解耦控制和抗干扰性的机械臂部分已知的动力外文文献翻译、中英文翻译、外文翻译.doc
解耦控制和抗干扰性的机械臂部分已知的动力外文文献翻译、中英文翻译、外文翻译
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附录一解耦控制和抗干扰性的机械臂部分已知的动力 摘要 众所周知，现在电机驱动机械臂的动力学是至关重要的，有必要考虑这个在实现提高的控制性能。与此同时,它也知道交叉耦合运动机械手动力学更为复杂,因此更难被控制时只考虑机械手的动力学。本文的动力传动装置和机械手一起被认为是和一个控制策略是实现简化控制器设计和开发高控制性能。通过使用本文算法开发,高度复杂的运动机械手动力学不仅可以有效地控制,但是汽车也使得传感敏感和补偿不确定动态代理在机械手关节。此外,作为一个重要结果,建模困难和控制可以显著减少机械手动力学的复杂性。1.介绍近年来,越来越多的关注已经支付给机械臂的控制问题使用更加“完整”模型和考虑非线性的影响。通常,一组强耦合、高度非线性二阶微分方程采用刚性机械手的动态行为特征1。扭矩(或力)驱动机械手的关节是输入这些方程式,而致动器动力学往往排除在系统方程考虑执行器为纯扭矩来源。然而,执行机构动力学并发挥重要作用在整个机械手动力学,特别是在大负载的情况下的变异和快速操作,尤其是直接传动机械手。最近,一些作者已经讨论了由于执行机构动力学的动态问题。实验评估由莱希和萨迪斯2(也在莱希等等 .3)表明,驱动系统的未建模动态交互可能占据主导地位的实际动力机械手具有高转矩放大驱动系统。对机械手控制太重要的影响得到有效补偿的PD反馈回路。高精度的高速工业机械手的控制依赖于一个完整的知识占主导地位的机械手的动力学及其驱动系统的链接。详细研究一个完整的机械轴包括电气和机械部分也由麦丽尔和理查德4。分析和实验结果表明,动态控制中使用的更完整的系统模型,更好的控制效率。陈5和良好的等等。6还考虑执行机构动力学的动态行为的重要性操纵者(一般三阶动态模型,包括汽车动力学提出了),表明发动机动力学可能主导机械手动力学。然而,他们的研究控制器设计仍基于简化的线性模型和线性单轴模型作为具体例子讨论动机在他们的论文。有一些通用的方法产生的控制驱动器机械手动力学由贝克曼和李7,冰斗湖等。8和朱等等。9。贝克曼和李7利用弗氏系统的非线性控制理论10更完整的控制系统,其中包含发动机机械手的动态交互。因为很难弗洛伊德的方法直接应用于整体机械手动态方程,大系统理论的分解原理是首次采用分解互动系统。另外,利用微分几何控制理论,塔恩等等。8提出了完善的算术的非线性反馈控制器集成了机械手动力学以及联合汽车动力学。对这两种方法，三阶动态模型的整体执行机械手系统应该首先被导出。在朱等等。9,线性化,解耦控制的一般方法包括电动机的机械臂动力学。三个系统的算法,没有必要麻烦推导出三阶模型,已经给出了。通过使用给定的分步算法,最初的强烈交叉耦合和高度非线性动态系统可以直接被转换为一组解耦和线性化子系统的形式简单纯粹的三重或双集成商。现有的最先进控制规律可以应用到生成的线性系统以实现期望的闭环性能和极大的缓解。如上所述,执行机构动力学既重要又复杂。工作提出了动机的考虑,在解决这样一个系统,由额外的复杂的传动机构动力学,我们可能会执行机构动力学充分利用时考虑。基于发动机机械手动力学解耦和线性化算法,使系统的耦合动力学进一步利用敏感外部扰动(和其他动态不确定性),和一种新型传感和补偿方案。生成的算法开发不仅有效而且简单。该算法的另一个令人鼓舞的结果是,更少的先验知识的完整动力学模型所需的机械臂控制合成,因此建模困难和复杂性在控制机械手动力学可以大大减少,即只有惯性条款需要建模和计算的控制目的。数值案例研究表明,该算法可以有效地解耦和线性化交叉耦合非线性动力学,和感觉,弥补外部扰动的影响(和模型的不确定性,和/或合并这两个因素的影响)。2.机械手执行器动力学系统典型的操纵者通常是由几个子系统,如广义控制器、功率放大器、致动器、传感器和机械臂本身。事实上,有各种各样的固有非线性在过去四个子系统和它们相互作用。自执行机构动态起着主导的作用比其他未建模动态和与机械手动力学的相互作用也更加复杂,只有机械手执行器互动的控制系统将被考虑。所有其他动态效果可以治疗以类似的方式使用文本算法开发。2.1.机械臂动力学 让我们首先考虑机械手的运动的动力学方程。机械手通常建模为一组n移动刚体连接的串行链一端固定另一端的基础和免费的。转动关节或柱状的身体贴合在一起,有一个转矩传动装置(传动器)在每个联合行动。一般来说,机械手的动力学模型包括n刚性连接在系列可以用一组高度非线性和强耦合的二阶微分方程编写的小型矩阵形式如下:在上面的两个方程,”加速度、速度和位置向量n的关节,分别;D(q)”是对称正定惯性矩阵;h(q,q)”是包含了向量的向量耦合力矩的离心,科里奥利,摩擦和重力力矩;VD Rn是外部扰动力矩的向量与其他可能的不确定性或未建模动态(在本文中,假设所有可能的干扰,电源波动等非线性功率放大器的影响和驱动系统,和负载变化,已经反映到机械手关节);VcRn的矢量控制扭矩;c()Rn是离心的向量和科里奥利扭矩;f(q,q)Rn的矢量驱动系统中摩擦力矩。和g(q)的矢量重力力矩。 动力学的复杂性,给出了方程(1),但它实际上并不是说很难找到一个可以线性化的非线性控制律和分离系统。然而,正如下面所示,执行机构和机械手关节之间的动态交互要解决的问题更加困难。2.2.致动器马达驱动器的动力学 致动器的功能是生成所需的扭矩(或力)移动机械手的关节。三种执行机构用于机械手系统:电气、液压和气动驱动器。到目前为止,最常见的是电动驱动器,因此只有这种执行机构将被认为是在这里。尽管一些操纵者使用步进电机或其他交流电机为执行机构,大多数工业电动机械手服务使用直流伺服电机。特别是,机械手由电枢可控直流电机电枢电压是输入将被考虑。动态方程n n关节马达执行机构可以派生通过应用基尔霍夫电压定律在n电机的电枢绕组电路。这个收益率以下电压方程ia,Ua、pRn是向量的电枢绕组电流,应用电枢电压,和n汽车的角位置,分别是La,Ra斜参数矩阵的电枢绕组电感和电阻,分别。左边的第三个任期的方程(2)反电动势。常量kEi 0(i = 1,E,n)对角矩阵反电动势是常量。由于磁场绕组电流恒定在一个电枢可控直流电机,电机轴的扭矩开发给出在弗吉尼亚州是矢量电机轴的扭矩发达和常量kTi 0(i = 1,n)对角矩阵KT是发动机的转矩常数。这两个位置向量,q和p(分别代表联合轴和电机轴),在方程(1)和(2)是由p =Nq.。这里N对角矩阵的高速电机轴之间的传动比率p和低速轴联合。直接传动机械手,矩阵N变成一个统一矩阵即假设没有反弹或弹性变形的传输子系统,然后在关节轴转矩矢量控制代理在以上推导,认为的有效惯性电机轴和驱动系统的摩擦效应已经反映到关节轴。方程(4)之间的关系表明,驾驶操纵和控制力矩电机电枢电流的特点是一个简单的代数方程。结果与上面给出的方程模型描述了机械臂的动态行为,更确切的说,可以用来设计更好的控制器。可以看出机械手的电气和机械子系统在闭环动态交互。制定问题的复杂性导致了下面的控制算法的发展。3.解耦和线性化交叉耦合非线性动力学 正如上面所示,马达调制器的动态系统是高度非线性和强交叉耦合。机械手系统的复杂性质呈现大多数现有的控制策略无效或无效的。在解决这样一个具有挑战性的问题,提出一种基于连续补偿方案在下文给出的解耦和线性化马达调制器交互系统。可以看到从上面给出的机械手的动力学模型,第一个障碍在我们的方式来控制高度复杂的马达调制器系统只是反馈机电动力学之间的交互,即。的负面反馈路径(KEN)。这可以消除如果下面的控制律使用是一个中间控制向量。这里,帽子的数量表示真正的植物参数的近似估计。观察到的变量的帽子,如q,是简单的省略。如果工厂是完全模仿,即()=,(在随后的派生,这种假设的语句将省略为简单起见),然后从方程(2)和(5),我们有方程(6)表明,不良相互作用的反馈路径(KEN)之间的电气和机械动力学已经完全弥补。造成的动力马达执行机构现在已经减少到一阶滞后和不受欢迎的两极(La-1 Ra)的控制ua给出方程(5),即传递函数从ub到ia现在一组延迟集成商。如果一些折衷的控制性能和控制简单是允许的,那么一个简单的算法与某些近似可以获得进一步的赔偿将如下所示。假设以下补偿器采用,ucbRn 是另一个中间控制向量,和Tf1Rnn是对角矩阵的时间常数补偿器。通过时间常数Tf1远小于方程(7),从本质上讲,一组及换挡器提供低频与高频波兰人零,将不良的波兰人的原始系统远离原点,即从()到(-Tfi -1)。方程(7)中给出的控制律是合成钢管取消。在替代方程(7)到(6),我们有以下近似的结果因忽视了动态由于新的两极(-Tfi 1)。这个简化基于控制精度和计算费用之间的权衡。自该算法推导来弥补内在耦合非线性动力学,因此为了方便,干扰(和不确定性)VD在方程(1)将暂时被忽视在下面讨论。因此,方程(1)现在让ucb在方程(7)设计和ucRn被设计成在uRn是闭环的矢量法确定。