含有采样反馈的二连杆系统稳定性

曹天成，张舒; Cao Tiancheng; Zhang Shu

引 en

含有采样反馈的二连杆系统稳定性^*

曹天成
机构：
同济大学航空航天与力学学院,上海 200092
×
，张舒
机构：
同济大学航空航天与力学学院,上海 200092
×

同济大学航空航天与力学学院,上海 200092；

Stability of Double Pendulum System with Sampling Feedback^*

Cao Tiancheng
Affiliation：
The School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092 , China
×
，Zhang Shu
Affiliation：
The School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092 , China
×

The School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092 , China；

通讯作者:

张舒,E-mail:zhangshu@tongji.edu.cn

中图分类号:O317

文献标识码:A

文章编号:1672-6553-2023-21(11)-010-009

DOI:10.6052/1672-6553-2022-062

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出版信息

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目录contents

摘要 Abstract
关键词 Keywords
1 二连杆混杂系统的动力学模型
1.1 不考虑采样反馈的动力学模型
1.2 考虑采样反馈的动力学模型
2 动态响应的离散映射
3 稳定性分析
4 结论
参考文献

摘要

当今主流的控制方法大都基于采样反馈得到的离散的状态量从而进行控制参数设计,然而对于采样反馈本身对系统的影响鲜有研究.本文以采样比例-微分控制反馈作用下的二连杆系统为研究对象,研究了采样周期对于混杂系统稳定性的影响.首先,通过数值模拟,发现了二连杆非线性混杂系统可以通过近似线性系统进行定性表征；其次,构造了状态量的离散映射,以反映二连杆混杂系统在目标位置附近的动态响应；最后,提出了二连杆混杂系统的稳定性判据,并以数值仿真的结果验证了判据的准确性.

Abstract

The majority of common control methods focus on control parameter design based on discrete state variables obtained by digital sampling currently. In this study, the influence of sampling period on the stability of hybrid systems is studied by studying a double pendulum system controlled with digital proportional differential feedback. Firstly, through numerical simulation, it is found that the double pendulum nonlinear hybrid system can be qualitatively characterized by an approximate linearized system. The discrete mapping of state variables is then constructed to reveal the dynamic behavior of double pendulum hybrid system near the target position. Furthermore, a stability criterion of the double pendulum hybrid system is purposed. The criteria’s accuracy is verified with numerical simulation results.

关键词

二连杆；混杂系统；稳定性；反馈控制

Keywords

double pendulum ； hybrid system ； stability ； feedback control

引言
机械臂由于能够提高生产过程中的自动化程度、改善工作环境降低生产事故风险等优势，在航空航天^[1-3]、金属冶炼^[4]等领域，得到了越来越广泛的应用，这也对机械臂的性能提出了越来越高的要求，其中末端执行器及其控制策略^[5，6]对机械臂的综合性能的影响尤为显著.然而，在很多连续时间的控制策略设计中，实际存在的各个传感器的采样反馈长期被忽略，对控制算法的有效性造成了影响.
在实际应用中，由于传感器每隔一个采样周期进行采样，反馈控制系统中控制器的输出是关于时间的分段常值，所用的数据相对于系统实际的状态量存在迟滞.定性地进行分析：若系统的采样周期为0，则输出的反馈是随采样数据实时变化的，与理想的连续时间采样控制系统相同；若系统的采样频率为0，则系统的控制器不会有输出.因此，必然存在某一临界的采样周期，使得系统的运动处于临界状态.
Habib等^[7-11]讨论了采样反馈PD控制的采样周期引起的输出迟滞对线性系统稳定性的影响，并进一步研究了存在采样系统的时滞^[7]、反馈系统的非线性^[7-11]的情况；王在华等^[12]研究了单自由度采样反馈反馈控制系统，并进一步讨论了位移信号与速度信号异步情况下的稳定性条件；王强等^[13]对倒立摆施加采样PD反馈控制，将闭环系统转化为一个差分方程，并给出了考虑采样反馈情况下的最优增益.然而，当前对采样反馈混杂系统的研究大都采用线性系统作为研究对象，对多自由度系统的讨论局限于能够直接解耦的机构.
本文以机械臂末端执行器中较常见的二连杆机构作为研究对象，建立了考虑采样反馈的动力学模型，并构造了考虑采样反馈的平衡点（即定位控制的目标位置）稳定性判据：第一节中，建立二连杆系统的动力学模型，并通过数值仿真得到采样周期造成的不同动态响应，与不考虑采样周期情况进行比较；第二节中，根据动力学方程，构造了状态量的离散映射，用于反映二连杆采样反馈反馈系统在平衡点（即目标位置）附近的动态响应；第三节，根据动态响应的离散映射，提出并验证了二连杆混杂系统稳定性的特征值判据，并通过数值仿真发现了采样周期引起的分岔现象；最后分析结果，得到结论.
1 二连杆混杂系统的动力学模型
考虑如下问题.如图1所示的二连杆系统，称连接底座的连杆为1号臂，另一根连杆为2号臂.两臂长度分别为l₁和l₂，质量分别为m₁和m₂，相对于各自质心的转动惯量分别为J₁和J₂.1号臂质心距离底座的距离为l_c1，2号臂质心距离两臂连接点的距离为l_c2.两臂都视为理想的刚性臂，在后续的动力学建模中不考虑杆的轴向位移与法向的柔性；两关节为理想关节，没有附加质量.对图1中的二连杆关节处施加的力矩为Q₁和Q₂.
使用机器人动力学分析中常用的拉格朗日法^[14-16]建模.两臂的质心坐标分别为

\{\begin{matrix} x_{1} = l_{c 1} s i n θ_{1} \\ y_{1} = - l_{c 1} c o s θ_{2} \\ x_{2} = l_{1} s i n θ_{1} + l_{c 2} s i n θ_{2} \\ y_{2} = - l_{1} c o s θ_{1} - l_{c 2} c o s θ_{2} \end{matrix}

(1)

图1 二连杆系统模型示意图
Fig.1 Sketch map of double pendulum model
根据式（1），可以得到系统的总动能和总势能为：

\begin{matrix} T = \frac{1}{2} m_{1} [{(\frac{d x_{1}}{d t})}^{2} + {(\frac{d y_{1}}{d t})}^{2}] + \\ \frac{1}{2} m_{2} [{(\frac{d x_{2}}{d t})}^{2} + {(\frac{d y_{2}}{d t})}^{2}] + \frac{1}{2} J_{1} {\dot{θ}}_{1}^{2} + \\ \frac{1}{2} J_{2} {\dot{θ}}_{2}^{2} = \frac{1}{2} (J_{1} + m_{1} l_{c 1}^{2} + m_{2} l_{c 2}^{2}) {\dot{θ}}_{1}^{2} + \\ m_{2} l_{1} l_{c 2} c o s (θ_{1} - c o s θ_{2}) {\dot{θ}}_{1} {\dot{θ}}_{2} + \frac{1}{2} (J_{2} + m_{2} l_{c 2}^{2}) {\dot{θ}}_{2}^{2} \end{matrix}

