{"defaultlang":"zh","titlegroup":{"articletitle":[{"lang":"zh","data":[{"name":"text","data":"基于二阶锥规划无标定参照物的手眼标定"}]},{"lang":"en","data":[{"name":"text","data":"Hand-eye calibration method without a calibration reference based on second-order cone programming"}]}]},"contribgroup":{"author":[{"name":[{"lang":"zh","surname":"李","givenname":"巍","namestyle":"eastern","prefix":""},{"lang":"en","surname":"LI","givenname":"Wei","namestyle":"western","prefix":""}],"stringName":[],"aff":[{"rid":"aff1","text":"1"}],"role":["corresp","first-author"],"corresp":[{"rid":"cor1","lang":"en","text":"LI Wei, E-mail:liweikilary@163.com","data":[{"name":"text","data":"LI Wei, E-mail:liweikilary@163.com"}]}],"bio":[{"lang":"zh","text":["李巍(1988-), 男, 河北承德人, 博士研究生, 2011年于北华航天工业学院获得学士学位, 2015年于北京信息科技大学获得硕士学位, 主要从事机器视觉及模式识别方面的研究。E-mail:liweikilary@163.com"],"graphic":[],"data":[[{"name":"bold","data":[{"name":"text","data":"李巍"}]},{"name":"text","data":"(1988-), 男, 河北承德人, 博士研究生, 2011年于北华航天工业学院获得学士学位, 2015年于北京信息科技大学获得硕士学位, 主要从事机器视觉及模式识别方面的研究。E-mail:"},{"name":"text","data":"liweikilary@163.com"}]]}],"email":"liweikilary@163.com","deceased":false},{"name":[{"lang":"zh","surname":"董","givenname":"明利","namestyle":"eastern","prefix":""},{"lang":"en","surname":"DONG","givenname":"Ming-li","namestyle":"western","prefix":""}],"stringName":[],"aff":[{"rid":"aff2","text":"2"}],"role":[],"deceased":false},{"name":[{"lang":"zh","surname":"娄","givenname":"小平","namestyle":"eastern","prefix":""},{"lang":"en","surname":"LOU","givenname":"Xiao-ping","namestyle":"western","prefix":""}],"stringName":[],"aff":[{"rid":"aff2","text":"2"}],"role":[],"deceased":false}],"aff":[{"id":"aff1","intro":[{"lang":"zh","label":"1","text":"北京邮电大学 信息光子学与光通信研究院, 北京 100876","data":[{"name":"text","data":"北京邮电大学 信息光子学与光通信研究院, 北京 100876"}]},{"lang":"en","label":"1","text":"Institute of Optical Communication & Optoelectronics, Beijing University of Posts & Telecommunications, Beijing 100876, China","data":[{"name":"text","data":"Institute of Optical Communication & Optoelectronics, Beijing University of Posts & Telecommunications, Beijing 100876, China"}]}]},{"id":"aff2","intro":[{"lang":"zh","label":"2","text":"北京信息科技大学 光电测试技术北京市重点实验室, 北京 100192","data":[{"name":"text","data":"北京信息科技大学 光电测试技术北京市重点实验室, 北京 100192"}]},{"lang":"en","label":"2","text":"Beijing Key Laboratory of Optoelectronics Measurement Technology, Beijing Information Science & Technology University, Beijing 100192, China","data":[{"name":"text","data":"Beijing Key Laboratory of Optoelectronics Measurement Technology, Beijing Information Science & Technology University, Beijing 100192, China"}]}]}]},"abstracts":[{"lang":"zh","data":[{"name":"p","data":[{"name":"text","data":"为了克服传统机器人手眼标定方法求解手眼关系及机器人坐标系与世界坐标系方位关系对标定参照物的依赖，提出一种基于二阶锥规划的无标定参照物手眼标定改进方法，并搭建相关实验系统进行验证。首先，利用运动恢复结构算法解算定义在一个尺度因子基础上的相机运动矩阵；然后，引入对偶四元数理论参数化标定方程中的旋转矩阵和平移向量；最后，通过二阶锥规划方法同时求解相机运动矩阵尺度因子、手眼关系及机器人坐标系与世界坐标系方位关系的最优解。仿真和实测结果表明，在没有标定参照物作为测量基准的情况下，标定结果中旋转参数相对误差为3.998%，平移参数相对误差为0.117%。