弹药传输机械臂固定时间终端滑模控制

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弹药传输机械臂固定时间终端滑模控制

姚来鹏,侯保林

(南京理工大学机械工程学院, 南京 210094)

摘要:

为提高坦克弹药传输机械臂在路面激励等外界扰动下位置控制的鲁棒性,设计一种固定时间终端滑模控制器（fixed-time terminal sliding mode controller, FTSMC）. 推导含垂直基础振动的弹药传输机械臂动力学方程,将系统的基础振动处理成干扰项. 采用新型固定时间收敛干扰观测器对系统不确定项进行补偿,改善了控制器的鲁棒性. 结合固定时间收敛双幂次趋近律和固定时间终端滑模面设计固定时间终端滑模控制器. 用Lyapunov理论证明了系统固定时间收敛特性. 3种工况下的对比实验表明,设计的复合控制器对不确定性干扰具有强鲁棒性,能够对外界扰动下的弹药传输机械臂进行准确定位控制.

关键词: 机械臂固定时间控制干扰观测器双幂次趋近律终端滑模

DOI：10.11918/201911102

分类号:TP273

文献标识码:A

基金项目:国家重点基础研究发展规划项目(973-61324901)；国家自然科学基金(51605344)

Fixed-time terminal sliding mode control for ammunition transfer manipulator

YAO Laipeng,HOU Baolin

(School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China)

Abstract:

To improve the robustness of position control of tank ammunition transfer manipulator under external disturbance such as road excitation, a fixed-time terminal sliding mode controller (FTSMC) was designed. Firstly, the dynamic model of the ammunition transfer manipulator with basic vibration was established, and the basic vibration of the system was processed into the model uncertain disturbance term. Then a new fixed-time convergence disturbance observer was designed to compensate the uncertainties of the system model and enhance the robustness of the controller, and a fixed-time terminal sliding mode controller was designed based on the fixed-time convergence double power reaching law and the fixed-time non-singular fast terminal sliding mode control surface. Finally, the fixed time convergence stability of closed-loop systems was proved by Lyapunov method. The comparison experimental results under three working conditions show that the designed composite controller is robust to model uncertain disturbance, and the accurate positioning of the ammunition transfer manipulator is realized. The experimental results verify the correctness and effectiveness of the algorithm in practical engineering applications.

Key words: manipulator fixed-time control disturbance observer double power reaching law terminal sliding mode

姚来鹏, 侯保林. 弹药传输机械臂固定时间终端滑模控制[J]. 哈尔滨工业大学学报, 2021, 53(1): 109-116. DOI: 10.11918/201911102. 复制到剪切板

YAO Laipeng, HOU Baolin. Fixed-time terminal sliding mode control for ammunition transfer manipulator[J]. Journal of Harbin Institute of Technology, 2021, 53(1): 109-116. DOI: 10.11918/201911102. 复制到剪切板

基金项目国家重点基础研究发展规划项目(973-61324901)；国家自然科学基金(51605344) 作者简介姚来鹏(1991—)，男，博士研究生;
侯保林(1965—)，男，教授，博士生导师通信作者侯保林，houbl@njust.edu.cn 文章历史收稿日期: 2019-10-31

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弹药传输机械臂固定时间终端滑模控制
姚来鹏, 侯保林
南京理工大学机械工程学院, 南京 210094

收稿日期: 2019-10-31
基金项目: 国家重点基础研究发展规划项目(973-61324901)；国家自然科学基金(51605344)
作者简介: 姚来鹏(1991—)，男，博士研究生; 侯保林(1965—)，男，教授，博士生导师
通信作者: 侯保林，houbl@njust.edu.cn

摘要: 为提高坦克弹药传输机械臂在路面激励等外界扰动下位置控制的鲁棒性，设计一种固定时间终端滑模控制器(fixed-time terminal sliding mode controller, FTSMC).推导含垂直基础振动的弹药传输机械臂动力学方程，将系统的基础振动处理成干扰项.采用新型固定时间收敛干扰观测器对系统不确定项进行补偿，改善了控制器的鲁棒性.结合固定时间收敛双幂次趋近律和固定时间终端滑模面设计固定时间终端滑模控制器.用Lyapunov理论证明了系统固定时间收敛特性. 3种工况下的对比实验表明，设计的复合控制器对不确定性干扰具有强鲁棒性，能够对外界扰动下的弹药传输机械臂进行准确定位控制.
关键词: 机械臂    固定时间控制    干扰观测器    双幂次趋近律    终端滑模
Fixed-time terminal sliding mode control for ammunition transfer manipulator
YAO Laipeng, HOU Baolin
School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China

Abstract: To improve the robustness of position control of tank ammunition transfer manipulator under external disturbance such as road excitation, a fixed-time terminal sliding mode controller (FTSMC) was designed. Firstly, the dynamic model of the ammunition transfer manipulator with basic vibration was established, and the basic vibration of the system was processed into the model uncertain disturbance term. Then a new fixed-time convergence disturbance observer was designed to compensate the uncertainties of the system model and enhance the robustness of the controller, and a fixed-time terminal sliding mode controller was designed based on the fixed-time convergence double power reaching law and the fixed-time non-singular fast terminal sliding mode control surface. Finally, the fixed time convergence stability of closed-loop systems was proved by Lyapunov method. The comparison experimental results under three working conditions show that the designed composite controller is robust to model uncertain disturbance, and the accurate positioning of the ammunition transfer manipulator is realized. The experimental results verify the correctness and effectiveness of the algorithm in practical engineering applications.
Keywords: manipulator    fixed-time control    disturbance observer    double power reaching law    terminal sliding mode
弹药传输机械臂是坦克炮弹药自动装填系统的重要组成部分，其作用是将弹药轴线协调到与待发射炮管轴线平行的位置^[1].坦克在运动时由于路面不平度的激励，车体会产生基础振动，基础振动对安装在车体上的弹药传输机械臂的定位精度产生严重影响.现有的弹药传输机械臂控制器设计大多基于固定参数的PID控制，鲁棒性较差^[2].因此，如何构造弹药传输机械臂的鲁棒控制器是一个亟待解决的难题.弹药传输机械臂是一种受扰动的不确定性系统.文献[2-3]根据一种特殊的Lyapunov函数构造了弹药传输机械臂的控制策略，仿真研究了有振动干扰的弹药传输机械臂位置控制，但是没有进行实验研究.

