一类欠驱动机械系统的非线性控制研究
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摘要
欠驱动机械系统是指控制输入数目少于系统自由度的机械控制系统,它广泛存在于机器人、航天航空和交通运输等各个领域,由于控制输入的缺失使得其控制问题成为控制领域富有挑战性的研究热点之一。吊车作为一个典型的欠驱动机械系统,研究其控制问题不但具有很强的实用价值,而且有助于促进欠驱动机械系统控制理论的发展。论文以欠驱动机械系统的非线性控制为理论研究主体,以欠驱动吊车系统的非线性控制为应用研究对象,结合哈尔滨工业大学实验设备开发基金项目“龙门吊车实物仿真技术研究”,展开一类欠驱动机械系统的非线性控制研究。
本文基于拉格朗日算子定义了简单机械系统,在此基础上建立了机械系统的动力学模型,讨论了机械控制系统的几种常见类别,即全驱动机械系统、欠驱动机械系统、平滑机械系统和非完整机械系统,并着重分析了欠驱动机械系统的动力学特征。另外,利用拉格朗日方程建立了三维欠驱动吊车系统的数学模型,并通过简化所建立的三维吊车系统模型得到二维和一维吊车系统的数学模型。
鉴于已有的非线性控制算法难于直接用来进行欠驱动机械系统的控制设计,在分析欠驱动机械系统动力学模型的基础上,根据其动能对称性的特点和形态变量的驱动情况,基于拉格朗日算子设计了闭环的坐标变换;其将欠驱动机械系统的动力学转换成具有结构特征的级联非线性系统。文中将这些具有结构特征的级联非线性系统称为欠驱动机械系统的级联规范型,分为三种,即严格反馈规范型、严格前馈规范型和非三角规范型。为了验证设计坐标变换的有效性,采用设计的坐标变换分别给出了TORA(Translational Oscillator with Rotational Actuator)系统、车摆系统和Pendubot系统的严格反馈规范型、严格前馈规范型和非三角规范型。
针对欠驱动机械系统的非线性级联规范型,给出了具有一般性的欠驱动机械系统的非线性控制方案。欠驱动机械系统的三角规范型可以通过已有的Backstepping步骤和嵌套饱和方案进行控制器的设计;针对非三角规范型的控制问题,本文重点研究了不动点控制方案,该方案将不动点方程的解作为系统的非线性状态反馈算法,即可通过递归的方法设计系统的控制器;同时文中还研究了不动点控制器的存在条件。分别采用Backstepping步骤、嵌套饱和方案和不动点控制方案设计了TORA系统、车摆系统和球棒系统的稳定控制器,并通过仿真实验证明了控制方案的有效性。
最后,针对吊车这一具体的欠驱动机械系统设计了易于工程实现的非线性控制器并进行了实物实验研究。对于变绳长的三维和二维吊车系统,选择模型中直接激励的自由度作为系统输出,经过配置部分反馈线性化对其进行轨迹跟踪控制;将欠驱动的自由度作为内部动态考虑并保证其稳定,从而实现吊车负载的定位。仿真结果表明,所设计的非线性控制器可有效实现吊车系统快速防摆和精确定位的要求;为了进一步验证该控制方案的实用性,在一个变绳长的二维吊车实验装置上实现了该算法,实验结果同样表明所设计的控制方案可有效实现负载的快速定位要求。
Underactuated mechanical systems (UMS) are mechanical control systems with fewer control inputs than the number of configuration variables. Control of underactuated mechanical systems is currently an active and challenging field of research due to the lack of control inputs and their broad applications in robotics, aerospace vehicles, and transportation vehicles. As a typical underactuated mechanical system, control of underactuated crane system is not only very practical but also helpful to the development of control theory on UMS. This dissertation is devoted to nonlinear control of a class of UMS, which is supported by the fund of Experimental Equipment Exploitation in Harbin Institute of Technology“Practical simulation technology of gantry crane system”.