与其他标定方法相比，该方法提高了无标定参照物模式下机器人手眼标定精度，扩展了手眼标定方法的应用范围。"}]}]},{"lang":"en","data":[{"name":"p","data":[{"name":"text","data":"In order to overcome the restrictions of traditional hand-eye methods for determining hand-eye correspondence and robot-world orientation with a calibration reference, an improved hand-eye calibration approach without a calibration reference is proposed based on second-order cone programming. A relevant experimental system is established for its validation. First, a structure-from-motion approach is used to recover the camera motion matrix up to scaling. Then, the rotation and translation matrix in the calibration equation is parameterized by dual quaternion theory. Finally, the second-order cone programming method is used to simultaneously determine the optimal solution for the scale factor of the camera motion matrix, the robot-world calibration and the hand-eye calibration. Both the simulation and experimental results indicate that, for the calibration precision, the relative error of rotation is 3.998% and the relative error of translation is 0.117% in the absence of a calibration reference as a benchmark. Compared with other calibration methods, the proposed method can effectively improve the accuracy of robot-world calibration and hand-eye calibration without a reference, and extend the range of applications of the hand-eye calibration method."}]}]}],"keyword":[{"lang":"zh","data":[[{"name":"text","data":"机器人"}],[{"name":"text","data":"标定参照物"}],[{"name":"text","data":"手眼标定"}],[{"name":"text","data":"二阶锥规划"}],[{"name":"text","data":"运动恢复结构"}]]},{"lang":"en","data":[[{"name":"text","data":"robot"}],[{"name":"text","data":"calibration reference"}],[{"name":"text","data":"hand-eye calibration"}],[{"name":"text","data":"second-order cone programming"}],[{"name":"text","data":"structure-from-motion"}]]}],"highlights":[],"body":[{"name":"sec","data":[{"name":"sectitle","data":{"label":[{"name":"text","data":"1"}],"title":[{"name":"text","data":"引言"}],"level":"1","id":"s1"}},{"name":"p","data":[{"name":"text","data":"自从1966年世界上第一台具有视觉传感器的智能机器人Shakey诞生以来，视觉传感器广泛应用于机器人自主避障和导航，医疗手术，汽车制造等领域"},{"name":"sup","data":[{"name":"text","data":"["},{"name":"xref","data":{"text":"1","type":"bibr","rid":"b1","data":[{"name":"text","data":"1"}]}},{"name":"text","data":", "},{"name":"xref","data":{"text":"2","type":"bibr","rid":"b2","data":[{"name":"text","data":"2"}]}},{"name":"text","data":"]"}]},{"name":"text","data":"。在利用机器人视觉系统进行自主作业时，需要将摄像机固定在机器人手臂的末端执行器上构成手眼系统，通过视觉传感器给机器人运动控制系统提供眼的功能。评估上述机器人视觉系统测量精度的关键步骤之一是准确地测算出机器人坐标系与视觉传感器坐标系间的相对方位关系"},{"name":"sup","data":[{"name":"text","data":"["},{"name":"xref","data":{"text":"3","type":"bibr","rid":"b3","data":[{"name":"text","data":"3"}]}},{"name":"text","data":"]"}]},{"name":"text","data":"，即Tsai"},{"name":"sup","data":[{"name":"text","data":"["},{"name":"xref","data":{"text":"4","type":"bibr","rid":"b4","data":[{"name":"text","data":"4"}]}},{"name":"text","data":"]"}]},{"name":"text","data":"所定义的机器人手眼标定过程。"