柔性线缆的机器人自动敷设关键技术与发展趋势

doi:10.3901/JME.2022.17.075

摘要/Abstract

摘要： 柔性线缆在机电产品中大量应用，其敷设工作是一项非常重要、但又繁琐而复杂的任务。随着机器人技术的发展和普遍应用，柔性线缆的机器人自动敷设技术逐渐引起了国内外学者的关注。柔性线缆的机器人自动敷设问题归属于柔性线缆的机器人自动规划问题，首先对柔性线缆的机器人自动规划技术的研究现状进行了总结；然后对柔性线缆的机器人自动敷设问题进行了分析，围绕平面上柔性线缆的机器人自动敷设过程，提出了柔性线缆机器人自动敷设的整体流程，介绍了其中的关键技术，包括柔性线缆的物性建模、运动学约束、碰撞接触响应、机器人单/双臂控制自动敷设和图像处理等的实现方法，并结合实例进行了仿真和实验验证；最后对柔性线缆的机器人自动敷设技术的未来发展趋势进行了总结与展望。

关键词: 柔性线缆, 机器人, 自动敷设, 物性建模, 碰撞接触响应

Abstract: Flexible cables are widely used in mechanical and electrical products. The laying of flexible cables is a very important but cumbersome and complex task. With the development and universal application of robot technology, the robot automatic laying technology of flexible cables has gradually attracted the attention of scholars at home and abroad. The robot automatic laying of flexible cables belongs to the robot automatic planning of flexible cables. Therefore, firstly, the research status of the robot automatic planning technology of flexible cables is summarized. Then the robot automatic laying technology of flexible cables is analyzed. Focusing on the robot automatic laying process of flexible cables on a plane, the overall process of the robot automatic laying of flexible cables is proposed, and the key technologies are introduced, including the implementation methods of the physical modeling, kinematic constraints, collision-contact response, robot single/dual-arm control automatic laying, and image processing of flexible cables. The simulation and experimental verification are also carried out with an example. Finally, the future development trend of the robot automatic laying technology of flexible cables is summarized and prospected.

Key words: flexible cable, robot, automatic laying, physical modeling, collision-contact response

中图分类号:

TP391

吕乃静, 刘检华. 柔性线缆的机器人自动敷设关键技术与发展趋势[J]. 机械工程学报, 2022, 58(17): 75-95.

Lü Naijing, LIU Jianhua. Key Technologies and Development Trend of Robot Automatic Laying of Flexible Cables[J]. Journal of Mechanical Engineering, 2022, 58(17): 75-95.

参考文献

[1] 刘佳顺. 复杂产品中的线缆自动布局设计与装配路径规划技术[D]. 北京：北京理工大学，2016. LIU Jiashun. Cable automatic routing and assembly path planning in complex products[D]. Beijing：Beijing Institute of Technology，2016.
[2] 王发麟，郭耀文，吉红伟. 复杂机电产品刚柔混合装配规划系统体系结构研究[J]. 科技创新与应用，2021(2)：31-35. WANG Falin，GUO Yaowen，JI Hongwei. Research on architecture of rigid-flexible hybrid assembly planning system for complex mechanical and electrical products[J]. Technology Innovation and Application，2021(2)：31-35.
[3] VAN DER VELDEN C，BIL C，YU X，et al. An intelligent system for automatic layout routing in aerospace design[J]. Innovations in Systems and Software Engineering，2007，3(2)：117-128.
[4] KIM S，CHOI T，KIM S，et al. Sequential graph-based routing algorithm for electrical harnesses，tubes，and hoses in a commercial vehicle[J]. Journal of Intelligent Manufacturing，2021，32(2)：917-933.
[5] GRAZIOSO S，DI GIRONIMO G，SICILIANO B. A geometrically exact model for soft continuum robots：The finite element deformation space formulation[J]. Soft Robotics，2019，6(6)：790-811.
