基于柔性机构捕捉卫星的空间机器人动态缓冲从顺控制$^{\bf 1)}$

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艾海平, 陈力^,²⁾福州大学机械工程及自动化学院, 福州 350116

BUFFER AND COMPLIANT DYNAMIC SURFACE CONTROL OF SPACE ROBOT CAPTURING SATELLITE BASED ON COMPLIANT MECHANISM$^{\bf 1)}$

Ai Haiping, Chen Li^,²⁾School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350116, China

通讯作者: 2)陈力, 教授, 主要研究方向: 空间机器人系统动力学与控制. E-mail:chnle@fzu.edu.cn

收稿日期:2019-03-5接受日期:2020-04-1网络出版日期:2020-07-18

基金资助:

1)国家自然科学基金.11372073
国家自然科学基金.51741502
福建省工业机器人基础部件技术重大研发平台.2014H21010011

Received:2019-03-5Accepted:2020-04-1Online:2020-07-18
作者简介 About authors

摘要
研究了空间机器人在轨捕获非合作卫星过程避免关节受碰撞冲击破坏的缓冲从顺控制问题, 为此在机械臂与关节电机之间配置了一种柔性机构, 其作用在于: (1)在接触、碰撞阶段可通过其内置弹簧的变形来吸收被捕获卫星对空间机器人关节产生的冲击力矩; (2)在镇定运动阶段, 结合与之配合的缓冲从顺控制策略来适时开、关关节电机, 以保证关节受到的冲击力矩受限在安全范围. 首先, 利用多刚体系统理论获得配置柔性机构空间机器人及目标卫星分体系统动力学方程; 之后, 结合整个系统动量守恒关系, 捕获操作后系统运动几何关系及力的传递规律, 建立了两者形成联合体系统的动力学方程, 并计算了碰撞过程的冲击效应与冲击力. 为了实现失稳联合体系统的镇定控制, 提出了一种基于动态面的缓冲从顺控制方案. 上述控制方案可在实现吸收捕获操作产生的冲击力矩的同时, 还能在冲击力矩过大时适时开启、关闭关节电机, 以避免关节电机发生破坏; 此外, 动态面的引入避免了反演法存在的计算膨胀问题, 有效减少了计算量. 基于Lyapunov函数法证明了系统的稳定性, 并通过系统数值仿真结果验证了上述缓冲从顺控制策略的正确性.
关键词： 柔性机构;空间机器人;捕获操作;缓冲从顺控制;动态面控制

Abstract
The buffer and compliant control for space robot to avoid joint damage during on-orbit capture non-cooperative satellite are studied. For the reason, a compliant mechanism is mounted between the joint motor and space manipulator, its functions are: first, the deformation of internal spring in compliant mechanism can absorb the impact torque of the captured satellite acting on the joint of the space robot; second, the joint impact torque can be limited to a safe range by reasonably designing the buffer and compliant control scheme. First of all, the dynamic models of the space robot system and the target satellite system before capture are derived by multi-body theory. After that, based on the law of conservation of momentum, the constraints of kinematics and the law of force transfer, the integrated dynamic model of the combined system is derived. At the same time, the impact effect and impact force are calculated. For the stabilization control of post-capture unstable combined system, a buffer and compliant control scheme based on dynamic surface is proposed. The proposed control scheme can not only effectively absorb the impact torque generated by the on-orbit capture process, but also timely open or close the joint motor when the impact torque is too large, which can avoid overload and damage of the joint motor. In addition, the dynamic surface control scheme is utilized to avoid calculation expansion caused by backstepping method and to reduce the calculation effectively. The stability of the system is proved by Lyapunov theorem, and numerical simulation verifies the effectiveness of the proposed buffer and compliant control method.
Keywords：compliant mechanism;space robot;capture satellite operation;buffer and compliant control;dynamic surface control

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本文引用格式
艾海平, 陈力. 基于柔性机构捕捉卫星的空间机器人动态缓冲从顺控制$^{\bf 1)}$. 力学学报[J], 2020, 52(4): 975-984 DOI:10.6052/0459-1879-20-068
Ai Haiping, Chen Li. BUFFER AND COMPLIANT DYNAMIC SURFACE CONTROL OF SPACE ROBOT CAPTURING SATELLITE BASED ON COMPLIANT MECHANISM$^{\bf 1)}$. Chinese Journal of Theoretical and Applied Mechanics[J], 2020, 52(4): 975-984 DOI:10.6052/0459-1879-20-068

引言

近年来随着空间技术的发展及人类对太空探索的进一步深入, 空间机器人被期望在太空服务中扮演更重要的角色并执行更复杂的任务: 如失效航天器的维修、在轨燃料加注、在轨装配和后勤支援等^[1-5], 以实现延长在轨航天器服务寿命、提升在轨服务性能的目的, 因此对其的研究引起了国内外****的诸多关注^[6-13]. 同时, 因发射失败及在轨故障导致失效的航天器日益增多, 进而影响了空间在轨服务任务的执行. 考虑太空轨道资源的宝贵及失效卫星回收的经济价值, 空间机器人对失效卫星的捕获操作研究具有重要意义^[14-18].

空间机器人执行在轨捕获操作时, 不可避免的要与目标卫星发生接触、碰撞, 在此过程, 其机械臂将会受到很大的碰撞冲击力矩^[19-20], 若冲击力矩过大, 很可能对最薄弱的关节处造成破坏. 考虑非合作卫星一般具备高速、旋转等特性, 空间机器人对其进行捕获操作时, 其遭受的冲击力矩将使关节处遭受损坏, 并导致空间任务的失败. 因此, 在捕获操作过程采取一定措施以避免空间机器人的关节电机受到冲击而破坏是极其必要的. 然而, 目前关于空间机器人避免关节遭受冲击破坏的研究却鲜见报道, 故对其的研究有着重要的探索价值和意义.

