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外文资料翻译Robust Tracking of a Lightweight Manipulator System AbstractA robust control design for high performance joint trajectory tracking of a flexible lightweight manipulator system is proposed. The design is based on a combined controller-observer scheme involving the sliding manifold approach and the optimal interpolation technique .This controller provides the designer with an enhanced joint tracking performance when the system is subject to parametric variations due to structural disturbances caused by link flexibility and load uncertainties. The parametric variations are handled by sliding control and the estimation of the nonlinearly excited elastic dynamics by an optimal interpolator of the structures dynamic responses. The design procedure is progressive, i.e., we start with a basic controller and then modify it in order to improve the performance. Closed loop simulations with the various designed controllers are used to validate the analytical results and to help choosing the most suitable one.Keywords: Flexible manipulator, sliding control, optimal interpolator. IntroductionDuring the last few years, there has been an increased interest in the area of modeling and controller design for flexible manipulator systems. This is mainly motivated by the growing need for designing faster and more dexterous manipulator systems. Throughout the paper, flexibility means structural elasticity at the manipulators links level.Recently, a large body of literature has discussed problems pertaining to the dynamic formulation and modeling of the open serial kinematics chains involving one or more flexible modes. Similarly, several authors have focused on dealing with the control aspects of such dynamic systems. This research activity has led to several interesting results. But while most of the existing literature has focused on using conventional control tools such as linearization about nominal trajectories, the inverse control technique or the virtual rigid manipulator based controller, only a few have dealt vigorously with the robustness issue. This is indeed crucial design concern that requires extensive studies, particularly when the system under consideration is subject to parametric variations due to structural induced disturbances or to changes in the dynamic configuration of the system.In this paper, we study the robust tracking control problem of a one-link flexible arm. The prototype chosen for the purpose of simulations is the manipulator in the Flexible Automation Laboratory at Georgia Institute of Technology. A combined controller-observer scheme is used as the control strategy. This method involves optimal interpolation technique combined with the sliding manifold approach.Consider the simpliflied model of a one-link flexible arm shown in Figure 1. =1,2,n is the n-vector of elastic modes, is the control torque applied to the joint,K=diag(k1,k2,kn)denotes the matrix of the constant flexural spring coefficients,B=diag(b1,b2bn)denotes the passive damping matrix which account for the internal viscous friction of the flexible manipulator.Assuming that all products of the form i, j andij,(i,j)=1,2,.,n are negligible, the flexible dynamics of Equation (2)can be expressed in the following bilinear form:And u=2 is an artificial input of the dynamics and v is a bias term which is a linear function of the angular acceleration. The matrices F and G(X) in Equation (3) are derived in Appendix A. The bilinear form of the flexible equation will be used for optimal interpolation purposes in Section 4. 3. Optimal interpolation Based ObserverIn case where the observation of high dimensional state vector is needed, the estimation task becomes quite heavy. In this situation, reliability and hardware cost become major limiting factors on the overall systems performance. The motivation here is to reduce the order of the bilinear dynamics, which corresponds in this case to the flexible dynamics of the manipulator, to a lower dimension bilinear model in which the input-output mapping can be expressed in number of training signals used in the initial learning stage. This off-line modeling procedure permits the system to deliver outputs, without the requirement for solving high dimensional ordinary differential equations. Moreover, this technique provides upper bounds on the state estimation errors. There are subsequently used in the control synthesis of the sliding control strategy. Having reduced the flexible dynamics of the arm to the bilinear form of Equation (3) by regarding the angular velocity and angular acceleration