自动化学报  2018, Vol. 44 Issue (10): 1771-1780   PDF    
无人机吊挂飞行系统的减摆控制设计
王诗章1, 鲜斌1, 杨森1     
1. 天津大学电气自动化与信息工程学院 天津 300072
摘要: 主要考虑了四旋翼无人机(Unmanned aerial vehicle,UAV)吊挂飞行系统的位置控制及负载摆动抑制的设计问题.在存在欠驱动特性以及未知系统参数的约束下,本文基于能量法设计了一种非线性控制策略,实现了对无人机位置的精确控制和飞行过程中负载摆动的快速抑制.基于Lyapunov方法的稳定性分析证明了闭环系统的稳定性,位置误差的收敛及摆动的抑制.实验结果表明本文提出的控制策略取得了较好的控制效果.
关键词: 四旋翼无人机     吊挂负载     未知参数     自适应控制     减摆控制    
Anti-swing Controller Design for an Unmanned Aerial Vehicle With a Slung-load
WANG Shi-Zhang1, XIAN Bin1, YANG Sen1     
1. School of Electrical and Information Engineering, Tianjin University, Tianjin 300072
Manuscript received : July 24, 2017, accepted: February 15, 2018.
Foundation Item: Supported by National Natural Science Foundation of China (91748121, 90916004, 60804004), Key Project of Tianjin Application and Research Program in Cutting-edge Technology (14JCZDJC31900), Key Project of Tianjin Science and Technology Support Program (15ZCZDGX00810)
Author brief: WANG Shi-Zhang   Master student at Tianjin University. Her main research interest is nonlinear control of unmanned aerial vehicle;
YANG Sen   Ph. D. candidate at Tianjin University. His main research interest is nonlinear control of unmanned aerial vehicle.
Corresponding author. XIAN Bin   Professor at Tianjin University. His research interest covers nonlinear control, unmanned aerial vehicle system, and real-time control system. Corresponding author of this paper.
Recommended by Associate Editor SUN Fu-Chun
Abstract: In this paper, position control and load swing suppression design for a quadrotor unmanned aerial vehicle (UAV) with a slung-load are investigated. Under the constraints of underactuated feature and unknown system parameters, a nonlinear control strategy is designed based on energy method, which achieves accurate position control of the UAV as well as payload swing fast suppression during the flight. The stability of the closed-loop system, convergence of position error and payload swing suppression are proved by Lyapunov-based stability analysis method. Experimental results show that the proposed control strategy has better control performance.
Key words: Quadrotor unmanned aerial vehicle (UAV)     slung-load     unknown parameter     adaptive control     anti-swing control    

近年来, 四旋翼无人机机(Unmanned aerial vehicle, UAV)在自然灾害勘探、农业施肥、商用及军用等领域发挥着重要的作用, 研究人员展开了很多相关研究[1-4].利用四旋翼无人机运送货物, 是四旋翼无人机应用研究的重要方向之一.

无人机吊挂飞行日益得到国内外研究人员的广泛关注, 与桥式吊车等运送系统[5-7]类似, 其控制目的在于将悬挂的负载平稳、安全、高效地运送到指定目标位置.针对四旋翼无人机吊挂飞行控制研究, 国内外研究团队提出了不同的控制方法.其中, 美国宾夕法尼亚大学的研究人员较早展开了相关研究, 取得了一些较好的研究成果.文献[8]中, 研究人员针对存在障碍物的环境, 设计了一种轨迹规划方法, 引导四旋翼吊挂系统避障飞行.文中采用了混合整数二次规划(Mixed integer quadratic program)方法对吊挂系统建模.为使负载顺利通过比较狭窄的障碍空间, 文中采用了两种不同的几何控制方法(Geometric control)对模型进行控制.文献[9]中, 研究人员针对四旋翼吊挂系统具有的微分平滑(Differential flatness)特性, 提出了一种轨迹生成方法, 并设计了一种控制器, 同时保证了无人机姿态、负载姿态及负载位置的稳定跟踪.文献[10]中, 研究人员利用几何控制和微分平滑方法对四旋翼无人机吊挂系统进行控制, 取得了位置和摆角几乎全局指数稳定的控制效果.另外, 美国新墨西哥大学也进行了关于四旋翼无人机吊挂控制的研究, 在文献[11]中, 研究人员使用近似值迭代(Approximate value iteration)的强化学习算法(Reinforcement learning)生成运动轨迹, 该方法对噪声具有较好的鲁棒性, 并进行了实验验证.文献[12]中, 在吊挂负载质量未知的情况下, 利用几何控制与最小二乘估计(Least-squares estimation)方法设计自适应控制器, 有效补偿了负载质量的不确定性, 实现了四旋翼无人机的安全起飞, 最后做了数值仿真对算法进行了验证.另外, 其他一些研究团队进行无人机吊挂飞行控制的研究, 文献[13]中, 研究人员提出了一种嵌套饱和(Nested saturation)的方法对吊挂系统进行控制, 并通过数值仿真验证了该方法的有效性.

关于无人机吊挂飞行系统的控制, 目前研究人员已经取得了一定的成果, 但仍然存在一些局限性: 1)一些已有的控制设计, 对被控对象的动态特性进行了较多的假设与简化, 例如假设负载摆角足够小, 然后在此基础上设计控制器.但在实际情况下, 应考虑当摆角出现大的波动时, 需设计必要的反馈控制来保证系统的稳定; 2)又如一些控制方法, 在平衡点附近对被控模型进行线性化处理, 但未考虑非平衡点处的稳定性与控制问题; 3)目前已有的多数控制方法, 建立在悬挂负载质量已知的情况下, 但在一些实际应用中, 存在载荷质量改变的情况, 有效的控制方法需要一定程度上不依赖于负载质量; 4)目前大多数的非线性控制方法, 只是进行了数值仿真, 未进行真实环境下的飞行验证, 对于实际应用仍存在一定的风险.

基于以上分析, 本文研究四旋翼无人机吊挂系统的二维动力学模型, 该模型考虑了系统所受的空气阻尼作用.针对存在未知负载质量和空气阻尼系数的情况, 本文设计的自适应控制器能实现四旋翼无人机的位置控制及负载摆角的快速抑制, 同时设计的参数更新律对模型中的未知对象参数进行在线估计.本文利用Lyapunov方法对控制器的稳定性进行了理论分析, 飞行实验验证了本文设计的自适应控制器的有效性.本文关于四旋翼无人机吊挂飞行系统研究的创新点主要包括: 1)尽管被控对象具有非线性、欠驱动、强耦合等特性, 本文未对被控对象的动态特性作过多的假设, 没有进行线性化处理, 也未对负载摆角进行严格限制; 2)本文在基于负载小球质量和空气阻尼系数未知的情况下, 通过一种参数自适应律, 对未知参数进行了在线估计, 补偿了未知负载质量和空气阻尼系数; 3)本文对提出的控制策略, 进行了实际飞行实验验证, 并与线性二次调节器(Linear quadratic regulator, LQR)进行对比, 实验结果表明本文设计的非线性控制器具有更好的控制效果.

本文内容安排为:第1节, 给出了四旋翼吊挂飞行系统的平面动力学模型; 第2节, 设计了一种参数自适应律, 在线更新未知参数, 接着设计了基于能量的自适应耦合非线性控制器; 第3节, 利用Lyapunov定理进行了稳定性分析; 第4节, 展示了实验结果, 并与LQR控制器的控制效果进行了对比; 最后, 在第5节中, 对本文设计进行了总结, 并对后续研究进行了展望.

1 动力学模型

四旋翼无人机吊挂飞行系统的结构简图如图 1所示, 其中, 图 1 (a)是其三维模型示意图, 图 1 (b)是其二维平面模型示意图.

