目前煤矸石分选方法存在诸多弊端，而从原煤中分选出矸石是煤炭资源高效、清洁利用中不可或缺的环节，因而煤矸分选难题制约着煤炭行业的发展。采用机器人进行煤矸分选是克服煤矸分选难题，实现煤矸石“快、准、稳”自动化分拣的有效途径。本文结合柔索驱动并联机器人的技术优势，提出柔索驱动拣矸机器人方案，对其进行分拣轨迹规划与轨迹跟踪控制研究。主要内容包括：

(1) 以柔索驱动拣矸机器人为研究对象，采用矢量封闭原理，基于逆运动学建立其运动学模型，仿真表明索长变化曲线符合末端抓斗运动轨迹的几何特征。采用牛顿-欧拉法，基于末端空间建立其动力学模型，指出了冗余驱动使得动力学模型无法直接求解柔索拉力的问题，需要对柔索拉力进行优化。研究为分拣轨迹规划与轨迹跟踪控制研究奠定了基础。

(2) 针对冗余驱动导致动力学模型不能直接求解索拉力问题，以柔索最小索拉力(预紧力)和最大承载力为线性约束条件，引入Moore-Penrose广义逆矩阵和零空间基底，以柔索拉力方差最小为优化目标，建立柔索驱动拣矸机器人柔索拉力优化模型。基于最小索拉力求解了最小索拉力等值曲面，末端抓斗在此曲面包络的空间内，柔索拉力均大于最小索拉力，能保证末端抓斗平稳运行。通过算例仿真表明，采用上述方法对柔索拉力优化有效且可靠，末端抓斗处于最小索拉力所形成的等值曲面内部时，柔索拉力均连续稳定变化，能满足稳定性要求。

(3) 由于矸石运动特点、矸石仓位置及柔索驱动并联机器人工作空间几何中心受力最佳等条件限制，使得末端抓斗分拣运动路径有特殊要求且精度要求较高。根据笛卡尔空间轨迹规划优点，从笛卡尔空间对柔索驱动拣矸机器人末端抓斗进行分拣轨迹规划。将末端抓斗的分拣轨迹规划为启动段、准备段、抓矸段和置矸段，并针对各段不同的运动特点，采用S型速度曲线和五次多项式组合的轨迹规化方法，对末端抓斗的四段运动进行规划。对规划的分拣轨迹进行仿真表明，分拣轨迹位于最小索拉力等值曲面内，四段轨迹光滑连接，分拣轨迹对应速度和加速度连续，索长变化光滑连续，且轨迹参数能适应矸石在皮带运输机上的不同分布情况。

(4) 考虑柔索驱动拣矸机器人柔索单向受力特性，使得其难以避免包括外部冲击载荷等在内的外部干扰影响的问题。针对外部干扰，以动力学模型为基础，根据柔索驱动拣矸机器人对末端位置要求高的特殊性，基于末端空间设计了柔索驱动拣矸机器人模糊鲁棒自适应轨迹跟踪控制系统。基于Lyapunov稳定性理论，证明了该控制系统是稳定的。通过算例对所设计的控制系统仿真表明，轨迹跟踪效果良好，柔索拉力均满足单向柔索单向受力特性，且均大于最小索拉力，控制策略可行且有效。

At present, there are many disadvantages in the coal gangue separation method, and the separation of the gangues from raw coals is an indispensable link in the efficient and clean utilization of coal resources. Therefore, separation of the coals and gangues restricts the development of the coal industry. Using robots for the separation of the coals and gangues is an effective way to solve the problem and to realize the automatic sorting of coals and gangues’ fast, accurately and stably. In this paper, combining with the advantages of cable-driven parallel robots, a cable-driven gangue sorting robot is proposed, and furthermore, the sorting trajectory planning and trajectory tracking control are studied. The main contents are as follows:

(1) The kinematics model of the cable-driven gangue sorting robot is established by using the vector closure principle, The simulation results show that the cable length curve conforms to the geometric characteristics of the end grab trajectory. The Newton-Euler method is used to establish the dynamic model of the robot, and it is pointed out that the dynamic model cannot solve the problem of flexible cable tension directly due to the redundant driving. It lays a foundation for the research on sorting trajectory planning and trajectory tracking control of cable-driven gangue sorting robot.