从方程(10)可以看出,控制谎言实际上是由两个组件:一个是形式的非线性反馈(第一学期),另一个是非线性前馈补偿(连任)。然后,根据上面给出的方程,我们有方程(11)意味着通过使用上述串行补偿算法,原始的紧耦合的非线性植物现在已经变成了一个解耦的线性系统包括纯双(二阶)集成商,即产生的解耦和线性化传递函数从u到q大约可以由一组二阶纯集成商。在这个阶段,我们现在只需要控制n个独立的单输入多输出子系统只是表现为开环双集成商与某些闭环控制。 上面的串行控制的补偿系统，下面的状态反馈控制律可以采用闭环控制:,Kv,KpRnn是对角的闭环状态反馈增益矩阵(分别对应速度和位置)。事实上,大多数基于状态方程的方法(如极点配置方法用于朱等。11)控制设计最终将导致这些增益矩阵的确定所需的性能。rRn是运动任务的参考信号。点对点控制,r是所需的设定值,而轨迹跟踪控制11 J r可以定义为即线性组合所需的关节轨迹qdRn的第一和第二差异。相应算法串行补偿和闭环控制开发的这一部分可以被概括算法1：方程（5）、（7）、（9）、（10）、（12）。不过可能注意到,上述控制系统是基于假设没有获得模型不匹配,没有未建模动力学(即。工厂是完全模仿和近似方程（8）或（11）是可以接受的),程序中，这是从来没有的情况，众所周知，依赖于模式的控制方法和基于模型的串行上面给出补偿算法也不例外，整个系统的性能可能会敏感植物模型的准确性。同时,在实践中,存在模型失配和其他不确定性,如机械手控制负载的变化和外部干扰,是不可避免的,通常计算控制之间的差异和实际需要一个然后出现。这些非理想因素可能显著降低任何相关模型算法的性能。因此,有必要将一些补偿装置控制系统不可避免的建模误差以及外部干扰。在下一节中,一种新型传感和补偿机制将被引入,以弥补这些非理想因素。4.干扰的估计和补偿 上述算法的性能已经比传统的基于模型的计算力矩方法忽略了致动器动力学。但正如上面指出的,给定的算法假设机械手系统的模型是清楚和不存在外部干扰。目的是实现更高的控制性能,一个观察者对所有可能的不确定性(即外部干扰、未建模动态和动力学可能由于模型不匹配),使现有的汽车合理的将在下面被开发。这允许另一个前馈补偿结构用于弥补观测到的不确定性。从方程(1)可以看出,的位置变量(及其衍生物)机械手可以被认为是一个函数的时间和外源输入(即风险投资与vD)的系统。因此,我们有以下功能在方程(13),不确定的词性病(连同其他不确定的动态,如模型失配和未建模动态)通常可以表示为一个逆函数的方程(13),即它可以表示为以下功能的位置向量(及其衍生物)和机械手的控制力矩向量或一个函数的位置向量(及其衍生物)和电动机电枢电流矢量这意味着与方程(15),不受欢迎的动力学vD现在可以被估计。然而,在实际应用中,加速度向量在方程(15)通常不能直接获得。在这种情况下,下面的近似估计从到 可以被用 表示的近似估计和参数对角的时间常数矩阵预过滤器可以类似于Tf1的决定。 现在,从方程(1),(4)和(16),我们有近似的观测VD对于方程(15)如下 可以从方程(17),估计干扰(连同其他不确定性,如参数变化和未建模动力学)是一个函数的位置向量(第一和二阶衍生品)和电动机电枢电流矢量的操纵者。 众所周知,控制系统与已知或估计可以干扰(负载)的质量控制通常可以改进的前馈控制,以弥补可能的干扰。然而,对于干扰排斥这里讨论算法1在最后一节中,必要的前馈补偿原来是非常简单的,和一些令人鼓舞的结果将在下面。使用前馈补偿,可以弥补现有干扰vD使用方程(17)中的估计签证官通过附加额外的前馈结构控制系统开发的最后部分。自估计扰动矢量是扭矩的形式,这意味着补偿信号可以直接被添加(美联储)转发给控制ic在方程(10)。换句话说,它可以指出,利用算法1在最后一节中,从控制传递函数u,扰动转矩vD的地方发生可以近似统一函数,这意味着所需的前馈补偿器实际上是一个统一的矩阵。这表明作为补偿元件,观察到的干扰力矩可以直接被注入到现有的控制系统开发的最后部分。 注意的是的形式,为简化计算,可以看出补偿可以更好的进行采用单独个人条件方程(17)。控制ucb和uc的方程(9)和(10)然后修改用方程(17)和第一个任期使用的第二个任期在方程(17)。可以指出,方程(19)相比显著简化了方程(10)。这意味着通过使用该传感和补偿方案,较少的先验知识机械臂的动力学模型是必需的,和h()的动力学方程(1),这是更加困难比惯性来看D(q),控制设计现在姿势没有什么困难。这意味着需求阶段的植物造型可以显著减少和更少的计算中需要控制的实现。这个显著的特性可以被认为是一位杰出的优势在这一节中给出的算法。算法2:方程（5）（7）（16）（18）（19）（12）. 可以看出结果控制系统是非常简单而高性能的解耦和扰动拒绝控制可以实现。上面两个控制系统开发的区别是复杂的非线性前馈补偿组件h()算法1是两个非常简单的补偿组件取代电机电枢电流ia和观察关节加速度。5.数值研究 为实际应用,可能会遇到重要的负载(参数)的变化以及大下面还考虑外部扰动。机械手的运动所需的任务是将第三个链接从初始位置q03 = 0有时 t0= 0的时候准确的最后位置qf3 = 2.0 rad 完全tf= 2.0s年代最初和最终的速度和加速度都是零。为了满足所有边界条件,五次多项式轨迹11作为期望的运动轨迹。 干扰的性能比较研究合理的系统C2(通过使用算法2)相比较与麻木的Cl(利用算法1)。结果跟踪误差和控制力矩的第三个链接分别图1和图2所示。这两个算法的鲁棒性验证的参数变化,植物控制设置为:P1, 图1所示.跟踪误差名义植物;P2,各种植物(载荷变化m= 5.0kg)。为了证明该算法的干扰抑制的有效性，外部干扰在联合行动是设置为：DI，脉冲扰动(打击)开始t = 0.5s，结束t = 0.51s和VD= 50 Nm);和D2,一步扰动(推)(开始t = 0.5s，结束在t = 1.0s与VD = 10 Nm)。表1给出了各种组合的研究。 图1(a),认为现有的这两个算法非常小的错误是由于忽视了动态连续地磁反转(错误的虚线是略小于实线)。虚线(C2)表明,遥感和补偿算法提出了甚至可以弥补这种小剩余未建模动态。图1(b-f),最低位虚线的错误反应证明该算法2的性能很好,给予优秀的赔偿未建模动态的影响,参数变化和各种外部干扰,甚至这些不确定性的综合影响。与这些不确定性,算法1的性能可以看到不一样的算法2。 控制扭矩的虚线图2所示(a-f)显示,算法2的高性能是由于快速、急剧反应生成的控制 图2.控制扭矩 表1 案例数值研究通过给定的传感和补偿控制系统,使该控制算法来弥补现有成功的不确定性。可以看出有快速、大幅的反应控制扭矩(虚线)的轨迹跟踪和抗干扰性。外部扰动的影响下,增加在该算法生成的控制力矩的震级约等于应用干扰。算法1的控制力矩响应,另一方面,可以看到被推迟,而不是这么快的算法2。6.结论 在许多先前的研究,重要的以及复杂的传动机构动力学是研究机械臂常被忽视。在本文中,这个因素是考虑到为了达到更高的性能和更精确的控制。通过使用本文中的控制算法开发,执行机构现在不仅仅是一个执行机构,但是一个敏感的致动器,不仅促动器,而且感觉和补偿不良干扰和动力学。此外,本文算法开发,较少的先验知识机械手模型所需的控制合成,这样植物造型非常显著降低的问题。工厂的困难的组件模型涉及耦合等方面所产生的离心,科里奥利,摩擦和重力的影响,不需要。这使得控制器设计了简化和降低所需的计算。附录二Decoupling control and disturbance rejection of mechanical manipulators with partially known dynamicsAbstract-It is known now that the dynamics of the motor drives of mechanical manipulators is of utmost importance and it is necessary to take this into account in achieving high control performance.At the same time, it is also known that the cross-coupled motor-manipulator dynamics is more complex and hence more difficult to be controlled than when only the manipulator dynamics is considered. In this paper, the dynamics of both the actuator and mechanical manipulator are considered together and a control strategy is developed for achieving ease of controller design and high control performance. By