[6] GRAZIOSO S，DI GIRONIMO G，SICILIANO B. From differential geometry of curves to helical kinematics of continuum robots using exponential mapping[C]// International Symposium on Advances in Robot Kinematics. Bologna，Italy：Springer，2018：319-326.
[7] GODBOLE H，CAVERLY R，FORBES J. Dynamic modelling and adaptive control of a single degree-of-freedom flexible cable-driven parallel robot[J]. Journal of Dynamic Systems Measurement & Control，2019，141(10)：101002 (101013 pages).
[8] TANG L，GOUTTEFARDE M，SUN H，et al. Dynamic modelling and vibration suppression of a single-link flexible manipulator with two cables[J]. Mechanism and Machine Theory，2021，162：104347.
[9] GUENERS D，BOUZGARROU B-C，CHANAL H. Cable behavior influence on cable-driven parallel robots vibrations：Experimental characterization and simulation[J]. Journal of Mechanisms and Robotics，2021，13(4)：041003 (041017 pages).
[10] KR GER J，SCHRECK G，SURDILOVIC D. Dual arm robot for flexible and cooperative assembly[J]. CIRP Annals - Manufacturing Technology，2011，60(1)：5-8.
[11] AIVALIOTIS，SOTIRIS，CHATZIGEORGIOU，et al. Dual arm robot in cooperation with humans for flexible assembly[J]. Cirp Annals，2017，66(1)：13-16.
[12] 秦富康. 基于六自由度机器人的线缆自动化装配关键技术研究[D]. 广州：华南理工大学，2020. QIN Fukang. Research on key technologies of automatic wire assembly based on six degrees of freedom robot[D]. Guangzhou：South China University of Technology，2020.
[13] 王一羽. 面向柔性线缆插线过程的自适应模糊力/位姿控制策略研究[D]. 广州：华南理工大学，2019. WANG Yiyu. Research on self-adaptive fuzzy force/posture control strategy for flexible cable insertion[D]. Guangzhou：South China University of Technology，2019.
[14] YAMANO M，KIM J S，KONNO A，et al. Cooperative control of a 3d dual-flexible-arm robot[J]. Journal of Intelligent & Robotic Systems，2004，39(1)：1-15.
[15] WANG W，LI R，CHEN Y，et al. Facilitating human-robot collaborative tasks by teaching-learning-collaboration from human demonstrations[J]. IEEE Transactions on Automation Science and Engineering，2018(99)：1-14.
[16] WANG W，LIU N，LI R，et al. Hucom：A model for human comfort estimation in personalized human-robot collaboration[C]//ASME 2018 Dynamic Systems and Control Conference. Atlanta，Georgia，USA：American Society of Mechanical Engineers，2018：V002T023A006 (006 pages).
[17] WANG W，LI R，DIEKEL Z M，et al. Robot action planning by online optimization in human–robot collaborative tasks[J]. International Journal of Intelligent Robotics and Applications，2018，2(2)：161-179.
[18] WANG W，LI R，DIEKEL Z M，et al. Controlling object hand-over in human–robot collaboration via natural wearable sensing[J]. IEEE Transactions on Human-Machine Systems，2018，49(1)：59-71.
[19] LAMIRAUX F，KAVRAKI L E. Planning paths for elastic objects under manipulation constraints[J]. The International Journal of Robotics Research，2001，20(3)：188-208.
[20] BURCHAN O, BAYAZIT J M L，NANCY M A. Probabilistic roadmap motion planning for deformable objects[C]// Proceedings of the IEEE International Conference on Robotics and Automation. Washington，DC，USA：IEEE，2002：2126-2133.
[21] RODRIGUEZ S，LIEN J M，AMATO N M. Planning motion in completely deformable environments[C]// Robotics and Automation，2006 ICRA 2006 Proceedings 2006 IEEE International Conference on. Orlando，FL，USA：IEEE，2006：2466-2471.
[22] MOLL M，KAVRAKI L E. Path planning for deformable linear objects[J]. IEEE Transactions on Robotics，2006，22(4)：625-636.