滑模控制是一类特殊的非线性控制方法，由于对匹配性模型不确定具有不变性，因而广泛应用在受扰动不确定系统中^[4-5].普通线性滑模在系统状态抵达滑模面后，只能无限逼近平衡点，不能有限时间收敛；而终端滑模可使系统状态有限时间收敛，且比线性滑模具有更好的收敛性能，但是普通终端滑模存在奇异问题.为了使绳驱动机械臂在复杂不确定工况下获得满意的控制性能，文献[6]设计了新型非奇异终端滑模方法，确保系统有限时间收敛，实现了两自由度绳驱动机械臂高精度控制.为了解决高超声速飞行器在受外界扰动下的飞行控制问题，文献[7]构造了一种时变滑模方法，理论推导证明了系统所有状态有限时间收敛，仿真验证了控制的有效性.

固定时间控制是一类特殊的有限时间控制，这类控制的收敛时间和系统初始条件情况无关，能够固定时间收敛.和一般的有限时间控制相比，固定时间控制中因为同时包含大于1和小于1的分数幂项，所以无论系统初始状态是在平衡点附近还是远离平衡点处，都能改善系统状态的收敛速度，保证系统在固定时间内收敛.文献[8]构造了新型固定时间快速终端滑模面，使滑模面的收敛时间不依赖系统初始条件，和传统的终端滑模控制器相比具有更好的收敛性能.文献[9]针对航天器受外界干扰下的位置姿态耦合控制问题，基于固定时间概念构造了终端滑模自适应控制策略，能较好地控制航天器的相对位置和姿态，干扰抑制能力好.文献[10]针对机动目标的末制导拦截问题，构造了一种新型固定时间终端滑模制导律，和有限时间制导律相比，收敛时间独立于系统初始条件，可以根据制导律预先给定，优化系统收敛速率，缩短拦截时间，提高拦截精度.文献[11]针对再入飞行器飞行环境复杂多变的特征，设计了全局滑模跟踪制导律，实现了固定时间收敛的鲁棒跟踪控制.文献[12]针对受海风海浪等扰动情况下潜射导弹姿态跟踪控制问题，设计了一种新型固定时间终端滑模控制器，典型导弹发射轨迹的仿真证明了控制策略鲁棒性良好.

在控制系统设计中联合干扰观测器技术形成复合控制，是解决不确定系统控制问题的有效方法之一.通过估计包含外界干扰及不确定性的复合扰动并加以补偿，可以有效抑制系统抖振.文献[13]针对无人机中存在不确定干扰问题，采用滑模干扰观测器对干扰进行估计，实现了无人机对期望指令的有限时间跟踪.文献[14]针对目标拦截过程中存在不确定性和建模误差的问题，构造了基于固定时间收敛干扰观测器的终端滑模制导律，仿真表明所设计的制导律有良好的收敛性能和抖振抑制能力.针对受有界扰动的动力学系统，文献[15]和文献[16]设计了一种基于超螺旋算法的固定时间干扰观测器，使得收敛时间不依赖系统初始状态，计算结果证明了控制策略有效.文献[17]针对含有参数摄动、外界干扰的某飞行器弹性模型含有强不确定性问题，采用固定时间超螺旋扰动观测器估计系统的不确定性并补偿到控制中，改善了控制器的鲁棒性.

为了解决弹药传输机械臂在基础振动情况下的准确定位问题，本文以某坦克弹药传输机械臂为研究背景，针对某弹药传输机械臂原理样机，设计了一种基于固定时间干扰观测器和固定时间终端滑模控制相结合的复合控制策略.固定时间干扰观测器用于估计和补偿系统总的干扰，改善系统的抗干扰性能.固定时间终端滑模用于设计弹药传输机械臂的主控制器，避免了传统终端滑模的奇异问题，并且使得系统状态的收敛时间存在和系统初始状态无关的可估计上界.采用Lyapunov稳定理论证明了系统固定时间收敛. 3种不同振动工况下的原理样机实验表明，与传统固定参数PID控制器相比，复合控制策略鲁棒性更强，精度更高，算法先进有效.

1 动力学模型图 1为弹药传输机械臂原理样机的工作原理.弹药传输机械臂主要由机架、升降部分、翻转部分组成.升降电机驱动主动链轮、传动链条、从动链轮转动，传动链条带动升降部分沿升降导轨进行升降运动.翻转部分由翻转电机驱动支臂进行回转运动.

Fig. 1
点击查看原图

1—升降电机；2—主动链轮；3—升降导轨；4—支臂；5—翻转电机；6—从动链轮；7—传动链条；8—机架图 1 弹药传输机械臂原理样机 Fig. 1 Prototype model of ammunition transfer manipulator

为便于分析，此处只考虑弹药传输机械臂在垂直平面内运动的动力学模型，如图 2所示.其中，XOY为笛卡尔坐标系；B₁、B₂、B₃分别表示安装机架、升降部分与翻转部分，C₂与C₃分别为B₂与B₃的质心；假设B₃的转轴过C₂点，y_r1为安装机架的基础振动位移；y_r2为B₂相对B₁的位移；θ₃为弹药传输机械臂支臂绕C₂的翻转角位移；L₃为C₂和C₃之间的距离.