Based on the Euler-Lagrange equation for mechanical control system, the important classes of mechanical control systems are discussed, namely, fully-actuated mechanical systems, UMS, flat mechanical systems, and nonholonomic mechanical systems, particularly, dynamics and control problems of UMS are focused. In addition, the dynamical model of three-dimensional crane system is derived by using Lagrange equations, which can be easily simplified into the dynamical models of two-dimensional and one-dimensional crane systems.
A procedure to reduce control design for original UMS is developed by considering that almost all real-life mechanical control systems possess kinetic symmetry properties, i.e. their kinetic energy only depends on a subset of configuration variables called shape variables. As a result, several closed-loop changes of coordinates are designed to transform several classes of UMS into cascade nonlinear systems with structural properties that are convenient for control design purposes. The obtained cascade normal forms are three classes of nonlinear systems, namely, systems in strict feedback form, feedforward form, and nontriangular form. The proposed changes of coordinates are successfully employed to transform the original dynamics of the Translational Oscillator with Rotational Actuator (TORA) system, the Cart-Pole system, and the Pendubot system into strict feedback normal form, feedforward normal form, and nontriangular normal form, respectively.
Once the reduced nonlinear cascade system is obtained, the special structural property of the normal forms is considered to the following control design of the UMS. The triangular normal forms of underactuated mechanical systems can be controlled using existing Backstepping procedures and nested saturation scheme. However, there is still no totally effective control design scheme for the nontriangular normal forms. In this work, the problem is addressed by introducing a fixed pointed controller based on the solutions of fixed-pointed equations as stabilizing nonlinear state feedback law. This controller can be obtained via a recursive method that is convenient for implementation, and also the sufficient conditions for global existence of the controller are provided. The stabilization control of the TORA system, the Cart-Pole system, and Beam-and-Ball system is realized by using Backstepping procedure, nested saturation scheme, and fixed pointed controller, repectively, and simulation results demonstrated the feasibility of the proposed approaches.
Finally, this dissertation focuses on control design and experiment implement of underactuated crane system. A nonlinear controller design scheme based on partial feedback linearization technique is presented for the three-dimensional and two-dimensional crane systems with various cord length. By choosing the actuated degrees corresponding to the freedom as the outputs, the controller is designed to track the trajectories of outputs while providing internal dynamics stability of unactuated degrees. The analysis of the internal dynamics shows that the stability of the zero dynamics guarantees the stability of the control system. Simulation results are presented to show the feasibility of the presented scheme. To verify its practicability, the nonlinear controller is used to realize the anti-swaying and positioning control of a two-dimensional scaled gantry crane system which can do hoisting and traveling motions simultaneously, and implemental results are given and discussed.