(2)

\begin{matrix} V = m_{1} g y_{1} + m_{2} g y_{2} \\ = - m_{1} g l_{c 1} c o s θ_{1} - m_{2} g (l_{1} c o s θ_{1} + l_{c 2} c o s θ_{2}) \end{matrix}

(3)

于是根据式（2）和式（3），可以得到拉格朗日量：

\begin{matrix} L = T - V = \frac{1}{2} (J_{1} + m_{1} l_{c 1}^{2} + m_{2} l_{c 2}^{2}) {\dot{θ}}_{1}^{2} + \\ m_{2} l_{1} l_{c 2} c o s (θ_{1} - c o s θ_{2}) {\dot{θ}}_{1} {\dot{θ}}_{2} + \\ \frac{1}{2} (J_{2} + m_{2} l_{c 2}^{2}) {\dot{θ}}_{2}^{2} + m_{1} g l_{c 1} c o s θ_{1} + \\ m_{2} g (l_{1} c o s θ_{1} + l_{c 2} c o s θ_{2}) \end{matrix}

(4)

根据式（4），有：
$\begin{matrix} \frac{\partial L}{\partial θ_{1}} = - g (m_{1} l_{c 1} + m_{2} l_{1}) s i n θ_{1} - \\ m_{2} l_{1} l_{c 2} s i n (θ_{1} - θ_{2}) {\dot{θ}}_{1} {\dot{θ}}_{2} \\ \frac{\partial L}{\partial θ_{2}} = - m_{2} g l_{c 2} s i n θ_{2} + m_{2} l_{1} l_{c 2} s i n (θ_{1} - θ_{2}) {\dot{θ}}_{1} {\dot{θ}}_{2} \end{matrix}$
$\begin{matrix} \frac{d}{d t} (\frac{\partial L}{\partial {\dot{θ}}_{1}}) = (J_{1} + m_{1} l_{c 1}^{2} + m_{2} l_{1}^{2}) {\ddot{θ}}_{1} + \\ m_{2} l_{1} l_{c 2} c o s (θ_{1} - θ_{2}) {\ddot{θ}}_{2} - \\ m_{2} l_{1} l_{c 2} s i n (θ_{1} - θ_{2}) ({\dot{θ}}_{1} {\dot{θ}}_{2} - {\dot{θ}}_{2}^{2}) \\ \frac{d}{d t} (\frac{\partial L}{\partial {\dot{θ}}_{2}}) = m_{2} l_{1} l_{c 2} c o s (θ_{1} - θ_{2}) {\ddot{θ}}_{1} + \\ (J_{2} + m_{2} l_{c 2}^{2}) {\ddot{θ}}_{2} - \\ m_{2} l_{1} l_{c 2} s i n (θ_{1} - θ_{2}) ({\dot{θ}}_{1}^{2} - {\dot{θ}}_{1} {\dot{θ}}_{2}) \end{matrix}$
代入第二类Lagrange方程中，经过化简，可以得到：

M (\begin{matrix} {\ddot{θ}}_{1} \\ {\ddot{θ}}_{2} \end{matrix}) + C (\begin{matrix} {\dot{θ}}_{1} \\ {\dot{θ}}_{2} \end{matrix}) + G = (\begin{matrix} Q_{1} \\ Q_{2} \end{matrix})

(5)

