On the control of a two-dimensional multi-link inverted pendulum: the form of the dynamic equations from choice of co-ordinate system

The inverted pendulum is a mechanical idealization which poses an interesting and, with an arbitrary level of degree of freedom, difficult problem in control. This paper, noting that the model may be taken as a simple two-dimensional approximation to a robot manipulator, proceeds to investigate how the relative form and complexity of the dynamic equations of motion, fundamental to the application of control theory, may be dictated by choice of co-ordinate system. Two such systems, both appropriate from the nature of the problem, are compared in a theoretical formulation using the method of Lagrange, and it is shown that one—the commonly accepted robotic kinematic notation of Denavit and Hartenberg—leads here to an unnecessarily complex set of closed form equations. It is also demonstrated how additional physical features may be incorporated into the equations to yield an improved approximation, and so a modified control problem.     

On the control of a two-dimensional uniform multi-link inverted pendulum: the enumeration and identification of equilibrium states

The standard control task of balancing a horizontally translating inverted pendulum is discussed within the wider context of other balancing-type control problems which may be set by the same system. Defining a ‘physically realizable’ state as one which contains, and provides a variant on, the standard control task associated with the system, an explicit general result is proposed for the computation of all such possible states. The result is validated by a concordant result arising from an interesting theoretical probabilistic analogy—the symmetric random walk of single dimension—which is also utilized to provide two direct proofs. The enumerative theory is then developed to accommodate further, rather more non-standard, control problems, and results given concerning the automatic generation and identification of states by computer.     

Some aspects of the inverted pendulum problem for modeling of locomotion systems

The inverted pendulum can be used as one of the simplest models of a biped locomotion system. Problems of stability, periodic motion, and locomotion of the inverted pendulum are considered in this paper. The development algorithms of this paper hopefully can be used for study, simulation, and analysis of more realistic models of biped locomotion systems.     

倒立摆系统的一种H∞优化控制

在运用H_∞环路成形设计方法时权函数的选择一般是凭经验试凑,不能保证所得控制器综合指标最优;且控制器阶次较高,不便于工程实现和参数调整.鉴于以上问题,提出了基于遗传算法的权函数选取方法,在遗传算法的目标函数里加入对鲁棒性能和时域性能指标的约束,从而得到使控制器综合性能指标最优的权函数,避免了凭经验试凑权函数的盲目性;然后,给出一种用多变量PID控制器逼近此控制器的方法,使所得控制器便于实际运用.最后用此方法对小车倒立摆系统进行仿真,结果表明了该方法的有效性.     

Intelligent backstepping control for wheeled inverted pendulum

The adaptive output recurrent cerebellar model articulation control (AORCMAC) is an adaptive system with simple computation, good generalization capability and fast learning property. The proposed AORCMAC has superior capability to the conventional cerebellar model articulation controller (CMAC) in efficient learning mechanism and dynamic response. In this study, an intelligent backstepping tracking control system is proposed for wheeled inverted pendulums (WIPs) with unknown system dynamics and external disturbance. In this control system, an ABORCMAC is used to copy an ideal backstepping control (IBC), and a compensated controller is designed to compensate for difference between the IBC law and AORCMAC. Moreover, all adaptation laws of the proposed system are derived based on the Lyapunov stability analysis, the Taylor linearization technique, so that the stability of the closed-loop system can be guaranteed.     

拟人控制二维单倒立摆

以二维单极倒立摆为被控对象，利用拟人控制的思想形成非线性控制律，并确定出反馈系数间的相对关系。通过在线调试实现定性控制律的量化，从而成功地控制了二维单级倒立摆的稳定。与传统的控制系统设计相比，不依赖于数学模型，不受线性约束；与模糊控制相比，不需要人类直接被控对象的经验。拟人控制方法设计简单易行，且得到的控制系统具有较强的鲁棒性。     

倒立摆系统的变结构控制方案

对单级倒立摆系统的稳定和鲁棒控制问题,提出了一种基于李雅普诺夫直接法的变结构控制方案,理论分析和计算机模拟都表明该控制方案不仅能镇定倒立摆,实现对台车的位移控制,而且当系统参数变化时也实现了对该系统的鲁棒控制.     

Evolving neurocontrollers for balancing an inverted pendulum

This paper introduces an evolutionary algorithm that is tailored to generate recurrent neural networks functioning as nonlinear controllers. Network size and architecture, as well as network parameters like weights and bias terms, are developed simultaneously. There is no quantization of inputs, outputs or internal parameters. Different kinds of evolved networks are presented that solve the pole-balancing problem, i.e. balancing an inverted pendulum. In particular, controllers solving the problem for reduced phase space information (only angle and cart position) use a recurrent connectivity structure. Evolved controllers of 'minimal' size still have a very good benchmark performance.     

基于模糊加权的倒立摆混合控制

针对小车倒立摆系统,提出了一种线性状态反馈控制和滑模控制模糊加权的控制方法.滑模控制器的作用是将摆角控制在零的一个邻域内,在此邻域内首先采用近似的线性化模型来描述倒立摆系统,然后采用基于极点配置的方法设计系统的线性状态反馈控制器以使系统的状态稳定在给定值,两个控制器的输出通过加权求和作为倒立摆的控制作用.仿真结果证实了该方法的有效性.     

