欠驱动机械系统的运动控制研究
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摘要
欠驱动机械系统是指驱动器个数少于自由度个数的机械系统。与全驱动机械系统相比,驱动器个数的减少使得这类系统具有重量轻、能耗低、运动灵活等特点。然而,这也将系统的状态量束缚在运动空间中一个不确定的位形上,从而大大增加了这类系统控制设计的难度。近二十年来,欠驱动机械系统的控制问题一直是工程控制领域中一个极具挑战性的课题。
本文对欠驱动机械系统的运动控制进行了深入的研究,其中包括系统的稳定控制,以及运动规划和跟踪控制研究。论文的主要研究成果和创新点如下：
(1)针对二自由度欠驱动机械系统,提出一种基于等价输入干扰的稳定控制方法。
具有一个控制输入和两个自由度的机械系统是一类最简单的欠驱动机械系统。根据这类系统惯性矩阵表现形式的不同,首先将它分为三种类型。然后为每一类构造一个微分同胚变换,将原系统变换为一个结构相对简单的新系统。在对新系统的能控、能观及零点分布等特性进行详尽分析后,提出一种基于等价输入干扰的稳定控制系统设计方法,以达到将新系统渐近稳定在零平衡点处的控制目标,并且由坐标变换的同胚特性来保证原系统在零平衡点处的渐近稳定性。基于等价输入干扰的稳定控制方法仅利用位置量信息便能实现系统的全局稳定控制目标,这不仅能降低控制系统成本,而且还能避免速度噪音对系统控制性能的影响。数值仿真结果显示这一方法在二自由度欠驱动机械系统的稳定时间及运动过程等性能方面,均能达到令人满意的控制效果。
(2)将基于等价输入干扰的控制方法应用到一类多自由度欠驱动机械系统的全局稳定控制中。
在基于等价输入干扰的二自由度欠驱动系统稳定控制的研究基础上,探讨这一控制方法在多自由度欠驱动系统中的应用。首先,针对一个三自由度欠驱动机械系统,分析系统的坐标变换以及变换后新系统的特性,构造基于等价输入干扰的稳定控制系统,实现系统在零平衡点处的全局渐近稳定。然后,分析基于等价输入干扰的控制方法在一类ｎ(ｎ>3)自由度欠驱动机械系统中的应用,将这一控制方法的适用范围进一步拓宽。
(3)针对一类多自由度欠驱动机械系统,提出一种基于力矩耦合的全局稳定控制策略。
通过分析多自由度欠驱动机械系统的结构特性,构造一个控制输入力矩间的耦合关系,将系统的某些状态变量进行解耦,并将原系统变换为一个由从动子系统和驱动子系统组成的级联系统。然后,对驱动子系统的稳定性以及从动子系统的无源特性进行分析,将原系统在零点处的渐近稳定问题转化为二自由度的从动子系统在零点处的渐近稳定问题。最后,运用基于无源性的方法设计控制器将从动子系统渐近稳定在零点处。将所提控制方法应用到一个平面三连杆机械臂的全局稳定控制中,数值仿真结果显示它能使机械臂快速而平滑的实现渐近稳定,这显示了方法的有效性与实用性。另外,这种基于力矩耦合的稳定控制方法将一个多自由度欠驱动机械系统的稳定控制问题转化为一个二自由度欠驱动系统的稳定控制问题,为一个对二自由度欠驱动系统有效的稳定控制方法在多自由度中的进一步推广搭建了桥梁。
(4)针对欠驱动Acrobot系统,提出期望运动轨迹的构造方法,并设计跟踪控制器使Acrobot沿期望运动轨迹达到渐近稳定。
两连杆机械臂Acrobot是一个典型的欠驱动机械系统,控制目标是将它从垂直向下位置摇起并最终稳定在垂直向上位置处。首先,基于Acrobot受重力作用的特性,通过利用一个虚拟摩擦及倒转的方法,为它在初始点和目标点之间设计一条用时最少的期望运动轨迹。然后,将系统的稳定控制问题转化为跟踪控制问题,并基于期望轨迹求取误差方程。最后,设计跟踪控制器使得Acrobot沿这条期望轨迹达到渐近稳定。本文为具有二阶非完整约束的非线性Acrobot系统构造了一条用时最少的期望运动轨迹,这为其它欠驱动非完整机械系统的运动规划起到了借鉴与启迪的作用。另外,本文无需对Acrobot的运动空间进行划分,仅用一个控制器便能实现系统的稳定控制目标,整个控制系统的结构比较简单。更为重要的是,Acrobot系统的整个稳定运动过程及稳定时间能够被准确预测,这为驱动器的选取及试验系统的构造提供了指导。
An underactuated mechanical system (UMS) is a mechanical system that has fewer actuators than degrees of freedom (DOF). Because there are fewer actuators, these kinds of systems are lighter, less energy-consuming, and more flexible than fully actuated ones. However, this characteristic also restricts the states of a UMS to an uncertain manifold of the motion space. As a result, the control design of such a system is very difficult; and the problem of controlling a UMS is a challenging task in the field of engineering control.