[23] GAYLE R，LIN M C，MANOCHA D. Constraint-based motion planning of deformable robots[C]// IEEE International Conference on Robotics and Automation. Barcelona，Spain：IEEE，2005：1046-1053.
[24] GAYLE R，REDON S，SUD A，et al. Efficient motion planning of highly articulated chains using physics-based sampling[C]// IEEE International Conference on Robotics and Automation. Rome，Italy：IEEE，2007：3319-3326.
[25] KABUL I，GAYLE R，LIN M C. Cable route planning in complex environments using constrained sampling[C]// Proceedings of the 2007 ACM Symposium on Solid and Physical Modeling. Beijing，China：ACM，2007：395-402.
[26] ZHENG Y F，PEI R，CHEN C. Strategies for automatic assembly of deformable objects[C]// IEEE International Conference on Robotics & Automation. Sacramento，CA，USA：IEEE，2002：2598-2603.
[27] ASANO Y，WAKAMATSU H，MORINAGA E，et al. Deformation path planning for manipulation of flexible circuit boards[C]// IEEE/RSJ International Conference on Intelligent Robots and Systems. Taipei，Taiwan，China：IEEE，2010：5386-5391.
[28] MAHONEY A，BROSS J，JOHNSON D. Deformable robot motion planning in a reduced-dimension configuration space[C]// IEEE International Conference on Robotics and Automation. Anchorage，AK，USA：IEEE，2010：5133-5138.
[29] HERMANSSON T，BOHLIN R，CARLSON J S，et al. Automatic assembly path planning for wiring harness installations[J]. Journal of Manufacturing Systems，2013，32(3)：417-422.
[30] BORUM A，BRETL T. The free configuration space of a kirchhoff elastic rod is path-connected[C]// IEEE International Conference on Robotics and Automation. Seattle，WA，USA：IEEE，2015：2958-2964.
[31] BRETL T，MCCARTHY Z. Quasi-static manipulation of a kirchhoff elastic rod based on a geometric analysis of equilibrium configurations[J]. International Journal of Robotics Research，2014，33(1)：48-68.
[32] MATTHEWS D，BRETL T. Experiments in quasi-static manipulation of a planar elastic rod[C]// 25th IEEE/RSJ International Conference on Robotics and Intelligent Systems. Vilamoura，Algarve，Portugal：Institute of Electrical and Electronics Engineers Inc.，2012：5420-5427.
[33] ROUSSEL O，BORUM A，TAIX M，et al. Manipulation planning with contacts for an extensible elastic rod by sampling on the submanifold of static equilibrium configurations[C]// IEEE International Conference on Robotics and Automation. Seattle，WA，USA：IEEE，2015：3116-3121.
[34] ROUSSEL O，TA X M，BRETL T. Efficient motion planning for quasi-static elastic rods using geometry neighborhood approximation[C]// IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM). Besacon，France. IEEE，2014：1024-1029.
[35] ROUSSEL O，TA X M，BRETL T. Motion planning for a deformable linear object[C]// European Workshop on Deformable Object Manipulation. Lyon，France，2014：153-158.
[36] MUKADAM M，BORUM A，BRETL T. Quasi-static manipulation of a planar elastic rod using multiple robotic grippers[C]// 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems，IROS 2014，September 14，2014 - September 18，2014. Chicago，IL，United states：Institute of Electrical and Electronics Engineers Inc.，2014：55-60.
[37] WANG W，BERENSON D，BALKCOM D J. An online method for tight-tolerance insertion tasks for string and rope[C]// 2015 IEEE International Conference on Robotics and Automation (ICRA). Seattle，WA，USA：IEEE，2015：2488-2495.
[38] WANG F，BURDET E，VUILLEMIN R，et al. Knot-tying with visual and force feedback for vr laparoscopic training[C]// 2005 IEEE Engineering in Medicine and Biology 27th Annual Conference. Shanghai，China：IEEE，2005：5778-5781.
[39] LADD A M，KAVRAKI L E. Using motion planning for knot untangling[J]. The International Journal of Robotics Research，2004，23(7-8)：797-808.