双轮共轴移动式倒立摆动力学建模与状态反馈控制

应用第一类拉格朗日方法对系统进行力学分析,建立了以电机转矩为输入且轮在轴向无滑移的非完整约束下系统的数学模型.双轮共轴移动式倒立摆的运动控制目标是移动式倒立摆在二维平面内按指定的方向和速度运动,同时保持摆杆平衡.利用状态反馈,构造闭环系统的状态方程,通过极点配置求得控制量.仿真结果验证了系统状态方程的正确性和控制方法的合理性.     

平行单级双倒立摆系统的建模与滑模变结构控制

从两摆间无弹簧连接的平行双倒立摆系统特定物理结构出发,结合控制目标进行合理建模与分析,验证所建模型可行性,讨论系统能控性和参数间的关系,并设计了滑模变结构控制律.该文的理论和方法对平行单级双倒立摆的进一步研究工作具有重要意义.     

二级直线倒立摆系统的实物控制

针对二级直线倒立摆系统,采用拉格朗日方程法建立其理论模型,分别使用线性二次最优控制（Linear Quadratic Regulator,LQR）及基于趋近律的滑模控制（Sliding Mode Control,SMC）算法来实现干扰存在情况下倒立摆的平衡控制。对于LQR算法,研究了矩阵[Q]和矩阵[R]与反馈控制矩阵[K]的定性关系,并经过反复多次实验,不断试凑,得到一组良好的控制参数,实现了倒立摆的稳定控制。SMC算法采用基于指数趋近律的控制方法进行了滑模变结构控制器的设计,并利用边界层法来进一步削弱抖振。最后通过仿真及实验,实现了倒立摆的实物平衡控制。     

倒立摆的局部反馈线性化控制方法

针对一级倒立摆系统,基于局部反馈线性化方法将系统的动力学转换为便于控制器设计的级联系统模型,并利用线性矩阵不等式设计了满意PID控制器.实验结果表明,该方案实现了系统输出时指令信号的精确跟踪且具有抗干扰能力.     

小车二级摆摆起控制器设计

研究了小车二级并行摆系统及小车二级串行摆系统的摆起控制器设计问题,并给出了这两种系统的实验结果.首先,针对上述两种系统,设计了两步控制器,即1)摆起双摆达到倒立稳摆位置的控制器,2)进行稳摆控制的控制器.其次,由于小车二级摆位移受轨道长度限制,又考虑了小车位移的控制问题.上述两种实际系统的摆起及稳摆成功,验证了所提出设计方法的有效性.     

倒立摆的可变增益迭代学习控制算法

对倒立摆系统的平衡控制问题进行研究。在建立系统数学模型的基础上,提出指数变增益迭代学习控制律,并设计了控制器。通过系统仿真实验,结果表明：与常规迭代学习控制律相比较,本文采用的方法收敛速度大大加快,系统动态性能得到很大改善。     

倒立摆的自适应积分反步控制策略

针对直线单级倒立摆在模型参数不确定和外部扰动情况下的稳定控制问题,提出一种自适应积分反步控制策略。采用拉格朗日方程建立倒立摆系统的运动学模型,为减少稳态误差,将误差积分项引入反步法,设计了倒立摆的控制器;对含有未知参数的系统非线性状态微分方程,设计适当的Lyapunov函数推导出系统未知参数的自适应更新律,削弱了参数不确定性的影响。将自适应积分反步控制与一般的反步法控制、模糊控制及神经网络控制的仿真结果进行了对比,并在LabVIEW开发环境下进行了实物实验。结果表明,自适应积分反步法可以较为迅速且精确地完成稳定控制,较好地克服系统参数不确定及外部扰动的影响,具有较强的鲁棒性。     

DRNN在倒立摆摆起控制中的研究

针对小车一级倒立摆的起摆控制,以DRNN神经网络作为辨识器,在线自适应调整PD控制器的两项参数。在起摆范围相同的情况下,DRNN神经网络控制的倒立摆系统其模型参数变化范围为-50%～30%,传统PD控制倒立摆系统其参数变化范围为-40%～20%。结果表明,基于DRNN神经网络的PD控制器比传统的PD控制器具有较强的抗干扰能力和自适应能力,系统鲁棒性增强,效果明显优于传统的PD控制器。     

倒立摆的迭代学习控制方法研究

针对倒立摆系统典型的非线性、多变量和自然不稳定的特点,在建立系统数学模型基础上,利用迭代学习控制方法,根据动态指标的要求设计了控制器,并完成了系统仿真,结果表明:与传统反馈控制相比较,所设计的迭代学习控制器使系统的动态性能得到了很大的改善。     

二级倒立摆的区间值模糊控制

为了提高倒立摆模糊控制的稳定性,研究了二级倒立摆的区间值模糊控制.根据区间值模糊集理论,由不同类型的隶属度函数构成区间结构,利用区间值模糊蕴涵建立控制规则,设计出二级倒立摆的区间值模糊控制器,并通过仿真与实时控制实验,对比分析了区间值模糊控制与点值模糊控制的性能.实验结果表明,区间值模糊控制具有更好的稳定性和鲁棒性.