图 1 四旋翼无人机吊挂飞行系统结构简图 Figure 1 Schematic of quadrotor UAV slung-load system

通过数学推导, 可获得四旋翼无人机吊挂飞行系统平面模型的动力学模型, 表达式为

$ \left\{ \begin{array} [c]{l} M_{c}(\pmb q)\ddot{\pmb q}+C(\pmb q, \dot{\pmb q})\dot{\pmb q}+\pmb G(\pmb q) = \pmb u+\pmb f_{d}\\ {J\ddot{\phi} = \tau} \end{array} \right. $ (1)

式中, $\pmb q(t) = [y(t), z(t), \theta(t)]^{\mathrm{T}}\in\mathbf{R}^{3} $, 表示系统的状态向量, $y(t)$是无人机$y$方向位移, $z(t)$是无人机$z$方向位移, $\theta (t)$是吊挂负载与机体竖直方向的夹角. $M_{c} (\pmb q)$, $C(\pmb q, \dot{\pmb q})\in \mathbf{R}^{3\times 3}$, $\pmb G(\pmb q)$, $\pmb u$, $\pmb f_{d}\in\mathbf{R}^{3}$分别表示系统的惯量矩阵, 向心力矩阵, 重力向量, 螺旋桨升力向量及空气阻力向量. $J$表示四旋翼无人机的转动惯量, $\phi(t) $表示四旋翼无人机的滚转角, $\tau $表示四旋翼无人机的姿态驱动力矩.此外, 在式(1)中, $M_{c}(\pmb q)$的表达式为

$ M_{c}(\pmb q) = \left[ \begin{array} [c]{ccc} m_{q}+m_{p}&0&m_{p}l\cos\theta\\ 0&m_{q}+m_{p}&m_{p}l\sin\theta\\ m_{p}l\cos\theta&m_{p}l\sin\theta&m_{p}l^{2} \end{array} \right] \label{1.2} $ (2)

$C(\pmb q, \dot{\pmb q})$的表达式为

$ C(\pmb q, \dot{\pmb q}) = \left[ \begin{array} [c]{ccc} 0&0&-m_{p}l\dot{\theta}\sin\theta\\ 0&0&m_{p}l\dot{\theta}\cos\theta\\ 0&0&0 \end{array} \right] \label{1.3} $ (3)

$\pmb G(\pmb q)$的表达式为

$ \pmb G(\pmb q) = \left[ \begin{array}{ccc} 0&(m_{q}+m_{p})g&m_{p}gl\sin \theta \end{array} \right] ^{\mathrm{T}} \label{1.4} $ (4)

$\pmb u$的表达式为

$ \pmb u = \left[ \begin{array} [c]{ccc} u_{y}&u_{z}&0 \end{array} \right] ^{\mathrm{T}} = \left[ \begin{array} [c]{ccc} -f\sin\phi&f\cos\phi&0 \end{array} \right] ^{\mathrm{T}} \label{1.5} $ (5)

$\pmb f_{d}$的表达式为

$ \pmb f_{d} = \left[ \begin{array} [c]{ccc} -d_{y}\dot{y}&-d_{z}\dot{z}&-c_{\theta}\dot{\theta} \end{array} \right] ^{\mathrm{T}} \label{1.6} $ (6)

式中, $d_{y}$, $d_{z} $, $c_{\theta}$是空气阻尼系数, 均为未知量, 根据已有文献[14-15], 可将空气阻尼系数设为常数.将式(2) $\sim$式(6)代入式(1)中, 可得到系统动力学模型的展开式如下:

$ \left\{ \begin{array}{l} (m_{q}+m_{p})\ddot{y}+m_{p}l\cos \theta \ddot{\theta}- \\ \;\;\;\;\;\;m_{p}l\dot{\theta}^{2}\sin \theta = u_{y}-d_{y}\dot{y} \\ (m_{q}+m_{p})\ddot{z}+m_{p}l\sin \theta \ddot{\theta}+m_{p}l\dot{\theta} ^{2}\cos \theta + \\ \;\;\;\;\;\;(m_{q}+m_{p})g = u_{z}-d_{z}\dot{z} \\ m_{p}l\cos \theta \ddot{y}+m_{p}l\sin \theta \ddot{z}+m_{p}l^{2}\ddot{\theta} + \\ \;\;\;\;\;\;m_{p}gl\sin \theta = -c_{\theta }\dot{\theta} \\ J\ddot{\phi} = \tau \end{array} \right. \label{1.7} $ (7)

另外, 四旋翼无人机的滚转角$\phi (t)$和无人机受到的总升力$f(t)$可根据$u_{y}(t)$$u_{z}(t)$计算得到, 表示如下:

$ \phi = \arctan(-\frac{u_{y}}{u_{z}}) \label{1.8} $ (8)
$ f = \sqrt{u_{y}^{2}+u_{z}^{2}} \label{1.9} $ (9)

可以证明系统的动力学模型具有如下性质:

1) 惯性矩阵$M_{c} (\pmb q)$是对称的正定矩阵, 且对于任意的向量$\pmb \chi \in\mathbf{R}^{3}$, 存在两个正数$\lambda_{m} $$\lambda_{M}$, 使得下式成立

$ \lambda_{m}\left\Vert \pmb \chi\right\Vert ^{2}\leq\pmb \chi^{\mathrm{T}}M_{c}(\pmb q)\pmb \chi\leq \lambda_{M}\left\Vert \pmb \chi\right\Vert ^{2} \label{2.0} $ (10)

2) 惯性矩阵$M_{c}(\pmb q)$和向心力矩阵$C(\pmb q, \dot{\pmb q})$具有如下关系.

$ \dot{M}_{c}(\pmb q) = C^{\mathrm{T}}(\pmb q, \dot{\pmb q})+C(\pmb q, \dot{\pmb q}) \label{2.2} $ (11)

本文研究基于负载小球质量未知的情况, 但负载小球质量的上下界已知, 如下式所示:

$ {\mathop m\limits_ -}_{p}\leq m_{p}\leq \bar{m}_{p} \label{2.3} $ (12)

根据实际飞行情况, 并依据已有研究成果[16-17], 可做出如下合理假设:

假设1.  负载小球始终在无人机的下方, 即负载小球相对于四旋翼无人机的摆角$\theta (t)$满足:

$ -\frac{\pi}{2}\leq\theta(t)\leq\frac{\pi}{2} \label{2.4} $ (13)

本文研究存在未知参数(负载小球质量、空气阻尼系数)的条件下, 四旋翼无人机吊挂飞行系统的自适应非线性控制算法设计.本文的研究目标是保证四旋翼无人飞行器纵向($y$方向)和垂直方向($z$方向)运动到目标位置, 同时负载摆角渐近收敛到0, 可用下述数学语言描述:

$ \underset{t\rightarrow\infty}{\lim}y(t) = y_{d}, \underset{t\rightarrow\infty }{\lim}z(t) = z_{d}, \underset{t\rightarrow\infty}{\lim}\theta(t) = 0 \label{2.5} $ (14)

式中, $y_{d}$, $z_{d} \in\mathbf{R}$表示无人机$y$, $z$方向的期望位置.