[40] WAKAMATSU H，ARAI E，HIRAI S. Knotting/unknotting manipulation of deformable linear objects[J]. The International Journal of Robotics Research，2006，25(4)：371-395.
[41] INABA M，INOUE H. Hand eye coordination in rope handling[J]. Journal of the Robotics Society of Japan，1985：538-547.
[42] BROWN J，LATOMBE J C，MONTGOMERY K. Real-time knot-tying simulation[J]. Visual Computer，2004，20(2)：165-179.
[43] MATSUNO T，TAMAKI D，ARAI F，et al. Manipulation of deformable linear objects using knot invariants to classify the object condition based on image sensor information[J]. IEEE/ASME Transactions on Mechatronics，2006，11(4)：401-408.
[44] SAHA M，ISTO P. Manipulation planning for deformable linear objects[J]. IEEE Transactions on Robotics，2007，23(6)：1141-1150.
[45] SPILLMANN J，TESCHNER M. An adaptive contact model for the robust simulation of knots[J]. Computer Graphics Forum，2008，27(2)：497-506.
[46] JAWED M K，BRUN P T，REIS P M. A geometric model for the coiling of an elastic rod deployed onto a moving substrate[J]. Journal of Applied Mechanics，2015，82(12)：121007.
[47] BERGOU M，AUDOLY B，VOUGA E，et al. Discrete viscous threads[J]. ACM Transactions on Graphics (TOG)，2010，29(4)：1-10.
[48] AUDOLY B，CLAUVELIN N，BRUN P T，et al. A discrete geometric approach for simulating the dynamics of thin viscous threads[J]. Journal of Computational Physics，2013，253(C)：18-49.
[49] Lü N，LIU J，XIA H，et al. Dynamic modeling and control of flexible cables for shape forming[C]// Dynamic Systems and Control Conference. American Society of Mechanical Engineers，2019：V001T003A006.
[50] WOLTER J，KROLL E. Toward assembly sequence planning with flexible parts[C]// Proceedings of IEEE International Conference on Robotics and Automation. Minneapolis，MN，USA：IEEE，1996：1517-1524.
[51] 刘检华，万毕乐，宁汝新. 虚拟环境下基于离散控制点的线缆装配规划技术[J]. 机械工程学报，2006，42(8)：125-130. LIU Jianhua，WAN Bile，NING Ruxin. Realization technology of cable harness process planning in virtual environment based on discrete control point modeling method[J]. Journal of Mechanical Engineering，2006，42(8)：125-130.
[52] LIU J，HOU W，SHANG W，et al. Integrated virtual assembly process planning system[J]. Chinese Journal Of Mechanical Engineering，2009，22(5)：717-728.
[53] 杨守勇. 三维线缆建模及布线设计的研究与实现[D]. 大连：大连理工大学，2012. YANG Shouyong. Research and realization of three-dimensional cable harness modeling and route designing[D]. Dalian：Dalian University of Technology，2012.
[54] XIA P，LOPES A M，RESTIVO M T. A review of virtual reality and haptics for product assembly：From rigid parts to soft cables[J]. Assembly Automation，2013，33(2)：157-164.
[55] LÜ N，LIU J，XIA H，et al. A review of techniques for modeling flexible cables[J]. Computer-Aided Design，2020，122：102826.
[56] HAUMANN D R，PARENT R E. The behavioral test-bed：Obtaining complex behavior from simple rules[J]. Visual Computer，1988，4(6)：332-347.
[57] PROVOT X. Deformation constraints in a mass-spring model to describe rigid cloth behavior[C]// Proceedings of the 1995 Graphics Interface Conference. Quebec，Que，Can：Canadian Information Processing Soc，1995：147-154.
[58] LOOCK A，SCH MER E，STADTWALD I. A virtual environment for interactive assembly simulation：From rigid bodies to deformable cables[C]// 5th World Multiconference on Systemics，Cybernetics and Informatics (SCI’01). Orlando，Florida USA：Citeseer，2001：325-332.