针对空间机器人在轨捕获操作的研究, 国内外****已经取得了一定的成果. Uyam等^[21]对空间机器人与自由漂浮卫星的接触效应进行了实验的评估. Rekleitis等^[22]研究了近距离捕获被动航天器的控制问题, 设计了基于模型的控制算法. 程靖等^[19]研究了空间机器人捕获卫星过程动力学演化模拟, 并设计了模糊H$\infty $控制方案以实现不稳定系统的镇定. Meng等^[23]针对含柔性构件空间机器人自主捕获目标前的弹性振动抑制问题, 基于动态耦合模型设计了一个闭环控制系统. Aghili等^[24]研究了空间机械臂捕获卫星的最优控制问题, 实现了联合体系统的最小时间镇定. Dong等^[25]研究了卫星自主交会对接过程位置及姿态的控制问题. Lampariello等^[26]设计了一种基于非线性优化的方法, 以实现有限时间内对翻滚目标的捕获. Stolfi等^[27]提出了一种基于PD控制的阻抗算法, 以实现对非合作目标的捕获. 值得一提的是, 以上研究成果主要关注的是捕获前轨迹规划及捕获后组合体的镇定控制上, 并未考虑减小空间机器人捕获卫星碰撞过程所受冲击力矩及实现镇定过程对关节电机的保护.

考虑到柔性机构 — RSEA (rotary series elastic actuator)在机器人与外界环境发生碰撞时, 在保护机器人关节执行器避免外部冲击破坏方面发挥了很好的作用^[28-29]. 为此, 本文尝试将RSEA引入到空间机器人系统中, 同时设计与之配合的开启、关闭电机策略以实现缓冲从顺控制. 然而由于RSEA装置存在缓冲弹簧, 因此也为系统带来了关节柔性. 此外, 空间机器人自身各构件间存在着强耦合作用, 捕获过程系统内部还会存在动量、动量矩及能量的传递变化. 以上多重问题的综合使得空间机器人在轨捕获卫星的动力学与控制研究大为复杂.

对于高性能的空间机械臂来说, 关节柔性是一个不可忽略的影响因素, 为了实现存在关节柔性的空间机器人的缓冲从顺控制, 本文基于奇异摄动思想, 将空间机器人及被捕获卫星形成的联合体系统分解为表征柔性部分的快变子系统及表征刚性部分的慢变子系统. 针对快变子系统, 设计了速度差值反馈控制器以主动抑制系统的弹性振动. 考虑星载计算机的运算能力有限, 而捕获操作将导致液体燃料晃动而产生扰动项. 鉴于自抗扰控制(active disturbance rejection control, ADRC)技术可对扰动项进行动态估计, 并实时补偿^[30-31]. 因此, 对慢变子系统提出了一种基于动态面的自抗扰控制器. 所提方法避免了反演法带来的计算膨胀问题, 有效减少了星载计算机的计算量^[32-33]. 同时, 有效提升了不稳定联合体系统的抗扰动能力, 并最终达到稳定控制及轨迹的精确跟踪. 最后, 通过对空间机器人捕获操作过程的仿真分析, 表明了柔性机构的抗冲击性能及所提缓冲从顺控制策略的有效性.

1 柔性机构结构及缓冲从顺策略

配置柔性机构空间机器人系统的RSEA装置安装在电机与机械臂之间, 通过其输入圆盘与电机相连, 机械臂则是与RSEA装置的扫臂负载空心轴相连. 空间机械臂总体结构简图如图1所示, 所设计RSEA装置的结构如图2所示. 图2中$R$为扫臂有效半径, $r$为弹簧的半径.

图1

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图1空间机械臂结构图

Fig. 1Structure of space manipulator

图2

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图2RSEA装置结构图

Fig. 2Structure of RSEA

在捕获阶段, 机械臂末端与目标卫星发生碰撞, 其关节处会受到很大的冲击力矩, 该力矩先作用在RSEA装置的输出扫臂上, 再传递到弹簧组, 通过内部弹簧对碰撞能量进行吸收、缓冲, 进而实现对关节的保护. 在镇定运动阶段, 受冲击效应的影响, 电机开启时也会受到冲击力矩, 若所受力矩超过电机所能承受的极限而不关停电机, 电机将遭受损坏. 因此, 需要根据关节所能承受的力矩极限设置一个关机力矩阈值让电机关停. 当检测到关节所受冲击力矩超过所设关机力矩阈值后电机关停, 此时, RSEA装置内部弹簧组会提供弹力以减小关节所受冲击力矩. 此外, 在实际操作中, 若是只设定关机力矩阈值, 将导致电机频繁开关机, 进而影响电机性能. 基于此, 本文所提的从顺控制策略设置了两个力矩阈值, 一个是电机关机力矩阈值, 另一个是电机开机力矩阈值. 当关节所受力矩超过关机力矩阈值时, 电机关停, 当关节所受力矩低于开机力矩阈值时, 电机再次开机.

2 动力学建模及冲击效应分析

以配置RSEA空间机器人捕获卫星操作过程为例, 如图3所示. 建立系统惯性坐标系$XOY$, 同时建立各分体的连体坐标系$x_i O_i y_i (i=0,1,2)$, 其中$O_0 $为基座质心; $O_i (i=1,2)$为连接各关节的转动铰中心. 设空间机器人系统基座$B_0 $质量、质心转动惯量及$O_0 $至$O_1 $长度分别为$m_0$, $I_0 $, $l_0 $; 机械臂质量、质心转动惯量及长度分别为$m_i$, $I_i $, $l_i(i=1,2)$; $d_i $为第$i$个关节转动铰中心到机械臂$i$质心的距离. 被捕获卫星的质量、质心转动惯量为$m_t $和$I_t $. 两关节电机转子转动惯量为$I_{im}(i=1,2)$; RSEA装置中弹簧的刚度为$k_{i\rm a}(i=1,2)$. $\theta _0$, $\theta _i(i=1,2)$和$\theta _t$分别为基座姿态、机械臂、被捕获卫星转动角; $\theta _{im}(i=1,2)$为电机转子转角. 系统总质心及基座、机械臂质心在惯性坐标系下位置矢量分别为${r}_{\rm c}$, ${r}_0 $, ${r}_1 $和${r}_2 $.