}]},{"name":"p","data":[{"name":"text","data":"目前大部分手眼标定方法都是通过精确标定的参照物(如棋盘格靶标)求解相机姿态变换矩阵，具有代表性的方法有Zhuang等"},{"name":"sup","data":[{"name":"text","data":"["},{"name":"xref","data":{"text":"5","type":"bibr","rid":"b5","data":[{"name":"text","data":"5"}]}},{"name":"text","data":"]"}]},{"name":"text","data":"的线性二分法、Horaud等"},{"name":"sup","data":[{"name":"text","data":"["},{"name":"xref","data":{"text":"6","type":"bibr","rid":"b6","data":[{"name":"text","data":"6"}]}},{"name":"text","data":"]"}]},{"name":"text","data":"的单位四元数法、Shah等"},{"name":"sup","data":[{"name":"text","data":"["},{"name":"xref","data":{"text":"7","type":"bibr","rid":"b7","data":[{"name":"text","data":"7"}]}},{"name":"text","data":"]"}]},{"name":"text","data":"的矩阵直积法、Daniilidis和Ulrich等"},{"name":"sup","data":[{"name":"text","data":"["},{"name":"xref","data":{"text":"8","type":"bibr","rid":"b8","data":[{"name":"text","data":"8"}]}},{"name":"text","data":", 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From Motion，SFM)方法，直接从自然场景中获得缺失尺度因子的相机姿态变换矩阵，将未知的尺度因子与手眼关系一同代入手眼标定方程进行求解。近年来，这种不需要标定参照物的手眼标定改进方法获得学者们的普遍重视和研究。例如，在Andreff的手眼标定模型基础上，Schmidt等"},{"name":"sup","data":[{"name":"text","data":"["},{"name":"xref","data":{"text":"17","type":"bibr","rid":"b17","data":[{"name":"text","data":"17"}]}},{"name":"text","data":"]"}]},{"name":"text","data":"提出基于对偶四元数扩展的非线性最优化法，Heller"},{"name":"sup","data":[{"name":"text","data":"["},{"name":"xref","data":{"text":"18","type":"bibr","rid":"b18","data":[{"name":"text","data":"18"}]}},{"name":"text","data":"]"}]},{"name":"text","data":"和陈明伟等"},{"name":"sup","data":[{"name":"text","data":"["},{"name":"xref","data":{"text":"19","type":"bibr","rid":"b19","data":[{"name":"text","data":"19"}]}},{"name":"text","data":"]"}]},{"name":"text","data":"提出基于二阶锥规划的全局优化方法、Pachtrachai等"},{"name":"sup","data":[{"name":"text","data":"["},{"name":"xref","data":{"text":"20","type":"bibr","rid":"b20","data":[{"name":"text","data":"20"}]}},{"name":"text","data":"]"}]},{"name":"text","data":"提出的基于计算机辅助设计(Computer Aided Design，CAD)导入模型的微创手术机器人手眼标定方法以及Wang等"},{"name":"sup","data":[{"name":"text","data":"["},{"name":"xref","data":{"text":"21","type":"bibr","rid":"b21","data":[{"name":"text","data":"21"}]}},{"name":"text","data":"]"}]},{"name":"text","data":"提出的微创手术机器人远程控制中心(Remote Center-of-Motion，RCM)机构视觉标定法。然而，这些方法只能计算手眼关系，不能同时求解出机器人坐标系与世界坐标系方位关系，且受特征点匹配误差和未知尺度因子的影响，基于SFM场景重建的无标定参照物手眼标定方法往往精度较低，不能满足移动机器人视觉导航和定位日益增长的精度需求。为解决这个问题，2017年，Park等"},{"name":"sup","data":[{"name":"text","data":"["},{"name":"xref","data":{"text":"22","type":"bibr","rid":"b22","data":[{"name":"text","data":"22"}]}},{"name":"text","data":"]"}]},{"name":"text","data":"以角度投影误差为优化模型提出基于分支定界法的全局优化方法，但该模型只适用于无平移量的云台摄像机旋转变换关系的优化，不能应用于移动机器人系统。"}]},{"name":"p","data":[{"name":"text","data":"针对上述问题，本文提出一种基于二阶锥规划无标定参照物手眼标定改进方法。首先利用SFM方法求解缺失尺度因子的摄像机姿态变换矩阵；然后推导了基于SFM的机器人手眼关系以及机器人坐标系与世界坐标系方位关系标定模型，并将标定模型中的非线性误差函数转化为具有凸可行域的二阶锥规划问题进行求解；最后以没有标志点的普通参照物进行对比实验。