[59] 王志斌，刘检华，刘佳顺，等. 面向电缆虚拟装配仿真的多分支弹簧质点模型[J]. 机械工程学报，2014，50(3)：174-183. WANG Zhibin，LIU Jianhua，LIU Jiashun，et al. A multi-branch mass-spring model for virtual assembly of cable harness[J]. Journal of Mechanical Engineering，2014，50(3)：174-183.
[60] Lü N，LIU J，DING X，et al. Physically based real-time interactive assembly simulation of cable harness[J]. Journal of Manufacturing Systems，2017：S0278612517300146.
[61] SELLE A，LENTINE M，FEDKIW R. A mass spring model for hair simulation[J]. ACM Transactions on Graphics，2008，27(3)：64 (11 pp.).
[62] BARAFF D，WITKIN A. Large steps in cloth simulation[C]// Proceedings of the 25th Annual Conference on Computer Graphics and Interactive Techniques. New York，NY，USA：ACM，1998：43-54.
[63] CHOI K J，KO H S. Stable but responsive cloth[J]. ACM Transactions on Graphics，2002，21(3)：604-611.
[64] ELBADRAWY A A，HEMAYED E E. Speeding up cloth simulation by linearizing the bending function of the physical mass-spring model[C]// International Conference on 3D Imaging，Modeling，Processing，Visualization and Transmission (3DIMPVT). Los Alamitos，CA，USA：IEEE Computer Society，2011：101-107.
[65] PATETE P，IACONO M I，SPADEA M F，et al. A multi-tissue mass-spring model for computer assisted breast surgery[J]. Medical engineering & physics，2013，35(1)：47-53.
[66] TESCHNER M，HEIDELBERGER B，MULLER M，et al. A versatile and robust model for geometrically complex deformable solids[C]// Proceedings Computer Graphics International. Crete，Greece：IEEE，2004：312-319.
[67] HERGENR THER E，D HNE P. Real-time virtual cables based on kinematic simulation[C]// Proceedings of WSCG. Plzen，Czech Republic：Univ. of West Bohemia，2000：402-409.
[68] ARISTIDOU A，LASENBY J. Fabrik：A fast，iterative solver for the inverse kinematics problem[J]. Graphical Models，2011，73(5)：243-260.
[69] REDON S，GALOPPO N，LIN M C. Adaptive dynamics of articulated bodies[J]. Acm Transactions on Graphics，2005，24(3)：936-945.
[70] GAYLE R，LIN M C，MANOCHA D. Adaptive dynamics with efficient contact handling for articulated robots[C]// Robotics：Science and systems. Philadelphia，Pennsylvania，USA：MIT Press，2006：231-238.
[71] HADAP S. Oriented strands：Dynamics of stiff multi-body system[C]// Proceedings of the 2006 ACM SIGGRAPH/Eurographics Symposium on Computer animation. Vienna，Austria，Eurographics Association Aire-la-Ville. 2006：91-100.
[72] BENDER J，SCHMITT A A. Fast dynamic simulation of multi-body systems using impulses[C]// 3rd Workshop on Virtual Reality Interactions and Physical Simulations. Madrid，Spain：Eurographics Association，2006：81-90.
[73] CHOE B，CHOI M G，KO H S. Simulating complex hair with robust collision handling[C]// Proceedings of the 2005 ACM SIGGRAPH/Eurographics symposium on Computer animation. Los Angeles，California，ACM. 2005：153-160.
[74] HADAP S，MAGNENAT-THALMANN N. Modeling dynamic hair as a continuum[J]. Computer Graphics Forum，2001，20(3)：329-338.
[75] 魏发远，陈新发，王峰军. 电缆虚拟布线及其逆运动学仿真[J]. 计算机辅助设计与图形学学报，2006，18(10)：1623-1627. WEI Fayuan，CHEN Xinfa，WANG Fengjun. Virtual wiring and simulation of cable layout with inverse kinematics[J]. Journal of Computer-Aided Design ＆ Computer Graphics，2006，18(10)：1623-1627.