图3

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图3空间机器人系统及目标卫星系统

Fig. 3Space robot and target satellite systems

将目标卫星视为均质刚体, 定义${q}_t =[x_t ,y_t ,\theta _t]^{\rm T}$为其广义坐标列向量, 则可通过牛顿-欧拉法获得被捕获卫星系统的动力学方程

(1)$\begin{eqnarray}{D}_t {\ddot{{q}}}_t ={J}_t^{\rm T} {F}'\end{eqnarray}$

其中, ${D}_t \in {\bf R}^{3\times 3}$为被捕获卫星系统广义质量阵, $x_t $和$y_t$为被捕获卫星质心位置坐标; ${J}_t \in {\bf R}^{3\times 3}$为被捕获卫星碰撞接触点对应的运动Jacobian矩阵, ${F}'\in {\bf R}^{3\times 1}$为被捕获卫星所受到的作用力.

根据图3中几何位置关系, 可得捕获前空间机器人各分体质心在惯性坐标系下的表达式为

(2)${r}_0 =(x_a \ \ y_a)^{\rm T}$

(3)$ {r}_i ={r}_0 +\sum\limits_{j=0}^{i-1} {l_i } {e}_i +d_i {e}_i\ \ (i=1,2)$

其中, $x_a $和$y_a $为载体质心位置坐标; ${e}_i (i=0,1,2)$为各连体坐标系$x_i $方向的基矢量.

对式(2)、式(3)进行求导, 可得到含柔性机构空间机器人系统总动能表达式为

(4)$\begin{eqnarray} T=\sum\limits_{i=0}^2 (\frac{1}{2} m_i \dot{{r}}_i^2 +\frac{1}{2}I_i {\omega }_i^2 )+\sum\limits_{j=1}^2 {\frac{1}{2}I_{jm} {\omega }_{jm}^2} \end{eqnarray}$

其中, ${\omega }_i (i=0,1,2)$表示载体及两机械臂杆的角速度; ${\omega }_{jm} (j=1,2)$表示电机转子的角速度.

忽略太空微弱重力影响, 可知空间机器人系统势能只来源于RSEA装置, 因而其总势能为

(5)$\begin{eqnarray}U=\sum\limits_{i=1}^2 {\frac{3}{2}} k_{i\rm a} (\Delta x_{i\rm L}^2 +\Delta x_{i\rm R}^2)\end{eqnarray}$

其中, $k_{i\rm a} (i=1,2)$为关节等效刚度, 其计算公式于仿真处给出; $\Delta x_{i\rm L} =R\sin (\alpha _i )$, $\Delta x_{i\rm R} =-R\sin (\alpha _i )$; $\alpha _i$为输入圆盘与扫臂之间的角度差.

基于上述动能、势能表达式, 可得到Lagrange函数, 结合第二类拉格朗日建模方法, 推导得捕获碰撞前载体位置不受控、姿态受控的空间机器人动力学方程为

(6)$\begin{eqnarray} \left.\begin{array}{l} D({q}){\ddot{{q}}}+{H}({q},{\dot{{q}}}){\dot{{q}}}={\tau }_{\rm c} +{J}^{\rm T}{F}\\ {J}_m \ddot{\theta}_m +{K}_m ({\theta }_m -{\theta })={\tau}_m\\ {K}_m ({\theta }_m -{\theta })={\tau }_\theta \\ \end{array}\right\} \end{eqnarray}$

其中, ${q}=[x_a ,y_a ,\theta _0 ,\theta _1 ,\theta _2 ]^{\rm T}$为空间机器人系统广义坐标; ${\theta }_m =[\theta _{1m} ,\theta _{2m} ]^{\rm T}$, ${\theta }=\left[ {\theta _1 ,\theta _2 } \right]^{\rm T}$; ${D}({q})\in {\bf R}^{5\times 5}$表示系统广义质量阵, ${H}({q},{\dot{{q}}}){\dot{{q}}}\in {\bf R}^{5\times 1}$为系统包含科氏力、离心力项; ${\tau }_{\rm c} =[{\tau }_a^{\rm T} ,\tau_0 ,{\tau }_\theta ^{\rm T} ]^{\rm T}$, ${\tau }_a ={\bf 0}_{2\times 1} $, $\tau _0 $为载体姿态控制力矩, ${\tau }_\theta $为关节输入力矩, ${\tau }_m =[\tau _{1m} ,\tau _{2m}]^{\rm T}$为电机输出力矩; ${J}_m ={\rm diag}(I_{1m} ,I_{2m} )$为电机转子转动惯量; ${K}_m ={\rm diag}(k_{1m} ,k_{2m} )$; ${J}\in {\bf R}^{3\times 5}$为机械臂末端对应的运动Jacobian矩阵; ${F}\in {\bf R}^{3\times 1}$为机械臂末端所受作用力.

在捕获操作过程, 空间机器人与被捕获卫星两者发生接触、碰撞, 其相互作用力满足牛顿第三定律, 即: ${F}=-{F}'$. 基于此并结合式(1)、式(6), 可得

(7)$\begin{eqnarray}{D}({q}){\ddot{{q}}}+{H}({q},{\dot{{q}}}){\dot{{q}}}={\tau }_{\rm c} -{J}^{\rm T}({J}_t^{\rm T} )^{-1}{D}_t {\ddot{{q}}}_t\end{eqnarray}$

由于捕获过程空间机器人与目标卫星形成的系统未受外力作用, 所以整个系统服从动量守恒关系; 此外, 为保护关节电机, 捕获阶段电机将处于关机状态, 即${\tau }_{\rm c} ={{\bf 0}}_{5\times 1} $, 因此, 对式(7)两端进行积分并整理得^[19]

(8)${D}({q})({\dot{{q}}}(t_0 +\Delta t)-{\dot{{q}}}(t_0 ))+ {J}^{\rm T}({J}_t^{\rm T} )^{-1}{D}_t ({\dot{{q}}}_t (t_0+\Delta t)-{\dot{{q}}}_t (t_0 ))=0$

捕获完成后, 空间机器人与目标卫星形成联合体系统, 两者末端接触点满足速度约束, 即自$t_0 +\Delta t$时刻恒有

(9)$\begin{eqnarray}{J\dot{{q}}}={J}_t {\dot{{q}}}_t\end{eqnarray}$

结合式(8)、式(9), 可得碰撞冲击对空间机器人运动状态的影响

(10)$\begin{eqnarray}{\dot{{q}}}(t_0 +\Delta t)={L}^{-1}[{D}({q}){\dot{{q}}}(t_0 )+{J}^{\rm T}({J}_t^{\rm T} )^{-1}{D}_t {\dot{{q}}}_t(t_0 )]\end{eqnarray}$

其中, ${L}={D}({q})+{J}^{\rm T}({J}_t^{\rm T} )^{-1}{D}_t {J}_t^{-1} {J}$.

