基于快速起摆的Furuta摆切换控制系统

针对由一个驱动臂和一个未驱动摆杆组成的欠驱动Furuta摆系统,设计了实现其稳定的切换控制系统.控制任务是将未驱动摆杆稳定在上方不稳定平衡点的同时,将驱动臂控制到零点,分为两部分:首先基于部分反馈线性化技术设计一个饱和的状态反馈控制器,将摆杆快速控制到上方不稳定平衡点附近;然后切换到一个线性的全状态反馈控制器,实现系统的稳定控制.仿真实验验证了控制器的有效性.     

A nonlinear hybrid controller for swinging-up and stabilizing the Furuta pendulum

The conventional switching strategy for solving the inverted pendulum control problem is based on two steps: swinging-up and stabilization. In this note, first, a new strategy for swinging the Furuta pendulum up towards the desired upright position is designed using the Speed-Gradient method, which uses only directly measured coordinates. Then, a nonlinear controller, based on the Forwarding approach, stabilizes the upright position. As a new contribution the latter leads to a nonlinear stabilizer around the upright position, whose Lyapunov function yields a larger size estimation of the domain of attraction than the one obtained with the traditional linear approach. This estimation allows us to use it in a global switching strategy in the practical implementation and guarantees almost-global asymptotic stability of the equilibrium. Successful experimental results are reported with the available laboratory Furuta pendulum.     

The direct Lyapunov method for the stabilisation of the Furuta pendulum

A nonlinear controller for the stabilisation of the Furuta pendulum is presented. The control strategy is based on a partial feedback linearisation. In a first stage only the actuated coordinate of the Furuta pendulum is linearised. Then, the stabilising feedback controller is obtained by applying the Lyapunov direct method. That is, using this method we prove local asymptotic stability and demonstrate that the closed-loop system has a large region of attraction. The stability analysis is carried out by means of LaSalle's invariance principle. To assess the controller effectiveness, the results of the corresponding numerical simulations are presented.     

Bifurcations of the horizontally forced spherical pendulum

Various planar motions of the horizontally forced damped spherical pendulum are considered and, in particular, their stability to non-planar perturbations. By making a careful choice of coordinates, all solutions of the planar pendulum can be considered including small amplitude periodic solutions, running oscillations and chaotic solutions. The full nonlinear equations in the chosen coordinates are derived and the symmetries of the system are described. Bifurcation diagrams for various types of solutions are presented. Stability of the chaotic solutions is determined by considering a normal Lyapunov exponent.     

Generating self-excited oscillations for underactuated mechanical systems via two-relay controller

A tool for the design of a periodic motion in an underactuated mechanical system via generating a self-excited oscillation of a desired amplitude and frequency by means of the variable structure control is proposed. First, an approximate approach based on the describing function method is given, which requires that the mechanical plant should be a linear low-pass filter–the hypothesis that usually holds when the oscillations are relatively fast. The method based on the locus of a perturbed relay systems provides an exact model of the oscillations when the plant is linear. Finally, the Poincaré map's design provides the value of the controller parameters ensuring the locally orbitally stable periodic motions for an arbitrary mechanical plant. The proposed approach is shown by the controller design and experiments on the Furuta pendulum.     

Stabilization of the Furuta pendulum around its homoclinic orbit

We present an energy based control approach to control the Furuta pendulum. A controller is proposed for swinging the pendulum and raise it to its uppermost unstable equilibrium position. The passivity properties of the system are used as a guideline in the control strategy. The stability analysis is carried out by using a Lyapunov technique.     

直线一级倒立摆的视觉伺服控制

把视觉引入倒立摆的控制中,建立了视觉伺服的倒立摆系统.利用针孔摄像机的成像几何模型建立与图像坐标有关的状态空间方程,采用基于图像的视觉伺服控制方法,对摆杆在像平面上的位置进行实时控制.在Matlab环境下用LQR(Linear Quadratic Regulator)和极点配置法进行仿真,对比仿真结果.验证了系统方程推导的正确性,以及控制算法的有效性.     

Dynamic analysis of a multibody open-chain system

The free plane motion of a compound pendulum is studied using the Lagrange equations of motion. The specific compound pendulum consists of a main plane disk from which are suspended two chains, each of which is composed of m and n rigid plane disks, respectively. A nonlinear system of m + n + 1 ordinary differential equations (ODEs) with respect to the properly selected generalized coordinates, is obtained. For small-amplitude swing the above system becomes a linear differential system. The eigenvalues and eigenvectors constitute the eigenfrequencies and the modeshapes, respectively, of the free swing of the considered compound pendulum. Also, relations are extracted which ensure a unique swing with no relative rotation between any two disks. Finally, several numerical results are given.     

