Control Of Underactuated Mechanical Systems Research Articles

Modeling and simulation of an underactuated system

One of the most active research areas in mechatronic systems is the control of mechanical systems controlled by electronic systems using computer programs. These programs execute algorithms called control laws. Our study focuses on the control of underactuated mechanical systems: Case of a reversed two-wheeled pendulum. It consists of developing a control law to stabilize this system. This class of system is rich in practical as well as theoretical applications (SEGWAY, Acrobot robots ...) and this is why the control synthesis for underactuated mechanical systems constitutes a very active research axis and still constitues an open domain for technological research.

Adaptive control schemes applied to a control moment gyroscope of 2 degrees of freedom

Control of underactuated mechanical systems has been an important trend in mechatronics and nonlinear systems. Typical examples are the Furuta pendulum and inertia wheel pendulum. On the other hand, gyroscopes are important in aerial and space systems. In this paper, a two-degree of freedom underactuated control moment gyroscope (CMG) is studied. This system is highly coupled of one joint to the another and is difficult to control, which make it an important benchmark system. The problem addressed in this paper is to achieve robust motion control of the underactuated CMG. Firstly, a trajectory tracking controller is developed by using the feedback linearization technique. Secondly, two new adaptive algorithms are introduced, which correspond to an adaptive neural network algorithm and an adaptive model regressor scheme. A real–time experimental comparison is carried out among a linear PD control law, a cascade PID-PID controller and the introduced schemes. The real-time experimental study validates the introduced theory, where the performance of the controllers is evaluated with and without a periodic disturbance at the input.

The variational discretization of the constrained higher-order Lagrange-Poincaré equations

In this paper we investigate a variational discretization for the class of mechanical systems in presence of symmetries described by the action of a Lie group which reduces the phase space to a (non-trivial) principal bundle. By introducing a discrete connection we are able to obtain the discrete constrained higher-order Lagrange-Poincaré equations. These equations describe the dynamics of a constrained Lagrangian system when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces. The equations, under some mild regularity conditions, determine a well defined (local) flow which can be used to define a numerical scheme to integrate the constrained higher-order Lagrange-Poincaré equations.Optimal control problems for underactuated mechanical systems can be viewed as higher-order constrained variational problems. We study how a variational discretization can be used in the construction of variational integrators for optimal control of underactuated mechanical systems where control inputs act soley on the base manifold of a principal bundle (the shape space). Examples include the energy minimum control of an electron in a magnetic field and two coupled rigid bodies attached at a common center of mass.

Sliding mode control for underactuated mechanical systems via nonlinear disturbance observer: stabilization of the rotational pendulum

This paper presents design of the sliding mode controller (SMC) via a nonlinear disturbance observer (NDO) for stabilization of the rotational pendulum. The mathematical model is depicted by four first-order nonlinear differential equations with two outputs and single input. The linear model is obtained by linearizing nonlinear equations. This linearized version is found to be highly unstable. The applied NDO is capable of estimating extremely nonlinear terms (disturbance) present in the model. Then, the NDO based SMC is designed using linear model and disturbance estimation, and hence is applied to the original nonlinear system. The developed controller is enough to drive a sliding surface (which holds both actuated and unactuated dynamics) attaining the sliding mode against severe nonlinearities existing in the model, in addition large parameter variations and/or external disturbance. The sliding coefficients are determined using the linear quadratic regulator (LQR) technique. The obtained controller parameters can establish the zero dynamics as of sliding mode dynamics asymptotically. Also, the proposed control algorithm has potential to alleviate the chattering phenomenon occurred in the control signal. Overall, the control scheme expands the stability region of an arm (in horizontal half-plane) and a pendulum (over vertical half-plane) both for large initial conditions. Lastly, numerical simulations and quantitative measurements are validated with an established controller in the literature.

Robust control of a class of under-actuated mechanical systems with model uncertainty

ABSTRACTRobust control of under-actuated mechanical systems (UMSs) with model uncertainty is still a challenging problem. For UMSs, the model parametric uncertainties make it difficult to precisely calculate the isolated equilibrium point corresponding to a fixed input. Without an accurate destination state, many set-point control methods cannot eliminate the positioning errors. An improved sliding mode control (ISMC) method is proposed to solve the robust control problem for a class of UMSs with model uncertainty and input disturbance. A balance variable is introduced in the sliding surface design to compensate for the disturbance caused by the inaccurate destination state, and the ISMC method is proposed to make the system state reach the sliding surface in finite time. Linear matrix inequality approach and particle swarm optimisation algorithm are applied to design the sliding mode surface parameters. The simulation results on an UMS are presented to show the effectiveness of the proposed scheme.

Control of a class of underactuated mechanical systems obviating matching conditions

The presently established techniques for control of underactuated mechanical systems are dependent on solving partial differential equations (PDEs) arising out of matching conditions. In this brief a novel controller design methodology for (asymptotic) stabilization of nonlinear systems is proposed that obviates the need of solving PDEs to obtain the control laws. The geometry based energy shaping methodology forms the fully constructive procedure for control of underactuated mechanical systems. The control methodology is based on manipulating symmetric structure of the system such that the power flow is established between the controller and unactuated part of the system. The modifications leads to identification of the two new passive outputs which are further utilized for kinetic and potential shaping and the desired controller is obtained using passivity based techniques. The theory is illustrated with two benchmark examples.