通过对式(6)第一项进行积分, 可得捕获阶段的碰撞冲量

(11)$\begin{eqnarray}{D}({q})({\dot{{q}}}(t_0 +\Delta t)-{\dot{{q}}}(t_0 ))={J}^{\rm T}{P}\end{eqnarray}$

其中, ${P}=\int_{t_0 }^{t_0 +\Delta t} {F} \mbox{d}t$为碰撞冲量. 结合式(10)、式(11), 可计算得碰撞冲量表达式为

(12)${P}=({J}^{\rm T}\mbox{)}^{+1}{D}({q})[{L}^{-1}({D}({q}){\dot{{q}}}(t_0 )+ {J}^{\rm T}({J}_t^{\rm T} )^{-1}{D}_t {\dot{{q}}}_t(t_0 ))-{\dot{{q}}}(t_0 )] $

其中, $({J}^{\rm T})^{+1}$为${J}^{\rm T}$的伪逆. 由于捕获碰撞时间$\Delta t$极小, 则碰撞力可以近似为

(13)$\begin{eqnarray}{F}=\frac{{P}}{\Delta t}\end{eqnarray}$

捕获操作完成后, 空间机器人与被捕获卫星两者形成联合体系统, 因此其末端满足式(9)的速度约束, 对式(9)进行求导, 并整理化简得

(14)$\begin{eqnarray}{\ddot{{q}}}_t ={J}_t^{-1} [{J\ddot{{q}}}+({\dot{{J}}}-{\dot{{J}}}_t {J}_t^{-1} {J}){\dot{{q}}}]\end{eqnarray}$

结合式(6)、式(7)、式(14), 可得联合体系统的综合系统动力学方程为

(15)$\begin{eqnarray} \left. \begin{array}{l} {D}_{\rm C} ({q}){\ddot{{q}}}+{H}_{\rm C} ({q},{\dot{{q}}}){\dot{{q}}}={\tau}_{\rm c}\\ {J}_m {\ddot{{\theta }}}_m +{K}_m ({\theta }_m -{\theta })={\tau }_m\\ {K}_m ({\theta }_m -{\theta })={\tau }_\theta \\ \end{array}\right\} \end{eqnarray}$

其中, ${D}_{\rm C} ({q})={D}({q})+{J}^{\rm T}({J}_t^{\rm T} )^{-1}{D}_t {J}_t^{-1} {J}$; ${H}_{\rm C} ({q},{\dot{{q}}})={H}({q},{\dot{{q}}})+{J}^{\rm T}({J}_t^{\rm T} )^{-1}{D}_t {J}_t^{-1} ({\dot{{J}}}-{\dot{{J}}}_t {J}_t^{-1} {J})$.

考虑空间机器人系统在轨服务寿命等原因, 载体位置处于不受控的状态. 基于此, 式(15) 表现为欠驱动形式, 其不利于控制的设计. 为将式(15)化为全驱动形式, 将其写成如下分块子矩阵形式

(16)$\begin{eqnarray} \left[ {{\begin{array}{ll} {{D}_{\rm C11} } & {{D}_{\rm C12} } \\ {{D}_{\rm C21} } & {{D}_{\rm C22} } \\ \end{array} }} \right]\left[ {{\begin{array}{l} {{\ddot{{q}}}_a } \\ {{\ddot{{q}}}_\theta } \\ \end{array} }} \right]+\left[ {{\begin{array}{ll} {{H}_{\rm C11} } & {{H}_{\rm C12} } \\ {{H}_{\rm C21} } & {{H}_{\rm C22} } \\ \end{array} }} \right]\left[ {{\begin{array}{l} {{\dot{{q}}}_a } \\ {{\dot{{q}}}_\theta } \\ \end{array} }} \right]=\left[ {{\begin{array}{l} {{\tau }_a } \\ {{\tau }_b } \\ \end{array} }} \right] \end{eqnarray}$

其中, ${q}_a =[x_a ,y_a ]^{\rm T}$, ${q}_\theta =[\theta _0 ,\theta _1 ,\theta _2 ]^{\rm T}$; ${\tau }_b =[\tau _0 \mbox{,}{\tau }_\theta ^{\rm T} ]^{\rm T}$. 通过观察发现${H}_{\rm C11}$和${H}_{\rm C21}$均为零矩阵, 基于此由式(16)第一行可解得${\ddot{{q}}}_a $的表达式, 并代入第二行, 可得到联合体系统全驱动形式动力学方程

(17)$\begin{eqnarray} \left.\begin{array}{l} {{D}_X {\ddot{{q}}}_\theta +{H}_X {\dot{{q}}}_\theta ={\tau }_b}\\ {{J}_m {\ddot{{\theta }}}_m +{K}_m ({\theta }_m -{\theta })={\tau }_m }\\ {{K}_m ({\theta }_m -{\theta })={\tau }_\theta }\\ \end{array}\right\} \end{eqnarray}$

其中, ${D}_X ={D}_{\rm C22} -{D}_{\rm C21} {D}_{\rm C11}^{-1} {D}_{\rm C12} $; ${H}_X ={H}_{\rm C22} -{D}_{\rm C21} {D}_{\rm C11}^{-1}{H}_{\rm C12} $.

3 控制器设计

3.1 快变子系统控制器设计

由于联合体系统存在RSEA装置, 其使得系统关节具备柔性, 为了抑制关节柔性引起的振动, 借助奇异摄动技术, 将联合体系统分解为快变子系统和慢变子系统分别进行控制设计, 因此系统的总控制律可写为如下形式

(18)$\begin{eqnarray}{\tau }_m ={\tau }_{\rm s} +{\tau }_{\rm f}\end{eqnarray}$

其中, ${\tau }_{\rm s} \in {\bf R}^{2\times 1}$为慢变子系统控制力矩, ${\tau }_{\rm f} \in {\bf R}^{2\times 1}$为快变子系统控制力矩.