其中，
$\begin{matrix} M = [\begin{matrix} m_{11} & m_{12} \\ m_{12} & m_{22} \end{matrix}], C = [\begin{matrix} 0 & c_{12} \\ c_{21} & 0 \end{matrix}] \\ G = (\begin{matrix} (m_{1} l_{c 1} + m_{2} l_{1}) g s i n θ_{1} \\ m_{2} g l_{c 2} s i n θ_{2} \end{matrix}) \\ m_{11} = J_{1} + m_{1} l_{c 1}^{2} + m_{2} l_{1}^{2} \\ m_{12} = m_{2} l_{1} l_{c 2} c o s (θ_{1} - θ_{2}) \\ m_{22} = J_{2} + m_{2} l_{c 2}^{2} \\ c_{12} = - m_{2} l_{1} l_{c 2} s i n (θ_{1} - θ_{2}) ({\dot{θ}}_{1} - {\dot{θ}}_{2}) \\ c_{21} = - m_{2} l_{1} l_{c 2} s i n (θ_{1} - θ_{2}) ({\dot{θ}}_{1} - {\dot{θ}}_{2}) \end{matrix}$
二连杆受到PD控制，目标位置为θ₁_d和θ₂_d。力矩由比例-微分控制力矩Q_PD和重力补偿力矩Q_C两部分组成，有：
$\begin{matrix} (\begin{matrix} Q_{1} \\ Q_{2} \end{matrix}) = Q_{P D} + Q_{c} \\ = (\begin{matrix} k_{p 1} (θ_{1 d} - {\hat{θ}}_{1}) \\ k_{p 2} (θ_{2 d} - θ_{1 d} - {\hat{θ}}_{2} + {\hat{θ}}_{1}) \end{matrix}) + \\ (\begin{matrix} - k_{d 1} {\hat{{\dot{θ}}_{1}}}_{} \\ - k_{d 2} ({\hat{{\dot{θ}}_{2}}}_{} - {\hat{{\dot{θ}}_{1}}}_{}) \end{matrix}) + (\begin{matrix} (m_{1} l_{c 1} + m_{2} l_{1}) g s i n θ_{1} \\ m_{2} g l_{c 2} s i n θ_{2} \end{matrix}) \end{matrix}$
其中， ${\hat{θ}}_{1}$ 和 ${\hat{θ}}_{2}$ 表示采样得到的角位移， $\hat{{\dot{θ}}_{1}}$ 和 $\hat{{\dot{θ}}_{2}}$ 分别表示采样得到的角速度，k_p₁和k_p₂为位置增益，k_d₁和k_d₂为速度增益，θ₁_d和θ₂_d分别为两臂的目标位置.
1.1 不考虑采样反馈的动力学模型
若不考虑采样反馈的影响，即连续时间PD控制，则有：
${\hat{θ}}_{1} = θ_{1}, {\hat{θ}}_{2} = θ_{2}, \hat{{\dot{θ}}_{1}} = {\hat{θ}}_{1}, \hat{{\dot{θ}}_{2}} = {\hat{θ}}_{2}$
引入误差：
$e_{1} = θ_{1} - θ_{1 d}, e_{2} = θ_{2} - θ_{2 d}$
则可将式（5）变为
$M (\begin{matrix} {\ddot{e}}_{1} \\ {\ddot{e}}_{2} \end{matrix}) + C (\begin{matrix} {\dot{e}}_{1} \\ {\dot{e}}_{2} \end{matrix}) + K_{p} (\begin{matrix} e_{1} \\ e_{2} \end{matrix}) + K_{d} (\begin{matrix} {\dot{e}}_{1} \\ {\dot{e}}_{2} \end{matrix}) = 0$
其中
$K_{p} = [\begin{matrix} k_{p 1} & 0 \\ - k_{p 2} & k_{p 2} \end{matrix}], K_{d} = [\begin{matrix} k_{d 1} & 0 \\ - k_{d 2} & k_{d 2} \end{matrix}]$
定义Lyapunov函数V_L：
$V_{L} = \frac{1}{2} {(\begin{matrix} e_{1} \\ e_{2} \end{matrix})}^{T} K_{p} (\begin{matrix} e_{1} \\ e_{2} \end{matrix}) + \frac{1}{2} {(\begin{matrix} {\dot{e}}_{1} \\ {\dot{e}}_{2} \end{matrix})}^{T} M (\begin{matrix} {\dot{e}}_{1} \\ {\dot{e}}_{2} \end{matrix})$
显然，V_L全局正定^[18，19] .可计算其对时间的导数为：
${\dot{V}}_{L} = {(\begin{matrix} {\dot{e}}_{1} \\ {\dot{e}}_{2} \end{matrix})}^{T} K_{p} (\begin{matrix} e_{1} \\ e_{2} \end{matrix}) + {(\begin{matrix} {\dot{e}}_{1} \\ {\dot{e}}_{2} \end{matrix})}^{T} M (\begin{matrix} {\ddot{e}}_{1} \\ {\ddot{e}}_{2} \end{matrix}) + \frac{1}{2} {(\begin{matrix} {\dot{e}}_{1} \\ {\dot{e}}_{2} \end{matrix})}^{T} \dot{M} (\begin{matrix} {\dot{e}}_{1} \\ {\dot{e}}_{2} \end{matrix})$
由于 $\dot{M}$ 和C主对角线上的元素均为0，因此有：
${(\begin{matrix} {\dot{e}}_{1} \\ {\dot{e}}_{2} \end{matrix})}^{T} (\dot{M} - 2 C) (\begin{matrix} {\dot{e}}_{1} \\ {\dot{e}}_{2} \end{matrix}) = 0$
于是得到：
$\begin{matrix} {\dot{V}}_{L} = {(\begin{matrix} {\dot{e}}_{1} \\ {\dot{e}}_{2} \end{matrix})}^{T} K (\begin{matrix} e_{1} \\ e_{2} \end{matrix}) + {(\begin{matrix} {\dot{e}}_{1} \\ {\dot{e}}_{2} \end{matrix})}^{T} M (\begin{matrix} {\ddot{e}}_{1} \\ {\ddot{e}}_{2} \end{matrix}) + {(\begin{matrix} {\dot{e}}_{1} \\ {\dot{e}}_{2} \end{matrix})}^{T} C (\begin{matrix} {\dot{e}}_{1} \\ {\dot{e}}_{2} \end{matrix}) \\ = {(\begin{matrix} {\dot{e}}_{1} \\ {\dot{e}}_{2} \end{matrix})}^{T} (M (\begin{matrix} {\ddot{e}}_{1} \\ {\ddot{e}}_{2} \end{matrix}) + C (\begin{matrix} {\dot{e}}_{1} \\ {\dot{e}}_{2} \end{matrix}) + K_{p} (\begin{matrix} e_{1} \\ e_{2} \end{matrix})) \\ = {(\begin{matrix} {\dot{e}}_{1} \\ {\dot{e}}_{2} \end{matrix})}^{T} (\begin{matrix} (m_{1} l_{c 1} + m_{2} l_{1}) g (s i n θ_{1 d} - s i n θ_{1}) \\ m_{2} l_{c 2} g (s i n θ_{2 d} - s i n θ_{2}) \end{matrix}) - \\ {(\begin{matrix} {\dot{e}}_{1} \\ {\dot{e}}_{2} \end{matrix})}^{T} K_{d} (\begin{matrix} {\dot{e}}_{1} \\ {\dot{e}}_{2} \end{matrix}) \end{matrix}$
由于 ${\dot{e}}_{1}$ 与sinθ₁-sinθ₁_d不同号， ${\dot{e}}_{2}$ 与sinθ₂-sinθ₂_d不同号，且K_d正定，故 ${\dot{V}}_{L}$ ＜0恒成立.
可见，不考虑采样反馈，对图1所示的二连杆系统施加带重力补偿的PD控制，二连杆的运动是收敛的.
1.2 考虑采样反馈的动力学模型
考虑采样反馈的影响，设采样周期为τ，则t∈[nτ，（n+1）τ）（n∈N）时，式（5）中的采样信号变为：
${\hat{θ}}_{1} (t) = θ_{1} [(n - 1) τ], {\hat{θ}}_{2} (t) = θ_{2} [(n - 1) τ]$
$\hat{{\dot{θ}}_{1}} (t) = {\dot{θ}}_{1} [(n - 1) τ], \hat{{\dot{θ}}_{2}} (t) = {\dot{θ}}_{2} [(n - 1) τ]$
此时，二连杆系统的动力学方程为：
$M (\begin{matrix} {\ddot{θ}}_{1} \\ {\ddot{θ}}_{2} \end{matrix}) + C (\begin{matrix} {\dot{θ}}_{1} \\ {\dot{θ}}_{2} \end{matrix}) = (\begin{matrix} Q_{P D 1} \\ Q_{P D 2} \end{matrix})$
其中
$\begin{matrix} Q_{P D 1} = k_{p 1} \{θ_{1 d} - θ_{1} [(n - 1) τ]\} - \\ k_{d 1} {\dot{θ}}_{1} [(n - 1) τ] \\ Q_{P D 2} = k_{p 2} \{θ_{2 d} - θ_{2} [(n - 1) τ]\} - \\ k_{p 2} \{θ_{1 d} - θ_{1} [(n - 1) τ]\} - \\ k_{d 2} \{{\dot{θ}}_{2} [(n - 1) τ] - {\dot{θ}}_{1} [(n - 1) τ]\} \end{matrix}$
这是一个存在非连续时变项的强耦合非线性系统，无法通过一般的处理连续时间动力系统的方式进行求解或讨论稳定性.通过Matlab进行数值仿真的结果如图2所示.二连杆系统的物理参数如表1所示，系统的控制增益如表2所示.
表1 二连杆系统的物理参数
Table1 Physical parameters of double pendulum system
表2 系统的控制增益
Table2 Control gains of system
可以发现，对于给定控制增益的二连杆系统，根据采样周期的变化，系统呈现出收敛于目标位置、在目标位置附近振荡、运动发散三种不同的动态响应：当采样周期较小时，在采样反馈PD控制下，二连杆系统收敛于目标位置，如图2（b）所示，与连续采样PD控制的情况类似，如图2（a）所示；当采样周期处于某一区间内时，在采样反馈PD控制下，二连杆系统会在包含目标位置的某一区间内，作有界的来回往复运动，如图2（c）所示；当采样周期较大时，二连杆系统的振荡会变得非常剧烈，呈现出发散的趋势，如图2（d）所示.
图2 不同采样周期下二连杆系统仿真结果
Fig.2 Simulation results of double pendulum system under different sampling time conditions
2 动态响应的离散映射
经过仿真验证，我们发现采样周期变化时，目标位置的线性化系统与原系统的动态响应具有类似的变化趋势，如图3所示.物理参数和控制增益如表1和表2所示.
由原系统与线性化系统的稳态响应可知，原系统在采样周期为0.0201秒附近出现平衡点失稳现象，线性化系统在采样周期为0.02秒附近出现平衡点失稳现象.可见，目标位置的线性化系统与原系统的动态响应类似，可以使用在目标位置的线性化系统替代原系统，研究采样周期对系统动态响应的影响.
图3 原系统与线性化系统的稳态响应
Fig.3 Steady-state response of original system and linearized system
我们通过构造相邻采样周期状态量之间离散映射的方式，讨论二连杆混杂系统在目标位置附近的稳定性.在平衡点（即目标位置）附近，有：
$\begin{matrix} M = {[\begin{matrix} m_{11} & m_{12} \\ m_{12} & m_{22} \end{matrix}]|}_{θ_{1} = θ_{1 d}, θ_{2} = θ_{2 d}, {\dot{θ}}_{1} = 0, {\dot{θ}}_{2} = 0} \\ = [\begin{matrix} m_{b 11} & m_{b 12} \\ m_{b 12} & m_{b 22} \end{matrix}] \\ C = {[\begin{matrix} 0 & c_{12} \\ c_{21} & 0 \end{matrix}]|}_{θ_{1} = θ_{1 d}, θ_{2} = θ_{2 d}, {\dot{θ}}_{1} = 0, {\dot{θ}}_{2} = 0} = 0 \\ G = {[\begin{matrix} (m_{1} l_{c 1} + m_{2} l_{1}) g s i n θ_{1} \\ m_{2} g l_{c 2} s i n θ_{2} \end{matrix})|}_{θ_{1} = θ_{1 d}, θ_{2} = θ_{2 d}, {\dot{θ}}_{1} = 0, {\dot{θ}}_{2} = 0} \\ = Q_{c} + [\begin{matrix} k_{1} & 0 \\ 0 & k_{2} \end{matrix}] [(\begin{matrix} θ_{1} \\ θ_{2} \end{matrix}) - (\begin{matrix} θ_{1 d} \\ θ_{2 d} \end{matrix})] \\ = Q_{c} + K_{b} [(\begin{matrix} θ_{1} \\ θ_{2} \end{matrix}) - (\begin{matrix} θ_{1 d} \\ θ_{2 d} \end{matrix})] \end{matrix}$
其中
$\begin{matrix} m_{b 11} = J_{1} + m_{1} l_{c 1}^{2} + m_{2} l_{1}^{2} \\ m_{b 12} = m_{2} l_{1} l_{c 2} c o s (θ_{1 d} - θ_{2 d}) \\ m_{b 22} = J_{2} + m_{2} l_{c 2}^{2} \end{matrix}$
$\begin{matrix} k_{1} = (m_{1} l_{c 1} + m_{2} l_{1}) g c o s θ_{1 d} \\ k_{2} = m_{2} g l_{c 2} c o s θ_{2 d} \end{matrix}$
得到线性化后的方程如下：