Fig. 2
点击查看原图

图 2 弹药传输机械臂的简化动力学模型 Fig. 2 Simplified dynamics model of ammunition transfer manipulator

设B₂与B₃的质量分别为m₂与m₃，B₃相对其质心的转动惯量为J₃.选择q=[y_r2, θ₃]^T为系统的广义坐标，则由第二类Lagrange方程可得弹药传输机械臂的动力学模型为

$\mathit{\boldsymbol{H}}(\mathit{\boldsymbol{q}})\mathit{\boldsymbol{\ddot q}} + \mathit{\boldsymbol{C}}(\mathit{\boldsymbol{q}},\mathit{\boldsymbol{\dot q}})\mathit{\boldsymbol{\dot q}} + \mathit{\boldsymbol{G}}(\mathit{\boldsymbol{q}}) = \mathit{\boldsymbol{U}} + \mathit{\boldsymbol{D}}.$ (1)

式中：H(q)为系统的惯性矩阵；$\mathit{\boldsymbol{C}}(\mathit{\boldsymbol{q}},\mathit{\boldsymbol{\dot q}})\mathit{\boldsymbol{\dot q}} $为系统的向心力向量；G(q)为系统的重力向量；U为系统的主动控制力向量；D为由于基础振动引起的不确定干扰向量；各项表达式分别为：

$\mathit{\boldsymbol{H}}(\mathit{\boldsymbol{q}}) = \left[ {\begin{array}{*{20}{c}}{{m_2} + {m_3}}&{{m_3}{L_3}\cos {\theta _3}}\\{{m_3}{L_3}\cos {\theta _3}}&{{m_3}L_3^2 + {J_3}}\end{array}} \right],$

$\mathit{\boldsymbol{C}}(\mathit{\boldsymbol{q}},\mathit{\boldsymbol{\dot q}}) = \left[ {\begin{array}{*{20}{c}}0&{ - {m_3}{L_3}{{\dot \theta }_3}\sin {\theta _3}}\\0&0\end{array}} \right],$

$\mathit{\boldsymbol{G}}(\mathit{\boldsymbol{q}}) = \left[ {\begin{array}{*{20}{l}}{({m_2} + {m_3})g}\\{{m_3}g{L_3}\cos {\theta _3}}\end{array}} \right],$

$\mathit{\boldsymbol{U}} = \left[ {\begin{array}{*{20}{l}}{{K_{T2}}{I_2}{i_2}{\eta _2}}\\{{K_{T3}}{I_3}{i_3}{\eta _3}}\end{array}} \right],$

$\mathit{\boldsymbol{D}} = \left[ {\begin{array}{*{20}{l}}{ - ({m_2} + {m_3}){{\ddot y}_{r1}}}\\{ - {m_3}{L_3}\cos {\theta _3}{{\ddot y}_{r1}}}\end{array}} \right].$

式中：K_T2、K_T3分别为升降电机和翻转电机的转矩系数；I₂、I₃分别为升降电机和翻转电机的控制电流；i₂、i₃分别为升降部分和翻转部分的传动比；η₂、η₃分别为升降部分和翻转部分的传动效率.

2 固定时间控制器设计令${\mathit{\boldsymbol{q}}_1} = \mathit{\boldsymbol{q}},{\mathit{\boldsymbol{q}}_2} = {{\mathit{\boldsymbol{\dot q}}}_2}$，则动力学模型式(1)可化为

$\left\{ {\begin{array}{*{20}{l}}{{{\mathit{\boldsymbol{\dot q}}}_1} = {\mathit{\boldsymbol{q}}_2},}\\{{{\mathit{\boldsymbol{\dot q}}}_2} = {\mathit{\boldsymbol{H}}^{ - 1}}(\mathit{\boldsymbol{q}})\mathit{\boldsymbol{U}} - {\mathit{\boldsymbol{H}}^{ - 1}}(\mathit{\boldsymbol{q}})[\mathit{\boldsymbol{G}}(\mathit{\boldsymbol{q}}) + \mathit{\boldsymbol{C}}(\mathit{\boldsymbol{q}},\mathit{\boldsymbol{\dot q}})\mathit{\boldsymbol{\dot q}}] + {\mathit{\boldsymbol{D}}_1}.}\end{array}} \right.$ (2)

式中：D₁为系统总的不确定扰动项，D₁=H^-1(q)D=diag(D₁₁、D₁₂)；${{\dot D}_{11}}, {{\dot D}_{12}}$的上界分别为正常数A₁、A₂，即|${{\dot D}_{11}}$|≤A₁, |${{\dot D}_{12}}$|≤A₂.