using the algorithm developed in this paper, not only can the highly complicated motor-manipulator dynamics be effectively controlled, but the motors are also made sensitive for sensing and compensating for the uncertain dynamics acting at manipulator joints. Furthermore, as a significant result, the modellingdifficulty and control complexity of the manipulator dynamics can be significantly reduced.1. INTRODUCTIONIn recent years, increasing attention has been paid to the control problems of mechanicalmanipulators using more complete models and with more nonlinear effectstaken into account. Usually, a set of strongly coupled and highly nonlinear secondorderdifferential equations is adopted to characterize the dynamic behavior of rigid manipulators 1. The torques (or forces) driving the joints of the manipulators are the inputs to these equations, while the actuator dynamics are often excluded from the system equations by considering the actuators as pure torque sources. However, actuator dynamics does play an important role in the overall manipulator dynamics,particularly in the case of large payload variation and quick operation, and especially for direct-drive manipulators. More recently, several authors have discussed the dynamic problem due to the actuator dynamics. Experimental evaluations made by Leahy and Saridis 2 (also in Leahy et al. 3) indicate that unmodelled dynamics of drive system interactions may dominate the actual dynamics of a manipulator with high torque amplification drive systems. Its influence on manipulator control is too significant to be effectively compensated for by a PD feedback loop. High-precision high-speed control of an industrial manipulator is dependent on a complete knowledge of the dominant dynamics of the manipulator links as well as their drive systems. A detailed study of a complete manipulator axis including the electrical and the mechanical parts was also made by Marilier and Richard 4. Both analytical and experimental results indicate that the more complete the system model used in dynamic control, the better the control efficiency. Chen 5 and Good et al. 6 also considered the importance of actuator dynamics in the dynamic behavior of manipulators (general third-order dynamic models that include the motor dynamics were presented) and showed that the motor dynamics may dominate the manipulator dynamics. However, their studies on the controller design were still based on simplified linear models and linear single-axis models served as specific examples for the discussion motivated in their papers. There are some general methods for the control of actuator-manipulator dynamics resulting from the work by Beekmann and Lee 7, Tarn et al. 8 and Zhu et al. 9. Beekmann and Lee 7 utilized Freunds systematic nonlinear control theory 10 for the control of a more complete system that contains the dynamics of the motormanipulator interaction. Since it is difficult to apply Freunds method directly to the overall-manipulator dynamic equations, the decomposition principle in large-scale system theory was first adopted to decompose the interacted system. Alternatively, by using differential geometric control theory, Tam et al. 8 proposed a mathematicallyperfect nonlinear-feedback controller that incorporates the manipulator dynamics as well as the joint motor dynamics. For both of the two methods, a third-order dynamic model of the overall actuator-manipulator system should first be derived. In Zhu et al. 9, a general method for the linearization and decoupling control of mechanical manipulators that include the motor dynamics is given. Three systematic algorithms, in which the troublesome derivation of the third-order model is not necessary, are presented. By using the given step-by-step algorithms, the original strongly crosscoupled