[76] 何大闯，闫静，左敦稳，等. 虚拟环境下基于逆运动学的电缆建模与仿真技术[J]. 机械设计与制造工程，2013，42(10)：25-28. HE Dachuang，YAN Jing，ZUO Dunwen，et al. The cable modeling and simulation technology in virtual environment based on inverse kinematics[J]. Machine Design and Manufacturing Engineering，2013，42(10)：25-28.
[77] SERVIN M，LACOURSIERE C. Rigid body cable for virtual environments[J]. IEEE Transactions on Visualization and Computer Graphics，2008，14(4)：783-796.
[78] PAI D K. Strands：Interactive simulation of thin solids using cosserat models[J]. Computer Graphics Forum，2002，21(3)：347-352.
[79] LIU J，ZHAO T，NING R，et al. Physics-based modeling and simulation for motional cable harness design[J]. Chinese Journal of Mechanical Engineering，2014，27(5)：1075-1082.
[80] BERTAILS F，AUDOLY B，CANI M P，et al. Super-helices for predicting the dynamics of natural hair[J]. Acm Transactions on Graphics，2006，25(3)：1180-1187.
[81] BERTAILS F. Linear time super-helices[J]. Computer Graphics Forum，2009，28(2)：417-426.
[82] GR GOIRE M，SCH MER E. Interactive simulation of one-dimensional flexible parts[J]. Computer-Aided Design，2007，39(8)：694-707.
[83] SPILLMANN J，TESCHNER M. Corde：Cosserat rod elements for the dynamic simulation of one-dimensional elastic objects[C]// Proceedings of the 2007 ACM SIGGRAPH/Eurographics Symposium on Computer Animation. San Diego，California，Assoc Computing Machinery. 2007：63-72.
[84] LANG H，LINN J. A second order semi-discrete cosserat rod model suitable for dynamic simulations in real time[C]// AIP Conference Proceedings. Rethymno，Crete (Greece)：AIP Publishing，2009：1057-1060.
[85] LANG H，LINN J，ARNOLD M. Multi-body dynamics simulation of geometrically exact cosserat rods[J]. Multibody System Dynamics，2011，25(3)：285-312.
[86] BERGOU M，WARDETZKY M，ROBINSON S，et al. Discrete elastic rods[J]. ACM Transactions on Graphics，2008，27(3)：63 (12 pp.).
[87] JAWED M K，NOVELIA A，O'REILLY O M. A primer on the kinematics of discrete elastic rods[M]. Springer，2018.
[88] 黄劲，沈中伟，王青，等. 基于优化的大步长准静态线缆模拟[J]. 计算机辅助设计与图形学学报，2011，23(1)：1-10. HUANG Jin，SHEN Zhongwei，WANG Qing，et al. Optimization based large step quasi static rod simulation[J]. Journal of Computer-Aided Design ＆ Computer Graphics，2011，23(1)：1-10.
[89] LINN J，DRE LER K. Discrete cosserat rod models based on the difference geometry of framed curves for interactive simulation of flexible cables[M]// GHEZZI L，H MBERG D，LANDRY C. Math for the digital factory. Cham; Springer International Publishing，2017：289-319.
[90] KIRCHHOFF G. Ueber das gleichgewicht und die bewegung eines unendlich dünnen elastischen stabes[J]. Journal für die reine und angewandte Mathematik，1859，56：285-313.
[91] 林海立，刘检华，唐承统，等. 基于cosserat弹性杆理论的柔性线缆物理建模方法[J]. 图学学报，2016，37(1)：34-42. LIN Haili，LIU Jianhua，TANG Chengtong，et al. Physical characteristic oriented modeling for flexible cable harness based on cosserat elastic rod theory[J]. Journal of Graphics，2016，37(1)：34-42.
[92] 林海立. 分支线缆物性建模与装配过程仿真技术研究[D]. 北京：北京理工大学，2016. LIN Haili. Research on the physically based modeling and assembly simulation of multi-branch cables[D]. Beijing：Beijing Institute of Technology，2016.