2.1 固定时间干扰观测器设计受文献[18]启发，固定时间干扰观测器可设计为：

$\left\{ {\begin{array}{*{20}{l}}{{{\mathit{\boldsymbol{\dot {\hat q}}}}_2} = {\mathit{\boldsymbol{H}}^{ - 1}}(\mathit{\boldsymbol{q}})\mathit{\boldsymbol{U}} - {\mathit{\boldsymbol{H}}^{ - 1}}(\mathit{\boldsymbol{q}})[\mathit{\boldsymbol{C}}(\mathit{\boldsymbol{q}},\mathit{\boldsymbol{\dot q}})\mathit{\boldsymbol{\dot q}} + \mathit{\boldsymbol{G}}(\mathit{\boldsymbol{q}})] + {{\mathit{\boldsymbol{\hat D}}}_1},}\\{\mathit{\boldsymbol{\sigma }} = {\mathit{\boldsymbol{q}}_2} - {{\mathit{\boldsymbol{\hat q}}}_2},}\\{{{\mathit{\boldsymbol{\hat D}}}_1} = {k_1} {\rm{sig}}{ ^{1/2}}(\mathit{\boldsymbol{\sigma }}) + {k_2} {\rm{sig}}{ ^p}(\mathit{\boldsymbol{\sigma }}) + {k_3}\int_0^t {{\rm{ sign }}} (\mathit{\boldsymbol{\sigma }}){\rm{d}}\tau .}\end{array}} \right.$ (3)

式中：${{\mathit{\boldsymbol{\dot q}}}_2}$为q₂的估计；σ为观测误差；${{\mathit{\boldsymbol{\dot D}}}_{1}}$为D₁的估计；k₁、k₂、k₃、p为常数，k₁>0, k₂>0, k₃>max(A₁, A₂), p>1.

定理1 ??对于系统(2)，若干扰观测器设计为式(3)，则观测器的估计误差σ固定时间收敛到原点，收敛时间T₁满足

$\begin{array}{l}{T_1} \le \max \left\{ {{T_{1i}} = \left( {\frac{{2{\varepsilon ^{1/2}}}}{{{k_1}}} + \frac{1}{{{k_2}(p - 1){\varepsilon ^{p - 1}}}}} \right) \times } \right.\\\left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left( {1 + \frac{1}{{{w_i}\left( {\frac{1}{{{W_i}}} - \frac{{{h_1}({k_1})}}{{{k_1}}}} \right)}}} \right),(i = 1,2)} \right\}.\end{array}$

式中：w_i=k₃-A_i, W_i=k₃+A_i, ${h_1}({k_1}) = 1/{k_1} + {\left( {2{\rm{e}}/{w_i}{k_1}} \right)^{\frac{1}{3}}}$, k₁h₁^-1k₁>W_i, (i=1, 2), ε=(k₁/k₂)^1/(p+1/2).

证明??对σ求导，代入式(2)和式(3)可得

$\begin{array}{*{20}{l}}{\mathit{\boldsymbol{\dot \sigma }} = {{\mathit{\boldsymbol{\dot q}}}_2} - {{\mathit{\boldsymbol{\dot {\hat q}}}}_2} = {\mathit{\boldsymbol{H}}^{ - 1}}(\mathit{\boldsymbol{q}})\mathit{\boldsymbol{U}} - {\mathit{\boldsymbol{H}}^{ - 1}}(\mathit{\boldsymbol{q}})[\mathit{\boldsymbol{C}}(\mathit{\boldsymbol{q}},\mathit{\boldsymbol{\dot q}})\mathit{\boldsymbol{\dot q}} + }\\{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{G}}(\mathit{\boldsymbol{q}})] + {\mathit{\boldsymbol{D}}_1} - \{ {\mathit{\boldsymbol{H}}^{ - 1}}(\mathit{\boldsymbol{q}})\mathit{\boldsymbol{U}} - {\mathit{\boldsymbol{H}}^{ - 1}}(\mathit{\boldsymbol{q}})[\mathit{\boldsymbol{C}}(\mathit{\boldsymbol{q}},\mathit{\boldsymbol{\dot q}})\mathit{\boldsymbol{\dot q}} + }\\{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{G}}(\mathit{\boldsymbol{q}})] + {{\mathit{\boldsymbol{\hat D}}}_1}\} = {\mathit{\boldsymbol{D}}_1} - {{\mathit{\boldsymbol{\hat D}}}_1}.}\end{array}$

代入式(3)可得

$\left\{ {\begin{array}{*{20}{l}}{\mathit{\boldsymbol{\dot \sigma }} = - {k_1} {\rm{sig}}{ ^{\frac{1}{2}}}(\mathit{\boldsymbol{\sigma }}) - {k_2} {\rm{sig}}{ ^p}(\mathit{\boldsymbol{\sigma }}) + \psi ,}\\{\mathit{\boldsymbol{\dot \psi }} = - {k_3} {\rm{sign}} (\mathit{\boldsymbol{\sigma }}) + {{\mathit{\boldsymbol{\dot D}}}_1}.}\end{array}} \right.$

由文献[18]可知，观测误差σ在固定时间T₁内收敛到原点，T₁满足

$\begin{array}{l}{T_1} \le \max \left\{ {{T_{1i}} = \left( {\frac{{2{\varepsilon ^{1/2}}}}{{{k_1}}} + \frac{1}{{{k_2}(p - 1){\varepsilon ^{p - 1}}}}} \right) \times } \right.\\{\kern 1pt} \left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left( {1 + \frac{1}{{{w_i}\left( {\frac{1}{{{W_i}}} - \frac{{{h_1}({k_1})}}{{{k_1}}}} \right)}}} \right),(i = 1,2)} \right\}.\end{array}$

即固定时间T₁后，${{\mathit{\boldsymbol{\dot D}}}_1} = {\mathit{\boldsymbol{D}}_1}$.证毕.