and highly nonlinear dynamic system can directly be transformed into a set of decoupled and linearized subsystems that are in the form of simply pure triple or double integrators. Most of the existing advanced control laws can then be applied to the resulting linear system to achieve desired closed-loop performance with great ease.As stated above, actuator dynamics is both important and complex. Work presented in this paper is motivated by the consideration that in tackling such a system, which is complicated by the additional actuator dynamics, we may turn the actuator dynamics to full advantage when it is taken into account. Based on the decoupling and linearization algorithm for the motor-manipulator dynamics, the coupled dynamics is further exploited to make the system sensitive to external disturbances (and otherd ynamic uncertainties), and a novel sensing and compensating scheme is developed. The resulting algorithm developed is not only effective but also simple. Another encouraging result of the proposed algorithm is that less a priori knowledge about thecomplete dynamic model of mechanical manipulators is required for control synthesis, thus the difficulty in modelling and the complexity in controlling the manipulator dynamics can be greatly reduced, i.e. only the inertial terms need to be modelled and to be computed for control purpose. Numerical case studies have shown that the proposed algorithm can effectively decouple and linearize the cross-coupled nonlinear dynamics, and sense and compensate for the effects of external disturbances (and model uncertainties, and/or the combined effects of the both factors). 2.DYNAMICS OF MANIPULATOR-ACTUATOR SYSTEMSA typical manipulator is usually composed of several subsystems, such as a generalized controller, power amplifier, actuator, sensor and the mechanical arm itself. As a matter of fact, there are various kinds of inherent nonlinearities in the last four subsystems and they interact with one another. Since actuator dynamics plays a dominant part over other unmodelled dynamics and its interaction with the manipulator dynamics is also more complex, only the control of the interacted manipulator-actuator system will be considered in this paper. All other dynamic effects can be treated in a similar way using the algorithm developed in this paper.2.1. Dynamics of mechanical manipulatorsLet us first consider the dynamic equations of motion for a mechanical manipulator. The manipulator is usually modelled as a set of n moving rigid bodies connected in a serial chain with one end fixed to the base and the other end free. The bodies are jointed together with revolute or prismatic joints, and there is a torque actuator (driver) acting at each joint. Generally, the dynamic model of a manipulator comprising n rigid links in series can be represented by a set of highly nonlinear and strongly coupled second-order differential equations written in the following compact matrix form:In the above two equations,are the acceleration, velocity, and position vectors of the n joints, respectively;is the symmetric positive-definite inertial matrix;is the vector of coupling torques that incorporates the vectors of centrifugal, Coriolis, frictional and gravitational torques;is the vector of external disturbance torques together with other possible uncertainties and/or unmodelled dynamics (in this paper, it is assumed that all possible disturbances, such as power-supply fluctuations, nonlinear effects of the power amplifiers and drive systems, and payload variations, have already been reflected to the manipulator joints);is the vector of control torques;is the vector of centrifugal and Coriolis torques;is the vector of frictional torques in the drive systems; and g(q) E I(8 is the vector of gravitational torques.The complexity