定义正比例因子$\varepsilon $及正定对角矩阵${K}_1 $, 并令其满足如下关系

(19)$\begin{eqnarray}{K}_m =\frac{{K}_1 }{\varepsilon ^2}\end{eqnarray}$

结合式(19), 则式(17)的后两项可重写为描写系统弹性振动的快变子系统方程

(20)$\begin{eqnarray}\varepsilon ^2{\ddot{{\tau }}}_\theta ={J}_m^{-1} {K}_1 ({\tau }_m -{J}_m {\ddot{{\theta }}}-{\dot{{\tau }}}_\theta )\end{eqnarray}$

设计如下速度差值反馈控制器对快变子系统进行控制

(21)$\begin{eqnarray} {\tau }_{\rm f} =-{K}_{\rm f} ({\dot{{\theta }}}_m -{\dot{{\theta }}}) \end{eqnarray}$

其中, ${K}_{\rm f} ={K}_{\rm 2} /\varepsilon $, ${K}_{\rm 2} \in {\bf R}^{2\times 2}$为正定、对角矩阵.

将式(18)、式(21)代入式(20), 可得

(22)$\begin{eqnarray} \varepsilon ^2{J}_m {\ddot{{\tau }}}_\theta ={K}_1 ({\tau }_{\rm s} -{J}_m {\ddot{{\theta }}}-{\tau }_\theta )-\varepsilon {K}_{\rm 2} {\dot{{\tau }}}_\theta \end{eqnarray}$

当$\varepsilon \to 0$时, 关节等效刚度${K}\to \infty $, 此时联合体系统等效为刚性模型; 则由式(17)、式(18)可得出慢变子系统的动力学方程

(23)$\begin{eqnarray}{D}_{S\theta } {\ddot{{q}}}_\theta +{H}_{S\theta } {\dot{{q}}}_\theta ={\tau }_{S\theta }\end{eqnarray}$

其中, ${D}_{S\theta } ={D}_X +{J}_X $, ${H}_{S\theta } $为${\dot{{\theta }}}={\dot{{\theta }}}_m $时对应的新矩阵; ${J}_X =\mbox{diag}(0,I_{1m} ,I_{2m} )$, ${\tau }_{S\theta } =[\tau _0 ,{\tau }_{\rm s}^{\rm T} ]^{\rm T}$.

3.2 慢变子系统控制器设计

考虑捕获操作将导致控制载体姿态的液体燃料晃动, 进而产生有界扰动项, 则慢变子系统动力学方程式(23)可写为

(24)$\begin{eqnarray}{D}_{S\theta } {\ddot{{q}}}_\theta +{H}_{S\theta } {\dot{{q}}}_\theta ={\tau }_{S\theta } +{\tau }_{\rm d}\end{eqnarray}$

其中, ${\tau }_{\rm d} \in {\bf R}^{3\times 1}$为扰动项.

为便于分析, 令${x}_1 ={q}_\theta $, ${x}_2 ={\dot{{q}}}_\theta $, 则式(24)可表示为如下形式状态方程

(25)$\begin{eqnarray} \left.\begin{array}{l} {{\dot{{x}}}_1 ={x}_2}\\ {{\dot{{x}}}_2 =-{Ax}_2 +{B}+{x}_3 }\\ \end{array} \right\} \end{eqnarray}$

其中, ${A}={D}_{S\theta }^{-1} {H}_{S\theta } $; ${B}={D}_{S\theta }^{-1} {\tau }_{S\theta } $; ${x}_3 ={D}_{S\theta }^{-1} {\tau }_{\rm d} $.

考虑星载计算机的运算能力有限, 借助动态面技术简化运算过程. 慢变子系统动态面控制设计步骤如下.

(1) 定义轨迹跟踪误差, 即第一个误差面

(26)$\begin{eqnarray}{s}_1 ={e}={x}_1 -{q}_{\theta\rm d}\end{eqnarray}$

其中, ${q}_{\theta \rm d} \in {\bf R}^{3\times 1}$为系统轨迹期望矢量.

对式(26)两边进行求导, 可整理得

(27)$\begin{eqnarray}{\dot{{s}}}_1 ={x}_2 -{\dot{{q}}}_{\theta\rm d}\end{eqnarray}$

定义虚拟控制量 , 其满足

(28)$\begin{eqnarray}{\bar{{x}}}_2 =-\eta _1 {s}_1 +{\dot{{q}}}_{\theta\rm d}\end{eqnarray}$

其中, $\eta _1 $为正常数.

同时, 选取为${\bar{{x}}}_2 $输入, 并采用一阶低通滤波器, 得到输出状态变量${x}_{\rm 2d} $

(29)$\begin{eqnarray} \left.\begin{array}{l} {\eta _2 {\dot{{x}}}_{\rm 2d} +{x}_{\rm 2d} ={\bar{{x}}}_2 }\\ {{x}_{\rm 2d} (0)={\bar{{x}}}_2 (0)} \\ \end{array}\right\} \end{eqnarray}$

其中, 时间$\eta _2 >0$.

(2) 为设计慢变子系统控制律, 定义第二个误差面

(30)$\begin{eqnarray}{s}_2 ={x}_2 -{x}_{\rm 2d}\end{eqnarray}$

对${s}_2 $两边进行求导, 并结合式(25), 可得

(31)$\begin{eqnarray}{\dot{{s}}}_2 =-{D}_{S\theta }^{-1} {H}_{S\theta } {x}_2 +{D}_{S\theta }^{-1} {\tau }_{S\theta } +{x}_3 -{\dot{{x}}}_{\rm 2d}\end{eqnarray}$

考虑有界扰动项将影响镇定运动的稳定性及轨迹跟踪精度, 因此采用扩张状态观测器对其进行动态估计, 扩张状态观测器设计如下

(32)$\begin{eqnarray} \left. \begin{array}{l} {{\dot{{\hat{{x}}}}}_1 ={\hat{{x}}}_2 +{\alpha }_1 {\tilde{{x}}}_1} \\ {{\dot{{\hat{{x}}}}}_2 ={\hat{{x}}}_3 +{D}_{S\theta }^{-1} ({\tau }_{S\theta } -{H}_{S\theta } {\dot{{q}}}_\theta )+{\alpha }_2 {\tilde{{x}}}_1}\\ {{\dot{{\hat{{x}}}}}_3 ={\alpha }_3 {\tilde{{x}}}_1}\\ \end{array}\right\} \end{eqnarray}$