M_{b} (\begin{matrix} {\ddot{θ}}_{1} \\ {\ddot{θ}}_{2} \end{matrix}) + K_{b} (\begin{matrix} θ_{1} - θ_{1 d} \\ θ_{2} - θ_{2 d} \end{matrix}) = (\begin{matrix} Q_{P D 1} \\ Q_{P D 2} \end{matrix})

(6)

令
$\begin{matrix} e_{1} = θ_{1} - θ_{1 d}, e_{2} = θ_{2} - θ_{2 d} \\ {\hat{e}}_{1} = {\hat{θ}}_{1} - θ_{1 d}, {\hat{e}}_{2} = {\hat{θ}}_{2} - θ_{2 d} \end{matrix}$
则可改写式（6）为：

M_{b} (\begin{matrix} {\ddot{e}}_{1} \\ {\ddot{e}}_{2} \end{matrix}) + K_{b} (\begin{matrix} e_{1} \\ e_{2} \end{matrix}) = (\begin{matrix} Q_{P D 1} \\ Q_{P D 2} \end{matrix})

(7)

设M^-1_bK_b的特征向量为（φ₁，1）^T和（φ₂，1）^T，可计算得到：
$\begin{matrix} φ_{1} = \frac{m_{b 11} k_{2} - m_{b 22} k_{1}}{2 m_{b 12} k_{1}} + \\ \frac{\sqrt{{(m_{b 11} k_{2} - m_{b 22} k_{1})}^{2} + 4 m_{b 12}^{2} k_{1} k_{2}}}{2 m_{b 12} k_{1}} \\ φ_{2} = \frac{m_{b 11} k_{2} - m_{b 22} k_{1}}{2 m_{b 12} k_{1}} - \\ \frac{\sqrt{{(m_{b 11} k_{2} - m_{b 22} k_{1})}^{2} + 4 m_{b 12}^{2} k_{1} k_{2}}}{2 m_{b 12} k_{1}} \end{matrix}$
设

(\begin{matrix} e_{1} \\ e_{2} \end{matrix}) = [\begin{matrix} φ_{1} & φ_{2} \\ 1 & 1 \end{matrix}] (\begin{matrix} z_{1} \\ z_{2} \end{matrix})

(8)

则可对动力学方程解耦：

\begin{matrix} [\begin{matrix} φ_{1} & 1 \\ φ_{2} & 1 \end{matrix}] M_{b} [\begin{matrix} φ_{1} & φ_{2} \\ 1 & 1 \end{matrix}] (\begin{matrix} {\ddot{z}}_{1} \\ {\ddot{z}}_{2} \end{matrix}) + \\ [\begin{matrix} φ_{1} & 1 \\ φ_{2} & 1 \end{matrix}] K_{b} [\begin{matrix} φ_{1} & φ_{2} \\ 1 & 1 \end{matrix}] (\begin{matrix} z_{1} \\ z_{2} \end{matrix}) = \\ [\begin{matrix} φ_{1} & 1 \\ φ_{2} & 1 \end{matrix}] (\begin{matrix} Q_{P D 1} \\ Q_{P D 2} \end{matrix}) \\ {\bar{M}}_{b} (\begin{matrix} {\ddot{z}}_{1} \\ {\ddot{z}}_{2} \end{matrix}) + {\bar{K}}_{b} (\begin{matrix} z_{1} \\ z_{2} \end{matrix}) = (\begin{matrix} f_{1} \\ f_{2} \end{matrix}) \end{matrix}

(9)

