报 |机械设计制造及其自动化 2025 , 56 (12) : 811 -818
2-RRC/1-RRS并联机器人运动学与动力学分析
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2.伦敦国王学院，伦敦　WC2R 2LS, bio={"content":"

刘善增（1977—），男，副教授，博士，主要从事机构学与机器人技术研究，E-mail：

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刘善增（1977—），男，副教授，博士，主要从事机构学与机器人技术研究，E-mail：

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刘善增1, 2 , 张著军1, 孟德超1
作者信息
  • 1.山东石油化工学院智能制造与控制工程学院，东营　257061
  • 2.伦敦国王学院，伦敦　WC2R 2LS
作者简介:

刘善增（1977—），男，副教授，博士，主要从事机构学与机器人技术研究，E-mail：

Kinematics and Dynamics Analysis of 2-RRC/1-RRS Parallel Manipulator
Shanzeng LIU1, 2 , Zhujun ZHANG1, Dechao MENG1
Affiliations
  • 1.School of Intelligent Manufacturing and Control Engineering, Shandong Institute of Petroleum and Chemical Technology, Dongying 257061, China
  • 2.King's College London, London WC2R 2LS, UK
出版时间: 2025-12-25 doi: 10.6041/j.issn.1000-1298.2025.12.075
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摘要
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针对农业物料筛分问题，本文提出了一种2-RRC/1-RRS并联机器人机构模型，对其运动学和动态特性进行了研究。基于2-RRC/1-RRS并联机器人机构各支链运动特点，开展了机器人结构特征与运动变量耦合关系分析。进行了2-RRC/1-RRS并联机器人机构运动学分析，给出了机器人运动学反解解析表达式。基于2-RRC/1-RRS并联机器人系统动能和势能分析，利用经典力学Lagrange方程法，推导了机器人动力学方程。通过算例探讨了2-RRC/1-RRS并联机器人机构运动参量、驱动力和能耗变化规律。研究结果对认知2-RRC/1-RRS并联机器人运动学特征、动态特性以及结构优化设计和系统的高精度控制等具有重要意义，为农业物料筛分装备设计提供了新思路。

并联机器人  /  运动学  /  动力学  /  Lagrange方程  /  物料筛分

A novel structure model of 2-RRC/1-RRS parallel manipulator mechanism was proposed for the screening of agricultural materials, and its kinematics and dynamic characteristics were studied. Firstly, based on the motion characteristics of each branch chain in the 2-RRC/1-RRS parallel manipulator mechanism, the manipulator structure characteristics and the coupling relationship between the motion variables were analyzed. Secondly, the kinematics of the 2-RRC/1-RRS parallel mechanism was deduced, and the analytical expression of the inverse kinematics of the robot was given. Then based on the discussion of kinetic energy and potential energy of the 2-RRC/1-RRS parallel manipulator, the dynamic equation of the manipulator was derived by Lagrange equation method of classical mechanics. Finally, the dynamic characteristics of the 2-RRC/1-RRS parallel manipulator mechanism, such as motion parameters, driving torque and energy consumption, were discussed by an example. The results showed that the parameters of moment of inertia in the dynamics equation of the 2-RRC/1-RRS parallel manipulator system were closely related to the structure size of the manipulator, the mass size and mass distribution of each component, and the motion configuration of the manipulator, so as to determine the kinematics and dynamics characteristics of the 2-RRC/1-RRS parallel manipulator. The research content was of great significance to deeply understand the kinematics and dynamic characteristics of 2-RRC/1-RRS parallel manipulator, as well as the structural optimization design and the high-precision control of the manipulator, which can provide an idea for the innovative design of new equipment for agricultural material screening.