of the dynamics given in equation (1) notwithstanding, it is actually not that difficult to find a nonlinear control law that can linearize and decouple the system. However, as will be shown below, the dynamic interaction between the actuators and manipulator joints makes the problem to be tackled much harder.2.2. Actuator dynamics of motor drivesThe function of the actuators is to generate the torques (or forces) needed to move the joints of the manipulator. Three kinds of actuators are used in manipulator systems: electric, hydraulic and pneumatic drives. By far the most common are the electric drives, and hence only this kind of actuator will be considered here. Although some manipulators use stepper motors or other AC motors for actuators, most industrial electric-driven manipulators in service use DC servomotors. In particular, the manipulators driven by armature-controlled DC motors with armature voltages being inputs will be considered in this paper. The dynamic equation of the n motor actuators for the n joints can be derived by applying Kirchhoffs voltage law around the armature winding circuits of the n motors. This yields the following voltage equationwhereare vectors of the armature winding current, the applied armature voltage, and the angular position of the n motors, respectively; andare diagonal parameter matrixes of the armature winding inductance and resistance, respectively. The third term on the left side of equation (2) is the back emf. The constants kEi 0 (i = 1,., n) of the diagonal matrix KE E ll8n are the back emf constants. Since the field winding current is constant in an armature-controlled DC motor, the torques developed at the motor shafts are given as whereis the vector of torques developed at the motor shafts, and the constants kTi 0 (i = 1,., n) of the diagonal matrixare the torque constants of the motors. The two position vectors, q and p (representing joint shafts and motor shafts, respectively), in equations (1) and (2) are related by p = Nq. Here N E IlBnxn is a diagonal matrix of the transmission ratios between the high-speed motor shafts p and the low-speed joint shafts q. For a direct-drive manipulator, the matrix N becomes a unity matrix I. Assuming that there is no backlash or elastic deformation in the transmission subsystems, then the control torque vector acting at joint shafts isIn above derivation, it is considered that the effective inertias of the motor shafts and the frictional effects in the drive systems have already been reflected to the joint shafts. Equation (4) shows that the relationship between the control torques for driving the manipulators and the motor armature currents is characterized by a simple algebraic equation. The resulting model with the equations given above describes the dynamic behavior of mechanical manipulators more exactly and can be used to design better controllers. It can be seen that the electrical and mechanical subsystems of a manipulator are dynamically interacting within a closed-loop. The complexity of the above formulated problem has led to the development of the following control algorithms.3.DECOUPLING AND LINEARIZATION OF THE CROSS-COUPLED NONLINEAR DYNAMICS As was shown above, the dynamics of the motor-manipulator system is highly nonlinear and strongly cross-coupled. The complicated nature of the manipulator systems renders most of the existing control strategies invalid or ineffective. In solving such a challenging problem, a model-based serial compensation scheme is given hereinbelow for the decoupling and linearization of the interacting motor-manipulator systems. As can be seen from the dynamic model of manipulators given above, the first obstacle met in our way to control the highly complicated motor-manipulator systems is just the feedback interaction between the