(2) It is well-known that the dynamic model of the robot cannot directly solve the cable tensions because of the redundant driving, And therefore the cable tension optimization for robot is established ,in which the minimum cable tension (preload) and the maximum bearing capacity of the cable are taken as linear constraints, and the minimum variance of the cable tensions is taken as the optimization objective. And furthermore the minimum cable tension equivalent surface is solved based on the minimum cable tensions. The cable tension is greater than the minimum cable tension in the envelope space of the end-grab, which can ensure the smooth operation of the end-grab. The simulation results show that the method is effective and reliable to optimize the cable tension. When the end-grab of the robot is in the equivalent surface formed by the minimum cable tension, the cable tension changes continuously and stably, which can meet the stability requirements.

(3) Due to the movement characteristics of the gangues, the position of the gangue bin and the optimal force of the geometric center of the workspace of the robot, the sorting path of the end-grab has special requirements and high accuracy requirements. According to the advantages of the Cartesian space trajectory planning, the sorting trajectory planning of the end-grab of the robot is carried out from the Cartesian space, and the sorting trajectory of the end-grab is planned as the starting section, the preparation section, the grab section and the gangue section, And moreover according to the different motion characteristics of each segment, the four-segment motion of the end grab is planned by using the S-type velocity curve and the five-degree polynomial trajectory planning method. The simulation of the planned trajectory scheme shows that the sorting trajectory is located in the minimum cable tension equivalent surface. The four segments of the trajectory are connected smoothly. The sorting trajectory corresponds to continuous velocity and acceleration, and the change of cable length is smooth and continuous. The trajectory parameters can adapt to the different distribution of gangue on the belt conveyor.

(4) Considering the unidirectional force characteristics of the cables, it is difficult to avoid the influence of external interference, including external impact load, a fuzzy robust adaptive trajectory tracking control system for the cable-driven gangue sorting robot is designed against the external interferences based on the dynamic model. Based on Lyapunov stability theory, it is proved that the control system is stable. The simulation of the designed control system shows that the trajectory tracking effect is good, and the cable tensions satisfy the unidirectional force characteristics of the cables. As a result, the control strategy is feasible and effective.

[1] 马骏，郁钟铭，舒仕海，等．煤矸石对矿区的环境危害及治理措施[J]．煤炭工程，2015，47(10):70-73．

[2] 李静，温鹏飞，何振嘉．煤矸石的危害性及综合利用的研究进展[J]．煤矿机械，2017，38(11)：128-130．

[3] Dan Ma, Hongyu Duan, Jiangfeng Liu, et al. The role of gangue on the mitigation of mining-induced hazards and environmental pollution: An experimental investigation[J]. Science of the Total Environment,2019,664436-448.

[4] Yuemin Zhao, Xuliang Yang, Zhenfu Luo, et al. Progress in developments of dry coal beneficiation[J]. International Journal of Coal Science & Technology,2014,1(001):103-112.