parallel manipulator  /  kinematics  /  dynamics  /  Lagrange equation  /  material screening
刘善增, 张著军, 孟德超. 2-RRC/1-RRS并联机器人运动学与动力学分析. 农业机械学报, 2025 , 56 (12) : 811 -818 . DOI: 10.6041/j.issn.1000-1298.2025.12.075
Shanzeng LIU, Zhujun ZHANG, Dechao MENG. Kinematics and Dynamics Analysis of 2-RRC/1-RRS Parallel Manipulator[J]. Transactions of the Chinese Society for Agricultural Machinery, 2025 , 56 (12) : 811 -818 . DOI: 10.6041/j.issn.1000-1298.2025.12.075
正文
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0　引言
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并联机器人相对于串联机器人具有诸多优点，引起了国际机构学和机器人技术学者们广泛关注[1-4]。由于在许多应用场合机器人只需部分自由度（2~5个自由度）就可满足使用要求，所以，20世纪80年代以来，世界各国学者相继开展了少自由度并联机器人探索和应用开发工作[5-8]，提出了Delta机器人、4-UPU并联机器人、5-TPS/PRPT并联机器人等，为机构学和机器人研究注入了新活力。HUNT[9]提出了能实现2个转动和1个移动的3-RPS并联机构，LEE等[10]对3-RPS并联机构进行了运动学和动力学分析，并将此机构应用为3自由度机器人操作器主臂。GOSSELIN等[11]分析了平面3自由度并联机器人运动学最优设计问题。FANG等[12]研究了3-RPS并联机构微分运动学问题，并采用螺旋理论分析了3-RPS、3-RRS和3-TPT等并联机构瞬时运动特性。CARRETERO等[13]分析了3-PRS并联机器人运动学问题，并利用非线性优化方法对系统有关参数进行了优化。WANG等[14]通过添加配重和弹簧研究了3-RRS并联机器人静平衡问题。赵铁石等[15]对3-RRC并联机器人位置分析、工作空间和运动学等问题进行了研究。文献[16-19]开展了3-RRC和3-RRS并联机器人刚性运动学和动力学研究以及考虑支链中构件柔性变形的机构刚柔耦合动力学建模和动态特性分析问题。李剑锋等[20]基于雅可比矩阵进行了一种适用于踝关节康复的3-UPS/RRR并联机构运动学、刚度特性和力矩传递性能分析。张俊等[21]以一类2UPR&2RPS型冗余驱动并联机器人为研究对象，进行了几何误差分析和运动学标定方法研究。LIU等[22]提出了一种用于评价四自由度高速并联机器人加速能力的性能指标，探究了运动/力的传递能力对机构加速能力的影响。张惠普等[23]针对4-RR（SS）2四自由度高速并联机器人，建立了以使机构动力学性能最优的机构尺度综合模型，开展了动力尺度综合方法研究。
农业物料筛分是机械化联合收获和农作物处理加工中的重要环节，筛分环节常常出现的设备物料堵塞是导致筛分效率降低、清选损失增加的主要因素；同时，目前筛分设备还存在振动较大、噪声污染、可靠性低等显著问题。针对一种能应用于振动筛分及医疗康复方面的2-RRC/1-RRS并联机器人，本文进行其运动学和动力学特性探究。开展机器人结构特征和运动学规律分析；基于Lagrange方程法建立系统形式简洁的解析式动力学方程；最后，探讨2-RRC/1-RRS并联机器人动态特性。
1　结构特征
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2-RRC/1-RRS并联机器人机构系统三维模型，如图1a所示，其机构运动简图如图1b所示。此并联机构动平台P1P2P3与基座B1B2B3之间由3个支链连接，其中支链B1C1P1的B1处和C1处皆为转动副，P1处为球面副。支链B2C2P2与B3C3P3结构组成相同，即Bi处和Ci处皆为转动副，Pi处为圆柱副（i=2，3）。建立如图1b所示的系统坐标系OXYZ以及与动平台P1P2P3相固结的局部坐标系Pxyz。
2-RRC/1-RRS并联机器人结构参数如图1b所示，其中BiCi长度为li1，CiPi长度为li2（i=1，2，3）以及l1=2b2和l2=2b4。在坐标系OXYZ中，点P坐标表示为（xP，yP，zP）T，同时给定点Bi（i=1，2，3）坐标矢量为B1=（b1，0，0）T、B2=（0，b2，0）T和B3=（0，-b2，0）T。在局部坐标系Pxyz中，点Pi（i=1，2，3）坐标矢量可表示为p1=（b3，0，0）T、p2=（-xP，b4，0）T和p3=（-xP，-b4，0）T。则在坐标系OXYZ中，点Pi（i=1，2，3）位置矢量Pi可表示为
