Controlled pendulum on a movable base

A plane motion of a multilink pendulum hinged to a movable base (a wheel or a carriage) is considered. The control torque applied between the base and the first link of the pendulum is independent of the base position and velocity and is bounded in absolute value. The coordinate determining the base position is cyclic. The mathematical model of the system permits one to single out the equations describing the pendulum motion alone, which differ from the well-known equations of motion of a pendulum with a fixed suspension point both in the structure and in the parameters occurring in these equations. The phase portrait of motions of a control-free one-link pendulum suspended on a wheel or a carriage is obtained. A feedback control ensuring global stabilization of the unstable upper equilibrium of the pendulum is constructed. Time-optimal control synthesis is outlined.     

Lyapunov-Based Controller for the Inverted Pendulum Cart System

A nonlinear control force is presented to stabilize the under-actuated inverted pendulum mounted on a cart. The control strategy is based on partial feedback linearization, in a first stage, to linearize only the actuated coordinate of the inverted pendulum, and then, a suitable Lyapunov function is formed to obtain a stabilizing feedback controller. The obtained closed-loop system is locally asymptotically stable around its unstable equilibrium point. Additionally, it has a very large attraction domain.Contributed by Prof. F. Pfeiffer.     

Stability and stabilization of the family of pseudolinear differential systems

New methods for the analysis of the robust stability of equilibrium states are developed for some classes of nonlinear differential systems. We formulate sufficient conditions for the stability of the trivial solutions of the families of pseudolinear controlled systems with undetermined coefficient matrices and feedback from the measured output. A method is developed for the analysis of stability according to the first approximation to the family of nonlinear systems. The application of the obtained results is reduced to the solution of systems of linear differential matrix inequalities. We present an example of a system of stabilization of a double inverted pendulum.     

Stability and instability of controlled motions of a two-mass pendulum of variable length

The problem of parametric control of plane motions of a two-mass pendulum (swing) is considered. The swing model is a weightless rod with two lumped masses one of which is fixed on the rod and the other slides along it within bounded limits. The control is the distance from the suspension point to the moving point. The proposed control law of swing excitation and damping consists in continuously varying the pendulumsuspension length depending on the phase state. The stability of various controlled motions, including the motions near the upper and lower equilibria, is studied. The Lyapunov functions that prove the asymptotic stability and instability of the pendulum lower position in the respective cases of the pendulum damping and excitation are constructed for the proposed control law. The influence of the viscous friction forces on the pendulum stable motions and the onset of stagnation regions in the case of its excitation is analyzed. The theoretical results are confirmed by graphical representation of the numerical results.     

Time-optimal swing-up feedback control of a pendulum

Time-optimal feedback control is obtained that brings a pendulum to the upper unstable equilibrium position. The solution is based on the maximum principle and involves analytical investigations combined with numerical computations. As a result, the switching and dispersal curves that bound the domains in the phase plane corresponding to different values of the optimal bang-bang control are constructed for various values of the maximal admissible control torque.     

基于修正型罗德里格斯参数的三维刚体摆姿态控制

3D 刚体摆是研究地球静止轨道航天器的一个力学简化模型, 它绕一个固定、无摩擦的支点旋转, 具有3 个转动自由度. 文章给出基于修正型罗德里格斯(Rodrigues) 参数描述的3D 刚体摆的姿态动力学方程, 针对3D 刚体摆姿态和角速度稳定的非线性控制设计问题, 基于无源性控制理论利用能量法设计了3D 刚体摆的系统控制器, 并证明了系统满足无源性. 构造了系统的李雅普诺夫(Lyapunov) 函数, 利用能量法设计出3D 刚体摆的姿态控制律, 并由拉萨尔(LaSalle) 不变集原理证明了该控制律的渐近稳定性. 仿真实验给出了3D 刚体摆在倒立平衡位置的姿态和角速度的渐近稳定性, 仿真实验结果表明基于能量方法的3D 刚体摆姿态控制是有效的.     

Equilibrium of an Inverted Mathematical Double-Link Pendulum with a Follower Force

A discrete model of an elastic pendulum with a follower force is studied. This model is an inverted mathematical two-link pendulum with viscoelastic hinges. It is shown that divergent bifurcations are possible for some absolute values of the follower force and the stiffness of the restraint of the pendulum's upper end. As a result, the vertical position of the equilibrium becomes unstable and two new nonvertical stable equilibrium states (fork bifurcation) occur.     