This dissertation concerns the motion control of UMSs, which includes system stabilization control, motion planning, and tracking control. The main results and innovations of this study are as follows:
(1) An equivalent-input-disturbance-(EID-) based stabilization control method was developed for2-DOF UMSs.
A mechanical system that has two DOFs and only one actuator is the simplest type of UMS. First,2-DOF UMSs were divided into three categories based on the inertia matrix of the system. And then, a homeomorphic coordinate transformation was devised for each category. It transforms the original system into a new one with a relatively simple structure. After the controllability, observability, arrangement of zero points, and other properties of the transformed system were thoroughly analyzed, the EID approach was used to globally stabilize the transformed system at the zero equilibrium point. The stabilizing control objective of the original system at the zero point is guaranteed by the homeomorphism of the coordinate transformation. The EID-based approach uses the state variables of position, but not velocity, to ensure global stabilization of2-DOF UMSs. This not only reduces the cost of the control system, but also avoids the influence of speed noise on control performance. Simulation results show that this method yields satisfactory results for stabilization time, motion process, and other measures of control performance for2-DOF UMSs.
(2) The EID-based control method was extended to the global stabilization of UMSs with multiple DOFs.
The results of the study on the EID-based stabilization control of2-DOF UMSs were used as a basis for examining the application of the EID-based approach to a multi-DOF UMS. First, a coordinate transformation for a3-DOF UMS was devised, and the properties of the transformed system were analyzed. Next, an EID-based control system was developed that stabilizes the3-DOF UMS at the zero equilibrium point. Then, the EID-based approach was applied to an n-DOF UMS (n>3), thereby further extending the applicability of this control method.
(3) A torque-coupling-based stabilization control strategy was devised for a class of multi-DOF UMSs.
An analysis of the structural characteristics of a class of multi-DOF UMSs was used to work out the coupling relationship between control torques. This relationship decouples some state variables of the system from others and changes the original system into a cascade-connected system consisting of a driving subsystem and a2-DOF driven subsystem. Next, the stability of the driving subsystem and the passivity of the driven subsystem were analyzed, and the results were used to convert the problem of asymptotically stabilizing the multi-DOF UMS at the zero equilibrium point into the problem of stabilizing the2-DOF driven subsystem at the zero equilibrium point. Finally, a passivity-based method was used to stabilize the driven subsystem at the zero point. The proposed method is employed on the global stabilization of a horizontal three-link manipulator. Simulation results show that the manipulator is quickly and smoothly stabilized at the target point. This demonstrates the validity and practicability of the method. In addition, this torque-coupled-based control strategy transforms the stabilization of an n-DOF UMS (n≥3) into that of a2-DOF UMS, and it might enable a control method for stabilizing a2-DOF UMS to be generalized to the stabilization of a multi-DOF one.
(4) For an underactuated Acrobot system, a method of constructing a time-optimal desired trajectory was devised; and a tracking controller was designed that asymptotically stabilizes the Acrobot at the target position along the desired trajectory.
A two-link Acrobot is a typical UMS. A common control objective is to swing it up from the straight-down position and stabilize it at the straight-up position. First, an artificial friction and rewinding approach was used to construct a time-optimal trajectory from the start position to the target position. Then, the stabilization control problem was converted into a tracking control problem; and an error dynamic equation was obtained based on the desired motion trajectory. Finally, a tracking controller was designed that asymptotically stabilized the Acrobot at the target position along the desired trajectory. In this study, a time-optimal trajectory for a nonlinear Acrobot system with a second-order nonholonomic constraint was constructed. It provides a guideline for the motion planning of other underactuated nonholonomic mechanical systems. Moreover, the trajectory-tracking-based stabilization control strategy is simple and efficient. It does not require division of the motion space of the Acrobot, and it uses just a single controller for motion control. Most importantly, the stabilizing movement and the settling time of the Acrobot can be accurately predicted from the desired trajectory. These can be used as guidelines for selecting actuators as well as for evaluating an experimental Acrobot system design.