其中, ${\hat{{x}}}_1 $, ${\hat{{x}}}_2 $, ${\hat{{x}}}_3 $分别为${x}_1 $, ${x}_2 $, ${x}_3 $的估计值; 观测误差${\tilde{{x}}}_1 ={x}_1 -{\hat{{x}}}_1 $, ${\tilde{{x}}}_2 ={x}_2 -{\hat{{x}}}_2 $, ${\tilde{{x}}}_3 ={x}_3 -{\hat{{x}}}_3 $. ${\alpha }_1 $, ${\alpha }_2 $, ${\alpha }_3$为观测器增益, 采用带宽化方式^[34]对其进行选取

(33)$\begin{eqnarray}{\alpha }_1 =3\lambda {I}_3 , \ \ {\alpha }_2 =3\lambda ^2{I}_3 , \ \ {\alpha }_3 =\lambda ^3{I}_3\end{eqnarray}$

其中, $\lambda >0$为观测器带宽.

在上述基础上, 慢变子系统控制律设计如下

(34)$\begin{eqnarray}{\tau }_{S\theta } ={D}_{S\theta } ({\dot{{x}}}_{\rm 2d} -{c}_2 {s}_2 )+{H}_{S\theta } {x}_2 -{D}_{S\theta } {\hat{{x}}}_3\end{eqnarray}$

其中, ${c}_2 $为正定、对角矩阵.

定义状态变量${\omega }_i ={\tilde{{x}}}_i /\lambda ^{i-1}(i=1,2,3)$, ${\omega }=[{\omega }_1^{\rm T} ,{\omega }_2^{\rm T} ,{\omega }_2^{\rm T} ]^{\rm T}$. 由式(25)、式(32), 可得

(35)$\begin{eqnarray}{\dot{{\omega }}}=\lambda {K}_v {\omega }+\frac{{K}_p {\dot{{x}}}_3 (t)}{\lambda ^2}\end{eqnarray}$

其中, ${K}_v =\left[ {{\begin{array}{c@{\quad }c@{\quad }c} {-3{I}_3 } & {{I}_3 } & {{{\bf 0}}_3 } \\ {-3{I}_3 } & {{{\bf 0}}_3 } & {{I}_3 } \\ {-{I}_3 } & {{{\bf 0}}_3 } & {{{\bf 0}}_3 } \\ \end{array} }} \right]$, ${K}_p =\left[ {{\begin{array}{l} {{{\bf 0}}_3 } \\ {{{\bf 0}}_3 } \\ {{I}_3 } \\ \end{array} }} \right]$.

定理 1 若${\dot{{x}}}_3 $是有界的, 任意选取观测器带宽$\lambda >0$, 观测误差服从一致有界性, 即: 满足存在常数$\gamma _i >0$, 使得${\tilde{{x}}}_i (i=1,2,3)$中所有元素在有限时间内满足$\vert {\tilde{{x}}}_{ij} \vert \leqslant \gamma _i (j=1,2,3)$.

证明假设观测器误差初始为零, 对式(35)进行求解, 可得

(36)${\omega }(t)=\int_0^t {{\rm e}^{\lambda {K}_v (t-\tau )}} \frac{{K}_p {\dot{{x}}}_3 (\tau )}{\lambda ^2}{\rm d}\tau = \\ \qquad\frac{1}{\lambda ^2}[\int_0^t {\lambda {K}_v {\rm e}^{\lambda {K}_v (t-\tau )}} {K}_p {x}_3 (\tau ){\rm d}\tau +{K}_p {x}_3 (t)] $

由于${\dot{{x}}}_3 (t)$是有界的, 故其元素全部有界, 即满足$\vert {\dot{{x}}}_3 (t)\vert \leqslant {x}_{\rm M} $, 其中${x}_{\rm M} $为正常向量, 则式(36)可有

(37)${\omega }(t)\leqslant \frac{1}{\lambda ^2}[\int_0^t {\lambda {K}_v {\rm e}^{\lambda {K}_v (t-\tau )}} {K}_p {x}_{\rm M} {\rm d}\tau +{K}_p {x}_{\rm M}]=\\ \qquad \frac{1}{\lambda ^2}{\rm e}^{\lambda {K}_v t}{K}_p {x}_{\rm M} $

参考文献^[35]相关推导可知, 若${K}_v $是Hurwitz的, 则存在时间$t_{\rm s} $, 在$t\geqslant t_{\rm s} $的任意时刻, 满足如下关系

(38)$\begin{eqnarray}{\vert}[{\rm e}^{\lambda {K}_v t}]_{ij} {\vert}\leqslant \frac{1}{\lambda ^9}\ \ \ (i,j=1,2,\cdots ,9)\end{eqnarray}$

因此, ${\omega }(t)$中各个元素满足如下关系

(39)$\begin{eqnarray}{\vert }[{\omega }_{ij} (t)]{\vert }\leqslant \frac{[{x}_{\rm M} ]_i }{\lambda ^{11}}\ \ \ (i,j=1,2,3)\end{eqnarray}$

结合${\omega }_i ={\tilde{{x}}}_i /\lambda ^{i-1}$可得

(40)$\begin{eqnarray}\vert {\tilde{{x}}}_{ij} \vert \leqslant \frac{[{x}_{\rm M} ]_i }{\lambda ^{12-i}}\leqslant \gamma _i\ \ \ (i,j=1,2,3)\end{eqnarray}$

综合以上分析, 可证明扩张状态观测器的观测误差服从一致有界性. 证毕.

定理 2 对于式(25)所示的慢变子系统, 基于所设计控制器(34), 可保证联合体系统半全局最终一致有界.

证明结合所设计动态面, 定义系统边界层误差如下

(41)$\begin{eqnarray}{z}_2 ={x}_{\rm 2d} -{\bar{{x}}}_2\end{eqnarray}$

结合式(29)及式(41), 可得

(42)$\begin{eqnarray}{\dot{{x}}}_{\rm 2d} =-{z}_2 /\eta _2\end{eqnarray}$

由此可解得${z}_2 $的一阶导数${\dot{{z}}}_2 $

(43)${\dot{{z}}}_2 =-{z}_2 /\eta _2 -\eta _1 {\dot{{s}}}_1 +{\ddot{{q}}}_{\theta\rm d} = -{z}_2 /\eta _2 +{ \varPhi }({s}_1 ,{s}_2 ,{q}_{\theta\rm d},{\dot{{q}}}_{\theta\rm d} ,{\ddot{{q}}}_{\theta\rm d} ,{z}_2 ) $

其中, ${ \varPhi }({s}_1 ,{s}_2 ,{q}_{\theta\rm d} ,{\dot{{q}}}_{\theta\rm d} ,{\ddot{{q}}}_{\theta\rm d} ,{z}_2)$为非负连续函数.