本文基于拉格朗日算子定义了简单机械系统,在此基础上建立了机械系统的动力学模型,讨论了机械控制系统的几种常见类别,即全驱动机械系统、欠驱动机械系统、平滑机械系统和非完整机械系统,并着重分析了欠驱动机械系统的动力学特征。另外,利用拉格朗日方程建立了三维欠驱动吊车系统的数学模型,并通过简化所建立的三维吊车系统模型得到二维和一维吊车系统的数学模型。
鉴于已有的非线性控制算法难于直接用来进行欠驱动机械系统的控制设计,在分析欠驱动机械系统动力学模型的基础上,根据其动能对称性的特点和形态变量的驱动情况,基于拉格朗日算子设计了闭环的坐标变换;其将欠驱动机械系统的动力学转换成具有结构特征的级联非线性系统。文中将这些具有结构特征的级联非线性系统称为欠驱动机械系统的级联规范型,分为三种,即严格反馈规范型、严格前馈规范型和非三角规范型。为了验证设计坐标变换的有效性,采用设计的坐标变换分别给出了TORA(Translational Oscillator with Rotational Actuator)系统、车摆系统和Pendubot系统的严格反馈规范型、严格前馈规范型和非三角规范型。
针对欠驱动机械系统的非线性级联规范型,给出了具有一般性的欠驱动机械系统的非线性控制方案。欠驱动机械系统的三角规范型可以通过已有的Backstepping步骤和嵌套饱和方案进行控制器的设计;针对非三角规范型的控制问题,本文重点研究了不动点控制方案,该方案将不动点方程的解作为系统的非线性状态反馈算法,即可通过递归的方法设计系统的控制器;同时文中还研究了不动点控制器的存在条件。分别采用Backstepping步骤、嵌套饱和方案和不动点控制方案设计了TORA系统、车摆系统和球棒系统的稳定控制器,并通过仿真实验证明了控制方案的有效性。
最后,针对吊车这一具体的欠驱动机械系统设计了易于工程实现的非线性控制器并进行了实物实验研究。对于变绳长的三维和二维吊车系统,选择模型中直接激励的自由度作为系统输出,经过配置部分反馈线性化对其进行轨迹跟踪控制;将欠驱动的自由度作为内部动态考虑并保证其稳定,从而实现吊车负载的定位。仿真结果表明,所设计的非线性控制器可有效实现吊车系统快速防摆和精确定位的要求;为了进一步验证该控制方案的实用性,在一个变绳长的二维吊车实验装置上实现了该算法,实验结果同样表明所设计的控制方案可有效实现负载的快速定位要求。
Underactuated mechanical systems (UMS) are mechanical control systems with fewer control inputs than the number of configuration variables. Control of underactuated mechanical systems is currently an active and challenging field of research due to the lack of control inputs and their broad applications in robotics, aerospace vehicles, and transportation vehicles. As a typical underactuated mechanical system, control of underactuated crane system is not only very practical but also helpful to the development of control theory on UMS. This dissertation is devoted to nonlinear control of a class of UMS, which is supported by the fund of Experimental Equipment Exploitation in Harbin Institute of Technology“Practical simulation technology of gantry crane system”.
Based on the Euler-Lagrange equation for mechanical control system, the important classes of mechanical control systems are discussed, namely, fully-actuated mechanical systems, UMS, flat mechanical systems, and nonholonomic mechanical systems, particularly, dynamics and control problems of UMS are focused. In addition, the dynamical model of three-dimensional crane system is derived by using Lagrange equations, which can be easily simplified into the dynamical models of two-dimensional and one-dimensional crane systems.
A procedure to reduce control design for original UMS is developed by considering that almost all real-life mechanical control systems possess kinetic symmetry properties, i.e. their kinetic energy only depends on a subset of configuration variables called shape variables. As a result, several closed-loop changes of coordinates are designed to transform several classes of UMS into cascade nonlinear systems with structural properties that are convenient for control design purposes. The obtained cascade normal forms are three classes of nonlinear systems, namely, systems in strict feedback form, feedforward form, and nontriangular form. The proposed changes of coordinates are successfully employed to transform the original dynamics of the Translational Oscillator with Rotational Actuator (TORA) system, the Cart-Pole system, and the Pendubot system into strict feedback normal form, feedforward normal form, and nontriangular normal form, respectively.
Once the reduced nonlinear cascade system is obtained, the special structural property of the normal forms is considered to the following control design of the UMS. The triangular normal forms of underactuated mechanical systems can be controlled using existing Backstepping procedures and nested saturation scheme. However, there is still no totally effective control design scheme for the nontriangular normal forms. In this work, the problem is addressed by introducing a fixed pointed controller based on the solutions of fixed-pointed equations as stabilizing nonlinear state feedback law. This controller can be obtained via a recursive method that is convenient for implementation, and also the sufficient conditions for global existence of the controller are provided. The stabilization control of the TORA system, the Cart-Pole system, and Beam-and-Ball system is realized by using Backstepping procedure, nested saturation scheme, and fixed pointed controller, repectively, and simulation results demonstrated the feasibility of the proposed approaches.