其中
$\begin{matrix} {\bar{M}}_{b} = [\begin{matrix} {\bar{m}}_{1} & 0 \\ 0 & {\bar{m}}_{2} \end{matrix}], {\bar{K}}_{b} = [\begin{matrix} {\bar{k}}_{1} & 0 \\ 0 & {\bar{k}}_{2} \end{matrix}] \\ f_{1} = φ_{1} Q_{P D 1} + Q_{P D 2}, f_{2} = φ_{2} Q_{P D 1} + Q_{P D 2} \\ {\bar{m}}_{1} = m_{b 22} + φ_{1} (2 m_{b 12} + m_{b 11} φ_{1}) \\ {\bar{m}}_{2} = m_{b 22} + φ_{2} (2 m_{b 12} + m_{b 11} φ_{2}) \\ {\bar{k}}_{1} = k_{1} φ_{1}^{2} + k_{2}, {\bar{k}}_{2} = k_{1} φ_{2}^{2} + k_{2} \end{matrix}$
至此，已将两自由度解耦.设
${\bar{ω}}_{1} = \sqrt{\frac{{\bar{k}}_{1}}{{\bar{m}}_{1}}}, {\bar{ω}}_{2} = \sqrt{\frac{{\bar{k}}_{2}}{{\bar{m}}_{2}}}, {\bar{f}}_{1} = \frac{f_{1}}{{\bar{m}}_{1}}, {\bar{f}}_{2} = \frac{f_{2}}{{\bar{m}}_{1}}$
可得到两个独立的方程：

{\ddot{z}}_{1} + {\bar{ω}}_{1}^{2} z_{1} = {\bar{f}}_{1}

(10a)

{\ddot{z}}_{2} + {\bar{ω}}_{2}^{2} z_{2} = {\bar{f}}_{2}

(10b)

在两个相邻采样时刻之间，即t∈[nτ，（n+1）τ）（n∈N）时，由式（10a）和式（10b）可得
$\begin{matrix} z_{1} (t) = \frac{{\bar{f}}_{1} (n τ)}{{\bar{ω}}_{1}^{2}} + \frac{{\dot{z}}_{1} (n τ)}{{\bar{ω}}_{1}} s i n [{\bar{ω}}_{1} (t - n τ)] + \\ [z_{1} (n τ) - \frac{{\bar{f}}_{1} (n τ)}{{\bar{ω}}_{1}^{2}}] c o s [{\bar{ω}}_{1} (t - n τ)] \\ z_{2} (t) = \frac{{\bar{f}}_{2} (n τ)}{{\bar{ω}}_{2}^{2}} + \frac{{\dot{z}}_{2} (n τ)}{{\bar{ω}}_{2}} s i n [{\bar{ω}}_{2} (t - n τ)] + \\ [z_{2} (n τ) - \frac{{\bar{f}}_{2} (n τ)}{{\bar{ω}}_{2}^{2}}] c o s [{\bar{ω}}_{2} (t - n τ)] \end{matrix}$
$\begin{matrix} {\dot{z}}_{1} (t) = \frac{d z_{1} (t)}{d t} = {\dot{z}}_{1} (n τ) c o s [{\bar{ω}}_{1} (t - n τ)] + \\ [z_{1} (n τ) {\bar{ω}}_{1} - \frac{{\bar{f}}_{1} (n τ)}{{\bar{ω}}_{1}}] s i n [{\bar{ω}}_{1} (t - n τ)] \\ {\dot{z}}_{2} (t) = \frac{d z_{2} (t)}{d t} = {\dot{z}}_{2} (n τ) c o s [{\bar{ω}}_{2} (t - n τ)] + \\ [z_{2} (n τ) {\bar{ω}}_{2} - \frac{{\bar{f}}_{2} (n τ)}{{\bar{ω}}_{2}}] s i n [{\bar{ω}}_{2} (t - n τ)] \end{matrix}$
因此，可以得到t=（n+1）τ（n∈N）时刻的状态量：
$\begin{matrix} z_{1} [(n + 1) τ] = \frac{{\bar{f}}_{1} (n τ)}{{\bar{ω}}_{1}^{2}} + \frac{{\dot{z}}_{1} (n τ)}{{\bar{ω}}_{1}} s i n ({\bar{ω}}_{1} τ) + \\ [z_{1} (n τ) - \frac{{\bar{f}}_{1} (n τ)}{{\bar{ω}}_{1}^{2}}] c o s ({\bar{ω}}_{1} τ) \\ z_{2} [(n + 1) τ] = \frac{{\bar{f}}_{2} (n τ)}{{\bar{ω}}_{2}^{2}} + \frac{{\dot{z}}_{2} (n τ)}{{\bar{ω}}_{2}} s i n ({\bar{ω}}_{2} τ) + \\ [z_{2} (n τ) - \frac{{\bar{f}}_{2} (n τ)}{{\bar{ω}}_{2}^{2}}] c o s ({\bar{ω}}_{2} τ) \\ {\dot{z}}_{1} [(n + 1) τ] = {\dot{z}}_{1} (n τ) c o s ({\bar{ω}}_{1} τ) + \\ [z_{1} (n τ) {\bar{ω}}_{1} - \frac{{\bar{f}}_{1} (n τ)}{{\bar{ω}}_{1}}] s i n ({\bar{ω}}_{1} τ) \\ {\dot{z}}_{2} [(n + 1) τ] = {\dot{z}}_{2} (n τ) c o s ({\bar{ω}}_{2} τ) + \\ [z_{2} (n τ) {\bar{ω}}_{2} - \frac{{\bar{f}}_{2} (n τ)}{{\bar{ω}}_{2}}] s i n ({\bar{ω}}_{2} τ) \end{matrix}$
根据PD控制律，可得：
$\begin{matrix} Q_{P D 1} [(n + 1) τ] = - k_{p 1} e_{1} (n τ) - k_{d 1} {\dot{e}}_{1} (n τ) \\ Q_{P D 2} [(n + 1) τ] = - k_{p 2} e_{2} (n τ) + \\ k_{p 2} e_{1} (n τ) - k_{d 2} {\dot{e}}_{2} (n τ) + k_{d 2} {\dot{e}}_{1} (n τ) \end{matrix}$
又由式（8），可得到物理坐标下，系统的状态量的离散映射.经过化简，可得物理指标下，系统的状态量和控制力满足如下离散映射关系：

{(\begin{matrix} e_{1} \\ e_{2} \\ {\dot{e}}_{1} \\ {\dot{e}}_{2} \\ Q_{P D 1} \\ Q_{P D 2} \end{matrix})|}_{(n + 1) τ} = {W (\begin{matrix} e_{1} \\ e_{2} \\ {\dot{e}}_{1} \\ {\dot{e}}_{2} \\ Q_{P D 1} \\ Q_{P D 2} \end{matrix})|}_{n τ}

(11)