Optimal feedback gains of a delayed proportional-derivative (PD) control for balancing an inverted pendulum

In the dynamics analysis and synthesis of a con-trolled system, it is important to know for what feedback gains can the controlled system decay to the demanded steady state as fast as possible. This article presents a sys-tematic method for finding the optimal feedback gains by taking the stability of an inverted pendulum system with a delayed proportional-derivative controller as an example. First, the condition for the existence and uniqueness of the stable region in the gain plane is obtained by using the D-subdivision method and the method of stability switch. Then the same procedure is used repeatedly to shrink the stable region by decreasing the real part of the rightmost charac-teristic root. Finally, the optimal feedback gains within the stable region that minimizes the real part of the rightmost root are expressed by an explicit formula. With the optimal feedback gains, the controlled inverted pendulum decays to its trivial equilibrium at the fastest speed when the initial val-ues around the origin are fixed. The main results are checked by numerical simulation.     

Controlling the inverted pendulum by means of a nested saturation function

A nonlinear controller is presented for the stabilization of the underactuated inverted pendulum mounted on a cart. The fact that this system can be expressed as a chain of integrators, with an additionally nonlinear perturbation, allows us to use a nested saturation control technique to bring the pendulum to the top position, with zero displacement of the cart. The obtained closed-loop system is semiglobal, asymptotically stable, and locally exponentially stable, under the assumption that the position of the angle is initialized above the upper half plane.     

一类带柔性附件充液航天器姿态机动控制

基于Lyapunov稳定性理论研究了用动量轮控制一类带轻质悬臂梁附件的充液航天器的姿态机动控制问题, 其中晃动液体用黏性力矩球摆模型代替, 悬臂梁附件用若干集中质量代替. 用动量矩定理和Lagrange方程分别推导得到航天器主刚体、等效球摆、等效集中质量的动力学方程, 所用反馈控制律包含了与动量轮角加速度密切相关的权重因子, 利用系统初、终状态和到达最终姿态所需时间解析确定此权重因子. 同时利用Lyapunov稳定性理论得到了实现最终姿态机动的稳定性判据. 数值仿真表明所用控制律的有效性, 分析附件的相对主刚体平面的转角、相对系统质心的高度、长度、刚度、质量、阻尼系数和到达最终姿态所需时间等因素对控制过程中航天器剩余章动角的影响大小.      

An extension of the Levi-Weckesser method to the stabilization of the inverted pendulum under gravity

Sufficient conditions are given for the stability of the upper equilibrium of the mathematical pendulum (inverted pendulum) when the suspension point is vibrating vertically with high frequency. The equation of the motion is of the form $$ \ddot{\theta}-\frac{1}{l}\bigl(g+a(t)\bigr) \theta=0, $$ where l,g are constants and a is a periodic step function. M. Levi and W. Weckesser gave a simple geometrical explanation for the stability effect provided that the frequency is so high that the gravity g can be neglected. They also obtained a lower estimate for the stabilizing frequency. This method is improved and extended to the arbitrary inverted pendulum not assuming even symmetricity between the upward and downward phases in the vibration of the suspension point.     

Stability and Hopf bifurcation in an inverted pendulum with delayed feedback control

In this paper, we investigate the dynamics of the inverted pendulum with delayed feedback control. The existence and stability of multiple equilibria depending on the control strengths are studied. Taking the time delay of the control terms as a parameter, periodic oscillations induced by delay are found. By using the method of multiple scales, the effect of the control gains and the relative mass of the pendulum on the stability and direction of Hopf bifurcations are discussed. Numerical simulations are employed to illustrate the obtained theoretical results.     

Stability domains of the vertical equilibrium state of a triple simple pendulum

The stability boundaries for the vertical equilibrium position of a triple simple pendulum subject to asymmetric follower force are analyzed. The effect of the upper spring and the magnitude of the follower force on the roots of the characteristic equation is studied Translated from Prikladnaya Mekhanika, Vol. 44, No. 10, pp. 122–134, October 2008.     

New analytical results of energy-based swing-up control for the Pendubot

In this paper, we revisit the energy-based swing-up control solutions for the Pendubot, a two-link underactuated planar robot with a single actuator at the base joint. The control objective is to swing the Pendubot up to its unstable equilibrium point (at which two links are in the upright position). We improve the previous energy-based control solutions by analyzing the motion of the Pendubot further. Our main contributions are threefold. First, we provide a bigger control parameter region for achieving the control objective. Specifically, we present a necessary and sufficient condition for avoiding the singular points in the control law. We obtain a necessary and sufficient condition on the control parameter such that the up–down equilibrium point (at which links 1 and 2 are in the upright and downward positions, respectively) is the only undesired closed-loop equilibrium point. Second, we prove that the up–down equilibrium point is a saddle via an elementary proof by using the Routh–Hurwitz criterion to show that the Jacobian matrix valued at the point has two and two eigenvalues in the open left- and right-half planes, respectively. We show that the Pendubot will eventually enter the basin of attraction of any stabilizing controller for all initial conditions with the exception of a set of Lebesgue measure zero provided that these improved conditions on the control parameters are satisfied. Third, we clarify the relationship between the swing-up controller designed via the partial feedback linearization and that designed by the energy-based approach. We present the simulation results for validation of these results.     