Finally, this dissertation focuses on control design and experiment implement of underactuated crane system. A nonlinear controller design scheme based on partial feedback linearization technique is presented for the three-dimensional and two-dimensional crane systems with various cord length. By choosing the actuated degrees corresponding to the freedom as the outputs, the controller is designed to track the trajectories of outputs while providing internal dynamics stability of unactuated degrees. The analysis of the internal dynamics shows that the stability of the zero dynamics guarantees the stability of the control system. Simulation results are presented to show the feasibility of the presented scheme. To verify its practicability, the nonlinear controller is used to realize the anti-swaying and positioning control of a two-dimensional scaled gantry crane system which can do hoisting and traveling motions simultaneously, and implemental results are given and discussed.
引文
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72 R. Fierro, F. L. Lewis, and A. Lowe. Hybrid Control for a Class of Underactuated Mechanical Systems. IEEE Transactions on Automatic Control. 1999, 29(6): 649~654
73 赖旭芝,蔡自兴,吴敏. 一类欠驱动机械系统的模糊与变结构控制. 自动化学报. 2001, 27(16): 850~854
74 T. Fukuda and Y. Hasegawa. Mechanism and Control of Mechatronic System with Higher Degrees of Freedom. Annual Reviews in Control. 2004, 28: 137~155
75 朱江滨, 易建强. 非线性欠驱动系统的实时控制. 自动化学报. 2004, 30(1): 131~136
76 S. Nair and N. E. Leonard. A Normal Form for Energy Shaping: Application to the Furuta Pendulum. IEEE Conference on Decision and Control, Las Vegas, Nevada USA. 2002: 516~521
77 K. J. Astrom and K. Furuta. Swinging up a Pendulum by Energy Control. Automatica. 2000, 36(2): 287~295
78 R. Teel. Using Saturation to Stabilize a Class of Single-Input Partially Linear Composite Systems. Proceedings of the IFAC Symposium on Nonlinear Control Systems Design, Bordeaux, France. 1992: 369~374
79 R. Teel. A Nonlinear Small Gain Theorem for the Analysis of Control Systems with Saturation. IEEE Transactions on Automatic Control. 1996, 41(9): 1256~1270
80 C. J. Wan, D. S. Bernstein, and V. T. Coppola. Global Stabilization of the Oscillating Eccentric Rotor. IEEE Conference on Decision and Control, Lake Buena Vista, FL USA. 1994:4024~4029
81 M. Jankovic, D. Fontaine, and P. V. Kokotovic. TORA Example: Cascade and Passivity-Based Control Designs. IEEE Transactions on Control Systems Technology. 1996, 4(3): 292~297
82 蒋怀伟,王石刚,徐威等. 纳米生物机器人研究与进展. 机器人. 2005, 27(6): 569~574
83 B. R. Donald, C. G. Levey, C. D. McGray, et al. An Untethered, Electrostatic, Globally Controllable MEMS Micro-Robot. Journal of Micromechanical Systems. 2006, 15(1): 1~15
84 D. C. D. Oguamanam and J. S. Hansen. Dynamics of a Three-Dimensional Overhead Crane System. Journal of Sound and Vibration. 2001, 242(3): 411~426
85 Y. S. Kim, H. S. Seo, and S. K. Sul. A New Anti-Sway Control Scheme for Trolley Crane System. IEEE Conference on Industry Applications, Chicago, Illinois USA. 2001: 548~552
86 A. Ciua, C. Sentzu, and C. Usai. Observer-Controller Design for Cranes Via Lyapunov Equivalence. Automatica. 1999, 35: 669~678