W的元素见附录.
3 稳定性分析
从式（11）可以发现，动态响应与控制力矩的离散映射中的线性映射矩阵W，决定了二连杆混杂系统在平衡点的稳定性.W是一个常值矩阵，除了采样周期，它的元素还和目标位置附近的近似线性系统以及控制率有关.因此，采样周期对于二连杆混杂系统平衡点稳定性的影响，体现在采样周期的取值影响线性映射矩阵W的特征值是否在复平面上的单位圆内.图4展示了通过W的特征值作为稳定性判据的准确性.二连杆系统的物理参数如表1所示，系统的控制增益如表2所示.
图4 特征值判据与仿真结果
Fig.4 Eigenvalue criterion and simulation results
设W阵的特征值为λ_i（i=1，2，3，4，5，6），I为单位阵，可得W阵的特征方程为：
$| W - λ I | = 0$
为研究六次方程根的分布，引入Mobius变换：
$λ = \frac{s + 1}{s - 1} \Leftrightarrow s = \frac{λ + 1}{λ - 1}$
对于复平面上单位圆内的点，有
$λ = ρ e^{θ i}, ρ < 1$
可进一步推导得到：
$R e (\frac{ρ e^{θ i} + 1}{ρ e^{θ i} - 1}) = \frac{- 1 + ρ^{2}}{(ρ - c o s θ)^{2} + {s i n}^{2} θ} < 0$
图5 二连杆系统分岔图
Fig.5 Bifurcation diagram of the double pendulum system
图6 倍周期分岔情况稳态响应相图
Fig.6 Steady-state response phase portraits in period-doubling bifurcation case
可见，Mobius变换将单位圆内的点（即ρ＜1的点）映射到左半平面上.因此，经过Mobius变换，离散映射的稳定性问题转化为了多项式方程根的实部符号判定问题.进而通过Routh判据，解得本例中等效线性系统的临界采样周期为0.020107秒.
此外，通过数值仿真，我们得到了二连杆采样反馈反馈控制系统的分岔曲线.二连杆系统的物理参数如表1所示，系统的控制增益如表2所示.
如图5所示，可以发现，随着采样周期的增加，二连杆系统出现失稳现象，临界采样周期约为0.0201秒，与线性化系统的临界采样周期极为接近.这进一步验证了将原系统等效为线性系统研究是可靠的.此外，在部分区间内，系统经由倍周期分岔产生多周期响应，对应的相图如图6所示.
4 结论
本文研究了含有采样反馈的二连杆系统定位控制的稳定性问题.首先，建立了考虑采样反馈的动力学模型，通过数值仿真得到采样周期变化时不同的动力学响应，在此基础上，构造了响应的离散映射，提出了二连杆混杂系统稳定性的特征值判据，最后通过数值仿真验证了判据的准确性并发现了采样周期引起的分岔现象.通过本文研究，可得出以下结论：
（1）传感器的采样周期会影响采样反馈反馈控制系统的稳定性，若采样频率过低，平衡点会发生失稳现象；
（2）可通过构造状态量与控制力的离散映射，判断映射稳定性的方法判断采样反馈是否造成平衡点失稳.
附录
W阵的元素如下：
$W = [\begin{matrix} w_{11} & w_{12} & w_{13} & w_{14} & w_{15} & w_{16} \\ w_{21} & w_{22} & w_{23} & w_{24} & w_{25} & w_{26} \\ w_{31} & w_{32} & w_{33} & w_{34} & w_{35} & w_{36} \\ w_{41} & w_{42} & w_{43} & w_{44} & w_{45} & w_{46} \\ - k_{p 1} & 0 & - k_{d 1} & 0 & 0 & 0 \\ k_{p 2} & - k_{p 2} & k_{d 2} & - k_{d 2} & 0 & 0 \end{matrix}]$
其中
$\begin{matrix} w_{11} = \frac{φ_{1} c o s ({\bar{ω}}_{1} τ) - φ_{2} c o s ({\bar{ω}}_{2} τ)}{φ_{1} - φ_{2}} \\ w_{12} = \frac{φ_{1} φ_{2} [- c o s ({\bar{ω}}_{1} τ) + c o s ({\bar{ω}}_{2} τ)]}{φ_{1} - φ_{2}} \\ w_{13} = \frac{φ_{1} {\bar{ω}}_{2} s i n ({\bar{ω}}_{1} τ) - φ_{2} {\bar{ω}}_{1} s i n ({\bar{ω}}_{2} τ)}{(φ_{1} - φ_{2}) {\bar{ω}}_{1} {\bar{ω}}_{2}} \\ w_{14} = \frac{φ_{1} φ_{2} [- {\bar{ω}}_{2} s i n ({\bar{ω}}_{1} τ) + {\bar{ω}}_{1} s i n ({\bar{ω}}_{2} τ)]}{(φ_{1} - φ_{2}) {\bar{ω}}_{1} {\bar{ω}}_{2}} \\ w_{15} = \frac{φ_{1}^{2} [1 - c o s ({\bar{ω}}_{1} τ)]}{{\bar{m}}_{1} {\bar{ω}}_{1}^{2}} + \frac{φ_{2}^{2} [1 - c o s ({\bar{ω}}_{2} τ)]}{{\bar{m}}_{2} {\bar{ω}}_{2}^{2}} \\ w_{16} = \frac{φ_{1} [1 - c o s ({\bar{ω}}_{1} τ)]}{{\bar{m}}_{1} {\bar{ω}}_{1}^{2}} + \frac{φ_{2} [1 - c o s ({\bar{ω}}_{2} τ)]}{{\bar{m}}_{2} {\bar{ω}}_{2}^{2}} \end{matrix}$
$\begin{matrix} w_{21} = \frac{c o s ({\bar{ω}}_{1} τ) - c o s ({\bar{ω}}_{2} τ)}{φ_{1} - φ_{2}} \\ w_{22} = \frac{- φ_{2} c o s ({\bar{ω}}_{1} τ) + φ_{1} c o s ({\bar{ω}}_{2} τ)}{φ_{1} - φ_{2}} \\ w_{23} = \frac{{\bar{ω}}_{2} s i n ({\bar{ω}}_{1} τ) - {\bar{ω}}_{1} s i n ({\bar{ω}}_{2} τ)}{(φ_{1} - φ_{2}) {\bar{ω}}_{1} {\bar{ω}}_{2}} \\ w_{24} = \frac{- φ_{2} {\bar{ω}}_{2} s i n ({\bar{ω}}_{1} τ) + φ_{1} {\bar{ω}}_{1} s i n ({\bar{ω}}_{2} τ)}{(φ_{1} - φ_{2}) {\bar{ω}}_{1} {\bar{ω}}_{2}} \end{matrix}$
$\begin{matrix} w_{25} = \frac{φ_{1} [1 - c o s ({\bar{ω}}_{1} τ)]}{{\bar{m}}_{1} {\bar{ω}}_{1}^{2}} + \frac{φ_{2} [1 - c o s ({\bar{ω}}_{2} τ)]}{{\bar{m}}_{2} {\bar{ω}}_{2}^{2}} \\ w_{26} = \frac{1 - c o s ({\bar{ω}}_{1} τ)}{{\bar{m}}_{1} {\bar{ω}}_{1}^{2}} + \frac{1 - c o s ({\bar{ω}}_{2} τ)}{{\bar{m}}_{2} {\bar{ω}}_{2}^{2}} \\ w_{31} = \frac{- φ_{1} {\bar{ω}}_{1} s i n ({\bar{ω}}_{1} τ) + φ_{2} {\bar{ω}}_{2} s i n ({\bar{ω}}_{2} τ)}{φ_{1} - φ_{2}} \\ w_{32} = \frac{φ_{1} φ_{2} [{\bar{ω}}_{1} s i n ({\bar{ω}}_{1} τ) - {\bar{ω}}_{2} s i n ({\bar{ω}}_{2} τ)]}{φ_{1} - φ_{2}} \\ w_{33} = \frac{φ_{1} c o s ({\bar{ω}}_{1} τ) - φ_{2} c o s ({\bar{ω}}_{2} τ)}{φ_{1} - φ_{2}} \end{matrix}$
$\begin{matrix} w_{34} = \frac{φ_{1} φ_{2} [- c o s ({\bar{ω}}_{1} τ) + c o s ({\bar{ω}}_{2} τ)]}{φ_{1} - φ_{2}} \\ w_{35} = \frac{φ_{1}^{2} [1 - c o s ({\bar{ω}}_{1} τ)]}{{\bar{m}}_{1} {\bar{ω}}_{1}^{2}} + \frac{φ_{2}^{2} [1 - c o s ({\bar{ω}}_{2} τ)]}{{\bar{m}}_{2} {\bar{ω}}_{2}^{2}} \\ w_{36} = \frac{φ_{1} s i n ({\bar{ω}}_{1} τ)}{{\bar{m}}_{1} {\bar{ω}}_{1}^{2}} + \frac{φ_{2} s i n ({\bar{ω}}_{2} τ)}{{\bar{m}}_{2} {\bar{ω}}_{2}^{2}} \\ w_{41} = \frac{- {\bar{ω}}_{1} s i n ({\bar{ω}}_{1} τ) + {\bar{ω}}_{2} s i n ({\bar{ω}}_{2} τ)}{φ_{1} - φ_{2}} \\ w_{42} = \frac{φ_{2} {\bar{ω}}_{1} s i n ({\bar{ω}}_{1} τ) - φ_{1} {\bar{ω}}_{2} s i n ({\bar{ω}}_{2} τ)}{φ_{1} - φ_{2}} \\ w_{43} = \frac{c o s ({\bar{ω}}_{1} τ) - c o s ({\bar{ω}}_{2} τ)}{φ_{1} - φ_{2}} \end{matrix}$