electrical and mechanical dynamics, i.e. the negative feedback path of (KEN). This can be eliminated if the following control law is used, whereis an intermediate control vector. Here, the quantities with hats (i) denote the approximate estimates of the real plant parameters. The hats for the observed variables, such as q, are omitted for simplicity. If the plant is perfectly modelled, i.e. (*) = (i), (in the ensuing derivation, statement of this assumption will be omitted for simplicity), then from equations (2) and (5), we haveEquation (6) indicates that the undesirable interaction due to the feedback path of (KEN) between the electrical and mechanical dynamics has been fully compensated for. The dynamics caused by the motor actuators has now been reduced to first-order lag terms with undesirable poles (-La 1 Ra) by the control ua given in equation (5), i.e. the transfer function from ub to ia is now a set of delayed integrators. If some tradeoff between the control performance and the control simplicity is permissible, then a simple algorithm with certain approximations can be obtained for further compensation as will be shown below. Suppose the following lead compensatoris adopted, where is another intermediate control vector, andis the diagonal time-constant matrix of the compensator. By making the time constants of 7ft much smaller thanequation (7) becomes, in essence, a set of pole-shifters which provides low-frequency zeros with high-frequency poles, shifting the undesirable poles of the original system further away from the origin, i.e. from The control law given in equation (7) is actually synthesized for pole cancellation. Upon substituting equation (7) into (6), we have the following approximate resultby neglecting the dynamics due to the new poles of (-Tfi 1). This simplification is based on the tradeoff between the control accuracy and computational expense. Since this algorithm is derived to compensate for the intrinsically coupled nonlinear dynamics, hence for the sake of convenience, the disturbance (and uncertainty) term VD in equation (1) will be neglected temporarily in the following discussion. Thus, equation (1) becomesNow let ucb in equation (7) be designed asand be designed aswhere is a vector of closed-loop law to be determined. It can be seen from equation (10) that the control lie is actually composed of two components: one is in the form of nonlinear feedback (i.e. the first term) and the other is of nonlinear feedforward compensation (the second term). Then, based on the equations given above, we haveEquation (11) means that by using the serial compensating algorithm given above, the original tightly-coupled nonlinear plant has now been transformed into a decoupled linear system comprising pure double (second-order) integrators, i.e. the resulting decoupled and linearized transfer function from u to q can approximately be represented by a set of second-order pure integrators. At this stage, we now merely have to control the n independent single input-single output subsystems which are simply behaving as open-loop double integrators with certain closed-loop control u. For the control of the above serial compensated system, the following state-feedback control law can be adopted as the closed-loop controlwhere are diagonal gain matrices for the closed-loop state feedback(corresponding to velocity and position, respectively). As a matter of fact, most ofthe state equation based methods (such as the pole-placement approach used in Zhu et al. 11) for control design will eventually result in the determination of these gain matrices for desired performance. is the reference signal of motion task. For point-to-point control, r is the desired set-points, while for trajectory-tracking control 11 , r can be defined asi.e. a linear combination of