87 T. Matsuo, R. Yoshino, Haruo Suemitsu, et al. Nominal Performance Recovery by PID+Q Controller and Its Application to Antisway Control of Crane Lifter with Visual Feedback. IEEE Transction on Control Systems Technology. 2004, 12(1): 156~166
88 A. Z. Al-Garni, K. A. F. Moustafa, and S. J. Nizami. Optimal Control of Overhead Cranes. Control Engineering Practice. 1995, 3(9): 1277~1284
89 A. Piazzi and A. Visioli. Optimal Dynamic-Inversion-Based Control of and Overhead Crane. IEE Proceedings: Control Theory Applications. 2002, 149(5): 405~411
90 Z. H. Wang and B. W. Surgenor. A Problem with the LQ Control of Overhead Cranes. Journal of Dynamic Systems, Measurment, and Control. 2006, 28: 436~440
91 M. Utierrez and R. Soto. Fuzzy Control of a Scale Prortotype Overhead Crane. IEEE Conference on Decision and Control, Tampa, Florida USA. 1998: 4266~4268
92 Y. C. Liang and K. K. Koh. Concise Anti-Swing Approach for Fuzzy Crane Control. Electronics Letters. 1997, 33(2): 167~168
93 A. Benhidjeb and G. L. Gissinger. Fuzzy Control of an Overhead Crane Performance Comparison with Classic Control. Control Engineering Practice. 1995, 3(12): 1687~1696
94 H. Butler, G. Honderd, and J. V. Amerongen. Model Reference Adaptive Control of a Gantry Crane Scale Model. IEEE Control Systems Magazine. 1991, 11(1): 57~62
95 T. Burg, D. Dawson, C. Rahn, et al. Nonlinear Control of an Overhead Crane via the Saturating Control Approach of Teel. Robotics and Automation, Proceedings of 1996 IEEE International Conference. 1996: 3155~3160
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97 J. Collado, R. Lozano, and I. Fantoni. Control of Convey-Crane Based on Passivity. Proceedings of the American Control Conference, Chicago, IllinoisUSA. 2000: 1260~1264
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99 G. Bartolini, A. Pisano, and E. Usai. Second Sliding-Mode Control of Container Cranes. Automatica. 2002, 38: 1783~1790
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71 D. T. Liu, J. Q. Yi, D. B. Zhao, et al. Adaptive sliding mode fuzzy control for a two-dimensional overhead crane. Mechatronics. 2005, 15: 505~522
72 R. Fierro, F. L. Lewis, and A. Lowe. Hybrid Control for a Class of Underactuated Mechanical Systems. IEEE Transactions on Automatic Control. 1999, 29(6): 649~654
73 赖旭芝,蔡自兴,吴敏. 一类欠驱动机械系统的模糊与变结构控制. 自动化学报. 2001, 27(16): 850~854
74 T. Fukuda and Y. Hasegawa. Mechanism and Control of Mechatronic System with Higher Degrees of Freedom. Annual Reviews in Control. 2004, 28: 137~155
75 朱江滨, 易建强. 非线性欠驱动系统的实时控制. 自动化学报. 2004, 30(1): 131~136
76 S. Nair and N. E. Leonard. A Normal Form for Energy Shaping: Application to the Furuta Pendulum. IEEE Conference on Decision and Control, Las Vegas, Nevada USA. 2002: 516~521
77 K. J. Astrom and K. Furuta. Swinging up a Pendulum by Energy Control. Automatica. 2000, 36(2): 287~295
78 R. Teel. Using Saturation to Stabilize a Class of Single-Input Partially Linear Composite Systems. Proceedings of the IFAC Symposium on Nonlinear Control Systems Design, Bordeaux, France. 1992: 369~374
79 R. Teel. A Nonlinear Small Gain Theorem for the Analysis of Control Systems with Saturation. IEEE Transactions on Automatic Control. 1996, 41(9): 1256~1270