为方便后面控制器的设计, 现定义误差信号如下:

$ \begin{align}& e_{y}(t) = y(t)-y_{d}, e_{z}(t) = z(t)-z_{d} \nonumber\\ &\pmb e(t) = \left[ \begin{array}{ccc} e_{y}(t)&e_{z}(t)&\theta (t) \end{array} \right] ^{\mathrm{T}} \label{2.6} \end{align} $ (15)

对式(15)分别求时间的一阶导数和二阶导数, 可得:

$ \begin{align}& \dot{e}_{y}(t) = \dot{y}(t), \dot{e}_{z}(t) = \dot{z}(t) \nonumber\\ &\dot{\pmb e}(t) = \left[ \begin{array}{lll} \dot{y}(t)&\dot{z}(t)&\dot{\theta}(t) \end{array} \right] ^{\mathrm{T}} \nonumber\\ & \ddot{e}_{y}(t) = \ddot{y}(t), \ddot{e}_{z}(t) = \ddot{z}(t) \nonumber\\ &\ddot{\pmb e}(t) = \left[ \begin{array}{ccc} \ddot{y}(t)&\ddot{z}(t)&\ddot{\theta}(t) \end{array} \right] ^{\mathrm{T}} \label{2.7} \end{align} $ (16)
2 控制器设计

本节考虑四旋翼无人机吊挂飞行系统在具有不确定对象参数的条件下, 包括负载质量未知、空气阻力系数未知, 设计了一种参数自适应律, 在线估计未知对象参数.然后提出了一种基于能量的自适应耦合非线性控制方法.

四旋翼无人机吊挂飞行系统的机械能$E(t)$定义为

$ E = \frac{1}{2}\dot{\pmb q}^{\mathrm{T}}M_{c}\dot{\pmb q}+m_{p}gl(1-\cos\theta) \label{3.1} $ (17)

对式(17)求一阶时间导数为

$ \begin{align} \dot{E} = \, &\dot{\pmb q}^{\mathrm{T}}\left[ \pmb u+\pmb f_{d}-\pmb G(\pmb q) \right] +m_{p}gl\sin \theta \dot{\theta} = \notag \\ &\dot{y}(u_{y}-\pmb\phi _{y}\pmb\omega _{y})+\dot{z}(u_{z}-\pmb\phi _{z}^{ \mathrm{T}}\pmb\omega _{z})-c_{\theta }\dot{\theta}^{2} \label{3.2} \end{align} $ (18)

在式(18)中, 一些参量定义为

$ \begin{array}{ll} \pmb\phi _{y} = \left[ \dot{y}\right] ,&\pmb\phi _{z} = \left[ \begin{array}{cc} 1&\dot{z} \end{array} \right] ^{\mathrm{T}} \\ \pmb\omega _{y} = \left[ d_{y}\right] ,&\pmb\omega _{z} = \left[ \begin{array}{cc} (m_{q}+m_{p})g&d_{z} \end{array} \right] ^{\mathrm{T}} \end{array} \label{3.3} $ (19)

根据系统能量的无源性特性, 可设计控制输入$u_{y} (t)$, $u_{z}(t)$如下[18]:

$ \left\{ \begin{array}{l} u_{y}(t) = -k_{py}e_{y}(t)-k_{dy}\dot{y}(t)+k_{d\theta }\dot{\theta}(t)+\pmb \phi _{y}\hat{\pmb\omega }_{y} \\ u_{z}(t) = -k_{pz}e_{z}(t)-k_{dz}\dot{z}(t)+\pmb\phi _{z}^{\mathrm{T}}\hat{\pmb\omega } _{z} \end{array} \right. \label{3.5} $ (20)

式中, $k_{py}$, $k_{dy} $, $k_{d\theta}$, $k_{pz}$, $k_{dz}\in\mathbf{R}^{+} $, 为控制增益. $\hat{\pmb \omega}_{y} $, $\hat{\pmb \omega}_{z}$分别是$\pmb \omega_{y}$, $\pmb \omega _{z}$的在线估计值, 其定义如下:

$ \begin{array}{cc} \hat{\pmb\omega }_{y} = \left[ \hat{d}_{y}\right] ,&\hat{\pmb\omega }_{z} = \left[ \begin{array}{cc} (m_{q}+\hat{m}_{p})g&\hat{d}_{z} \end{array} \right] ^{\mathrm{T}} \end{array} \label{3.7} $ (21)

将控制器式(23)代入式(18)中, 可得:

$ \begin{align} \dot{E} = \, &-k_{dy}\dot{y}^{2}-k_{dz}\dot{z}^{2}-c_{\theta }\dot{\theta} ^{2}+k_{d\theta }\dot{\theta}\dot{y}- \notag \\ &k_{py}\dot{y}e_{y}-k_{pz}\dot{z}e_{z}-\pmb\phi _{y}\tilde{\pmb\omega }_{y} \dot{y}-\pmb\phi _{z}^{\mathrm{T}}\tilde{\pmb\omega }_{z}\dot{z} \label{3.8} \end{align} $ (22)

其中, $\tilde{\pmb \omega}_{y}$, $\tilde{\pmb \omega}_{z} $分别是$\pmb \omega_{y}$, $\pmb \omega_{z}$的在线估计误差, 定义如下:

$ \begin{array}{cc} \tilde{\pmb\omega} _{y} = \pmb\omega _{y}-\hat{\pmb\omega } _{y},&\tilde{\pmb \omega }_{z} = \pmb\omega _{z}-\hat{\pmb\omega } _{z} \end{array} \label{3.9} $ (23)

对式(23)求导, 可得:

$ \begin{array}{cc} \overset{\cdot }{\tilde{\pmb\omega }}_{y} = -\overset{\cdot }{\hat{\pmb\omega }}_{y},&\overset{\cdot }{\tilde{\pmb\omega }}_{z} = -\overset{\cdot }{\hat{ \pmb\omega}} _{z} \end{array} \label{3.10} $ (24)

$\hat{\pmb \omega}_{y}$$\hat{\pmb \omega}_{z}$的参数自适应律设计如下:

$ \left\{ \begin{array}{l} \overset{\cdot }{\hat{\pmb\omega }}_{y} = -\Gamma _{y}\pmb\phi _{y}\left[ \alpha \dot{y}+\rho \left(\dfrac{e_{y}}{2}\right)\right] \\ \overset{\cdot }{\hat{\pmb\omega }}_{z} = -\Gamma _{z}\pmb\phi _{z}\left[ \alpha \dot{z}+\rho \left(\dfrac{e_{z}}{2}\right)\right] \end{array} \right. \label{3.11} $ (25)

其中, $\Gamma_{y} = \gamma_{y}>0$, $\Gamma _{z} = $diag$\{\gamma _{m_{p}}, \gamma _{z}\}>0$, $\alpha\in\mathbf{R}^{+} $是正实数参数.式(25)中$\rho(s)$是一个微分饱和函数, 其定义如下:

$ \rho(s) = \begin{cases} 1, &s>\dfrac{\pi}{2}\\ \sin(s), &\left\vert s\right\vert \leq\dfrac{\pi}{2}\\[2mm] -1, &s<-\dfrac{\pi}{2} \end{cases} \label{3.13} $ (26)

可以证明, 式(26)中的$\rho(s)$函数具有如下性质:

1) 对任意$s\in\mathbf{R}$, $\left\vert \rho(s)\right\vert \leq1$;

2) 定义$\rho(s)$关于$s$的导数为

$ \rho_{s}(s) = \frac{\partial\rho(s)}{\partial s} = \begin{cases} 0, &\left\vert s\right\vert >\dfrac{\pi}{2}\\[2mm] \cos(s), &\left\vert s\right\vert \leq\dfrac{\pi}{2} \end{cases} \label{3.14} $ (27)

且对任意$s\in\mathbf{R}$, $\left\vert \rho _{s}(s)\right\vert \leq1$;

3) 对任意$s\in\mathbf{R}$, $\rho ^{2}(s)\leq s\rho(s)$.