2.2 固定时间终端滑模控制律设计定义e₁=q₁-q_d, ${\mathit{\boldsymbol{e}}_2} = {\mathit{\boldsymbol{q}}_2} - {{\mathit{\boldsymbol{\dot q}}}_{\rm{d}}}, {\mathit{\boldsymbol{q}}_{\rm{d}}}$和${{\mathit{\boldsymbol{\dot q}}}_{\rm{d}}}$分别为期望轨迹及其导数.根据文献[8]，新型非奇异固定时间终端滑模面可设计为

$\mathit{\boldsymbol{s}} = \mathit{\boldsymbol{\varphi }}({\mathit{\boldsymbol{e}}_1}){\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{e}}_1} + {\rm{sig}}{ ^{{r_2}}}({\mathit{\boldsymbol{e}}_2}).$ (4)

式中：φ(e₁)=diag(φ(e₁₁), φ(e₁₂))为对称矩阵，其元素为$\varphi \left( {{e_{1i}}} \right) = {\left( {\alpha {{\left| {{e_{1i}}} \right|}^{{g_1} - 1/\left( {{k_2}} \right)}} + \beta {{\left| {{e_{1i}}} \right|}^{{g_2} - 1/\left( {k{r_2}} \right)}}} \right)^{k{r_2}}}$，i=1, 2；α、β、k、r₂、g₁、g₂为常数，且α、β>0, k>1, 2>r₂>1, 1/r₂ < g₁k < 1, g₂k>1.

定义非负函数?(x)为

$\mathit{\boldsymbol{\phi}} (x) = \left\{ {\begin{array}{*{20}{l}}{\sin ({\rm{ \mathsf{ π} }}{x^2}/(2{\tau ^2})),}&{|x| \le \tau ;}\\{1,}&{|x| > \tau .}\end{array}} \right.$

式中：τ为常数，τ>0，函数?(x)满足x→0, (?(x))/x→0.

结合固定时间收敛双幂次趋近律^[19]，则固定时间终端滑模控制律可设计为

$\begin{array}{l}\mathit{\boldsymbol{U}} = - \mathit{\boldsymbol{H}}(\mathit{\boldsymbol{q}})\{ (1/{r_2})[\mathit{\boldsymbol{\bar \varphi }}({\mathit{\boldsymbol{e}}_1}) + \mathit{\boldsymbol{\varphi }}({\mathit{\boldsymbol{e}}_1})] {\rm{sig}}{ ^{2 - {r_2}}}({\mathit{\boldsymbol{e}}_2}) + \\{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}}{(1/{r_2}) {\rm{diag}} (\mathit{\boldsymbol{\phi}} (|{\mathit{\boldsymbol{e}}_2}{|^{{r_2} - 1}})) {\rm{diag}} (|{\mathit{\boldsymbol{e}}_2}{|^{1 - {r_2}}}) \times }\\{[{l_1} {\rm{sig}}{ ^{{a_1}}}(\mathit{\boldsymbol{s}}) + {l_2} {\rm{sig}}{ ^{{a_2}}}(\mathit{\boldsymbol{s}})] - }\\{{\mathit{\boldsymbol{H}}^{ - 1}}(\mathit{\boldsymbol{q}})[\mathit{\boldsymbol{C}}(\mathit{\boldsymbol{q}},\mathit{\boldsymbol{\dot q}})\mathit{\boldsymbol{\dot q}} + \mathit{\boldsymbol{G}}(\mathit{\boldsymbol{q}})] + }\\{{{\mathit{\boldsymbol{\hat D}}}_1} + \eta {\rm{sign}} (\mathit{\boldsymbol{s}}) - {{\mathit{\boldsymbol{\ddot q}}}_d}\} .}\end{array}\end{array}$ (5)

式中：l₁、l₂、a₁、a₂、η为常数，且l₁、l₂>0, a₁>1, 0 < a₂ < 1, η>0；φ(e₁)=diag(φ(e₁₁), φ(e₁₂))为对称矩阵，其元素为

$\begin{array}{l}\bar \varphi ({e_{1i}}) = k{r_2}{(\alpha |{e_{1i}}{|^{{g_1} - 1/(k{r_2})}} + \beta |{e_{1i}}{|^{{g_2} - 1/(k{r_2})}})^{k{r_2} - 1}} \times \\{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left[ {\begin{array}{*{20}{l}}{\alpha ({g_1} - 1/(k{r_2}))|{e_{1i}}{|^{{g_1} - 1/(k{r_2})}} + }\\{\beta ({g_2} - 1/(k{r_2}))|{e_{1i}}{|^{{g_2} - 1/(k{r_2})}}}\end{array}} \right],i = 1,2.\end{array}$

定理2 ??对于式(2)，若采用固定时间观测器(3)、滑模面式(4)和控制律(5)，则系统误差在时间T内收敛到原点，收敛时间T满足

$T \le {T_1} + {T_2} + {T_3}.$

式中：${T_2} \le \frac{2}{{{2^{\frac{{{a_1} + 1}}{2}}}{l_1}\left( {{a_1} - 1} \right)}} + \frac{2}{{{2^{\frac{{1 + {a_2}}}{2}}}{l_2}\left( {1 - {a_2}} \right)}} + \frac{{{\tau ^{1/\left( {{r_2} - 1} \right)}}}}{\eta }$; ${T_3} \le \frac{1}{{{\alpha ^k}\left( {1 - {g_1}k} \right)}} + \frac{1}{{{\beta ^k}\left( {{g_2}k - 1} \right)}}$.