联立式(27)$\sim\!$式(30)及式(42), 有

(44)$\begin{eqnarray}{\dot{{s}}}_1 ={s}_2 +{z}_2 -\eta _1 {s}_1\end{eqnarray}$

定义如下Lyapunov函数

(45)$\begin{eqnarray} V=\frac{1}{2}({s}_1^{\rm T} {s}_1 +{s}_2^{\rm T} {s}_2 +{z}_2^{\rm T} {z}_2 ) \end{eqnarray}$

对式(45)进行求导, 可得

(46)$\begin{eqnarray}\dot{{V}}={s}_1^{\rm T} {\dot{{s}}}_1 +{s}_2^{\rm T} {\dot{{s}}}_2 +{z}_2^{\rm T} {\dot{{z}}}_2\end{eqnarray}$

将式(34)代入式(31)中, 得到

(47)$\begin{eqnarray}{\dot{{s}}}_2 =-{c}_2 {s}_2 +{\tilde{{x}}}_3\end{eqnarray}$

根据式(43)、式(44)、式(47), 将其代入式(46)可有

(48)$\dot{{V}}={s}_1^{\rm T} ({s}_2 +{z}_2 -\eta _1 {s}_1 )+{s}_2^{\rm T} (-{c}_2 {s}_2 +{\tilde{{x}}}_3 )+ \\ \qquad{z}_2^{\rm T} (-{z}_2 /\eta _2 +{ \varPhi }({s}_1,{s}_2,{q}_{\theta\rm d} ,{\dot{{q}}}_{\theta\rm d} ,{\ddot{{q}}}_{\theta\rm d},{z}_2 )) $

利用Young不等式得

$ \left\{\begin{array}{l} {s}_1^{\rm T} {s}_2 \leqslant \dfrac{1}{2}{s}_1^{\rm T} {s}_1 +\dfrac{1}{2}{s}_2^{\rm T} {s}_2 \\ {s}_1^{\rm T} {z}_2 \leqslant \dfrac{1}{2}{s}_1^{\rm T} {s}_1 +\dfrac{1}{2}{z}_2^{\rm T} {z}_2 \\ {s}_2^{\rm T} {\tilde{{x}}}_3 \leqslant \dfrac{1}{2}{s}_2^{\rm T} {s}_2 +\dfrac{1}{2}{\tilde{{x}}}_3^{\rm T} {\tilde{{x}}}_3 \\ {z}_2^{\rm T} { \varPhi }({s}_1 ,{s}_2 ,{q}_{\theta\rm d},{\dot{{q}}}_{\theta\rm d} ,{\ddot{{q}}}_{\theta\rm d} ,{z}_2 )\leqslant \dfrac{{z}_2^{\rm T} {z}_2 { \varPhi }^{\rm T}{ \varPhi}}{2k}+\dfrac{k}{2} \end{array}\right. $

进而, 式(48)满足

(49)$ \dot{{V}}\leqslant -{s}_1^{\rm T} \eta _1 {s}_1 -{c}_2 {s}_2^{\rm T} {s}_2 -{z}_2^{\rm T} {z}_2 /\eta _2 +{s}_2^{\rm T} {s}_2 + \\ \quad \frac{1}{2}{z}_2^{\rm T} {z}_2 +\frac{1}{2}{\tilde{{x}}}_3^{\rm T} {\tilde{{x}}}_3 +\frac{{z}_2^{\rm T} {z}_2 { \varPhi }^{\rm T}{ \varPhi }}{2k}+\frac{k}{2}\\ \quad \leqslant (1-\eta _1 ){s}_1^{\rm T} {s}_1 +(1-{c}_2 ){s}_2^{\rm T} {s}_2 +\\ \quad {z}_2^{\rm T} (\frac{1}{2}-\frac{1}{\eta _2 }+\frac{{ \varPhi }^{\rm T}{ \varPhi }}{2k}){z}_2 +\frac{1}{2}{\tilde{{x}}}_3^{\rm T} {\tilde{{x}}}_3 +\frac{k}{2} $

参考文献^[36]可知, $\vert \vert { \varPhi }\vert \vert \leqslant \psi ,(\psi >0)$, 选取${1}/{\eta _2 }\geqslant {1}/{2}+{\psi ^2}/({2k})+a_0$, 可得到

(50)$\begin{eqnarray}\dot{{V}}\leqslant -\varsigma V+\rho\end{eqnarray}$

其中$\varsigma =\min \{2(\eta _1 -1),2({c}_2 -1),2a_0 \}$, $\rho ={\tilde{{x}}}_3^{\rm T} {\tilde{{x}}}_3/2 +{k}/{2}$.

通过对$\varsigma $适当选取, 使其满足$\varsigma >\rho /\phi $, 则当$V=\phi $时, $\dot{{V}}\leqslant 0$是一个不变集, 即存在$V(0)\leqslant \phi $, 则对$t>0$时, 恒有$V(t)\leqslant \phi $.

求解式(50)可得

(51)$\begin{eqnarray}0\leqslant V\leqslant \rho /\varsigma +[V(0)-\rho /\varsigma ]{\rm e}^{-\varsigma t}\end{eqnarray}$

基于上述条件, 并结合Lyapunov稳定性定理, 可知该系统以$\rho /\varsigma $为界, 联合体系统半全局最终一致有界. 因此, 联合体系统轨迹跟踪误差${e}$可收敛到零的任意小邻域. 证毕.