本文对欠驱动机械系统的运动控制进行了深入的研究,其中包括系统的稳定控制,以及运动规划和跟踪控制研究。论文的主要研究成果和创新点如下：
(1)针对二自由度欠驱动机械系统,提出一种基于等价输入干扰的稳定控制方法。
具有一个控制输入和两个自由度的机械系统是一类最简单的欠驱动机械系统。根据这类系统惯性矩阵表现形式的不同,首先将它分为三种类型。然后为每一类构造一个微分同胚变换,将原系统变换为一个结构相对简单的新系统。在对新系统的能控、能观及零点分布等特性进行详尽分析后,提出一种基于等价输入干扰的稳定控制系统设计方法,以达到将新系统渐近稳定在零平衡点处的控制目标,并且由坐标变换的同胚特性来保证原系统在零平衡点处的渐近稳定性。基于等价输入干扰的稳定控制方法仅利用位置量信息便能实现系统的全局稳定控制目标,这不仅能降低控制系统成本,而且还能避免速度噪音对系统控制性能的影响。数值仿真结果显示这一方法在二自由度欠驱动机械系统的稳定时间及运动过程等性能方面,均能达到令人满意的控制效果。
(2)将基于等价输入干扰的控制方法应用到一类多自由度欠驱动机械系统的全局稳定控制中。
在基于等价输入干扰的二自由度欠驱动系统稳定控制的研究基础上,探讨这一控制方法在多自由度欠驱动系统中的应用。首先,针对一个三自由度欠驱动机械系统,分析系统的坐标变换以及变换后新系统的特性,构造基于等价输入干扰的稳定控制系统,实现系统在零平衡点处的全局渐近稳定。然后,分析基于等价输入干扰的控制方法在一类ｎ(ｎ>3)自由度欠驱动机械系统中的应用,将这一控制方法的适用范围进一步拓宽。
(3)针对一类多自由度欠驱动机械系统,提出一种基于力矩耦合的全局稳定控制策略。
通过分析多自由度欠驱动机械系统的结构特性,构造一个控制输入力矩间的耦合关系,将系统的某些状态变量进行解耦,并将原系统变换为一个由从动子系统和驱动子系统组成的级联系统。然后,对驱动子系统的稳定性以及从动子系统的无源特性进行分析,将原系统在零点处的渐近稳定问题转化为二自由度的从动子系统在零点处的渐近稳定问题。最后,运用基于无源性的方法设计控制器将从动子系统渐近稳定在零点处。将所提控制方法应用到一个平面三连杆机械臂的全局稳定控制中,数值仿真结果显示它能使机械臂快速而平滑的实现渐近稳定,这显示了方法的有效性与实用性。另外,这种基于力矩耦合的稳定控制方法将一个多自由度欠驱动机械系统的稳定控制问题转化为一个二自由度欠驱动系统的稳定控制问题,为一个对二自由度欠驱动系统有效的稳定控制方法在多自由度中的进一步推广搭建了桥梁。
(4)针对欠驱动Acrobot系统,提出期望运动轨迹的构造方法,并设计跟踪控制器使Acrobot沿期望运动轨迹达到渐近稳定。
两连杆机械臂Acrobot是一个典型的欠驱动机械系统,控制目标是将它从垂直向下位置摇起并最终稳定在垂直向上位置处。首先,基于Acrobot受重力作用的特性,通过利用一个虚拟摩擦及倒转的方法,为它在初始点和目标点之间设计一条用时最少的期望运动轨迹。然后,将系统的稳定控制问题转化为跟踪控制问题,并基于期望轨迹求取误差方程。最后,设计跟踪控制器使得Acrobot沿这条期望轨迹达到渐近稳定。本文为具有二阶非完整约束的非线性Acrobot系统构造了一条用时最少的期望运动轨迹,这为其它欠驱动非完整机械系统的运动规划起到了借鉴与启迪的作用。另外,本文无需对Acrobot的运动空间进行划分,仅用一个控制器便能实现系统的稳定控制目标,整个控制系统的结构比较简单。更为重要的是,Acrobot系统的整个稳定运动过程及稳定时间能够被准确预测,这为驱动器的选取及试验系统的构造提供了指导。
An underactuated mechanical system (UMS) is a mechanical system that has fewer actuators than degrees of freedom (DOF). Because there are fewer actuators, these kinds of systems are lighter, less energy-consuming, and more flexible than fully actuated ones. However, this characteristic also restricts the states of a UMS to an uncertain manifold of the motion space. As a result, the control design of such a system is very difficult; and the problem of controlling a UMS is a challenging task in the field of engineering control.