证明??取Lyapunov函数为V=0.5s^Ts，对其求导得

$\begin{array}{l}\mathit{\boldsymbol{\dot V}} = {\mathit{\boldsymbol{s}}^{\rm{T}}}\mathit{\boldsymbol{\dot s}} = \\{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}}{{\mathit{\boldsymbol{s}}^{\rm{T}}}\left[ {\frac{{{\rm{d}}\mathit{\boldsymbol{\varphi }}({\mathit{\boldsymbol{e}}_1})}}{{{\rm{d}}t}}{\mathit{\boldsymbol{e}}_1} + \mathit{\boldsymbol{\varphi }}({\mathit{\boldsymbol{e}}_1}){\mathit{\boldsymbol{e}}_2} + {r_2} {\rm{diag}} (|{\mathit{\boldsymbol{e}}_2}{|^{{r_2} - 1}}){{\mathit{\boldsymbol{\dot e}}}_2}} \right] = }\\{{\mathit{\boldsymbol{s}}^{\rm{T}}}[\mathit{\boldsymbol{\bar \varphi }}({\mathit{\boldsymbol{e}}_1}){\mathit{\boldsymbol{e}}_2} + \mathit{\boldsymbol{\varphi }}({\mathit{\boldsymbol{e}}_1}){\mathit{\boldsymbol{e}}_2} + {r_2} {\rm{diag}} (|{\mathit{\boldsymbol{e}}_2}{|^{{r_2} - 1}})({{\mathit{\boldsymbol{\dot q}}}_2} - {{\mathit{\boldsymbol{\ddot q}}}_{\rm{d}}})] = }\\{{\mathit{\boldsymbol{s}}^{\rm{T}}}\{ - {\rm{diag}} (\mathit{\boldsymbol{\phi}} (|{\mathit{\boldsymbol{e}}_2}{|^{{r_2} - 1}}))[{l_1} {\rm{sig}}{ ^{{a_1}}}(\mathit{\boldsymbol{s}}) + {l_2} {\rm{sig}}{ ^{{a_2}}}(\mathit{\boldsymbol{s}})] + }\\{{r_2} {\rm{diag}} (|{e_2}{|^{{r_2} - 1}})[{\mathit{\boldsymbol{D}}_1} - {{\mathit{\boldsymbol{\hat D}}}_1} - \eta {\rm{sign}} (\mathit{\boldsymbol{s}})]\} .}\end{array}\end{array}$ (6)

经过T₁时间后，由定理1知，${{\mathit{\boldsymbol{\dot D}}}_1} = {\mathit{\boldsymbol{D}}_1}$，代入式(6)得

$\begin{array}{l}\mathit{\boldsymbol{\dot V}} = {\mathit{\boldsymbol{s}}^{\rm{T}}}\{ - {\rm{diag}} (\mathit{\boldsymbol{\phi}} (|{\mathit{\boldsymbol{e}}_2}{|^{{r_2} - 1}}))[{l_1} {\rm{sig}}{ ^{{a_1}}}(\mathit{\boldsymbol{s}}) + {l_2} {\rm{sig}}{ ^{{a_2}}}(\mathit{\boldsymbol{s}})] - \\{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}}{{r_2} {\rm{diag}} (|{\mathit{\boldsymbol{e}}_2}{|^{{r_2} - 1}})\eta {\rm{sign}} (\mathit{\boldsymbol{s}})\} = }\\{ - \sum\limits_{i = 1}^2 \mathit{\boldsymbol{\phi}} (|{\mathit{\boldsymbol{e}}_{2i}}{|^{{r_2} - 1}})({l_1}|{s_i}{|^{{a_1} + 1}} + {l_2}|{s_i}{|^{{a_2} + 1}}) - }\end{array}\\{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}}{{r_2}\eta \sum\limits_{i = 1}^2 | {e_{2i}}{|^{{r_2} - 1}}|{s_i}| \le - \min (\mathit{\boldsymbol{\phi}} (|{e_{2i}}{|^{{r_2} - 1}})) \cdot }\\{\sum\limits_{i = 1}^2 {({l_1}|{s_i}{|^{{a_1} + 1}} + {l_2}|{s_i}{|^{{a_2} + 1}})} ,}\end{array}\end{array}$

即

$\mathit{\boldsymbol{\dot V}} \le - \min (\mathit{\boldsymbol{\phi}} (|{e_{2i}}{|^{{r_2} - 1}}))({2^{\frac{{{a_1} + 1}}{2}}}{l_1}{\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{V}}^{1 + \frac{{{a_1} - 1}}{2}}} + {2^{\frac{{1 + {a_2}}}{2}}}{l_2}{\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{V}}^{1 - \frac{{1 - {a_2}}}{2}}}).$

如果e_2i≠0则?(e_2i^r₂-1)>0, (i=1, 2)，系统状态空间被划分如下两个区域：

${{\varOmega _1} = \{ ({e_{1i}},{e_{2i}})||{e_{2i}}{|^{{r_2} - 1}} > \tau ,i = 1,2\} ,}$

${{\varOmega _2} = \{ ({e_{1i}},{e_{2i}})||{e_{2i}}{|^{{r_2} - 1}} \le \tau ,i = 1,2\} .}$

当系统状态在区域Ω₁时，?(e_2i^r₂-1)=1，$\dot V \le - \left( {{2^{\frac{{{a_1} + 1}}{2}}}{l_1}{V^{1 + \frac{{{a_1} - 1}}{2}}} + {2^{\frac{{1 + {a_2}}}{2}}}{l_2}{V^{1 - \frac{{1 - {a_2}}}{2}}}} \right)$，由文献[20]可知，系统在固定时间

${T_s} = [2/({2^{\frac{{{a_1} + 1}}{2}}}{l_1}({a_1} - 1))] + [2/({2^{\frac{{1 + {a_2}}}{2}}}{l_2}(1 - {a_2}))]$

内到达滑模面^[12].