A new swing-up law for the Furuta pendulum

In this paper the swing-up problem for the Furuta pendulum is solved applying Fradkov's speed-gradient (SG) method to a dimension 4 model of the system. The new law is compared with the conventional Åström-Furuta strategy, based on a dimension 2 model. A comparative analysis, including simulations and experiments, whereby the advantages and effectiveness of the new law for swinging the pendulum up are shown, is included.     

Swing-up and LQR stabilization of a rotary inverted pendulum

In this article, we consider the swing-up and linear quadratic regulator (LQR) stabilization of a rotary inverted pendulum. A DC motor rotates a rigid arm. At the end of the rigid arm is a joint with a pendulum suspended from it. Two encoders check the degree of the rigid arm and the pendulum every 0.5 ms. This article describes a modified bang-bang control which swings the pendulum up safely and fast. In order to solve the stabilization problem, we used a linear quadratic regulator. When the user gives a large disturbance to the pendulum so that it loses its position, it quickly recovers to the upright position. Experimental results showed that the proposed bang-bang and LQR controllers can stabilize a rotary inverted pendulum system within 3.0 s from any starting point. The system also showed robustness to a large disturbance.     

Nonlinear controller tuning based on a sequence of identifications of linearized time-varying models

A novel algorithm for tuning controllers for nonlinear plants is presented. The algorithm iteratively minimizes a criterion of the control performance. In each iteration one experiment is performed with a reference signal slightly different from the previous reference signal. The input–output signals of the plant are used to identify a linear time-varying model of the plant which is then used to calculate an update of the controller parameters. The algorithm requires an initial feedback controller that stabilizes the closed loop for the desired reference signal and in its vicinity, and that the closed-loop outputs are similar for the previous and current reference signals. The tuning algorithm is successfully tested on a laboratory set-up of the Furuta pendulum.     

Total Energy Shaping Control of Mechanical Systems: Simplifying the Matching Equations Via Coordinate Changes

Total energy shaping is a controller design methodology that achieves (asymptotic) stabilization of mechanical systems endowing the closed-loop system with a Lagrangian or Hamiltonian structure with a desired energy function - that qualifies as Lyapunov function for the desired equilibrium. The success of the method relies on the possibility of solving two PDEs which identify the kinetic and potential energy functions that can be assigned to the closed loop. Particularly troublesome is the partial differential equation (PDE) associated to the kinetic energy which is nonlinear and inhomogeneous and the solution, that defines the desired inertia matrix, must be positive-definite. In this note, we prove that we can eliminate or simplify the forcing term in this PDE by modifying the target dynamics and introducing a change of coordinates in the original system. Furthermore, it is shown that, in the particular case of transformation to the Lagrangian coordinates, the possibility of simplifying the PDEs is determined by the interaction between the Coriolis and centrifugal forces and the actuation structure. The examples of pendulum on a cart and Furuta's pendulum are used to illustrate the results.     

Design of optimized fuzzy cascade controllers by means of Hierarchical Fair Competition-based Genetic Algorithms

In this study, we present a design of an optimized fuzzy cascade controller based on Hierarchical Fair Competition-based Genetic Algorithms (HFCGA) for a rotary inverted pendulum system. In this system, one controls the movement of a pendulum through the adjustment of a rotating arm. The objective is to control the position of the rotating arm and to make the pendulum maintain the unstable equilibrium point at vertical position. To control the system, we design a fuzzy cascade controller scheme which consists of two fuzzy controllers arrange in a cascaded topology. The parameters of the controller are optimized by means of the HFCGA algorithm. The fuzzy cascade scheme comprises two controllers located in two loops. An inner loop controller governs the position of the rotating arm while an outer controller modifies a set point of the inner controller implied by the changes of the angle of pendulum. The HFCGA being a computationally effective scheme of the Parallel Genetic Algorithm (PGA) has been developed to eliminate an effect of premature convergence encountered in Serial Genetic Algorithms (SGA). It has emerged as an effective optimization vehicle to deal with very large search spaces. A comparative analysis involving computing simulations and practical experiment demonstrates that the proposed HFCGA based fuzzy cascade controller comes with superb performance in comparison with the conventional Linear Quadratic Regulator (LQR) controller as well as HFCGA-based PD cascade controller.     