as the artificial inputs, the optimal interpolator can be applied now to generate estimates of the elastic modes and their corresponding upper bounds, and hence for generating the required control efforts.4. Numerical Simulation results In a preliminary stage, the optimal interpolator inset up for the purpose of generating the first elastic modes. The simulations were run using the first flexible mode and for a smooth commanded joint trajectory specified by an angular motion from Where, T=2sec. the total simulation time was 4 sec. Figure 2 shows the results of applying this controller to the one-link flexible arm assuming an initial angular error of 9 degrees. As Expected, the controller attempts to prevent variation of the sliding variables Figure3.Comparison of the correction torques for controllers #2, #3, and#4. and drives its derivative (and as a result 1 ) to zero. Obviously, this controller is not suitable for angular trajectory tracking. Therefore, an additional correction torque (which is a static function of the sliding variables) is necessary. Figure 3 shows the correction torque 1u as a function of for controllers #2,#3 and#4. Controller #2 has the ideal VSC form and because of the sign(s) term, the correction effort is discontinuous at s =0.Controller #3 uses the boundary layer approach and comprises the term sat (s/), where has the optimal value derived in Equation (16). Although the correction term is a continuous function of s in this case, its first time derivative is not. Controller #4 uses the term tanh(gs) instead of sat(s/)，which is infinitely differentiable. The value of the gain g was chosen for the best match with the correction term in controller #3. Figures 4 and 5 show the results of applying these controllers when I (0)=9deg. 4) The following can be readily drawn form these simulations: Although controller #2(ideal VSC) has best performance in terms of tracking accuracy and error decay, it nevertheless displays high level of chattering, a dynamical behavior that is usually undesiradle. As expected, controllers #3 and #4 eliminate the chattering behavior at the expense of tolerating slower decay rate of tracking errors. Note that the correction control effort has better profile (in term of less chattering). When using controller #4 , the correction profile is smoother compared to that of controller #3. This is an expected behavior of this controller as discussed earlier. According to the simulation results, controller #4 has relatively the most attractive behavior of all other controllers. This is in terms of smaller error and smaller control bandwidth. 5. Conclusion In this paper, a robust control strategy based on the variable structure control combined with an optimal interpolator is proposed. Used here for the purpose of improving trajectory tracking of a one link flexible arm subject to induced disturbances and variation of the dynamic configuration, this control technique might equally be applied to a wider range of systems in which the dynamical model is subject to multiplicative as well as additive modeling uncertainties. The elastic mode estimates and their upper bound errors are generated through the optimal interpolator output. The controller design procedure proposed is progressive. It starts with a basic controller with average overall performance and ends up with a modified boundary-layer VSC controller that exhibits a better performance than the others. Lately, there have been efforts for efficiently modeling multi-link flexible manipulators. In that direction, the control strategy outlined here could well be extended for the robust control of such systems.具有鲁棒输出跟踪的轻量级机械手系统 摘要 机械手系统，是在鉴于鲁棒控制设计中高性能关节具有的灵活、轻便的轨迹跟踪控制问题而提出的。该系统设计是借助滑动流行方法和最优插值方法完成的，此外，设计师还提供了一个共同跟踪的性能。该系统运行参数由于结构紊乱引起的灵活性和负载不确定的因素会产生相应的变化。这个参数变化可以引起滑膜控制和弹性动力学中非线性兴奋的结构优化的圆弧插补其产生动态响应。然后逐次修改此次设计，结合众多设计控制人员的封闭环境模拟来验证分析的结果，最终选择最佳的方案。关键词：灵活的机械手，滑动控制，最佳的圆弧插补器1. 介绍最近，大量的有关探讨拥有一个或更多的灵活的方式的机械手系统问题的动态配方和建模以及串行运动学链的文献相继发表。与此同时，有些专家聚焦在处理控制方面的动态系统。该研究活动已经诞生了几个有趣的结论。但是大多数现有文献主要集中在使用常规控制工具，如线性化约命名的轨迹反控制技术和虚拟僵化的机械手，只有少数是基础控制的鲁棒性问题。这确实是一个至关重要的设计，需要广泛的研究，特别是由于结构参数变化引起的干扰或变动态配置的系统设计。在本论文中，我们阐述了鲁棒轨迹跟踪控制问题的机械手控制系统。该系统实际与柔性佐治亚理工学院的自动化实验室的模拟的机械，应用优插值方法、结合理论与方法进行研究的。2.柔性臂的动态模型 简化模型的one-link柔性臂如图1所示。其中，是严格的对应的方式角位置， =1,2,n是n-vector弹性的模式，u是用于控制力矩的联合，K=diag(k1,k2,kn)中的矩阵的恒弯曲弹簧系数，B=diag(b1,b2bn)的矩阵表示被动阻尼占内部粘滞摩擦柔性臂的。假设所有产品的形式i，j和ij,(i,j)=1,2,.n，是可以忽略不计，弹性（15）的动态方程可以用一下的双线性形式。其中，X是一种存在状态向量的维，2n代表第n弹性模式。其衍生工具如下：和u=2是一个人工输入的动力学，v是一种偏见的术语，是线性的，具有角加速度的功能。通过这个矩阵F和克（X）可以导出（3）。3.基于最优插值的观察如果有必要观察高位状态向量，那么估计的任务会变得很艰巨。在这种情况下，系统的可靠性和硬件成
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