[5] 陈俊涛，于恒江，徐德永．龙湖选煤厂原煤车间排矸方式的改造[J]．煤矿机械，2003(08)：87-88．

[6] 康利，黄金辉，刘畅．GDRT型γ射线智能干法分选系统在六家煤矿的应用[J]．煤炭加工与综合利用，2017(3)：22-24．

[7] 康利，黄金辉，刘畅．GDRT型γ射线智能干法分选系统在六家煤矿的应用[J]．煤炭加工与综合利用，2017(3)：22-24．

[8] 赵宏霞，丁芳亮．KRS-智能干法分选系统在矿物分选中的应用[J]．煤炭加工与综合利用，2016(9)：24-26．

[9] 高明．塔山选煤厂原煤准备车间技术改造[J]．选煤技术，2015(3)：43-45．

[10] 吕鹏飞，何敏，陈晓晶，等．智慧矿山发展与展望[J]．工矿自动化，2018．44(09)：84-88．

[11] 王国法，杜毅博．智慧煤矿与智能化开采技术的发展方向[J]．煤炭科学技术，2019，47(01)：1-10．

[12] 夏晶，张昊，周世宁，等．煤矸分拣机器人动态拣取避障路径规划[J]．煤炭学报，2021，46(S1)：570-577．

[13] 曹现刚，吴旭东，王鹏，等．面向煤矸分拣机器人的多机械臂协同策略[J]．煤炭学报，2019，44(S2)：763-774．

[14] 訾斌，段宝岩，黄进．大射电望远镜馈源柔索支撑系统的建模与控制[J]．西安交通大学学报，2006，(06)：681-685．

[15] 于亮亮，仇原鹰，苏宇．高速柔索牵引摄像机器人动力工作空间研究[J]．工程力学，2013，30(11)：245-250．

[16] 宋达，张立勋，王炳军，等．柔索牵引式力觉交互机器人控制策略[J]．机器人，2018，40(04)：440-447．

[17] 郑亚青，林麒，刘雄伟，等．用于低速风洞飞行器气动导数试验的绳牵引并联支撑系统[J]．航空学报，2009，30(08)：1549-1554．

[18] Gorman J J, Jablokow K W, Canon D J. The cable array robot: theory and experiment[C]. //2001 IEEE International Conference on Robotics and Automation (ICRA 2001),vol.III.2001:2804-2810.

[19] Rosati G, Masiero S, Rossi A. On the Use of Cable-Driven Robots in Early Inpatient Stroke Rehabilitation[M]. //Springer International Publishing,2017,53(17):92-102.

[20] 刘鹏，马宏伟，乔心州，等．柔索驱动拣矸机器人最小索拉力等值曲面研究[J]．西安科技大学学报，2020，40(05)：797-804．

[21] Barbazza Luca, Zanotto Damiano, Rosati Giulio, et al. Design and Optimal Control of an Underactuated Cable-Driven Micro–Macro Robot[J]. IEEE Robotics & Automation Letters,2017,2(2):896-903.

[22] Albus James, Bostelman Roger, Dagalakis Nicholas. The NIST RoboCrane[J]. Journal of Robotic System,1993,10(5):709-724.

[23] Xiaoqiang Tang. An Overview of the Development for Cable-Driven Parallel Manipulator[J]. Advances in Mechanical Engineering,2014,2014(1):1-9.

[24] Lytle Alan M, Saidi Kamel S, Bostelman Roger V, et al. Adapting a teleoperated device for autonomous control using three-dimensional positioning sensors: experiences with the NIST RoboCrane[J]. Automation in construction,2004,13(1):p.101-118.

[25] Merlet Jean-pierre. Analysis of Wire Elasticity for Wire-driven Parallel Robots[C]. //Proceedings of EUCOMES 08.:Springer Science + Business Media B. V., 2008:471-478.

[26] Tanaka M,Seguchi Y, Shimada S. Kineto-statics of skycam-type wire transport system[C] //Proceedings of USA-Japan Symposium on Flexible Automation, Crossing Bridges: Advances in Flexible Automation and Robotics Minneapolis, Minnesot a, 1988:689-694.

[27] Agrawal Sunil K, Dubey Venketsh N, Gangloff, Jr John J, et al. Design and optimization of a cable driven upper arm exoskeleton[J].Journal of Medical Devices-Transactions of the ASME,2009,3(031004):1-8.

[28] Banala, Sai K, Agrawal, Sunil K, Kim seokhun, et al. Novel Gait Adaptation and Neuromotor Training Results Using an Active Leg Exoskeleton[J]. IEEE/ASME transactions on mechatronics: A joint publication of the IEEE Industrial Electronics Society and the ASME Dynamic Systems and Control Division,2010,15(2):216-225.

[29] Jensen Finn v, Kjerulff Uffe, Kristiansen Brian. The SACSO methodology for troubleshooting complex systems[J]. Artificial intelligence for engineering design, analysis and manufacturing: AI EDAM,2001,15(4):321-333.

[30] 吴鑫基．观天巨眼400年系列之二十三：世界上口径最大的阿雷西博射电望远镜[J].太空探索，2004(01)：40-42．

[31] Ferraresi Carlo, Paoloni Marco, Pastorelli Stefano, et al. A new 6-DOF parallel robotic structure actuated by wires: The wiro-6.3[J]. Journal of robotic systems,2004,21(11):581-595.