Periodic motion planning and control for underactuated mechanical systems

ABSTRACTWe consider the problem of periodic motion planning and of designing stabilising feedback control laws for such motions in underactuated mechanical systems. A novel periodic motion planning method is proposed. Each state is parametrised by a truncated Fourier series. Then we use numerical optimisation to search for the parameters of the trigonometric polynomial exploiting the measure of discrepancy in satisfying the passive dynamics equations as a performance index. Thus an almost feasible periodic motion is found. Then a linear controller is designed and stability analysis is given to verify that solutions of the closed-loop system stay inside a tube around the planned approximately feasible periodic trajectory. Experimental results for a double rotary pendulum are shown, while numerical simulations are given for models of a spacecraft with liquid sloshing and of a chain of mass spring system.

Discontinuous H ∞ control of underactuated mechanical systems with friction and backlash

Nonlinear H∞-control is extended to discontinuous mechanical systems with degree of underactuation one, where nonlinear phenomena such as Coulomb friction and backlash are considered. The problem in question is to design a feedback controller via output measurements so as to obtain the closed-loop system in which all trajectories are locally ultimate bounded, and the underactuated link is regulated to a desired position while also attenuating the influence of external perturbations and nonlinear phenomena. It is considered that positions are the only measurements available for feedback in the system. Performance issues of the discontinuous H∞-regulation controller are illustrated in an experimental study made for a rectilinear plant with friction modified to have a gap in the point of contact between bodies.

Shaping the Energy of Mechanical Systems Without Solving Partial Differential Equations

Control of underactuated mechanical systems via energy shaping is a well-established, robust design technique. Unfortunately, its application is often stymied by the need to solve partial differential equations (PDEs). In this technical note a new, fully constructive, procedure to shape the energy for a class of mechanical systems that obviates the solution of PDEs is proposed. The control law consists of a first stage of partial feedback linearization followed by a simple proportional plus integral controller acting on two new passive outputs. The class of systems for which the procedure is applicable is identified imposing some (directly verifiable) conditions on the systems inertia matrix and its potential energy function. It is shown that these conditions are satisfied by three benchmark examples.

Collocated Adaptive Control of Underactuated Mechanical Systems

Collocated adaptive control of underactuated mechanical systems is still a concern for the control community. The main difficulty comes from the nonlinearity of the collocated inverse dynamics with respect to the base parameters, which forbids the direct application of classical adaptive control schemes. This paper extends and encompasses the Slotine's adaptive control, which was developed for fully actuated mechanical systems, to stabilize the collocated state space of an underactuated mechanical system. The key point is to define the sliding variable as the difference between the system's velocity and an exogenous state whose dynamics is considered as control input. We first revisit the Slotine's result in view of this definition and then show how to extend it to the underactuated case. Stability and convergence of time-varying reference trajectories for the collocated dynamics are shown to be in the sense of Lyapunov. Global well-posedness of the control laws is achieved by means of a new algebraic property of the mass matrix. Simulations, comparisons to existing control strategies, and experimental results on a two-link manipulator verify the soundness of the proposed approach.

Online payload estimation for the control of underactuated mechanical systems

This paper presents a payload estimation scheme for underactuated robotic manipulators with passive joints that are not driven by actuators. In the proposed scheme, only the payload, which can be quite uncertain when a robot performs various tasks, is estimated, because the manipulator’s electrical and other mechanical parameters are generally known in advance. In comparison to other adaptive schemes for underactuated robotic manipulators, the proposed scheme produces satisfactory transient performance and also reduces the computational burden in real-time implementation. The proposed estimation law is also based on the theory of Variable Structure Systems. In contrast to existing adaptation laws that have an integral form, the proposed law estimates uncertain payload using lowpass filtering of a switching signal that is always bounded, which avoids the parameter-drifting problem that is often encountered when using the previous integral laws. Real-time experiments are conducted using an inverted pendulum and the experimental results demonstrate the feasibility and effectiveness of the proposed scheme.

Unified formalism for higher-order variational problems and its applications in optimal control

In this paper, we consider an intrinsic point of view to describe the equations of motion for higher-order variational problems with constraints on higher-order trivial principal bundles. Our techniques are an adaptation of the classical Skinner–Rusk approach for the case of Lagrangian dynamics with higher-order constraints. We study a regular case where it is possible to establish a symplectic framework and, as a consequence, to obtain a unique vector field determining the dynamics. As an interesting application we deduce the equations of motion for optimal control of underactuated mechanical systems defined on principal bundles.