结果表明：与其他基于SFM的标定方法相比，该方法可以提高手眼关系和机器人坐标系与世界坐标系方位关系求解精度。具有操作简单、成本低廉、适用范围广等优点。"}]}]},{"name":"sec","data":[{"name":"sectitle","data":{"label":[{"name":"text","data":"2"}],"title":[{"name":"text","data":"手眼标定模型"}],"level":"1","id":"s2"}},{"name":"p","data":[{"name":"text","data":"如"},{"name":"xref","data":{"text":"图 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LP)的推广，可以看作是半定规划(SDP)的一种特例，在求解非光滑凸规划问题时，兼具解的最优性和计算高效性双重特性。广泛应用于电力系统和金融信息等领域的组合优化问题"},{"name":"sup","data":[{"name":"text","data":"["},{"name":"xref","data":{"text":"23","type":"bibr","rid":"b23","data":[{"name":"text","data":"23"}]}},{"name":"text","data":"]"}]},{"name":"text","data":"。"}]},{"name":"p","data":[{"name":"text","data":"为了最大限度确保式(8)中求解的平移向量"},{"name":"bold","data":[{"name":"italic","data":[{"name":"text","data":"t"}]}]},{"name":"sub","data":[{"name":"bold","data":[{"name":"italic","data":[{"name":"text","data":"X"}]}]}]},{"name":"text","data":", "},{"name":"bold","data":[{"name":"italic","data":[{"name":"text","data":"t"}]}]},{"name":"sub","data":[{"name":"bold","data":[{"name":"italic","data":[{"name":"text","data":"Y"}]}]}]},{"name":"text","data":"和尺度因子"},{"name":"italic","data":[{"name":"text","data":"s"}]},{"name":"text","data":"每次优化结果都能收敛到全局最优解，需要结合手眼标定理论与SOCP松弛方法，将原最优化问题转化为标准的二阶锥规划问题求解。式(8)手眼标定方程目标函数可以表示为L"},{"name":"sub","data":[{"name":"text","data":"2"}]},{"name":"text","data":"范数和最小的形式："}]},{"name":"p","data":[{"name":"dispformula","data":{"label":[{"name":"text","data":"11"}],"data":[{"name":"text","data":" "},{"name":"text","data":"  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2","type":"fig","rid":"Figure2","data":[{"name":"text","data":"图 2"}]}},{"name":"text","data":"所示(彩图见期刊电子版)，首先，将原问题非凸可行域"},{"name":"bold","data":[{"name":"italic","data":[{"name":"text","data":"U"}]}]},{"name":"sub","data":[{"name":"text","data":"original"}]},{"name":"text","data":"松弛成为一个凸二阶锥可行域"},{"name":"bold","data":[{"name":"italic","data":[{"name":"text","data":"U"}]}]},{"name":"sub","data":[{"name":"text","data":"soc"}]},{"name":"text","data":"，此时原标定方程已转换到SOCP的凸空间上，再利用高效的内点算法求解SOCP的全局最优解。由于二阶锥松弛的引入，可行域"},{"name":"bold","data":[{"name":"italic","data":[{"name":"text","data":"U"}]}]},{"name":"sub","data":[{"name":"text","data":"soc"}]},{"name":"text","data":"中得到的最优解("},{"name":"xref","data":{"text":"图 2(b)","type":"fig","rid":"Figure2","data":[{"name":"text","data":"图 2(b)"}]}},{"name":"text","data":"红点G)是原问题的一个下界解，若该最优解("},{"name":"xref","data":{"text":"图 2(a)","type":"fig","rid":"Figure2","data":[{"name":"text","data":"图 2(a)"}]}},{"name":"text","data":"红点G)是原可行域"},{"name":"bold","data":[{"name":"italic","data":[{"name":"text","data":"U"}]}]},{"name":"sub","data":[{"name":"text","data":"original"}]},{"name":"text","data":"中的点，则其便是原问题的最优解。"