[32] 杜芝茂．“中国天眼”的追赶与超越——FAST建成记[J]．中学科技，2021(09)：8-13．

[33] 段学超，仇原鹰，段宝岩．柔性支撑Stewart平台动力学建模与轨迹跟踪控制[J]．系统工程与电子技术，2009，31(04)：895-900．

[34] 段清娟，殷成熙，段宝岩．FAST 50m模型馈源支撑系统的结构参数优化[J]．机械工程学报，2017，53(17)：31-35．

[35] 韩旺，段学超，仇原鹰，等．基于线驱动并联机构的FAST馈源平台位姿测量方法[J]．机械工程学报，2017，53(17)：43-49．

[36] 王从思，肖岚，项斌斌，等．大型射电望远镜天线主动面补偿研究进展[J]．中国科学：物理学 力学 天文学，2017,47(05)：19-34．

[37] 訾斌，段宝岩，杜敬利．柔索驱动并联机器人动力学建模与数值仿真[J]．机械工程学报，2007，43(11)：82-88．

[38] 訾斌，段宝岩，黄进. 大射电望远镜馈源柔索支撑系统的建模与控制[J]. 西安交通大学学报，2006，40(6)：681-685．

[39] 訾斌，段宝岩，仇原鹰．大射电望远镜馈源指向系统轨迹跟踪免疫PID控制[J]．高技术通讯，2005，15(9)：29-33．

[40] Tang X, Liu Z, Shao Z, et al. Self-excited Vibration Analysis for the Feed Support System in FAST[J]. International Journal of Advanced Robotic Systems,2014,11(1):1-13.

[41] 韦慧玲，仇原鹰，盛英．高速绳牵引摄像机器人的运动稳定控制[J]．西安电子科技大学学报，2016，43(5)：70-77．

[42] 路光达，张秋月，安宁，等．一种柔索驱动手指康复机器人设计与动力学分析[J]．科学技术与工程，2020，20(33)：13725-13729．

[43] 郑亚青，吴建坡．用于提高修船效率的绳牵引并联机器人之设计[J]．中国修船，2009，22(1)：11-15．

[44] 雷扎•N.贾扎尔(Reza N.Jazar)，周高峰等译，应用机器人学：运动学、动力学与控制技术[M]．北京：械工业出版社，2017：19-20．

[45] 黄海灵，吴洪涛，陈柏．基于柔索驱动的踝关节康复机器人的研究[J]．机械设计与制造工程，2012，41(3)：27-29+33．

[46] 崔海，沈博侃．3T柔索驱动并联空中拍摄云台设计、建模与分析[J]．机械强度，2015，37(04)：689-694．

[47] Leijie Jiang, Bingtuan Gao, Zhenyu Zhu. Dynamic modeling and control of a cable-driven parallel mechanism with a spring spine[J]. Proceedings of the Institution of Mechanical Engineers, Part C. Journal of mechanical engineering science,2017,231(21):1-19.

[48] 殷家宁，姜鹏，陈明，等．FAST索驱动并联机器人与Stewart平台结合的动力学建模方法[J/OL]．清华大学学报(自然科学版)：1-8[2022-03-09]．

[49] Thibaut Paty, Nicolas Binaud, Stéphane Caro, et al. Cable-driven parallel robot modelling considering pulley kinematics and cable elasticity[J]. Mechanism and Machine Theory: Dynamics of Machine Systems Gears and Power Trandmissions Robots and Manipulator Systems Computer-Aided Design Methods,2021,159.

[50] Erika Ottaviano, Andrea Arena, Vincenzo Gattulli. Geometrically exact three-dimensional modeling of cable-driven parallel manipulators for end-effector positioning[J]. Mechanism and Machine Theory,2021,155.

[51] 訾斌，朱真才，曹建斌．混合驱动柔索并联机器人的设计与分析[J]．机械工程学报，2011，47(17)：1-8．

[52] 雷静桃，蒋运旗，吴启帆．柔索驱动仿生机体弯曲动力学建模与仿真[J]．高技术通讯，2017，27(08)：735-743．

[53] 朱春涛，段鑫，胡小春．一种柔索驱动关节控制系统及参数优化研究[J]．机械工程师，2019(01)：14-20．

[54] 陈桥，訾斌，孙智，等．柔索驱动并联腰部康复机器人设计、分析与试验研究[J]．机械工程学报，2018，54(13)：126-134．

[55] Yanlin Wang, Keyi Wang, Wanli Wang, et al. Appraisement and Analysis of Dynamical Stability of Under-Constrained Cable-Driven Lower-Limb Rehabilitation Training Robot[J]. Robotica: International journal of information, education and research in robotics and artificial intelligence,2021,39(6):1023-1036.