3 稳定性证明

定理1.  对于如式(7)所示的四旋翼无人机吊挂飞行系统, 在未知负载质量$m_{p} $, 及空气阻力项$\pmb f_{d}$系数$d_{y}$, $d_{z}$, $c_{\theta}$均未知的情况下, 式(25)中的负载质量估计方法, 能使负载质量估计值渐近收敛到真实值; 式(20)中的非线性控制器可以使四旋翼无人机从起始位置渐近收敛到目标位置, 同时使负载摆角较快地渐近收敛到0, 即

$ \left\{ \begin{array}{l} \underset{t\rightarrow \infty }{\lim }\left[ \begin{array}{ccc} y&z&\theta \end{array} \right] = \left[ \begin{array}{ccc} y_{d}&z_{d}&0 \end{array} \right] \\[2mm] \underset{t\rightarrow \infty }{\lim }\left[ \begin{array}{ccc} \dot{y}&\dot{z}&\dot{\theta} \end{array} \right] = \left[ \begin{array}{ccc} 0&0&0 \end{array} \right] \\ \underset{t\rightarrow \infty }{\lim }\hat{m}_{p} = m_{p} \end{array} \right. \label{p} $ (28)

证明.选取如下Lyapunov候选函数$V(t)$:

$ \begin{align} V(t) = \, &\frac{1}{2}\alpha \dot{\pmb q}^{\mathrm{T}}M_{c}(\pmb q)\dot{\pmb q} +\alpha m_{p}gl(1-\cos \theta )+ \notag \\ &\frac{1}{2}\alpha k_{py}e_{y}^{2}+\frac{1}{2}\alpha k_{pz}e_{z}^{2}+\pmb \rho _{e}^{\mathrm{T}}\left(\frac{\pmb e}{2}\right)M_{c}(\pmb q)\dot{\pmb q}+ \notag \\ &\frac{1}{2}\tilde{\pmb\omega }_{y}^{\mathrm{T}}\Gamma _{y}^{-1}\tilde{\pmb \omega }_{y}+\frac{1}{2}\tilde{\pmb\omega }_{z}^{\mathrm{T}}\Gamma _{z}^{-1} \tilde{\pmb\omega }_{z} \label{4.1} \end{align} $ (29)

式(29)中, $\pmb \rho_{e}({\pmb e}/{2})$定义如下:

$ \pmb \rho_{e}\left(\frac{\pmb e}{2}\right) = \left[ \begin{array} [c]{ccc} \rho\left(\dfrac{e_{y}}{2}\right) & \rho\left(\dfrac{e_{z}}{2}\right) & \rho\left(\dfrac{\theta}{2}\right) \end{array} \right] ^{\mathrm{T}} \label{4.2} $ (30)

对于$V(t)$中的$\pmb \rho_{e}^{\mathrm{T}}({\pmb e}/{2})M_{c} (\pmb q)\dot{\pmb q}$项, 根据式(27)可知, 其符号与$\dot{\pmb q}^{\mathrm{T}}M_{c}(\pmb q)\dot{\pmb q}$符号一致, 所以$\pmb \rho _{e}^{\mathrm{T}}(\frac{\pmb e}{2})M_{c}(\pmb q)\dot{\pmb q}\geq0$, 易得$V(t)\geq0$, 可知Lyapunov候选函数是半正定的.

对式(29)求一阶时间导数, 可得:

$ \begin{align} \dot{V}(t) = \, &-\alpha k_{dy}\dot{y}^{2}-\alpha k_{dz}\dot{z}^{2}-\alpha c_{\theta }\dot{\theta}^{2}+\alpha k_{d\theta }\dot{\theta}\dot{y}- \notag \\ &\alpha \pmb\phi _{y}\tilde{\pmb\omega }_{y}\dot{y}-\alpha \pmb\phi _{z}^{ \mathrm{T}}\tilde{\pmb\omega }_{z}\dot{z}+\pmb\rho _{e}^{\mathrm{T}} \left(\frac{\pmb e }{2}\right)M_{c}(\pmb q)\ddot{\pmb q}+ \notag \\ &\pmb\rho _{e}^{\mathrm{T}}\left(\frac{\pmb e}{2}\right)\dot{M}_{c} (\pmb q)\dot{\pmb q}+ \dot{\pmb\rho }_{e}^{\mathrm{T}}\left(\frac{\pmb e}{2}\right)M_{c}(\pmb q)\dot{\pmb q}+ \notag \\ &\tilde{\pmb\omega }_{y}^{\mathrm{T}}\Gamma _{y}^{-1}\overset{\cdot }{ \tilde{\pmb\omega }}+\tilde{\pmb\omega }_{z}^{\mathrm{T} }\Gamma _{z}^{-1}\overset{\cdot }{\tilde{\pmb\omega }}_{z} \label{4.3} \end{align} $ (31)

对于式(31)中的$\pmb \rho_{e} ^{\mathrm{T}}({\pmb e}/{2})M_{c}(\pmb q)\ddot{\pmb q}$项, 由式(1)可得:

$ M_{c}(\pmb q)\ddot{\pmb q} = \pmb u+\pmb f_{d}-C(\pmb q, \dot{\pmb q})\dot{\pmb q}-\pmb G(\pmb q) = \pmb Q \label{4.4} $ (32)

其中, $\pmb Q$的表达式为

$\small \pmb Q = \left[ \begin{array}{l} -k_{py}e_{y}-k_{dy}\dot{y}+k_{d\theta }\dot{\theta}(t)-\pmb\phi _{y}\tilde{ \pmb\omega }_{y}-m_{p}l\dot{\theta}^{2}\sin \theta \\ -k_{pz}e_{z}-k_{dz}\dot{z}-\pmb\phi _{z}^{\mathrm{T}}\tilde{\pmb\omega } _{z}-m_{p}l\dot{\theta}^{2}\cos \theta \\ -c_{\theta }\dot{\theta}-m_{p}gl\sin \theta \end{array} \right] \label{4.4.1} $ (33)

根据式(32), 可得:

$ \begin{eqnarray} &&\pmb\rho _{e}^{\mathrm{T}}\left(\frac{\pmb e}{2}\right)M_{c}(\pmb q)\ddot{\pmb q} = -k_{py}e_{y}\rho \left(\frac{e_{y}}{2}\right)-\notag \\ &&\qquad k_{dy}\dot{y}\rho \left(\frac{e_{y}}{2}\right)+ k_{d\theta }\dot{\theta}(t)\rho \left(\frac{e_{y}}{2}\right)-\notag \\ &&\qquad\pmb\phi _{y} \tilde{\pmb \omega }_{y}\rho \left(\frac{e_{y}}{2}\right)- m_{p}l\dot{\theta}^{2}\sin \theta \rho \left(\frac{e_{y}}{2}\right)- \notag \\ &&\qquad k_{pz}e_{z}\rho ( \frac{e_{z}}{2})- k_{dz}\dot{z}\rho \left(\frac{e_{z}}{2}\right)- \notag \\ &&\qquad \pmb\phi _{z}^{\mathrm{T}}\tilde{\pmb \omega }_{z}\rho \left(\frac{e_{z}}{2}\right)- m_{p}l\dot{\theta}^{2}\cos \theta \rho \left(\frac{e_{z}}{2}\right)-\notag \\ &&\qquad c_{\theta }\dot{ \theta}\rho \left(\frac{\theta }{2}\right)- m_{p}gl\sin \theta \rho \left(\frac{\theta }{2}\right) \label{4.5} \end{eqnarray} $ (34)

对于式(31)中的$\pmb \rho_{e} ^{\mathrm{T}}({\pmb e}/{2})\dot{M}_{c}(\pmb q)\dot{\pmb q}$项, 将式(11)代入, 可得:

$ \begin{align} &\pmb \rho_{e}^{\mathrm{T}}\left(\frac{\pmb e}{2}\right)\dot{M}_{c}(\pmb q)\dot{\pmb q} = \pmb \rho_{e}^{\mathrm{T}}\left(\frac{\pmb e} {2}\right)C(\pmb q, \dot{\pmb q})\dot{\pmb q}+\nonumber\\&\qquad\dot{\pmb q}^{\mathrm{T}} C(\pmb q, \dot{\pmb q})\pmb \rho_{e}\left(\frac{\pmb e} {2}\right)\label{4.6} \end{align} $ (35)

