1、The inverted pendulum Key words: inverted pendulum, modeling, PID controllers, Fuzzy controllers, state space controllers What is an Inverted Pendulum? Remember when you were a child and you tried to balance a broom-stick or baseball bat on your index finger or the palm of your hand? You had to cons
2、tantly adjust the position of your hand to keep the object upright. An Inverted Pendulum does basically the same thing. However, it is limited in that it only moves in one dimension, while your hand could move up, down, sideways, etc. Check out the video provided to see exactly how the Inverted Pend
3、ulum works. 倒立摆是什么 ？ 还记得当你是个孩子时你曾用你的食指或者掌心设法去平衡一把扫帚柄或者棒球棍吗？你必须不断地调整你 的手的位置以保持对象的垂直。一个倒立摆在本质上就是做相同的事情。然而，它会受限制因为它只能在一 定范围内移动，虽然你的手可以上升 、 下降、斜向一边等等。检查录象提供的画面来观察倒立摆是如何确切 地 工作的。 An inverted pendulum is a physical device consisting in a cylindrical bar (usually of aluminum) free to oscillate around a fix
4、ed pivot. The pivot is mounted on a carriage, which in its turn can move on a horizontal direction. The carriage is driven by a motor, which can exert on it a variable force. The bar would naturally tend to fall down from the top vertical position, which is a position of unsteady equilibrium. 一个倒立摆是
5、个物理设备它包括一个圆柱体的棒子（通常是铝的）可以在一个支点周围振荡。这个支点是安在一个车架上，它的转动方向是水平的偏转。小车是由一个马达控制的，它可以运 用于一个变力。棒子会有自然的趋势从最高的竖直位置下落，那是一个不稳定的平衡位置。 The goal of the experiment is to stabilize the pendulum (bar) on the top vertical position. This is possible by exerting on the carriage through the motor a force which tends to con
6、trast the free pendulum dynamics. The correct force has to be calculated measuring the instant values of the horizontal position and the pendulum angle (obtained e.g. through two potentiometers). 实验的目标是使摆（棒子）稳定在最高的竖直位置。这是有可能的只要运用通过马达的小车一个力该力可以与 “ 自由 ” 摆的动力学抵消。这个正确的力必须通过计算测量水平偏转的瞬时值和摆的 角度（获得两个电位计）。 T
7、he system pendulum+cart+motor can be modeled as a linear system if all the parameters are known (masses, lengths, etc.), in order to find a controller to stabilize it. If not all the parameters are known, one can however try to reconstruct the system parameters using measured data on the dynamics of
8、 the pendulum. 系统摆 +车 +马达可以建模成一个线形系统如果所有的参数都是已知的（质量、长度等），为了寻找一个控制 器去稳定住它。如果不是所有的参数都已知，可以用多种方法去 “ 推想 ” 系统参数像用摆的动力学的测量数据。 What is it used for?Just like the broom-stick, an Inverted Pendulum is an inherently unstable system. Force must be properly applied to keep the system intact. To achieve this, pro
9、per control theory is required. The Inverted Pendulum is essential in the evaluating and comparing of various control theories. 倒立摆是干什么的？就好象扫帚柄，一个倒立摆是一个 天生的不稳定系统。力度必须被严格地应用以保持系统的完整性。为了实现它，严格的控制理论是必须的。倒立摆在求数值和各种控制理论的比较中是必要的。 The inverted pendulum is a traditional example (neither difficult nor trivia
10、l) of a controlled system. Thus it is used in simulations and experiments to show the performance of different controllers (e.g. PID controllers, state space controllers, fuzzy controllers.). 倒立摆是一个控制器系统中的一个传统的例子（既不困难也不是没有价值）。尽管它是仿真和实验来显示不同控制器的性能（举例来说 PID 控制器，状态空间控制器，模糊控制器）。 The Real-Time Inverted P
11、endulum is used as a benchmark, to test the validity and the performance of the software underlying the state-space controller algorithm, i.e. the used operating system. Actually the algorithm is implement form the numerical point of view as a set of mutually co-operating tasks, which are periodical
12、ly activated by the kernel, and which perform different calculations. The way how these tasks are activated (e.g. the activation order) is called scheduling of the tasks. It is obvious that a correct scheduling of each task is crucial for a good performance of the controller, and hence for an effect
13、ive pendulum stabilization. Thus the inverted pendulum is very useful in determining whether a particular scheduling choice is better than another one, in which cases, to which extent, and so on. 实时倒立摆被作为一个基准，去测试软件在状态空间控制器运算法则下 的有效性和性能，也就是实用的操作系统。事实上运算法则是通过数值点实现的该数值点看作一组互助的协同操作的任务，它是周期性的通过核心的活动，它执行不
14、同的计算。这些任务如何活动的方法（举例来说激活命令）被称作任务的时序安排。很明显每个任务的时序安排对控制器的一个好的性能是至关紧要的，因此对一个摆的稳定性是有效的。如此倒立摆是非常有用的在决定是否一个特殊的时序安排的选择比另一个好，在哪个情形下，在什么程度内等等。 Modeling an inverted pendulum.Generally the inverted pendulum system is modeled as a linear system, and hence the modeling is valid only for small oscillations of the
15、pendulum. 为倒立摆建模 。 通常倒立摆系统建模成一个线形系统，因此模型只对小幅度摆动的摆才有效。 Prescribed trajectory tracking with certain accuracy is a main task of robotic control. The control is often based on a mathematical model of the system. This model is never an exact representation of reality, since modeling errors are inevitable
16、. Moreover, one can use a simplified model on purpose. In this paper, the structured and unstructured uncertainties are of primary interest, i.e., the modeling error due to the parameters variation and unmodeled modes, especially the friction and sensor dynamics, neglected time delays, etc. 法定轨道通过确定
17、的精确性是机器控制的一个主要任务。控制通常是基于一个系统的数学模型。模型不是一 个准确的实体表现，模型的误差是不可避免的。此外，我们可以特意使用一个简化的模型。在这篇论文中， 构造好的和未构造好的不确定因素是主要的兴趣所在，也就是说模型的误差导致参数变化和未模型化的模式 ，尤其是摩擦力和敏感元件的力度，被忽视的时间延迟等等。 The erroneous model and the demand for high performance require the controller to be robust. The sliding mode controllers(SMC) based on
18、variable structure control can be used if the inaccuracies in the model structure are bounded with known bounds. However, an SMC has some disadvantages, related to chattering of the control input signal. Often this phenomenon is undesirable, since it causes excessive control action leading to increase wear of the actuators and to excitation of unmodeled dynamics.