4 仿真算例分析

4.1 捕获碰撞过程RSEA抗冲击性能模拟

采用图3所示的空间机器人及目标卫星系统进行数值仿真试验. 模型参数选取如下: $m_0 =80$ kg, $I_0 =40$ kg$\cdot$m$^{2}$, $l_0=1$ m; $m_i =5$ kg, $I_i =3$ kg$\cdot$m$^{2}$, $l_i =2$ m, $d_i =1$ m $(i=1,2)$; $m_t =30$ kg, $I_t =15$ kg$\cdot$m$^2$, $l_t =0.5$ m; $I_{1m} =I_{2m}=0.05$ kg$\cdot$m$^2$, $k_{1a} =k_{2a} =1000$ N/m. 关节等效刚度的计算公式^[28]为

(52)$\begin{eqnarray}{K}_m =2{K}_a (3R^2+r^2)(2\cos ^2{\varphi }-1)\end{eqnarray}$

其中, ${K}_a ={\rm diag}(k_{1a} ,k_{2a} )$, $R=0.1$ m, $r=0.01$ m, ${\varphi }$为机械臂末端施加${\tau }_{\rm F} =[20 {\rm Nm},\ 20 {\rm Nm},\ 0 {\rm Nm}]^{\rm T}$的载荷时扫臂的转角, 仿真时选取${\varphi }=\mbox{diag}(3^\circ ,2^\circ )$.

为了验证空间机器人碰撞过程的抗冲击性能, 采用配置/未配置RSEA装置空间机器人系统对不同初速度卫星进行捕获模拟试验, 选取空间机器人系统的初始构型为${q}=[0,0,90^\circ ,45^\circ ,45^\circ ]^{\rm T}$, 碰撞所受冲击力矩结果如表1所示. 表1中, 第二及第三列前、后项分别为未配置与配置RSEA装置关节所受冲击力矩; 第四列为冲击力矩最大降低百分比.

Table 1
表 1
表 1卫星不同初速度下RSEA的抗冲击性能模拟
Table 1RSEA impact resistance at different satellite initial velocities

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由表1可看出, 针对捕获不同初速度卫星的操作过程, 配置RSEA装置较未配置RSEA装置都能有效的减小空间机器人关节所受碰撞冲击力矩, 进而实现了对关节的保护.

4.2 镇定运动过程缓冲从顺控制策略性能模拟

为验证镇定运动过程缓冲从顺控制策略的有效性, 运用本文第三部分所提控制方案对图3所示系统进行数值仿真试验. 所提控制方案的控制参数选取如下: ${K}_2 ={\rm diag}(5,5)$, ${c}_2 ={\rm diag}(5,5,5)$, $\varepsilon =0.5$, ${\alpha}_1={\rm diag}(9,9,9)$, ${\alpha}_2 ={\rm diag}(27,27,27)$, ${\alpha}_3 ={\rm diag}(27,27,27)$. 由捕获产生的扰动项为${\tau}_{\rm d} =[2\sin (\pi t/3)-2\cos (\pi t/3),2\sin (\pi t/3),2\cos (\pi t/3)]^{\rm T}$. 假设在$t_0=0$时空间机器人对卫星进行捕获操作, 此时卫星速度为$v_t=[0.1 \mbox{m/s},0.1 \mbox{m/s},0.35 \mbox{rad/s}]^{\rm T}$. 捕获完成后空间机器人与目标卫星形成的联合体系统期望位置选取为${q}_{\theta\rm d} =[100^\circ, 30^\circ, 60^\circ]^{\rm T}$. 仿真时间选取为$t=20$ s. 仿真结果如图4$\sim\!$图8所示.

图4

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图4未采用开、关机策略关节所受冲击力矩

Fig. 4Joint impact torque without switching strategy

图5

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图5采用开、关机策略关节所受冲击力矩

Fig. 5Joint impact torque with switching strategy

图6

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图6关节电机开、关机信号

Fig. 6Switch signal of joint motor

图7

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图7开启缓冲从顺控制时镇定运动轨迹

Fig. 7Trajectory tracking of stabilization under buffer and compliant control

图8

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图8关闭快变控制器轨迹

Fig. 8Trajectory without fast controller

图4为未开启主动开、关电机策略时, 关节所受冲击力矩. 假设关节电机正常工作时, 所能承受的冲击力矩极限为80 N$\cdot$m. 可发现此时的冲击力矩虽然较未配置柔性机构得到降低, 但依然超出安全阈值. 因此需结合开、关机策略进行控制, 以实现对关节电机的保护; 选取电机关机阈值为${\tau }_{\rm O} =48$ N$\cdot$m, 开机阈值为$\tau_{\rm I} =9$ N$\cdot$m. 图5及图6分别为开启所提开、关机策略时, 关节所受冲击力矩及关节电机开关机情况. 对比图4与图5可知, 结合缓冲从顺控制, 可使得关节所受冲击力矩限制在安全范围内, 有效实现了对关节电机的保护.

图7为采用上述缓冲从顺控制时的镇定轨迹. 其中, 实线为结合自抗扰补偿的动态面控制方法时的镇定轨迹, 虚线为关闭自抗扰补偿项时的镇定轨迹. 通过两种方法对比可知, 所提基于动态面的自抗扰控制方法可有效实现对扰动项的补偿, 并更快实现了对受扰动联合体系统的镇定控制, 有效提升了失稳联合体系统抗扰动的能力.

图8为关闭其快变子系统速度差值反馈控制器时, 所得的跟踪轨迹情况; 比较图7与图8可知, 所提速度差值反馈控制器, 可实现对系统关节弹性振动的主动抑制, 进而达到轨迹的精确跟踪.

5 结论

考虑空间机器人捕获操作过程, 其机械臂关节处将会受到巨大的碰撞冲击力矩的影响. 为了避免该冲击力矩对关节电机造成破坏, 本文设计了一种含RSEA装置的空间机器人, 并提出了一种与之配合的适时开、关机控制策略. 通过数值仿真可知, 所提方案在捕获接触、碰撞阶段, 最大可减小63.2%关节所受碰撞冲击力矩, 最小也能减小49.9%, 体现了良好的抗冲击性能. 在镇定运动阶段, 借助奇异摄动技术, 实现了对系统弹性振动的主动抑制, 并保证了关机所受冲击力矩限定在安全范围内, 从而避免了关节电机的过载、破坏. 此外, 所提基于动态面的自抗扰控制方案不仅简化了计算过程; 同时, 还提高了系统的抗扰动性能, 保证了镇定运动的精确性和稳定性.

参考文献原文顺序
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