式中
      T——变换矩阵
由于2-RRC/1-RRS并联机器人机构中支链BiCiPi（i=1，2，3）皆分别受到2个相互平行的转动副约束。所以，支链上点P1运动轨迹应位于OXZ平面内，支链中点P2和点P3运动轨迹应位于OYZ平面内。
根据前述分析，由式（1）可得
式中
      ni、nj——坐标系Pxyz中x轴单位主矢相对于坐标系OXYZ中X、Y轴的方向余旋
      oi——坐标系Pxyz中x轴单位主矢相对于坐标系OXYZ中X轴的方向余旋
取Z-Y-X型欧拉角（α，β，γ）[24]表示动平台P1P2P3姿态，根据支链BiCiPi（i=1，2，3）各点Pi运动轨迹特点，化简式（2）可得
因此，可得到2-RRC/1-RRS并联机器人机构运动特征为：动平台6个位姿变量xP、yP、zP、α、β、γ中，存在α≡0、β≡0和yP≡0。所以，此并联机构拥有1个转动自由度和2个移动自由度，共3个自由度；并联机构3个自由度间无驱动约束关系，即γ、zP和xP之间无驱动耦合，有利于系统高精度控制。
2　机构运动学
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设点Pi（i=1，2，3）在坐标系OXYZ中坐标为（xi，yi，zi）T，则由式（1）可得
同时，若给定构件BiCi和CiPi（i=1，2，3）运动转角为θi、αi，则点Pi（i=1，2，3）坐标（xi，yi，zi）T也可通过支链BiCiPi（i=1，2，3）运动位姿关系表示为
联立式（4）~（6）和式（7）~（9）可得
求解式（10）~（12）可得
其中
从式（13）~（15）可以看出，当已知2-RRC/1-RRS并联机器人机构动平台运动位姿变量γ、zP和xP时，此并联机构存在8种运动学反解位形。可知，对于相同机器人操作任务或操作端运动轨迹，可以通过机器人机构驱动位形优选或轨迹优化达到改善机器人运动特性的目的。
3　机构动力学
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假定2-RRC/1-RRS并联机器人机构中构件BiCi和CiPi（i=1，2，3）质量为mi1和mi2，各构件质心皆处于其几何中心且质心坐标表示为（xi1，yi1，zi1）T和（xi2，yi2，zi2）T。在坐标系OXYZ中，构件BiCi、CiPi质心坐标以及动平台质心点P坐标可表达为（未给出的坐标值恒为0）
2-RRC/1-RRS并联机器人中构件BiCi和CiPi（i=1，2，3）质心速度分别为和，以及绕各自质心转动惯量分别为Ji1和Ji2；动平台质心线速度为，角速度为，动平台质量为m0，动平台相对于坐标系Pxyz的惯性矩阵为IP，对其质心主转动惯量分别为Jx、Jy和Jz。取图1b中OXY面为重力零势能面，重力加速度为g，则系统动能T和势能V可表示为
式（16）~（22）分别对时间t求导，代入式（23）可得
其中
式中
      ——机器人系统转动惯量
将式（24）、（25）代入Lagrange方程
式中
      τ——第i个广义坐标的广义力
并化简，可以得到
同时，2-RRC/1-RRS并联机器人系统驱动能耗E可以表示为
式中
      t0——机器人运动起始时刻
      tf——机器人运动终止时刻
4　算例
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给定2-RRC/1-RRS并联机器人中所有构件材质为合金钢，材料密度ρ为7 700 kg/m3；支链中构件长度li1=li2=0.20 m（i=1，2，3），矩形截面尺寸为0.02 m×0.01 m；基座及动平台结构参数b1=0.22 m，b2=b4=0.12 m，b3=0.15 m；动平台质量参数m0=8.5 kg，Jx=4.5×10-2 kg·m2，Jy=1.2×10-1 kg·m2，Jz=1.7×10-1 kg·m2；支链下连杆质量参数为m11=m21=m31=0.35 kg，J11=J21=J31=1.3×10-3 kg·m2；支链上连杆质量参数为m12=0.34 kg，m22=m32=0.38 kg，J12=1.2×10-3 kg·m2，J22=J32=1.6×10-3 kg·m2；时间T1=20 s。
机器人操作端运动规律为
其中
式中
      s（t）——摆线运动函数
对于给定的机器人操作端运动规律，2-RRC/1-RRS并联机器人动平台运动参量位移变化曲线如图2所示。xP运动区间为[-50 mm，50 mm]，zP运动区间为[180 mm，300 mm]，γ运动区间为[-0.17 rad，0.17 rad]。由于采用摆线运动函数操作运动轨线，机器人运动起始和终止时刻轨迹曲线都比较平滑，避免了机器人运行过程中产生刚性冲击或柔性冲击。