$\begin{matrix} w_{44} = \frac{- φ_{2} c o s ({\bar{ω}}_{1} τ) + φ_{1} c o s ({\bar{ω}}_{2} τ)}{φ_{1} - φ_{2}} \\ w_{45} = \frac{φ_{1} s i n ({\bar{ω}}_{1} τ)}{{\bar{m}}_{1} {\bar{ω}}_{1}^{2}} + \frac{φ_{2} s i n ({\bar{ω}}_{2} τ)}{{\bar{m}}_{2} {\bar{ω}}_{2}^{2}} \\ w_{46} = \frac{\sin ({\bar{ω}}_{1} τ)}{{\bar{m}}_{1} {\bar{ω}}_{1}^{2}} + \frac{\sin ({\bar{ω}}_{2} τ)}{{\bar{m}}_{2} {\bar{ω}}_{2}^{2}} \end{matrix}$
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- [2] 王百争.新舟60飞机C级飞行训练模拟器建模与仿真 [C/OL]//大型飞机关键技术高层论坛暨中国航空学会2007年学术年会,中国,深圳.2007：79-94 [2022-11-15].https://d.wanfangdata.com.cn/conference/6390397.
- [3] 尤鑫烨,陈力.外部扰动下空间机器人基于扰动观测器的鲁棒控制 [J].动力学与控制学报,2021,19(2):37-42.YOU X Y,CHEN L.Robust control of space robot based on disturbance observer under external disturbance [J].Journal of Dynamics and Control,2021,19(2):37-42.(in Chinese)
- [4] 范振宇.KUKA机器人在合金自动生产线上的运用 [J].自动化技术与应用,2020,39(9):25-27.FAN Z Y.Application of KUKA robot in automatic alloy production line [J].Techniques of Automation and Applications,2020,39(9):25-27.(in Chinese)
- [5] WILSON W J,WILLIAMS HULLS C C,BELL G S.Relative end-effector control using Cartesian position based visual servoing [J].IEEE Transactions on Robotics and Automation,1996,12(5):684-696.
- [6] SHOME R,TANG W N,SONG C,et al.Towards robust product packing with a minimalistic end-effector [C]//2019 International Conference on Robotics and Automation(ICRA)，Montreal,Canada.Piscataway，America：IEEE,2019:9007-9013.
- [7] HABIB G,REGA G,STEPAN G.Delayed digital position control of a single-DOF system and the nonlinear behavior of the act-and-wait controller [J].Journal of Vibration and Control,2016,22(2):481-495.
- [8] HABIB G,REGA G,STEPAN G.Nonlinear bifurcation analysis of a single-DOF model of a robotic arm subject to digital position control [J].Journal of Computational and Nonlinear Dynamics,2013,8(1):011009.
- [9] HABIB G,REGA G,STEPAN G.Bifurcation analysis of a two-DOF mechanical system subject to digital position control.Part I:theoretical investigation [J].Nonlinear Dynamics,2014,76(3):1781-1796.
- [10] HABIB G,REGA G,STEPAN G.Bifurcation analysis of a two-DOF mechanical system subject to digital position control.Part II.Effects of asymmetry and transition to chaos [J].Nonlinear Dynamics,2013,74(4):1223-1241.
- [11] HABIB G,REGA G,STEPAN G.Stability analysis of a two-degree-of-freedom mechanical system subject to proportional-derivative digital position control [J].Journal of Vibration and Control,2015,21(8):1539-1555.
- [12] 王在华,胡海岩.具有采样反馈的力控制系统稳定性 [J].力学学报,2016,48(6):1372-1381.WANG Z H,HU H Y.Stability of a force control system with sampled-data feedback [J].Chinese Journal of Theoretical and Applied Mechanics,2016,48(6):1372-1381.(in Chinese)
- [13] 王强,梁松,王在华.基于采样PD反馈的倒立摆控制系统的稳定性 [J].动力学与控制学报,2018,16(4):377-384.WANG Q,LIANG S,WANG Z H.Stability of an inverted pendulum with a sampled-data pd feedback control [J].Journal of Dynamics and Control,2018,16(4):377-384.(in Chinese)
- [14] PEDRAMMEHR S,DANAEI B,ABDI H,et al.Dynamic analysis of Hexarot:axis-symmetric parallel manipulator [J].Robotica,2018,36(2):225-240.
- [15] MURALIDHARAN V,MAMIDI T K,GUPTAS-ARMA S,et al.A comparative study of the configuration-space and actuator-space formulations of the Lagrangian dynamics of parallel manipulators and the effects of kinematic singularities on these [J].Mechanism and Machine Theory,2018,130:403-434.
- [16] 郭丁旭,姜乃晶,张舒,等.含有LuGre摩擦并联机械臂的自适应控制 [J].动力学与控制学报,2019,17(4):362-368.GUO D X,JIANG N J,ZHANG S,et al.Adaptive control of parallel manipulators with lugre friction compensation [J].Journal of Dynamics and Control,2019,17(4):362-368.(in Chinese)
- [17] 霍伟.机器人动力学与控制 [M].北京:高等教育出版社,2005.HUO W.Robot dynamics and control [M].Beijing:Higher Education Press,2005.(in Chinese)
- [18] 刘金琨.机器人控制系统的设计与MATLAB仿真:基本设计方法 [M].北京:清华大学出版社,2016.LIU J K.Robot control system design and MATLAB simulation [M].Beijing:Tsinghua University Press,2016.(in Chinese)