[93] 薛纭，陈立群，刘延柱. 受曲面约束弹性细杆的平衡问题[J]. 物理学报，2004，53(7)：2040-2045. XUE Yun，CHEN Liqun，LIU Yanzhu. Problems on equilibrium of a thin elastic rod constrained on a surface[J]. Acta Physica Sinica，2004，53(7)：2040-2045.
[94] COSSERAT E，COSSERAT F. Théorie des corps déformables[M]. Paris：A. Hermann et fils，1909.
[95] ANTMAN S S. Nonlinear problems of elasticity. 2nd，revised and extended ed[M]. 2005.
[96] BONANNI U，KMOCH P，MAGNENAT-THALMANN N. Haptic interaction with one-dimensional structures[C]// Proceedings of the 16th ACM Symposium on Virtual Reality Software and Technology. Kyoto，Japan：ACM，2009：75-78.
[97] LI H，LEOW W K，CHIU I S. Elastic tubes：Modeling elastic deformation of hollow tubes[J]. Computer Graphics Forum，2010，29(6)：1770-1782.
[98] WANG C S，NING R X，LIU J H，et al. Dynamic simulation and disturbance torque analyzing of motional cable harness based on kirchhoff rod model[J]. Chinese Journal of Mechanical Engineering，2012，25(2)：346-354.
[99] OLSON S D，LIM S，CORTEZ R. Modeling the dynamics of an elastic rod with intrinsic curvature and twist using a regularized stokes formulation[J]. Journal Of Computational Physics，2013，238(Complete)：169-187.
[100] TERZOPOULOS D，QIN H. Dynamic nurbs with geometric constraints for interactive sculpting[J]. ACM Transactions on Graphics (TOG)，1994，13(2)：103-136.
[101] QIN H，TERZOPOULOS D. D-nurbs：A physics-based framework for geometric design[J]. IEEE Transactions on Visualization and Computer Graphics，1996，2(1)：85-96.
[102] TERZOPOULOS D，FLEISCHER K. Deformable models[J]. Visual Computer，1988，4(6)：306-331.
[103] TERZOPOULOS D，PLATT J，BARR A，et al. Elastically deformable models[J]. ACM SIGGRAPH Computer Graphics，1987，21(4)：205-214.
[104] NOCENT O，REMION Y. Continuous deformation energy for dynamic material splines subject to finite displacements[C]// Eurographic Workshop on Computer Animation and Simulation. Manchester，UK：Springer，2001：87-97.
[105] LENOIR J，COTIN S，DURIEZ C，et al. Interactive physically-based simulation of catheter and guidewire[J]. Computers & Graphics，2006，30(3)：416-422.
[106] LENOIR J，GRISONI L，CHAILLOU C，et al. Adaptive resolution of 1d mechanical b-spline[C]// GRAPHITE Conference. Dunedin，New Zealand：ACM，2005：395-403.
[107] THEETTEN A，GRISONI L，ANDRIOT C，et al. Geometrically exact dynamic splines[J]. Computer-Aided Design，2008，40(1)：35-48.
[108] ECHEGOYEN Z，VILLAVERDE I，MORENO R，et al. Linked multi-component mobile robots：Modeling，simulation and control[J]. Robotics & Autonomous Systems，2010，58(12)：1292-1305.
[109] VALENTINI P P，PENNESTR E. Modeling elastic beams using dynamic splines[J]. Multibody System Dynamics，2011，25(3)：271-284.
[110] VALENTINI P P. Interactive cable harnessing in augmented reality[J]. International Journal on Interactive Design and Manufacturing (IJIDeM)，2011，5(1)：45-53.
[111] L ON J-C，GANDIAGA U，DUPONT D. Modelling flexible parts for virtual reality assembly simulations which interact with their environment[C]// Proceedings International Conference on Shape Modeling and Applications. Genova，Italy：IEEE，2001：335-344.
[112] ANDREU A，GIL L，ROCA P. A new deformable catenary element for the analysis of cable net structures[J]. Computers & Structures，2006，84(29/30)：1882-1890.