驱动构件BiCi（i=1，2，3）运动转角变化曲线，如图3所示，对应的运动角速度变化曲线，如图4所示。θ1运动变化范围为[0.580 rad，1.320 rad]，而θ2、θ3运动变化范围分别为[0.420 rad，0.936 rad]和[0.517 rad，0.766 rad]。构件B1C1运动角速度ω1最大值为0.070 9 rad/s，构件B2C2、B3C3运动角速度ω2与ω3最大值为0.051 9、0.024 4 rad/s。
机器人驱动构件BiCi（i=1，2，3）转矩变化曲线，如图5所示，各个驱动转矩见表1。驱动转矩τ1最大值为0.565 9 N·m，最小值为0.166 7 N·m，波动率为3.395倍；驱动转矩τ2最大值为18.960 5 N·m，最小值为11.272 4 N·m，波动率为1.677倍；驱动转矩τ3最大值为15.174 1 N·m，最小值为12.885 0 N·m，波动率为1.178倍。对于给定的机器人运动规律，驱动构件B1C1电机能耗为0.284 3 J，驱动构件B2C2电机能耗为7.475 2 J，驱动构件B3C3电机能耗为3.596 0 J，系统总能耗为11.355 5 J。可见，通过了解2-RRC/1-RRS并联机器人驱动特性，可为机器人系统电机选型和工程应用的优化提供科学依据。
由式（26）~（28）可知，2-RRC/1-RRS并联机器人各个驱动构件驱动转矩不仅与变量θ1、θ2和θ3及其导数有关，而且与转动惯量、、、、和及波动密切相关。分析及优化前述转动惯量参量变化特性，对改善机器人系统动态特性以及降低驱动能耗都具有重要意义。对于给定机器人操作端运动规律，转动惯量、、、、和变化曲线如图6所示。转动惯量如表2所示。变化比较剧烈，且其最大值达到0.752 kg·m2，这主要源于受到动平台质量m0影响以及机器人运动位形的作用。和取值及波动都比较小，表明变量θ1与θ2、θ3及其对应支链之间的耦合作用较弱。显然，这些转动惯量与机器人结构尺寸、各构件质量及其分布等密切相关，且随着机构运动轨迹位形改变而变化，从而决定了机器人运动学和动力学特性。所以，在机器人设计以及运动规划时，需充分考虑机器人结构质量分布和运动轨迹问题，以便有效达到优化机器人动态特性的目的。
5　结束语
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对2-RRC/1-RRS并联机器人结构特性进行了分析，揭示了机构中各运动参量间的耦合关系；分析了2-RRC/1-RRS并联机器人机构运动学参量关系，给出了机器人运动学反解解析表达式；利用Lagrange方程法推导了2-RRC/1-RRS并联机器人机构动力学模型，为机器人动力学控制奠定了基础；通过算例讨论了2-RRC/1-RRS并联机器人运动学和动力学特性，探明了机器人动态特征及其变化规律，为2-RRC/1-RRS并联机器人产品开发奠定了理论依据。
基金
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  • 国家自然科学基金项目(51975571)
  • 东营市科学发展基金项目(DJB2023050)
文献
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2025年第56卷第12期
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doi: 10.6041/j.issn.1000-1298.2025.12.075
  • 接收时间：2024-08-25
  • 首发时间：2026-09-30
  • 出版时间：2025-12-25
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  • 收稿日期：2024-08-25
  • 修回日期：2024-10-13
基金
国家自然科学基金项目(51975571)
东营市科学发展基金项目(DJB2023050)
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    1.山东石油化工学院智能制造与控制工程学院，东营　257061
    2.伦敦国王学院，伦敦　WC2R 2LS
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2种不同金属材料的力学参数
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Number of
species		 占总种数比例
Percentage of
total species (%)		 属
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Number of
species		 占总种数比例
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鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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