This dissertation concerns the motion control of UMSs, which includes system stabilization control, motion planning, and tracking control. The main results and innovations of this study are as follows:
(1) An equivalent-input-disturbance-(EID-) based stabilization control method was developed for2-DOF UMSs.
A mechanical system that has two DOFs and only one actuator is the simplest type of UMS. First,2-DOF UMSs were divided into three categories based on the inertia matrix of the system. And then, a homeomorphic coordinate transformation was devised for each category. It transforms the original system into a new one with a relatively simple structure. After the controllability, observability, arrangement of zero points, and other properties of the transformed system were thoroughly analyzed, the EID approach was used to globally stabilize the transformed system at the zero equilibrium point. The stabilizing control objective of the original system at the zero point is guaranteed by the homeomorphism of the coordinate transformation. The EID-based approach uses the state variables of position, but not velocity, to ensure global stabilization of2-DOF UMSs. This not only reduces the cost of the control system, but also avoids the influence of speed noise on control performance. Simulation results show that this method yields satisfactory results for stabilization time, motion process, and other measures of control performance for2-DOF UMSs.
(2) The EID-based control method was extended to the global stabilization of UMSs with multiple DOFs.
The results of the study on the EID-based stabilization control of2-DOF UMSs were used as a basis for examining the application of the EID-based approach to a multi-DOF UMS. First, a coordinate transformation for a3-DOF UMS was devised, and the properties of the transformed system were analyzed. Next, an EID-based control system was developed that stabilizes the3-DOF UMS at the zero equilibrium point. Then, the EID-based approach was applied to an n-DOF UMS (n>3), thereby further extending the applicability of this control method.
(3) A torque-coupling-based stabilization control strategy was devised for a class of multi-DOF UMSs.
An analysis of the structural characteristics of a class of multi-DOF UMSs was used to work out the coupling relationship between control torques. This relationship decouples some state variables of the system from others and changes the original system into a cascade-connected system consisting of a driving subsystem and a2-DOF driven subsystem. Next, the stability of the driving subsystem and the passivity of the driven subsystem were analyzed, and the results were used to convert the problem of asymptotically stabilizing the multi-DOF UMS at the zero equilibrium point into the problem of stabilizing the2-DOF driven subsystem at the zero equilibrium point. Finally, a passivity-based method was used to stabilize the driven subsystem at the zero point. The proposed method is employed on the global stabilization of a horizontal three-link manipulator. Simulation results show that the manipulator is quickly and smoothly stabilized at the target point. This demonstrates the validity and practicability of the method. In addition, this torque-coupled-based control strategy transforms the stabilization of an n-DOF UMS (n≥3) into that of a2-DOF UMS, and it might enable a control method for stabilizing a2-DOF UMS to be generalized to the stabilization of a multi-DOF one.
(4) For an underactuated Acrobot system, a method of constructing a time-optimal desired trajectory was devised; and a tracking controller was designed that asymptotically stabilizes the Acrobot at the target position along the desired trajectory.
A two-link Acrobot is a typical UMS. A common control objective is to swing it up from the straight-down position and stabilize it at the straight-up position. First, an artificial friction and rewinding approach was used to construct a time-optimal trajectory from the start position to the target position. Then, the stabilization control problem was converted into a tracking control problem; and an error dynamic equation was obtained based on the desired motion trajectory. Finally, a tracking controller was designed that asymptotically stabilized the Acrobot at the target position along the desired trajectory. In this study, a time-optimal trajectory for a nonlinear Acrobot system with a second-order nonholonomic constraint was constructed. It provides a guideline for the motion planning of other underactuated nonholonomic mechanical systems. Moreover, the trajectory-tracking-based stabilization control strategy is simple and efficient. It does not require division of the motion space of the Acrobot, and it uses just a single controller for motion control. Most importantly, the stabilizing movement and the settling time of the Acrobot can be accurately predicted from the desired trajectory. These can be used as guidelines for selecting actuators as well as for evaluating an experimental Acrobot system design.
引文
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