对于式(31)中的$\tilde {\pmb \omega}_{y}^{\mathrm{T}}\Gamma_{y}^{-1}\overset{\cdot}{\tilde{\pmb \omega}}_{y} +\tilde{\pmb \omega}_{z}^{\mathrm{T}}\Gamma_{z}^{-1}\overset{\cdot}{\tilde{\pmb \omega}}_{z} $项, 将式(25)代入, 可得:

$ \begin{align} &\tilde{\pmb\omega }_{y}^{\mathrm{T}}\Gamma _{y}^{-1}\overset{\cdot }{\tilde{ \pmb\omega }}_{y}+\tilde{\pmb\omega }_{z}^{\mathrm{T}}\Gamma _{z}^{-1} \overset{\cdot }{\tilde{\pmb\omega }}_{z} = \alpha \tilde{\pmb\omega }_{y}^{ \mathrm{T}}\pmb\phi _{y}\dot{y}+\tilde{\pmb\omega }_{y}^{\mathrm{T}}\pmb\phi _{y}\rho \left(\frac{e_{y}}{2}\right)+ \notag \\ &\qquad\alpha \tilde{\pmb\omega }_{z}^{\mathrm{T}}\pmb\phi _{z}\dot{z}+\tilde{\pmb \omega }_{z}^{\mathrm{T}}\pmb\phi _{z}\rho \left(\frac{e_{z}}{2}\right) \label{4.7} \end{align} $ (36)

将式(32) $\sim$式(36)代入式(31)中, 并消去同类项, 可得:

$ \begin{align} \dot{V}(t) = \, &-\alpha k_{dy}\dot{y}^{2}-\alpha k_{dz}\dot{z}^{2}-\alpha c_{\theta }\dot{\theta}^{2}+ \notag \\ &\alpha k_{d\theta }\dot{\theta}\dot{y}-k_{py}e_{y}\rho \left(\frac{e_{y}}{2}\right)-k_{pz}e_{z}\rho \left(\frac{e_{z}}{2}\right)- \notag \\ &k_{dy} \dot{y}\rho \left(\frac{e_{y}}{2}\right)- k_{dz}\dot{z}\rho \left(\frac{e_{z}}{2}\right)+\notag \\ &k_{d\theta }\dot{\theta}(t)\rho \left(\frac{ e_{y}}{2}\right)-c_{\theta }\dot{\theta}\rho \left(\frac{\theta }{2}\right)- \notag \\ &m_{p}gl\sin \theta \rho \left(\frac{\theta }{2}\right)+\dot{\pmb q}^{\mathrm{T}}C(\pmb q, \dot{\pmb q})\pmb\rho _{e}\left(\frac{\pmb e}{2}\right)+ \notag \\ &\dot{\pmb\rho }_{e}^{\mathrm{T}}\left(\frac{\pmb e}{2}\right)M_{c}(\pmb q)\dot{\pmb q} \end{align} $ (37)

对于式(37)中的$-k_{py} e_{y}\rho({e_y}/{2})-k_{pz}e_{z}\rho({e_z}/{2})$项, 由$\rho(s)$函数的性质(3)可得:

$ \begin{align} &-k_{py}e_{y}\rho\left(\frac{e_{y}}{2}\right)-k_{pz}e_{z} \rho\left(\frac{e_{z}}{2}\right)\leq\nonumber\\ &\quad -2k_{py}\rho^{2}\left(\frac{e_{y}}{2}\right)-2k_{pz}\rho^{2} \left(\frac{e_{z}}{2}\right) \end{align} $ (38)

对于式(37)中的$-k_{dy} \dot{y}\rho({e_y}/{2})-k_{dz}\dot{z}\rho({e_z}/{2})+k_{d\theta}\dot{\theta}(t) \rho({e_y}/{2})$项, 有下列不等式成立:

$ \begin{align} &-k_{dy}\dot{y}\rho \left(\frac{e_{y}}{2}\right)-k_{dz}\dot{z}\rho \left(\frac{e_{z}}{2} \right)+k_{d\theta }\dot{\theta}(t)\rho \left(\frac{e_{y}}{2}\right) \leq \nonumber\\ &\quad W(e_{y}, e_{z}, \dot{y}, \dot{z} , \dot{\theta}) \label{4.10} \end{align} $ (39)

其中, $W(e_{y}, e_{z}, \dot{y}, \dot{z}, \dot{\theta})$的表达式为

$ \begin{align} W(e_{y}, e_{z}, \dot{y}, \dot{z}, \dot{\theta}) = \, & \frac{k_{dy}^{2}}{2} \dot{y}^{2}+\frac{ 1}{2}\rho ^{2}\left(\frac{e_{y}}{2}\right)+ \notag \\ &\frac{k_{dz}^{2}}{2}\dot{z}^{2}+\frac{1}{2}\rho ^{2}\left(\frac{e_{z}}{2}\right)+ \notag \\ &\frac{k_{d\theta }^{2}}{2}\dot{\theta}^{2}+\frac{1}{2}\rho ^{2}\left(\frac{e_{y} }{2}\right) \label{4.10.1} \end{align} $ (40)

对于式(37)中的$-m_{p} gl\sin\theta\rho({\theta}/{2})$项, 有下列不等式成立:

$ \begin{align} -m_{p}gl\sin\theta\rho\left(\frac{\theta}{2}\right)\leq\ -2m_{p}gl\left(\sin\frac{\theta}{2} \cos\frac{\theta}{2}\right)^{2} \label{4.11} \end{align} $ (41)

对于式(37)中的$\dot{\pmb q} ^{\mathrm{T}}C(\pmb q, \dot{\pmb q})\pmb \rho_{e}({\pmb e}/{2})$项, 代入式(3)和式(30), 可得:

$ \dot{\pmb q}^{\mathrm{T}}C(\pmb q, \dot{\pmb q})\pmb \rho_{e} \left(\frac{\pmb e}{2}\right)\leq\frac{m_{p}^{2}}{4}\dot{y} ^{2}+\frac{m_{p}^{2}}{4}\dot{z}^{2}+2l^{2}\dot{\theta}^{2}\label{4.12} $ (42)

对于式(37)中的$\dot{\pmb \rho }_{e}^{\mathrm{T}}({\pmb e}/{2})M_{c}(\pmb q)\dot{\pmb q}$项, 由式(10)可得:

$ M_{c}(\pmb q)\dot{\pmb q}\leq\dot{\pmb q}^{\mathrm{T}}M_{c}(\pmb q)\dot{\pmb q}\leq\lambda_{M}\left\Vert \dot {\pmb q}\right\Vert ^{2} \label{4.13} $ (43)

$N(\cdot) = \dot{\pmb \rho }_{e}^{\mathrm{T}}({\pmb e}/{2})M_{c}(\pmb q)\dot{\pmb q}$, 则由式(43)和$\rho(s)$函数的性质(2)可得:

$ \begin{align} N(\cdot ) = ~&\dot{\pmb\rho }_{e}^{\mathrm{T}}\left(\frac{\pmb e}{2}\right)M_{c}(\pmb q) \dot{\pmb q}\leq \notag \\ &\frac{\max \left\{ \begin{array}{ccc} \rho _{\frac{e_{y}}{2}}\left(\frac{e_{y}}{2}\right) & \rho _{\frac{e_{z}}{2}}\left(\frac{ e_{z}}{2}\right)&\rho _{\frac{\theta }{2}}\left(\frac{\theta }{2}\right) \end{array} \right\} }{2}\times\notag \\ &\lambda _{M}\left\Vert \dot{\pmb q}\right\Vert ^{2} \leq \frac{1}{2}\lambda _{M}(\dot{y}^{2}+\dot{z}^{2}+\dot{\theta}^{2}) \label{4.15} \end{align} $ (44)

将式(38) $\sim$式(42)和式(44)代入式(37)中, 可得:

$ \begin{eqnarray} \dot{V}(t) \!& \!\leq \!& \!-\left(\alpha k_{dy}-\frac{1}{2}\lambda _{M}-\frac{m_{p}^{2}}{4} \frac{k_{dy}^{2}}{2}\right)\dot{y}^{2}- \notag \\ &&\left(\alpha k_{dz}-\frac{1}{2}\lambda _{M}-\frac{m_{p}^{2}}{4}\frac{k_{dz}^{2} }{2}\right)\dot{z}^{2}- \notag \\ &&\left(\alpha c_{\theta }-\frac{1}{2}\lambda _{M}-2l^{2}+\frac{k_{d\theta }^{2}}{ 2}\right)\dot{\theta}^{2}- \notag \\ &&(2k_{py}-1)\rho ^{2}\left(\frac{e_{y}}{2}\right)- \notag \\ &&\left(2k_{pz}-\frac{1}{2}\right)\rho ^{2}\left(\frac{e_{z}}{2}\right) -c_{\theta }\dot{\theta} \rho \left(\frac{\theta }{2}\right)- \notag \\ &&2m_{p}gl\left(\sin \frac{\theta }{2}\cos \frac{\theta }{2}\right)^{2}+\alpha k_{d\theta }\dot{\theta}\dot{y} \label{4.16} \end{eqnarray} $ (45)

$\dot{V}(t)$可表示为

$ \dot{V}(t)\leq -\Lambda (e_{y}, e_{z}, \theta , \dot{y}, \dot{z}, \dot{\theta} )-c_{\theta }\dot{\theta}\rho \left(\frac{\theta }{2}\right) \label{4.16.0} $ (46)

其中, $\Lambda(e_{y}, e_{z}, \theta, \dot{y}, \dot{z}, \dot{\theta})$定义如下:

$ \begin{align} \Lambda = \,&\left(\alpha k_{dy}-\frac{1}{2}\lambda _{M}- \frac{m_{p}^{2}}{4}\frac{ k_{dy}^{2}}{2}\right)\dot{y}^{2}+\notag \\ &(2k_{py}-1)\rho ^{2} \left(\frac{e_{y}}{2}\right)+ \notag \\ &\left(\alpha k_{dz}-\frac{1}{2}\lambda _{M}-\frac{m_{p}^{2}}{4}\frac{k_{dz}^{2} }{2}\right)\dot{z}^{2}+\notag \\ &\left(2k_{pz}-\frac{1}{2}\right)\rho ^{2}\left(\frac{e_{z}}{2}\right)+ \notag \\ &\left(\alpha c_{\theta }-\frac{1}{2}\lambda _{M}-2l^{2}+\frac{k_{d\theta }^{2}}{ 2}\right)\dot{\theta}^{2}+\alpha k_{d\theta }\dot{\theta}\dot{y}+ \notag \\ &2m_{p}gl\left(\sin \frac{\theta }{2}\cos \frac{\theta }{2}\right)^{2} \label{4.16.1} \end{align} $ (47)

当参数$\alpha$, $k_{dy}$, $k_{dz}$, $k_{py}$, $k_{pz}$, $k_{d\theta}$满足以下条件时,

$ \begin{align} \alpha = \, &k_{dy} = k_{dz}>\nonumber\\ &\max \left\{ \dfrac{\sqrt{2}}{2}\sqrt{2\lambda _{M}+\bar{m}_{p}}, \qquad\dfrac{\lambda _{M}+4l^{2}}{2c_{\theta }}\right\}\nonumber\\ &k_{py}>\dfrac{1}{2}, k_{pz}>\dfrac{1}{4}, k_{d\theta }<\dfrac{2\sqrt{ \epsilon r_{3}}}{\alpha } \label{4.17.1} \end{align} $ (48)

$-\Lambda(e_{y}, e_{z}, \theta, \dot{y}, \dot{z}, \dot{\theta})$是负定的.式(48)中, $\epsilon$$r_{3}$的表达式为

$ \left\{ \begin{array} [c]{c} \epsilon = \dfrac{1}{2}\alpha^{2}-\dfrac{1}{2}\lambda_{M}-\dfrac{\bar{m} _{p}^{2}}{4}\\[2mm] r_{3} = \alpha c_{\theta}-\dfrac{1}{2}\lambda_{M}-2l^{2} \end{array} \right. $ (49)

下面将进一步分析如何获得式(48)中的参数条件,

$ \begin{align} -\Lambda = \, &-f_{1}\dot{y}^{2}-f_{2}\dot{z}^{2}-f_{3}\dot{\theta} ^{2}-f_{4}\rho ^{2}\left(\dfrac{e_{y}}{2}\right)- \notag \\ &f_{5}\rho ^{2}\left(\dfrac{e_{z}}{2}\right)+\alpha k_{d\theta }\dot{\theta}\dot{y}- \notag \\ &2m_{p}gl\left(\sin \dfrac{\theta }{2}\cos \dfrac{\theta }{2}\right)^{2} \label{4.18} \end{align} $ (50)

其中, $f_{1}, f_{2}, f_{3}, f_{4}, f_{5}$的定义如下:

$ \left\{ \begin{array}{l} f_{1} = \alpha k_{dy}-\dfrac{1}{2}\lambda _{M}-\dfrac{m_{p}^{2}}{4}\dfrac{ k_{dy}^{2}}{2} \\[2mm] f_{2} = \alpha k_{dz}-\dfrac{1}{2}\lambda _{M}-\dfrac{m_{p}^{2}}{4}\dfrac{ k_{dz}^{2}}{2} \\[2mm] f_{3} = \alpha c_{\theta }-\dfrac{1}{2}\lambda _{M}-2l^{2} \\ f_{4} = 2k_{py}-1 \\[2mm] f_{5} = 2k_{pz}-\dfrac{1}{2} \end{array} \right. \label{4.18.1} $ (51)

现定义函数$S(t)$, 其表达式为

$ \begin{eqnarray} S \!&\! = \!&\!-f_{1}\dot{y}^{2}-f_{2}\dot{z}^{2}-f_{3}\dot{\theta}^{2}+\alpha k_{d\theta }\dot{\theta}\dot{y} = \notag \\ &&-\left[ \begin{array}{ccc} \dot{y}&\dot{z}&\dot{\theta} \end{array} \right] P\left[ \begin{array}{c} \dot{y} \\ \dot{z} \\ \dot{\theta} \end{array} \right] \label{4.19} \end{eqnarray} $ (52)

其中, 三阶矩阵$P$的表达式为:

$ P = \left[ \begin{array} [c]{ccc} f_{1}&0&-\dfrac{1}{2}\alpha k_{d\theta}\\ 0&f_{2}&0\\ -\dfrac{1}{2}\alpha k_{d\theta}&0&f_{3} \end{array} \right] \label{4.20} $ (53)

则式(52)可写成

$ S = -\dot{\pmb e}^{\mathrm{T}}(t)P\dot{\pmb e}(t) \label{4.20.1} $ (54)

将式(54)代入式(50), 可得:

$ \begin{eqnarray} -\Lambda \!&\! = \!&\!-\dot{\pmb e}^{\mathrm{T}}(t)P\dot{\pmb e}(t)-f_{4}\rho ^{2}\left( \frac{e_{y}}{2}\right)- \notag \\ &&f_{5}\rho ^{2}\left(\frac{e_{z}}{2}\right)-2m_{p}gl\left(\sin \frac{\theta }{2}\cos \frac{\theta }{2}\right)^{2} \label{4.19.1} \end{eqnarray} $ (55)

要保证函数$-\Lambda(e_{y}, e_{z}, \theta, \dot {y}, \dot{z}, \dot{\theta})$负定, 则需保证矩阵$P$正定, 由此可得:

$ \left\{ \begin{array} [c]{l} f_{1}>0\\ f_{1}f_{2}>0\\ {f_{1}f_{2}f_{3}-\left(\dfrac{1}{2}\alpha k_{d\theta}\right)^{2}f_{2}>0} \end{array} \right. \label{4.211} $ (56)

由式(56)可得:

$ \left\{ \begin{array} [c]{l} f_{1}>0\\ f_{3}>0\\ {k_{d\theta}<\dfrac{2\sqrt{f_{1}f_{3}}}{\alpha}} \end{array} \right. \label{4.21} $ (57)

由于已知负载小球质量的上下界, 根据式(12), 则可得:

$ k_{d\theta}<\frac{2\sqrt{\epsilon r_{3}}}{\alpha} \label{4.22} $ (58)

在式(58)中, $\epsilon$$r_{3}$定义为

$ \left\{ \begin{array} [c]{c} \epsilon = \dfrac{1}{2}\alpha^{2}-\dfrac{1}{2}\lambda_{M}-\dfrac{\bar{m} _{p}^{2}}{4}\\[2mm] r_{3} = \alpha c_{\theta}-\dfrac{1}{2}\lambda_{M}-2l^{2} \end{array} \right. \label{4.23} $ (59)

另外, 为使$-\Lambda(e_{y}, e_{z}, \theta, \dot{y}, \dot{z}, \dot{\theta})$负定, 需满足:

$ \left\{ \begin{array} [c]{c} f_{4} = 2k_{py}-1>0\\ f_{5} = 2k_{pz}-\dfrac{1}{2}>0 \end{array} \right. $

进而得到:

$ \begin{array}{cc} k_{py}>\dfrac{1}{2},&k_{pz}>\dfrac{1}{4} \end{array} \label{4.24} $ (60)

综上, 由式(57) $\sim$式(60)可以得到关于参数$\alpha $, $k_{dy}$, $k_{dz}$, $k_{py}$, $k_{pz} $, $k_{d\theta}$的限定条件, 即式(48)和式(49).

对式(45)两边同时积分, 可得:

$ \begin{align} V(t)-V(0) \leq &-\int_{0}^{t}\Lambda \mathrm{d}\tau +\notag \nonumber\\ &2c_{\theta }\left[ \cos \frac{\theta (\tau )}{2}-\cos \frac{\theta (0)}{2}\right] \leq 4c_{\theta } \label{4.25} \end{align} $ (61)

根据式(61)可得:

$ V(t)\leq V(0)+4c_{\theta}\ll+\infty\label{4.26} $ (62)

则由式(29)和式(62)可知, $\dot{y}$, $\dot{z}$, $\dot {\theta}$, $e_{y}$, $e_{z}$, $\tilde{\pmb \omega}_{y} $, $\tilde{\pmb \omega}_{z}\in L_{\infty}$.进而结合式(20)可知, $u_{y}$, $u_{z}\in L_{\infty} $.结合系统模型, 即式(7)第1 $\sim$ 3行, 可得, $\ddot{y} $, $\ddot{z}$, $\ddot{\theta}\in L_{\infty}$.另外, 由于$V(t)\in L_{\infty} $, 则可得$\int_{0}^{t}\Lambda(e_{y}, e_{z}, \theta, \dot{y}, \dot{z}, \dot{\theta})\mathrm{d}\tau\in L_{\infty}$, 根据式(50), 可知,

$ \dot{y}, \dot{z}, \dot{\theta}, \rho\left(\frac{e_{y}}{2}\right), \rho\left(\frac{e_{z}}{2} \right), \sin\theta\in L_{2} \label{4.27} $ (63)

则由芭芭拉定理[19]可得:

$ \begin{array} [c]{ccc} e_{y}, e_{z}, \theta, \dot{y}, \dot{z}, \dot{\theta}\rightarrow0,& & t\rightarrow\infty \end{array} \label{4.28} $ (64)

将式(64)代入式(20)第1行可知,

$ \underset{t\rightarrow \infty }{\lim }u_{y} = 0 \label{4.28.1} $ (65)

将式(63)和式(64)代入式(7)第1行和第3行中, 可得:

$ \begin{align} &\underset{t\rightarrow \infty }{\lim }\left[ (m_{q}+m_{p})\ddot{y}+m_{p}l \ddot{\theta}\right] = 0 \nonumber\\ &\underset{t\rightarrow \infty }{\lim }\left[ m_{p}l\ddot{y}+m_{p}l^{2}\ddot{ \theta}\right] = 0 \label{4.29} \end{align} $ (66)

可推得

$ \begin{array}{cc} \underset{t\rightarrow \infty }{\lim }\ddot{y} = 0, & \underset{t\rightarrow \infty }{\lim }\ddot{\theta} = 0 \end{array} \label{4.31} $ (67)

由式(7)第2行和式(20)第2行可得

$ (m_{q}+m_{p})\ddot{z} = -\tilde{m}_{p}g+\Xi \label{4.36} $ (68)

其中, $\Xi$的定义如下:

$ \Xi = -k_{pz}e_{z}-k_{dz}\dot{z}+\dot{z}\hat{d}_{z}-\dot{z}d_{z}-m_{p}l(\sin \theta \ddot{\theta}+\dot{\theta}^{2}\cos \theta ) \label{4.37} $ (69)

由式(64)和式(67)可得

$ \lim\limits_{t\rightarrow\infty}\Xi = 0 \label{4.38} $ (70)

由式(25)第2行知

$ \overset{\cdot}{\tilde{m}}_{p}g = -\gamma_{m_{p}}\left[ \alpha\dot{z} +\rho\left(\frac{e_{z}}{2}\right)\right] \in L_{\infty} \label{4.39} $ (71)

则由式(68)、式(70)和式(71), 根据扩展芭芭拉定理, 可得

$ \begin{array}{cc} \lim\limits_{t\rightarrow \infty }\ddot{z} = 0, & \lim\limits_{t\rightarrow \infty }\tilde{m}_{p}g = 0 \end{array} \label{4.40} $ (72)

由此可得

$ \lim\limits_{t\rightarrow\infty}\hat{m}_{p} = m_{p} \label{4.41} $ (73)
4 实验验证

为了验证本文设计的非线性控制器对四旋翼无人机吊挂飞行系统的实际控制效果, 本文采用如图 2所示的实验平台进行了实验.实验平台的相关参数为: $m_{q} = 1.0082$ kg、$m_{p} = 0.076$ kg、$l = 1.085$ m及$g = 9.81$ m/s$^{2}$.设定的吊挂飞行系统的初始位置和目标位置为: $y_{0} = 1.5$ m、$z_{0} = -1.6$ m、$y_{d} = -1.0$ m及$z_{d} = -1.7$ m.