80 C. J. Wan, D. S. Bernstein, and V. T. Coppola. Global Stabilization of the Oscillating Eccentric Rotor. IEEE Conference on Decision and Control, Lake Buena Vista, FL USA. 1994:4024~4029
81 M. Jankovic, D. Fontaine, and P. V. Kokotovic. TORA Example: Cascade and Passivity-Based Control Designs. IEEE Transactions on Control Systems Technology. 1996, 4(3): 292~297
82 蒋怀伟,王石刚,徐威等. 纳米生物机器人研究与进展. 机器人. 2005, 27(6): 569~574
83 B. R. Donald, C. G. Levey, C. D. McGray, et al. An Untethered, Electrostatic, Globally Controllable MEMS Micro-Robot. Journal of Micromechanical Systems. 2006, 15(1): 1~15
84 D. C. D. Oguamanam and J. S. Hansen. Dynamics of a Three-Dimensional Overhead Crane System. Journal of Sound and Vibration. 2001, 242(3): 411~426
85 Y. S. Kim, H. S. Seo, and S. K. Sul. A New Anti-Sway Control Scheme for Trolley Crane System. IEEE Conference on Industry Applications, Chicago, Illinois USA. 2001: 548~552
86 A. Ciua, C. Sentzu, and C. Usai. Observer-Controller Design for Cranes Via Lyapunov Equivalence. Automatica. 1999, 35: 669~678
87 T. Matsuo, R. Yoshino, Haruo Suemitsu, et al. Nominal Performance Recovery by PID+Q Controller and Its Application to Antisway Control of Crane Lifter with Visual Feedback. IEEE Transction on Control Systems Technology. 2004, 12(1): 156~166
88 A. Z. Al-Garni, K. A. F. Moustafa, and S. J. Nizami. Optimal Control of Overhead Cranes. Control Engineering Practice. 1995, 3(9): 1277~1284
89 A. Piazzi and A. Visioli. Optimal Dynamic-Inversion-Based Control of and Overhead Crane. IEE Proceedings: Control Theory Applications. 2002, 149(5): 405~411
90 Z. H. Wang and B. W. Surgenor. A Problem with the LQ Control of Overhead Cranes. Journal of Dynamic Systems, Measurment, and Control. 2006, 28: 436~440
91 M. Utierrez and R. Soto. Fuzzy Control of a Scale Prortotype Overhead Crane. IEEE Conference on Decision and Control, Tampa, Florida USA. 1998: 4266~4268
92 Y. C. Liang and K. K. Koh. Concise Anti-Swing Approach for Fuzzy Crane Control. Electronics Letters. 1997, 33(2): 167~168
93 A. Benhidjeb and G. L. Gissinger. Fuzzy Control of an Overhead Crane Performance Comparison with Classic Control. Control Engineering Practice. 1995, 3(12): 1687~1696
94 H. Butler, G. Honderd, and J. V. Amerongen. Model Reference Adaptive Control of a Gantry Crane Scale Model. IEEE Control Systems Magazine. 1991, 11(1): 57~62
95 T. Burg, D. Dawson, C. Rahn, et al. Nonlinear Control of an Overhead Crane via the Saturating Control Approach of Teel. Robotics and Automation, Proceedings of 1996 IEEE International Conference. 1996: 3155~3160
96 K. Yoshida and H. Kawabe. A Design of Saturating Control with a Guaranteed Cost and Its Application to the Crane Control. IEEE Transactions on Automation Control. 1992, 37(1): 121~127
97 J. Collado, R. Lozano, and I. Fantoni. Control of Convey-Crane Based on Passivity. Proceedings of the American Control Conference, Chicago, IllinoisUSA. 2000: 1260~1264
98 B. Vikramaditya and R. Rajamani. Nonlinear Control of a Trolley Crane System. Proceedings of the American Control Conference, Chicago, Illinois USA. 2000: 1032~1036
99 G. Bartolini, A. Pisano, and E. Usai. Second Sliding-Mode Control of Container Cranes. Automatica. 2002, 38: 1783~1790
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