the desired joint trajectory with its first and second differentials. The corresponding algorithm of the serial compensation and closed-loop control developed in this section can be summed up as:Algorithm 1 : equations (5), (7), (9), (10), (12). It may be noticed, however, that the above control system is obtained based on the assumption that there is no model mismatch, no unmodelled dynamics i.e. the plant is perfectly modelled and the approximation in equation (8) or (11) is acceptable and no external disturbances (i.e. Vp = 0). In practical applications, this is never the case and it is well known that for model-dependent control approaches, and our model-based serial compensating algorithm given above is no exception, the performance of the overall system may be sensitive to the accuracy of the plant model. Also, in practice, the presence of model mismatch and other uncertainties, e.g. the payload variations and external disturbances in manipulator control, is inevitable and usually a discrepancy between the computed control and the real required one then arises. These non-ideal factors may significantly degrade the performance of any model-related algorithm. Hence, it is necessary to incorporate some compensating mechanism into the control system for the inevitable modelling errors as well as external disturbances. In the next section, a novel sensing and compensating mechanism will be introduced to compensate for these non-ideal factors.4.ESTIMATION AND COMPENSATION FOR THE DISTURBANCESThe performance of the algorithm given above is already much better than that of the conventional model-based computed-torque method which ignores the actuator dynamics. But as is pointed out above, the given algorithm assumes that the model of the manipulator system is known exactly and no external disturbance exists. With the aim of achieving even higher control performance, an observer for all possible uncertainties (i.e. external disturbances, unmodelled dynamics, and dynamics due to possible model mismatch) which makes the existing motor sensible will be developed below. This then allows another feedforward compensating structure to be used to compensate for the observed uncertainties. It can be seen from equation (1) that the position variables (and their derivatives) of the manipulators can be considered as a function of time and the exogenous inputs (i.e. vc and vD) of the system. Thus, we have the following general function In equation (13), the uncertain term vD (together with other uncertain dynamics, such as model mismatch and unmodelled dynamics) can generally be represented as an inverse function of that in equation (13), i.e. it can be expressed as the following function of the position vector (and its derivates) and the control-torque vector of the manipulatoror a function of the position vector (and its derivatives) and the motor armature-current vector This means that with equation (15), the undesirable dynamics vD can now be estimated. However, in practical applications, the acceleration vector q in equation (15) usually cannot be obtained directly. In such cases the following approximate estimator for q from q can be usedwhere denotes the approximate estimate of q, and parameters of the diagonal time-constant matrix of the pre-filters can be determined similar to those of Tfl. Now, from equations (1), (4) and (16), we have the approximate observation of vD for equation (15) as followsIt can be seen from equation (17) that the estimated disturbance vD (together with other uncertainties, such as parameter variation and unmodelled dynamics) is really a function of the position vector (and its first- and second-order derivatives) and motor armature-current vector of the manipulator.It is common knowledge that for a control system with known or can-be-estimated