}]},{"name":"fig","data":{"id":"Figure2","caption":[{"lang":"zh","label":[{"name":"text","data":"图2"}],"title":[{"name":"text","data":"二阶锥规划示意图"}]},{"lang":"en","label":[{"name":"text","data":"Fig 2"}],"title":[{"name":"text","data":"Second-Order cone programming schematic drawing"}]}],"subcaption":[],"note":[],"graphics":[{"print":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=1711838&type=","small":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=1711838&type=small","big":"http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=1711838&type=middle"}]}},{"name":"p","data":[{"name":"text","data":"需要说明的是如果最终结果不需要求解尺度因子"},{"name":"italic","data":[{"name":"text","data":"s"}]},{"name":"text","data":"，可以在式(3)平移方程两端都乘以反对称矩阵"},{"name":"bold","data":[{"name":"italic","data":[{"name":"text","data":"t"}]}]},{"name":"sub","data":[{"name":"bold","data":[{"name":"italic","data":[{"name":"text","data":"A"}]}]}]},{"name":"text","data":"，消去尺度因子"},{"name":"italic","data":[{"name":"text","data":"s"}]},{"name":"text","data":"如式(13)所示，再利用上述二阶锥规划方法求解平移向量。"}]},{"name":"p","data":[{"name":"dispformula","data":{"label":[{"name":"text","data":"13"}],"data":[{"name":"text","data":" "},{"name":"text","data":"  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"},{"name":"italic","data":[{"name":"text","data":"then"}]},{"name":"text","data":"不存在可行的解，删除第"},{"name":"italic","data":[{"name":"text","data":"i"}]},{"name":"text","data":"次相机运动数据，重新计算"},{"name":"bold","data":[{"name":"italic","data":[{"name":"text","data":"x"}]}]},{"name":"text","data":";"}]}],[{"data":[{"name":"text","data":"      6. "},{"name":"italic","data":[{"name":"text","data":"until δ"}]},{"name":"text","data":" ≤ "},{"name":"italic","data":[{"name":"text","data":"e"}]},{"name":"text","data":";"}]}],[{"style":"class:table_bottom_border","data":[{"name":"text","data":"      7.求解式(12)SOCP问题."}]}]],"foot":[]}]}},{"name":"p","data":[{"name":"text","data":"首先利用线性方法求解标定方程(7), 计算每组相机和机器人运动位姿数据对应的误差值"},{"name":"bold","data":[{"name":"italic","data":[{"name":"text","data":"C"}]}]},{"name":"sub","data":[{"name":"bold","data":[{"name":"italic","data":[{"name":"text","data":"i"}]}]}]},{"name":"bold","data":[{"name":"italic","data":[{"name":"text","data":"x"}]}]},{"name":"text","data":"-"},{"name":"bold","data":[{"name":"italic","data":[{"name":"text","data":"d"}]}]},{"name":"sub","data":[{"name":"bold","data":[{"name":"italic","data":[{"name":"text","data":"i"}]}]}]},{"name":"text","data":"，如果误差值大于数据筛选的阈值"},{"name":"italic","data":[{"name":"text","data":"e"}]},{"name":"text","data":"，则删除此次相机运动数据重新计算，直到每组数据都满足筛选阈值后，再利用SOCP优化方法计算式(12)的最优问题。"}]},{"name":"p","data":[{"name":"text","data":"具体实验中可以结合鲁棒方法(如随机抽样一致性和M值估计法)进行数据筛选。"}]}]}]},{"name":"sec","data":[{"name":"sectitle","data":{"label":[{"name":"text","data":"5"}],"title":[{"name":"text","data":"测量实验与结果"}],"level":"1","id":"s5"}},{"name":"sec","data":[{"name":"sectitle","data":{"label":[{"name":"text","data":"5.1"}],"title":[{"name":"text","data":"仿真实验结果及分析"}],"level":"2","id":"s5-1"}},{"name":"p","data":[{"name":"text","data":"为模拟手眼标定现场的真实过程，在Inter 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