[56] Saracino Arianna, Oude-vrielink Timo J.C., Menciassi Arianna, et al. Haptic intracorporeal palpation using a cable-driven parallel robot: a user study[J]. IEEE Transactions on Biomedical Engineering, 2020, PP(99):1-12.

[57] Hussein Hussein, Santos Joao Cavalcanti, Izard Jean-Baptiste, et al. Smallest Maximum Cable Tension Determination for Cable-Driven Parallel Robots[J]. IEEE Transactions on Robotics, 2021, PP(99):1-20.

[58] Yu Su, Yuanying Qiu, Peng Liu. Optimal Cable Tension Distribution of the High-Speed Redundant Driven Camera Robots Considering Cable Sag and Inertia Effects[J]. Advances in Mechanical Engineering, 2014, 2014(4):1-11.

[59] 刘欣．两种并联机器人的机构性能分析与运动控制研究[D]．西安：西安电子科技大学, 2009．

[60] 李冉．柔索驱动并联机器人的建模与控制[D]．太原：太原科技大学，2015．

[61] Verhoeven R, Hiller M, Tadokoro S. Workspace, stiffness, singularities and classification of tendon driven stewart platforms[J]. Advances in Robot Kinematics Analysis & Control, 1998:105-114.

[62] 朱冠亚．三自由度绳驱并联机器人运动控制研究[D]．武汉：武汉工程大学，2019．

[63] Javier Garrido, Yu Wen, Xiaoou Li. Robot trajectory generation using modified hidden Markov model and Lloyd's algorithm in joint space[J]. Engineering Applications of Artificial Intelligence: The International Journal of Intelligent Real-Time Automation, 2016(53):32-40.

[64] 温贻芳，孙立宁，徐朋．表面改性冗余机器人关节空间的轨迹优化算法[J]．机械科学与技术，2018，37(12)：1870-1874．

[65] Valero Francisco, Mata Vicente, Besa Antonio. Trajectory planning in workspaces with obstacles taking into account the dynamic robot behaviour[J]. Mechanism and Machine Theory, 2005,41(5):525-536.

[66] Andulkar Mayur V., Chiddarwar Shital S., Marathe Akshay S.. Novel integrated offline trajectory generation approach for robot assisted spray painting operation[J]. Journal of Manufacturing Systems, 2015,37:201-216.

[67] 谢少荣，刘思淼，罗均等．一种混合驱动柔索并联仿生眼的轨迹规划[J]．机器人，2015，37(04)：395-402．

[68] Rasheed Tahir, Philip Long, Roos Adolfo Suarez, et al. Optimization based Trajectory Planning of Mobile Cable-Driven Parallel Robots[C] //2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE,2020.

[69] Yifan Hong, Jianping Tan. Trajectory Planning of a Planar Cable-Driven Robot for Industrial Detection[J]. Journal of Physics Conference Series,2020,1570:012037.

[70] 范伟，彭光正，高建英，等．气动人工肌肉驱动球面并联机器人的位置控制研究[J]．北京理工大学学报，2004(06)：516-519．

[71] 洪逸凡，谭建平．基于柔索驱动的锅炉检测机器人轨迹规划研究[J]．传感器与微系统，2021，40(07)：25-27+31．

[72] Sen Qian, Kunlong Bao, Bin Zi, et al. Dynamic Trajectory Planning for a Three Degrees-of-Freedom Cable-Driven Parallel Robot Using Quintic B-Splines[J]. Journal of Mechanical Design, 2020, 142(7):073301.

[73] Tao Zhao, Bin Zi, Sen Qian, et al. Algebraic Method-Based Point-to-Point Trajectory Planning of an Under-Constrained Cable-Suspended Parallel Robot with Variable Angle and Height Cable Mast[J]. Chinese Journal of Mechanical Engineering,2020,33(4):45-62.

[74] Jiajun Xu, Jyoung-Su Park. Moving obstacle avoidance for cable-driven parallel robots using improved RRT[J]. Microsystem Technologies, 2021(27):2281-2292.