disturbance (load), the quality of control can often be improved by the addition of feedforward control to compensate for the possible disturbance. For disturbance rejection discussed here however, with Algorithm 1 given in last section, the necessary feed forward compensation turns out to be very simple, and some encouraging results will be given in what follows. Using feed forward compensation, the existing disturbance vD can be compensated for using the estimated Vo in equation (17) by attaching an additional feed forward structure to the control system developed in last section. Since the estimated disturbance vector D is in the form of torques, this means that the compensating signal can directly be added (fed forward) to the control 1Ic in equation (10). In other words, as it can be noted that by using Algorithm 1 given in last section, the transfer function from control u, to the place where the disturbance torque vD occurs can be approximated to a unity function, this means that the feedforward compensator required is actually a unity matrix. This indicates that as a compensating component, the observed disturbance torque vp can directly be injected into the existing control system developed in last section. By noting the form of vD, for simplified computation, it can be seen that the compensation can better be carried out by adopting the individual terms in equation (17) separately. The controls lick and uc of equations (9) and (10) are then modified aswith use of the first term in equation (17) andwith use of the second term o in equation (17). It may be noted that equation (19) has been significantly simplified as compared with equation (10). It implies that by using the proposed sensing and compensating scheme, less a priori knowledge about the dynamic model of mechanical manipulators is required, and the dynamics of h(q, q) in equation (1), which is much more difficult to be modelled than the inertial term D(q), poses no difficulty for the control design now. This means that the requirements at the stage of plant modelling can be significantly reduced and less computation is required in control implementation. This salient feature can be deemed as an outstanding advantage of the algorithm given in this section. The algorithm derived in this section can be summarized into: Algorithm 2: equations (5), (7), (16), (18), (19), (12). It can be seen that the resulting control system is really very simple while high performance of decoupling and disturbance rejecting control can be achieved. The difference between the two control systems developed above is that the complex nonlinear feedforward compensating component h(, q) in Algorithm 1 is replaced by the two very simple compensating components of the motor armature current ia and the observed joint acceleration q.5.NUMERICAL STUDIESIn order to verify and demonstrate the effectiveness and performance of the proposed control algorithm, detailed comparative studies of a particular manipulator 11 will be conducted. As may be encountered in practical applications, significant payload (parameter) variations as well as large external disturbances are also considered below. The desired motion task of the manipulator is to move its third link from the initial position q03 = 0 at time to = 0 to the final position qf3 = 2.0 rad in exactly tf = 2.0 s with its initial and final velocities and accelerations all being zero. To satisfy all of the boundary conditions, a quintic polynomial trajectory 11 is taken as the desired motion trajectory.For comparison studies, the performance of the disturbance sensible system C2 (by using Algorithm 2) is compared with that of the insensible one Cl (by using Algorithm 1). The resulting tracking-error and control-torque of the
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