图 2 四旋翼无人机吊挂飞行系统实验平台 Figure 2 Experiment testbed of quadrotor UAV slung-load system

本文设计的自适应非线性控制器的相关参数为: $\sigma _{y} = 2.0$$\sigma_{m} = 1.2$$\sigma_{z} = 10.0$$\alpha = 0.5$$k_{py} = 4.9$$k_{pz} = 16.0$$k_{dy} = 8.0$$k_{dz} = 9.0$$k_{d\theta} = 0.7$.另外, 本文选择了LQR控制器进行了实验对比验证.需对四旋翼无人机吊挂系统进行线性化处理, 其状态空间描述为

$ \left\{ \begin{array} [c]{c} \dot{\pmb x}(t) = A\pmb x(t)+B\pmb v(t)\\ \pmb r(t) = C\pmb x(t)+D\pmb v(t) \end{array} \right. \label{lqr1} $ (74)

其中, $\pmb x(t) = [\dot{y}, \dot{z}, y, z, \dot{\theta}, \theta]^{\mathrm{T}}$为状态量, $\pmb v(t) = [u_{y}, u_{z}]^{\mathrm{T}}$为输入量, $\pmb r(t) = [y, z]^{\mathrm{T}}$为输出量, $A\in\mathbf{R} ^{6\times6}, B\in\mathbf{R}^{6\times2}, C\in\mathbf{R}^{2\times6}, D\in \mathbf{R}^{2\times2}$.借助MATLAB线性化工具, 可获得关于无人机吊挂系统的状态空间矩阵$A, B, C, D$的值, 分别为

$ \left\{ \begin{array}{l} A = \left[ \begin{array}{cccccc} 0&0&0&0&0&0.5785 \\ 0&0&0&0&0&0 \\ 1&0&0&0&0&0 \\ 0&1&0&0&0&0 \\ 0&0&0&0&0&-9.565 \\ 0&0&0&0&1&0 \end{array} \right] \\ \\ B = \left[ \begin{array}{cc} 0.9223&0 \\ 0&0.9223 \\ 0&0 \\ 0&0 \\ -0.8501&0 \\ 0&0 \end{array} \right] \\ \\ C = \left[ \begin{array}{cccccc} 0&0&1&0&0&0 \\ 0&0&0&1&0&0 \end{array} \right] \\[5mm] D = \left[ \begin{array}{cc} 0&0 \\ 0&0 \end{array} \right] \end{array} \right. \label{lqr22} $ (75)

设计线性二次最优指标$J(t)$

$ J = \frac{1}{2} %TCIMACRO{\dint } %BeginExpansion {\displaystyle\int} %EndExpansion [\pmb x^{\mathrm{T}}(t)Q\pmb x(t)+\pmb v^{\mathrm{T}}(t)R\pmb v(t)]\mathrm{d}t\label{lqr3} $ (76)

其中, $Q\in\mathbf{R}^{6\times6}, R\in\mathbf{R}^{2\times2} $均设为单位矩阵.通过MATLAB可获得式(76)中最优控制系统信号$\pmb v(t) = -K\pmb x(t)$.基于仿真结果, 在实际飞行实验中, 依据实际飞行状态, 对控制器参数做了一些相应调整, 最终确定的LQR控制器中矩阵$K$的表达式为

$ {\small\begin{align} &K = \nonumber\\ &\left[ \begin{array} [c]{cccccc} 4.354&0.077&3.743&0.412&0.213&0.683\\ -2.392&9.543&-0.930&9.642&0.037&-0.874 \end{array} \right] \label{lqr4} \end{align}} $ (77)

图 3描述了在两种控制器下系统的状态$y(t)$, $z(t)$, $\theta(t)$随时间变化的情况. 图 4描述了在两种控制器下$y$, $z$方向的控制输入信号量$u_{y} (t)$, $u_{z}(t)$随时间变化的情况. 图 5描述了飞行器的滚转角$\phi (t)$随时间变化的情况.

图 3 无人机位置$y(t), z(t)$及负载摆角$\theta(t)$ Figure 3 $y(t), z(t)$ of UAV and payload swing $\theta(t)$
图 4 无人机控制输入$u_{y} (t), u_{z}(t)$ Figure 4 Control inputs $u_{y}(t), u_{z}(t)$ of UAV
图 5 无人机滚转角$\phi(t)$ Figure 5 Roll angle $\phi(t)$ of UAV

本文中, 定义当四旋翼无人机到达目标位置并保持在规定误差带(目标位置的$\pm 5 \, \%$)内时, 则系统进入稳态.定义调节时间为四旋翼无人机进入稳态所需的最短时间.通过对图 3的动态过程进行具体分析, 可得到表 1, 表 1中是非线性控制器和LQR控制器调节时间对比结果.在非线性控制器下, $z$方向的位移$y(t)$, $z(t)$和负载摆角$\theta(t)$的调节时间均远远小于LQR控制器.且在LQR控制器下, $y(t)$, $z(t)$振荡较为剧烈, 尤其是$z$方向位移, 超调量达$0.19$ m.对于负载摆角$\theta(t)$, 尽管非线性控制器的振幅大, 约$7\, ^{\circ }$左右, LQR控制器的负载摆角相对较小, 约$5\, ^{\circ }$左右, 但非线性控制器比LQR控制器的收敛速度快, 摆角较快地收敛到较小值, 且振荡次数少.根据图 4图 5可知, 对于控制输入量$u_{y} (t)$和飞行器的滚转角$\phi(t)$, 从动态调节过程的前半部分来看, 非线性控制器振幅比LQR控制器大, 但后半部分, 非线性控制器振幅迅速大幅衰减.整体来看, 非线性控制器的动态过程收敛速度快, 振动幅度较快实现衰减, 且振荡次数相对较少.对于控制输入量$u_{z} (t)$, 非线性控制器和LQR控制器的动态过程均较快.总的来看, 对于控制器产生的控制输入量$u_{y} (t)$, $u_{z}(t)$及飞行器的滚转角$\phi(t)$, 非线性控制器的效果比LQR控制器好一些.

表 1 非线性控制器和LQR控制器调节时间对比 Table 1 Comparison of the settling time between nonlinear controller and LQR controller

图 3中的状态量的稳态过程进行定量分析, 选取第20秒$\sim$第80秒的数据, 分别对图中各个量求取了误差均值, 均方误差(Mean square error, MSE)和最大偏差. 表 2 $\sim$ 表 4分别是非线性控制器和LQR控制器稳态后误差均值、均方误差和最大偏差的对比.对于无人机$y$方向位移$y(t)$, 非线性控制器的误差均值、均方误差和最大偏差均小于LQR控制器, 能实现高精度地$y$方向位置控制, 且稳态过程的稳定性较好; 对于无人机$z$方向位移$z(t)$, $z$方向位置控制精度优于LQR控制器, 但非线性控制器的均方误差和最大偏差均略微大于LQR控制器; 负载摆角$\theta (t)$, 非线性控制器的误差均值和均方误差与LQR控制器基本相同, 但LQR控制器的最大偏差较大, 总的来看, 非线性控制器和LQR控制器的稳态性能均较好. 图 4中控制输入量$u_{y}(t)$, $u_{z} (t)$的稳态性能和$y$, $z$方向的位移$y(t)$, $z(t)$的稳态性能对应, 合理地解释了$y$, $z$方向的位移$y(t)$, $z(t)$的变化曲线, 间接体现了非线性控制器的优良性能.分析图 5可知, 非线性控制器下飞行器的滚转角$\phi (t)$的稳态过程好于LQR控制器.

表 2 非线性控制器和LQR控制器稳态误差均值对比 Table 2 Comparison of the steady-state mean error between nonlinear controller and LQR controller
表 3 非线性控制器和LQR控制器稳态均方误差对比 Table 3 Comparison of the steady-state mean square error between nonlinear controller and LQR controller
表 4 非线性控制器和LQR控制器稳态最大偏差对比 Table 4 Comparison of the steady-state maximum deviation between nonlinear controller and LQR controller
5 总结与展望

针对四旋翼无人机吊挂飞行系统, 本文基于未知对象参数的系统二维动力学模型, 设计了未知参数在线估计器, 补偿负载质量和空气阻尼作用.同时, 设计了自适应耦合非线性控制器, 该控制器能实现四旋翼吊挂飞行系统的位置控制并具有较好的减摆效果.最后, 在真实飞行实验下, 将本文设计的自适应耦合非线性控制器与LQR控制器的控制效果进行了对比, 结果表明本文设计的非线性控制器有更好的准确定位和快速减摆作用.

后续工作会考虑四旋翼无人机吊挂飞行系统的三维动态特性, 不仅仅考虑飞行器的滚转方向, 还会扩展到俯仰方向, 并进行相关的建立动态模型, 稳定性证明和实验验证等工作.另外, 后续将考虑四旋翼无人机吊挂飞行系统在受外界较强干扰的影响下的控制问题.

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