Radial-basis-functions neural network sliding mode control for underactuated mechanical systems

This paper presents an indirect adaptive neural network sliding mode Control (IANSMC) technique and a neural network sliding mode control (NNSMC) for underactuated robot manipulators. The adaptive neural network (NN) based on radial basis functions (RBF) is used to estimate the equivalent control and to compensate model uncertainties. In IANSMC, the adaptive learning algorithms are derived using Lyapunov stability analysis. Sliding mode control and indirect adaptive technique are combined to deal with modeling parameter uncertainties and bounded disturbances. The stability of the mixed controller is then proved. NN parameters are tuned on-line, without an off-line learning phase. For the NNSMC, the NN control is used to learn the equivalent control due to the unknown nonlinear system dynamics and the robust sliding mode control (SMC) is designed for a trajectory tracking control. Simulation results show that the NNSMC and IANSMC are better than the classical SMC to control underactuated manipulators. Although the proposed controllers can eliminate the chattering phenomena and estimate matching uncertainties. The IANSMC can also reject mismatched perturbations. Discussions and comparisons between proposed controllers are presented.

Analysis and Control of Underactuated Mechanical Systems [On the Shelf

This book consists of five chapters and four appendixes. Each chapter ends with many excellent references. The first chapter is an introduction and historical literature review. There have been few attempts like this to assemble the literature and present a systematic framework for the control of underactuated mechanical systems (UMSs). Although this chapter does not cover all of the vast number of techniques that are available, it does provide many references for those not described. The second chapter covers generalities and the state of the art on the control of UMSs. The third chapter presents UMSs from the Lagrangian formalism. who are experts in IEEE fields of interest or in history can request a guest account for editing access. The fourth chapter details the two main methods for classifying the UMSs and explores the possibility of a link between them. The key steps of each method are laid out, and several examples are provided, further aiding in cementing the reader,s understanding. In the fifth and final chapter, the authors describe control approaches to address all of the classifications within the CFD method.

A survey of underactuated mechanical systems

An underactuated mechanical system (UMS) is a system which has fewer independent control actuators than degrees of freedom to be controlled. Control of UMS is considered as one of the most active fields of research because of its diverse engineering applications. This survey reviews UMS from its history to the state-of-the-art research on modelling, classification, control, and to some extent, provides some unique insights for bottleneck issues and future research directions.

Passivity based control of under-actuated mechanical systems with nonlinear dynamic friction

Passivity-based control of under-actuated mechanical systems with nonlinear friction effects in the generalized coordinates of motion is analyzed in this paper. Nonlinear friction is modeled with a modified LuGre dynamic friction model. The internal states of the dynamic friction model are incorporated as generalized coordinates in a port-controlled Hamiltonian formulation for the complete mechanical system in such a way that all passivity properties of this formulation are preserved for the extended generalized coordinates system. Interconnection and damping assignment passivity-based control laws are developed for the models of two case studies: a building with a magneto-rheological damper and a double pendulum. Simulation results are also presented.

Optimal control of underactuated mechanical systems: A geometric approach

In this paper, we consider a geometric formalism for optimal control of underactuated mechanical systems. Our techniques are an adaptation of the classical Skinner and Rusk approach for the case of Lagrangian dynamics with higher-order constraints. We study a regular case where it is possible to establish a symplectic framework and, as a consequence, to obtain a unique vector field determining the dynamics of the optimal control problem. These developments will allow us to develop a new class of geometric integrators based on discrete variational calculus.

Dynamics and control of underactuated mechanical systems: analysis and simple experimental verification

AbstractUnderactuated mechanical systems are systems with fewer control inputs than the degrees of freedom, m < n, the relevant technical examples being e.g. cranes, aircrafts and flexible manipulators. The determination of an input control strategy that forces an underactuated system to complete a set of m specified motion tasks (servo‐constraints) is a demanding problem. The solution is conditioned to differential flatness of the problem, denoted that all 2n state variables and m control inputs can algebraically be expressed, at least theoretically, in terms of the desired m outputs and their time derivatives up to a certain order. A more practical formulation, motivated hereafter, is to pose the problem as a set of differential‐algebraic equations, and then obtain the solution numerically. The theoretical considerations are illustrated by a simple two‐degree‐of‐freedom underactuated system composed of two rotating discs connected by a flexible rod (torsional spring), in which the pre‐specified motion of the first disc is actuated by the torque applied to the second disc, n = 2 and m = 1. The determined control strategy is then verified experimentally on a laboratory stand representing the two‐disc system. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Control of underactuated mechanical systems via passive velocity field control: Application to snakeboard and 3D rigid body

In this paper we consider the application of a new formulation of passive velocity field control (PVFC) to the control problem of a class of underactuated mechanical systems (UAMS). In our previous report synthesizing the desired velocity vector fields for PVFC by the decoupling vector fields, we were able to prove that control torques may be successfully obtained and the solvability of the synthesizing problem of PVFC for the UAMS may be guaranteed. We newly develop this control design methodology and propose an explicit feedback solution to the control problems of the UAMS, such as snakeboard, underactuated 3D rigid bodies. Several simulations show the validity of the proposed control methodology.

Non-linear control of under-actuated mechanical systems

In this paper, non-linear controls of three kinds of under-actuated mechanical systems are summarised. The systems are: inverted pendulum systems, Acrobot system and helicopter experimental system. In particular, concerning the inverted pendulum system, a parallel cart-type double inverted pendulum and a serial cart-type double inverted pendulum are introduced. For the systems, different controller design schemes are considered. Experimental results on the systems are also shown.