Shaping stable periodic motions of inertia wheel pendulum: theory and experiment

We consider an underactuated two‐link robot called the inertia wheel pendulum. The system consists of a free planar rotational pendulum and a symmetric disk attached to its end, which is directly controlled by a DC‐motor. The goal is to create stable oscillations of the pendulum, which is not directly actuated. We exploit a recently proposed feedback‐control design strategy based on motion planning via virtual holonomic constraints. This strategy is shown to be useful for design of regulators for achieving orbitally exponentially stable oscillatory motions. The main contribution is a step‐by‐step procedure on how to achieve oscillations with pre‐specified amplitude from a given range and an arbitrary independently chosen period. The theoretical results are verified via experiments with a real hardware setup. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Topology of real algebraic space curves

In this paper we give a new projection-based algorithm for computing the topology of a real algebraic space curve given implicitly by a set of equations. Under some genericity conditions, which may be reached through a linear change of coordinates, we show that a plane projection of the given curve, together with a special polynomial in the ideal of the curve contains all the information needed to compute its topological shape. Our method is also designed in such a way to exploit important particular cases such as complete intersection curves or curves contained in nonsingular surfaces.     

Finding the principal bifurcation value of the maximum control torque in the problem of optimal control synthesis for a pendulum

The principal bifurcation value in the problem of synthesis of a time-optimal control for damping the oscillations of a nonlinear pendulum is found. A one-link pendulum controlled by a torque of forces produced at the suspension point is considered. The magnitude that bounds the modulus of the control torque is the only essential parameter of the investigated system. It is supposed that an admissible control may exceed the moment of the gravitational force. By the principal bifurcation value, we mean the value of the maximum admissible control torque for which a trajectory with two switches of relay optimal control arises in the synthesis pattern. This publication is a logical complement to papers [1–3], where the mentioned synthesis patterns were obtained numerically with the use of the maximum principle. In contrast to [1–3], we formulate a system of explicit relations of integral type for the numerical construction of the boundaries of domains of the phase plane inside which the trajectories with two switches start. Based on these relations, an analytical upper estimate of the principal bifurcation value is obtained. This value is also found numerically. In a special coordinate system, the synthesis patterns are constructed that illustrate the specific features of the optimal control for large angular velocities of pendulum rotations.     

Use of a coordinate transformation in the incremental phase plane

A coordinate transformation in the incremental phase plane is introduced which facilitates the design of sampled-data control systems in which there is a simple non-linearity. Use of these coordinates permits the slope of a boundary specified by the nonlinearity to be directly related to a simple gain term in the transformed system. A realizable form of compensation is shown to evolve in terms of the system variables. By using the transformed coordinates, it is possible to identify intersample oscillations which are undetectable in the original coordinate system. Examples are presented to illustrate the method.     

A procedure to find equivalences among dynamic models of planar biped robots

In this paper, a method to find equivalences among dynamic models is presented. The models are given in different generalized coordinates for the same mechanical system. This novel method is valid for planar biped robots, i.e., those whose motions are executed only in a plane. We show that no matter which generalized coordinates are used to get the dynamic models, there is always an equivalence among them by using a particular input matrix. We exhibit some advantages of getting the dynamic model using absolute coordinates instead of relative coordinates, and we show how to calculate the input matrix for getting the equivalence between these models. Because of its simplicity, a compass-like biped robot model is used as example to explain in detail the novel procedure. In order to appreciate the benefits of the proposed procedure in biped robots of high degrees of freedom, we also present the equivalence between two dynamic models for the single support phase of a 5 degrees of freedom biped robot using absolute coordinates and relative ones.     

Interconnection and damping assignment passivity‐based control of a class of underactuated mechanical systems with dynamic friction

This paper considers the extension of the interconnection and damping assignment passivity‐based control methodology for a class of underactuated mechanical systems with dynamic friction. We present a new damping assignment approach to compensate friction by means of a nonlinear observer. Friction at the actuated joints is assumed to be captured by a bristle deflection model: the Dahl model. Based on the Lyapunov direct method we show that, under some conditions, the overall closed‐loop system is stable and, by invoking the theorem of Barbashin–Krasovskii, we arrive to asymptotic stability conditions. Experiments with an underactuated mechanical system, the Furuta pendulum, show the effectiveness of the proposed scheme when friction is compensated. Copyright © 2010 John Wiley & Sons, Ltd.