基本信息

中图分类号: O317
文献标识码: A
DOI: 10.6052/1672-6553-2022-062
文章编号: 1672-6553-2023-21(11)-010-009

基金信息

国家自然科学基金资助项目(11872277,12072237,11932015)；
National Natural Science Foundation of China (11872277,12072237,11932015)；

引用信息

稿件历史

收稿日期: 2022-11-15

参考文献

[1] 刘宏,蒋再男,刘业超.空间机械臂技术发展综述 [J].载人航天,2015,21(5):435-443.LIU H,JIANG Z N,LIU Y C.Review of space manipulator technology [J].Manned Spaceflight,2015,21(5):435-443.(in Chinese)
[2] 王百争.新舟60飞机C级飞行训练模拟器建模与仿真 [C/OL]//大型飞机关键技术高层论坛暨中国航空学会2007年学术年会,中国,深圳.2007：79-94 [2022-11-15].https://d.wanfangdata.com.cn/conference/6390397.
[3] 尤鑫烨,陈力.外部扰动下空间机器人基于扰动观测器的鲁棒控制 [J].动力学与控制学报,2021,19(2):37-42.YOU X Y,CHEN L.Robust control of space robot based on disturbance observer under external disturbance [J].Journal of Dynamics and Control,2021,19(2):37-42.(in Chinese)
[4] 范振宇.KUKA机器人在合金自动生产线上的运用 [J].自动化技术与应用,2020,39(9):25-27.FAN Z Y.Application of KUKA robot in automatic alloy production line [J].Techniques of Automation and Applications,2020,39(9):25-27.(in Chinese)
[5] WILSON W J,WILLIAMS HULLS C C,BELL G S.Relative end-effector control using Cartesian position based visual servoing [J].IEEE Transactions on Robotics and Automation,1996,12(5):684-696.
[6] SHOME R,TANG W N,SONG C,et al.Towards robust product packing with a minimalistic end-effector [C]//2019 International Conference on Robotics and Automation(ICRA)，Montreal,Canada.Piscataway，America：IEEE,2019:9007-9013.
[7] HABIB G,REGA G,STEPAN G.Delayed digital position control of a single-DOF system and the nonlinear behavior of the act-and-wait controller [J].Journal of Vibration and Control,2016,22(2):481-495.
[8] HABIB G,REGA G,STEPAN G.Nonlinear bifurcation analysis of a single-DOF model of a robotic arm subject to digital position control [J].Journal of Computational and Nonlinear Dynamics,2013,8(1):011009.
[9] HABIB G,REGA G,STEPAN G.Bifurcation analysis of a two-DOF mechanical system subject to digital position control.Part I:theoretical investigation [J].Nonlinear Dynamics,2014,76(3):1781-1796.
[10] HABIB G,REGA G,STEPAN G.Bifurcation analysis of a two-DOF mechanical system subject to digital position control.Part II.Effects of asymmetry and transition to chaos [J].Nonlinear Dynamics,2013,74(4):1223-1241.
[11] HABIB G,REGA G,STEPAN G.Stability analysis of a two-degree-of-freedom mechanical system subject to proportional-derivative digital position control [J].Journal of Vibration and Control,2015,21(8):1539-1555.
[12] 王在华,胡海岩.具有采样反馈的力控制系统稳定性 [J].力学学报,2016,48(6):1372-1381.WANG Z H,HU H Y.Stability of a force control system with sampled-data feedback [J].Chinese Journal of Theoretical and Applied Mechanics,2016,48(6):1372-1381.(in Chinese)
[13] 王强,梁松,王在华.基于采样PD反馈的倒立摆控制系统的稳定性 [J].动力学与控制学报,2018,16(4):377-384.WANG Q,LIANG S,WANG Z H.Stability of an inverted pendulum with a sampled-data pd feedback control [J].Journal of Dynamics and Control,2018,16(4):377-384.(in Chinese)
[14] PEDRAMMEHR S,DANAEI B,ABDI H,et al.Dynamic analysis of Hexarot:axis-symmetric parallel manipulator [J].Robotica,2018,36(2):225-240.
[15] MURALIDHARAN V,MAMIDI T K,GUPTAS-ARMA S,et al.A comparative study of the configuration-space and actuator-space formulations of the Lagrangian dynamics of parallel manipulators and the effects of kinematic singularities on these [J].Mechanism and Machine Theory,2018,130:403-434.
[16] 郭丁旭,姜乃晶,张舒,等.含有LuGre摩擦并联机械臂的自适应控制 [J].动力学与控制学报,2019,17(4):362-368.GUO D X,JIANG N J,ZHANG S,et al.Adaptive control of parallel manipulators with lugre friction compensation [J].Journal of Dynamics and Control,2019,17(4):362-368.(in Chinese)
[17] 霍伟.机器人动力学与控制 [M].北京:高等教育出版社,2005.HUO W.Robot dynamics and control [M].Beijing:Higher Education Press,2005.(in Chinese)
[18] 刘金琨.机器人控制系统的设计与MATLAB仿真:基本设计方法 [M].北京:清华大学出版社,2016.LIU J K.Robot control system design and MATLAB simulation [M].Beijing:Tsinghua University Press,2016.(in Chinese)

含有采样反馈的二连杆系统稳定性^*

Stability of Double Pendulum System with Sampling Feedback^*

摘要

Abstract

关键词

Keywords

1 二连杆混杂系统的动力学模型

(1)

(2)

(3)

(4)

(5)

1.1 不考虑采样反馈的动力学模型

1.2 考虑采样反馈的动力学模型

2 动态响应的离散映射

(6)

(7)

(8)

(9)

(10a)

(10b)

(11)

3 稳定性分析

4 结论

参考文献

基本信息

基金信息

引用信息

稿件历史

参考文献

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手机版网站二维码

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使用微信“扫一扫”功能。

含有采样反馈的二连杆系统稳定性*

Stability of Double Pendulum System with Sampling Feedback *

摘要

Abstract

关键词

Keywords

1 二连杆混杂系统的动力学模型

(1)

(2)

(3)

(4)

(5)

1.1 不考虑采样反馈的动力学模型

1.2 考虑采样反馈的动力学模型

2 动态响应的离散映射

(6)

(7)

(8)

(9)

(10a)

(10b)

(11)

3 稳定性分析

4 结论

参考文献

基本信息

基金信息

引用信息

稿件历史

参考文献

微信公众号二维码

手机版网站二维码

含有采样反馈的二连杆系统稳定性^*

Stability of Double Pendulum System with Sampling Feedback^*