基于Gauss伪谱法的3D刚体摆姿态最优控制

研究Gauss伪谱法求解3D刚体摆姿态最优控制问题．针对其最优姿态控制问题,既要满足由任意位置运动到平衡位置姿态运动规划问题,又要满足系统含有动力学约束的力学模型问题,提出基于四元数来描述3D刚体摆的数学模型,建立3D刚体摆姿态的动力学和运动学方程,为了解决3D刚体摆在平衡位置处的姿态最优控制问题,设计基于Gauss伪谱算法的最优姿态开环控制器,得到了3D刚体摆的姿态最优控制轨迹,得到满足的可行解,通过仿真实验验证了其开环解在平衡位置的控制姿态最优性．     

Influence of the Nonlinearity of the Elastic Elements on the Stability of a Double Pendulum with Follower Force in the Critical Case

A generalized mathematical theory of a double mathematical pendulum with follower force is used to analyze the stability of the vertical equilibrium position of the pendulum with both linear and nonlinear (hard and soft) elastic elements in the critical case of one zero root of the characteristic equation. The influence of the parameters of these elements on the safe and dangerous sections of the stability boundary is demonstrated__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 4, pp. 133–142, April 2005.     

Output feedback stabilization of the inverted pendulum system: a Lyapunov approach

In this work, we present an output feedback stabilization method for the Inverted Pendulum Cart (IPC) system around its unstable equilibrium point. The pendulum is initialized in the upper-half plane, and the position of the cart and the pendulum angular positions are always available. Our strategy was accomplished introducing a suitable coordinate change to obtain a nonlinear version of the original system, which is affine in the unmeasured velocities state. This fact allows us to adapt an observer based controller devoted to render the closed-loop system to the origin. The proposed observer based controller was designed using the direct Lyapunov method. This allows estimating the corresponding attraction domain for the whole system, which can be as large or as small as desired. While the corresponding closed-loop stability analysis was made using the LaSalle Invariance Theorem. Convincing numerical simulations were included to show the performance of the closed-loop system.     

Using the delayed feedback control and saturation control to suppress the vibration of the dynamical system

In the present paper, the delayed feedback control is applied to suppress or stabilize the vibration of the primary system in a two degree-of-freedom dynamical system with parametrically excited pendulum. The case of a 1:2 internal resonance between pendulum and primary system is studied. The method of multiple scales is applied to obtain second-order approximations of the response of the system. The system stability and bifurcations of equilibrium point of the averaged equations are computed. It is shown that the delayed feedback control can be used to suppress the vibration or stabilize the system when the saturation control is invalid. The vibration of the primary system can be suppressed by the delayed feedback control when the original system is in the single-mode motion. The effect of gain and delay on the vibration suppression is discussed. As the delay varies at a fixed value of the gain, the vibration of the primary system can be suppressed at some values of the delay. The vibration suppression performance of the system is improved at a large value of the gain. The vibration of the primary system could be suppressed about 56% compared with the original system by choosing the appropriate values of gain and delay. The delayed feedback control also can be used to stabilize the system when the original system is unstable. The gain and delay could be chosen as the controlling parameters. Numerical simulation is agreement with the analytical solutions well.     

Steering performance of an inverted pendulum vehicle with pedals as a personal mobility vehicle

From the viewpoints of environmental protection, support for the aged and ensuring the right to mobility, there is a need to develop a new type of mobility vehicle that provides more effective transportation. The authors propose an inverted pendulum vehicle with pedals as one of the forms of personal mobility vehicles (PMVs). In this paper, the steering performance of the inverted pendulum vehicle with pedals is discussed based on experiments on a prototype. From the experimental results, it was confirmed that the errors from the five subjects for the target trajectory and the five-grade evaluation of the maneuverability were similar. Finally, we created an inverted pendulum vehicle with pedals to which was added a reaction actuator for the steering system. From the experimental results, it was found that setting appropriate feedback gains for the handle steering angle and its rate of rotation, which control the right and left wheel driving torques, resulted in greatly improved maneuverability.     

Adaptive control of rotary inverted pendulum system with time-varying uncertainties

In this paper, an adaptive controller is proposed to balance a rotary inverted pendulum with time-varying uncertainties. The goal of the control is to bring the pendulum close to the upright position regardless of the various uncertainties and disturbances. Its underactuated dynamics is first decoupled by Olfati’s transformation into a cascade form, and then an adaptive controller is designed to deal with the uncertainties in the new space. Based on the Lyapunov-like theory, the closed loop stability and boundedness of all internal signals can be proved. The simulation results show that the proposed scheme is capable of giving good performance, as desired.