当系统状态在区域Ω₂时，代入控制律(5)到系统(2)中可得

$\begin{array}{l}{{\mathit{\boldsymbol{\dot e}}}_2} = - \frac{1}{{{r_2}}}[\mathit{\boldsymbol{\bar \varphi }}({\mathit{\boldsymbol{e}}_1}) + \mathit{\boldsymbol{\varphi }}({\mathit{\boldsymbol{e}}_1})] {\rm{sig}}{ ^{2 - {r_2}}}({\mathit{\boldsymbol{e}}_2}) - \\{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}}{\mathit{\boldsymbol{r}}_2^{ - 1} {\rm{diag}} (\mathit{\boldsymbol{\phi}} (|{\mathit{\boldsymbol{e}}_2}{|^{{r_2} - 1}})) {\rm{diag}} (|{\mathit{\boldsymbol{e}}_2}{|^{1 - {r_2}}}) \times }\\{[{l_1} {\rm{sig}}{ ^{{a_1}}}(\mathit{\boldsymbol{s}}) + {l_2} {\rm{sig}}{ ^{{a_2}}}(\mathit{\boldsymbol{s}})] - \eta {\rm{sign}} (\mathit{\boldsymbol{s}}) \cdot }\end{array}\end{array}$ (7)

当e_2i→0，式(7)可化简为${{\mathit{\boldsymbol{\dot e}}}_2} = - \eta {\mathop{\rm sign}\nolimits} (\mathit{\boldsymbol{s}})$，当s_i>0时, e_2i < 0，当s_i < 0时, e_2i>0，因此e_2i=0不是系统在滑模到达阶段的吸引子，系统状态将在时间T_f=τ^1/(r₂-1)/η内穿越区域Ω₂进入区域Ω₁或者到达滑模面^{[10, 12]}.综上，系统在固定时间T₂=T_s+T_f内收敛到滑模面.

当系统状态到达滑模面后，由滑模面(4)可知，

${\mathit{\boldsymbol{\dot e}}_1} = - {\rm{sig}}{ ^k}(\alpha {\rm{sig}}{ ^{{g_1}}}({\mathit{\boldsymbol{e}}_1}) + \beta {\rm{sig}}{ ^{{g_2}}}({\mathit{\boldsymbol{e}}_1})).$ (8)

定义辅助变量χ_i=|e_1i^1－g₁k|, (i=1, 2)，则式(8)可以改写成

${\dot \chi _i} = - (1 - {g_1}k){(\alpha + \beta \chi _i^\mu )^k}.$ (9)

式中μ=(g₂-g₁)/(1-g₁k).

对式(9)进行积分，可得

$\begin{array}{l}\begin{array}{*{20}{l}}{{T_3} = \mathop {\lim }\limits_{{\chi _i} \to \infty } \frac{1}{{1 - {g_1}k}}\int_0^{{\chi _i}} {\frac{1}{{{{(\alpha + \beta \chi _i^\mu )}^k}}}} {\rm{d}}{\chi _i} = }\\{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathop {\lim }\limits_{{\chi _i} \to \infty } \frac{1}{{1 - {g_1}k}}(\int_1^{{\chi _i}} {\frac{1}{{{{(\alpha + \beta \chi _i^\mu )}^k}}}} {\rm{d}}{\chi _i} + }\end{array}\\{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}}{{\kern 1pt} \int_0^1 {\frac{1}{{{{(\alpha + \beta \chi _i^\mu )}^k}}}} {\rm{d}}{\chi _i}) \le }\\{\mathop {\lim }\limits_{{\chi _i} \to \infty } \frac{1}{{1 - {g_1}k}}\left( {\int_1^{{\chi _i}} {\frac{1}{{\beta \chi _i^{\mu k}}}} {\rm{d}}{\chi _i} + \int_0^1 {\frac{1}{{{\alpha ^k}}}} {\rm{d}}{\chi _i}} \right) = }\\{\frac{1}{{1 - {g_1}k}}\left[ {\frac{1}{{{\beta ^k}(\mu k - 1)}} + \frac{1}{{{\alpha ^k}}}} \right] = \frac{1}{{{\alpha ^k}(1 - {g_1}k)}} + }\\{\frac{1}{{{\beta ^k}({g_2}k - 1)}}.}\end{array}\end{array}$

故系统状态到达滑模面后在固定时间T₃内到达原点.

综上所述，系统状态在固定时间T≤T₁+T₂+T₃内收敛到原点.证毕.

为了说明本文所采用滑模面的优越性，选取传统非奇异终端滑模面和文献[21]的非奇异快速终端滑模面与本文的非奇异固定时间终端滑模面进行仿真对比研究.

1) 传统非奇异终端滑模面为

${s_1} = x + {\sigma _1} {\rm{sig}}{ ^{{r_2}}}(\dot x).$

式中：σ₁为大于0的常数.

2) 文献[21]的非奇异快速终端滑模面为

${s_2} = x + {\sigma _2} {\rm{sig}}{ ^{{r_3}}}(x) + {\sigma _1} {\rm{sig}}{ ^{{r_2}}}(\dot x).$

式中：σ₂、r₃为大于0的常数，且r₃>r₂.

为了便于公平比较，仿真中选取系数α=β=1, k=2, r₂=5/3, g₁=3/10, g₂=7/10, σ₁=σ₂=1, r₃=7/3.则本文的非奇异固定时间终端滑模面可化为

$s = {(|x{|^{\frac{3}{{10}}}} + |x{|^{\frac{7}{{10}}}})^{\frac{{10}}{3}}} {\rm{sign}} (x) + {\rm{sig}}{ ^{\frac{5}{3}}}(\dot x),$

传统非奇异终端滑模面可化为

${s_1} = x + {\rm{sig}}{ ^{\frac{5}{3}}}(\dot x),$

文献[21]的非奇异快速终端滑模面可化为

${s_2} = x + {\rm{sig}}{ ^{\frac{7}{3}}}(x) + {\rm{sig}}{ ^{\frac{5}{3}}}(\dot x).$

仿真选取初始状态x(0)=0.1 (状态变量距离平衡点较近)和x(0)=10 (状态变量距离平衡点较远)两种情况进行对比.仿真结果如图 3所示.