[75] 黄佳怡，陈柏，胡忠文，等．一种柔索驱动太空舱外搬运机器人研究[J]．机械科学与技术，2012，31(11)：1748-1753．

[76] 张波，赵明扬，房立金．一种6自由度柔索并联机器人的动力学研究[J]．机械科学与技术，2004(06)：735-738．

[77] Efe Levent Oyman, Muhammed Yusuf Korkut, Cuneyt Ylmaz, et al. Design and control of a cable-driven rehabilitation robot for upper and lower limbs[J]. Robotica, 2021:1-37.

[78] S A Khalilpour, R Khorrambakht, H Damirchi, et al. Tip-trajectory tracking control of a deployable cable-driven robot via output redefinition[J]. Multibody System Dynamics, 2021(52):31-58.

[79] Qianqian Yang, Chenglin Xie, Rongrong Tang, et al. Hybrid Active Control With Human Intention Detection of an Upper-Limb Cable-Driven Rehabilitation Robot[J]. IEEE Access, 2020, 8:195206-195215.

[80] Bin Zi. Fuzzy Control System Design and Analysis for Completely Restrained Cable-Driven Manipulators[M]. InTech, 2012,59-80.

[81] Atal Anil Kumar, Jean-Francois Antoine, Gabriel Abba. Control of an Underactuated 4 Cable-Driven Parallel Robot using Modified Input-Output Feedback Linearization[J]. IFAC-PapersOnLine, 2020, 53(2):8777-8782.

[82] 张立勋，李来禄，姜锡泽，等．柔索驱动的宇航员深蹲训练机器人力控与实验研究[J]．机器人，2017，39(05)：733-741．

[83] 陆兴华，叶铭铭，陈俊祥，等．柔索牵引式机器人的姿态自适应调节控制优化[J]．传感器与微系统，2018，37(12)：38-41．

[84] 訾斌，朱真才，杜敬利．柔索驱动并联机器人运动控制研究[J]．振动与冲击，2009，28(09)：48-51+73+213．

[85] Niloufar Sadat Seyfi, Ali Keymasi Khalaji. Robust control of a cable-driven rehabilitation robot for lower and upper limbs[J]. ISA Transactions, 2021. https://doi.org/10.1016/j.isatra.2021.07.016.

[86] Ronghuai Qi, Amir Khajepour, William W. Melek Modeling, Vibration Control, and Trajectory Tracking of a Kinematically Constrained Planar Hybrid Cable-Driven Parallel Robot[J]. 2020. https://doi.org/10.48550/arXiv.2012.14029

[87] Weiwei Shang, Bin Zhang, Shuang Cong, et al. Dual-space adaptive synchronization control of redundantly-actuated cable-driven parallel robots[J]. Mechanism and Machine Theory, 2020(152):103954.

[88] Bosscher Paul, Imme Ebert-Uphoff. A Stability Measure for Under constrained Cable-Driven robots[C] //2004 IEEE International Conference on Robotics and Automation (ICRA 2004), vol.5. 2004:4943-4949.

[89] Bosscher Paul M. Disturbance Robustness Measures and Wrench-Feasibile Workspace Generation Techniques for Cable-Driven Robots[D]. Atlanta:PHD. Dissertation of Georgia Institute of Technology, 2004.

[90] 北京起重运输机械设计研究院武汉丰凡科技开发有限公司．DTⅡ(A)型带式输送机设计手册.第2版[M]. 北京:冶金工业出版社，2013：16．

[91] 李振娜，王涛，王斌锐，等．基于带约束S型速度曲线的机械手笛卡尔空间轨迹规划[J]．智能系统学报，2019，14(4)：655-661．

[92] 蔡自兴．智能控制原理与应用第3版[M]．北京：清华大学出版社，2019：83-98．

[93] Kook Yoo Byung, Ham Chul Woon. Adaptive control of robot manipulator using fuzzy compensator[J]. IEEE Transactions on Fuzzy Systems: A Publication of the IEEE Neural Networks Council,2000,8(2):186-199.

[94] 刘金琨．机器人控制系统的设计与MATLAB仿真：基本方法[M]．北京：清华大学出版社，2019：73-93．