[113] YANG M G，CHEN Z Q，HUA X G. A new two-node catenary cable element for the geometrically non-linear analysis of cable-supported structures[J]. ARCHIVE Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science 1989-1996 (vols 203-210)，2010，224(6)：1173-1183.
[114] KAUFMANN P，MARTIN S，BOTSCH M，et al. Flexible simulation of deformable models using discontinuous galerkin fem[J]. Graphical Models，2009，71(4)：153-167.
[115] WANG Q，FANG H，LI N，et al. An efficient fe model of slender members for crash analysis of cable barriers[J]. Engineering Structures，2013，52(9)：240-256.
[116] PRESS W H，TEUKOLSKY S A，VETTERLING W T，et al. Numerical recipes 3rd edition：The art of scientific computing[M]. Cambridge University Press，2007.
[117] NEALEN A，M LLER M，KEISER R，et al. Physically based deformable models in computer graphics[J]. Computer Graphics Forum，2010，25(4)：809-836.
[118] LIU T，BARGTEIL A W，O'BRIEN J F，et al. Fast simulation of mass-spring systems[J]. ACM Transactions on Graphics，2013，32(6)：1-7.
[119] HILDE L，MESEURE P，CHAILLOU C. A fast implicit integration method for solving dynamic equations of movement[C]// Proceedings of the ACM symposium on Virtual reality software and technology. Baniff，Alberta，Canada：ACM，2001：71-76.
[120] MARTIN S，THOMASZEWSKI B，GRINSPUN E，et al. Example-based elastic materials[J]. ACM Transactions on Graphics (TOG)，2011，30(4)：1-8.
[121] HAHN F，MARTIN S，THOMASZEWSKI B，et al. Rig-space physics[J]. ACM Transactions on Graphics，2012，31(4)：1-8.
[122] ASCHER U M，RUUTH S J，WETTON B. Implicit-explicit methods for time-dependent partial differential equations[J]. SIAM J Numer Anal，1993，32(1)：797-823.
[123] HAUTH M，ETZMUSS O，STRASSER W. Analysis of numerical methods for the simulation of deformable models[J]. Visual Computer，2003，19(7-8)：581-600.
[124] BRIDSON R，MARINO S，FEDKIW R. Simulation of clothing with folds and wrinkles[C]// ACM Siggraph/eurographics Symposium on Computer Animation. Los Angeles，California，USA：ACM，2003：28-36.
[125] STERN A，GRINSPUN E. Implicit-explicit variational integration of highly oscillatory problems[J]. International Journal of Theoretical Physics，2012，51(6)：1663-1673.
[126] BERGEN G V D. Efficient collision detection of complex deformable models using aabb trees[J]. Journal of Graphics Tools，1997，2(4)：1-13.
[127] GOTTSCHALK S，LIN M C，MANOCHA D. Obbtree：A hierarchical structure for rapid interference detection[C]// Proceedings of the 23th Annual Conference on Computer Graphics and Interactive Techniques. ACM，1996：171-180.
[128] PANKAJ，AGARWAL，AND，et al. Collision detection for deforming necklaces[J]. Computational Geometry，2004，28(2-3)：137-163.
[129] KLOSOWSKI J T，HELD M，MITCHELL J S B，et al. Efficient collision detection using bounding volume hierarchies of k-dops[J]. IEEE Transactions on Visualization and Computer Graphics，1998，4(1)：21-36.
[130] EHMANN S A，LIU M C. Accurate and fast proximity queries between polyhedra using convex surface decomposition[J]. Computer Graphics Forum，2001，20(3)：C/500-C/510.
[131] XU H，ZHAO Y，BARBIC J. Implicit multibody penalty-baseddistributed contact[J]. Visualization & Computer Graphics IEEE Transactions on，2014，20(9)：1266-1279.
[132] LÜ N，LIU J，JIA Y. Coordinated control of flexible cables with human-like dual manipulators[J]. Journal of Dynamic Systems Measurement and Control，2021，143：081006-081001.