Fig. 3
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图 3 不同滑模面下状态变量x的收敛曲线 Fig. 3 Convergence curves of state variable x under three sliding mode surfaces

从图 3(a)和3(b)可以看出，与滑模面s₁、s₂相比，本文采用的固定时间终端滑模面无论系统初始状态是在平衡点附近还是远离平衡点处，都能改善系统状态的收敛速度，具有更快的收敛速度.证明了本文固定时间终端滑模的优越性.

3 实验验证为检验提出的控制器的有效性，搭建了如图 4所示的弹药传输机械臂样机实验装置.基础振动台用来模拟路面产生的激励.

Fig. 4
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图 4 弹药传输机械臂实验装置 Fig. 4 Experimental setup of ammunition transfer manipulator

图 5为弹药传输机械臂系统控制框图.实验控制算法在上位机中采用LabVIEW编写，执行电机选用Maxon公司的EC45型无刷直流电机，控制器采用70/10型EPOS2数字位置控制器，控制器通过USB和上位机进行通讯交换数据，编码器采用HEDL9140型光电编码器，电机工作在电流模式下.

Fig. 5
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图 5 弹药传输机械臂系统控制框图 Fig. 5 Control block diagram of ammunition transfer manipulator system

弹药传输机械臂样机参数如下：m₂=8.6 kg, m₃=3.6 kg, J₃=1.6 kg·m², L₃=0.25 m，g=9.8 m/s²，K_T2=K_T3=45.5 mN·m/A, η₂=η₃=0.9, i₂=113, i₃=353.控制器参数设置为：k₁=20, k₂=15, k₃=5, p=3, α=1, β=1, k=2, r₂=16/15, g₁=5/12, g₂=1, l₁=1, l₂=2, a₁=2, a₂=1/2, η=1, τ=0.005.

为说明所设计方法的实用性和优越性，选取工程上弹药传输机械臂系统常用的固定参数PID控制方法进行对比实验.升降部分的PID控制参数取值为K_p1=4 400, K_i1=1 200, K_d1=1 800；翻转部分的PID控制参数取值为K_p2=290, K_i2=40, K_d2=40.

进行考虑3种不同正弦垂直振动工况下的定位实验，工况1：振动台振幅为10 mm，频率为1 Hz；工况2：振动台振幅为20 mm，频率为1 Hz；工况3：振动台振幅为50 mm，频率为0.5 Hz.升降定位目标为0.3 m，翻转定位目标为π/3·rad. 3种工况实验结果分别如图 6~8所示.

Fig. 6
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图 6 工况1实验结果 Fig. 6 Experimental results under working condition 1

Fig. 7
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图 7 工况2实验结果 Fig. 7 Experimental results under working condition 2

Fig. 8
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图 8 工况3下的实验结果 Fig. 8 Experimental results under working condition 3

由图 6(a)和6(b)可以看出，工况1振动下，FTSMC算法和PID算法都能使得弹药传输机械臂快速准确定位到位.这是因为PID算法的控制参数是在工况1下整定得到，因而PID算法也有较好的控制精度.但是，结合固定时间收敛干扰观测器的FTSMC算法的升降和翻转定位时间都要明显小于PID算法相应的定位时间，表明结合固定时间收敛干扰观测器的FTSMC算法收敛速度更快.

对比3种振动工况的实验结果可知，3种振动工况下FTSMC算法的升降和翻转定位误差都要小于PID算法相应的定位误差，并且随着基础振动的加强，FTSMC算法的升降和翻转定位误差都没有明显增加，说明设计的固定时间收敛干扰观测器能够有效地估计和补偿外界的振动激励影响，表明结合固定时间收敛干扰观测器的FTSMC算法具有很强的鲁棒性.相比之下，由图 7(a)、7(b)、8(a)和8(b)可知，PID算法在基础振动加强后，升降和翻转的定位误差都有明显增加，特别是图 7(b)和8(b)翻转部分的定位误差已趋于恶化，分别为0.016 rad和0.021 rad，远远高于FTSMC算法的0.005 rad和0.003 rad.说明当外界振动激励增强后，PID算法控制精度大大下降，抗干扰能力较差.同时，3种工况下结合固定时间收敛干扰观测器的FTSMC算法的升降和翻转定位时间都要小于PID算法相应的定位时间，再次验证了结合固定时间收敛干扰观测器的FTSMC算法收敛速度更快.升降滑模面和翻转滑模面都能在固定时间内收敛，验证了定理2的结论.

综上所述，本文提出的FTSMC算法在定位误差精度和时间上都要明显优于传统固定参数的PID控制算法，所设计的固定时间控制律能够使弹药传输机械臂在更短的时间内收敛到期望位置，振动抑制效果明显优于传统固定参数PID算法，鲁棒性强，更有利于实际工程应用。

4 结论1) 针对弹药传输机械臂在路面激励外界扰动下的位置控制问题，基于拉格朗日方程，建立了考虑垂直基础振动的弹药传输机械臂动力学方程.

2) 提出了固定时间观测器和固定时间终端滑模相结合的复合控制策略，固定时间观测器用于估计和补偿外界的总干扰项，固定时间终端滑模用于设计弹药传输机械臂的主控制器，采用Lyapunov稳定理论证明了系统固定时间收敛.

3) 实验结果验证了所设计算法的有效性和强鲁棒性.与固定参数PID控制算法相比，本文提出的算法具有更高的控制精度和干扰抑制能力，对其他具有基础振动的机械臂控制器设计有工程实际参考意义.

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