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

Underactuated 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)     

Inverse dynamics of underactuated multibody systems

The present work deals with controlled mechanical systems subject to holonomic constraints. In particular, we focus on underactuated systems, defined as systems in which the number of degrees of freedom exceeds the number of inputs. The governing equations of motion can be written in the form of differential-algebraic equations (DAEs) with a mixed set of holonomic and control constraints. The rotationless formulation of multibody dynamics will be considered [1]. To this end, we apply a specific projection method to the DAEs in terms of redundant coordinates. A similar projection approach has been previously developed in the framework of generalized coordinates by Blajer & Kołodziejczyk [2]. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Control of a class of underactuated mechanical systems

In this paper a new control methodology for underactuated mechanical systems is proposed. The basic idea of this method is to combine passive velocity field control with decoupling vector field. In order that the underactuated mechanical systems can be stabilized at the desired position after settling on the desired velocity vector field, novel control strategies are proposed. Proposed control strategies are applied to the underactuated planar three-link manipulator and underactuated planar body. Simulations demonstrate the usefulness and effectiveness of the proposed control methodology.     

Orbital stabilization of a class of underactuated mechanical systems

Typically, the aim of a controller is to stabilize an equilibrium or a reference trajectory. For underactuated mechanical systems this objective might not be achievable because of the occurance of limit cycles due to discontinuous effects like actuator backlash or Coulomb friction. Therefore, an alternative approach is to explicitly impose a stable periodic orbit on the system by a suited controller. This way at least amplitude, frequency and settling time can be influenced directly. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A numerical method for the servo constraint problem of underactuated mechanical systems

Servo constraints are used in inverse dynamics simulations of discrete mechanical systems, especially for trajectory tracking control problems [1], whose desired outputs are represented by state variables and treated as servo constraints [2]. Servo constraint problems can be classified into fully actuated and underactuated multibody systems, and the equations of motion take the form of differential algebraic equations (DAEs) including holonomic and servo constraints. For fully actuated systems, control inputs can be solved from the equations by model inversion, as the input distribution matrix is nonsingular and invertible. However, underactuated systems have more degrees of freedom than control inputs. The input distribution matrix is not invertible, and in contrast to passive constraints, the realization of servo constraints with the use of control forces can range from orthogonal to tangential [3]. Therefore, it is challenging for the determination of control inputs which force the underactuated system to realize the partly specified motion. For differentially flat underactuated systems, the differentiation index of DAEs may exceed three. Hence we need to apply specific index reduction techniques, such as the projection approach applied in [3], [4], and [6]. The present work applies index reduction by minimal extension [5] to differentially flat underactuated crane systems and shows that the index can be reduced from five to three and even to one. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Tanker chartering: A system dynamics case study

This paper describes a practical system dynamics case study. It attempts to show how this OR technique can be applied to a practical problem and the results which it can, and cannot, be expected to give.Although the paper is based on a real project, the details have been altered for commercial secrecy, and the emphasis in this paper is, for the same reason, rather different from that of the original work. The aim of the paper is, therefore, to give enough of a “real” background to contrast the industrial application of SD with some of its other applications.For the sake of reasonable brevity, the emphasis in this paper is on the managerial viewpoint, and the extent to which an SD model provides helpful answers to managerial questions. The model paper is described only briefly as full details, and a model listing are given by Coyle. Some other aspects of the system are treated by Winch.     

Urban dynamics and multiple-objective programming: A case study of Beijing

This study serves as a primary application of the integrated system dynamics and multiple-objective programming (ISDMOP) model for strategic planning of Beijing city, which is here divided into six subsystems as population, resources, energy, economy, environment and ecosystem, with the planning horizon spanning from 2003 to 2020. Comparison between the original system dynamics (ORSD) model based on the existing economic structure of Beijing and the optimized system dynamics (OPSD) model adjusted according to the solutions of the multiple-objective programming (MOP) are conducted. The developing trend of each subsystem is simulated and illuminated, based on which constructive suggestions are provided for urban strategic planning of Beijing. The ISDMOP model is proved effective for investigating urban dynamics and realizing the multiple-objective programming.     

Computing statistics for Hamiltonian systems: A case study

We present the results of a set of numerical experiments designed to investigate the appropriateness of various integration schemes for molecular dynamics simulations. In particular, we wish to identify which numerical methods, when applied to an ergodic Hamiltonian system, sample the state-space in an unbiased manner. We do this by describing two Hamiltonian system for which we can analytically compute some of the important statistical features of its trajectories, and then applying various numerical integration schemes to them. We can then compare the results from the numerical simulation against the exact results for the system and see how closely they agree. The statistic we study is the empirical distribution of particle velocity over long trajectories of the systems. We apply four methods: one symplectic method (Störmer–Verlet) and three energy-conserving step-and-project methods. The symplectic method performs better on both test problems, accurately computing empirical distributions for all step-lengths consistent with stability. Depending on the test system and the method, the step-and-project methods are either no longer ergodic for any step length (thus giving the wrong empirical distribution) or give the correct distribution only in the limit of step-size going to zero.     

On some simple examples of mechanical systems with hyperbolic chaos

Examples of mechanical systems with hyperbolic chaos are discussed, including the Thurston–Weeks–Hunt–MacKay hinge mechanism, in which conservative Anosov dynamics is realized, and dissipative systems with Smale–Williams type attractors (a particle on a plane under periodic kicks, interacting particles sliding on two alternately rotating disks, and a string with parametric excitation by modulated pump). The examples considered in the paper are interesting from the viewpoint of filling hyperbolic theory, as a well-developed field of the mathematical theory of dynamical systems, with physical content. The results of computer tests for hyperbolicity of the systems are presented that are based on the analysis of the statistics of intersection angles of stable and unstable manifolds.     

Forced nonlinear oscillation of underactuated systems: the balance board example

In this contribution we consider a mechanical model of the so-called balance board or “indoboard” system, which is an unstable underactuated mechanical system. Instead of stabilizing one equilibrium point we investigate the asymptotic stabilization of a closed orbit, which results in a nonlinear oscillation. To this end the system is considered in coordinates of the real Jordan normal form, which facilitates the construction of a Control-Lyapunov function. On this basis, a nonlinear feedback law which renders the periodic orbit asymptotically stable is easily derived via Sontags formula. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mixing methodologies for information systems development and strategy: A higher education case study

Although informed approaches from an interpretivist perspective have been widely promoted over the last two decades information systems development is largely informed by a functionalist perspective, into which human-centred or interpretivist issues are incorporated as they are seen to be required. By treating such developments as critically informed social systems, this paper offers an alternative viewpoint. Firstly, a theoretical justification for such an approach is offered, drawing on developments in Critical Management Science, themselves based on a strand of Critical Social Theory. Secondly, an action research-based intervention is detailed, demonstrating the practical applicability of such an approach, an applicability that has hitherto been questioned. The outcome is a synthesised approach to operational and strategic information systems development, within a critical framework, which, it is argued, offers a greater potential for success than the currently promoted approaches.     

Verification of hyperbolicity for attractors of some mechanical systems with chaotic dynamics

Computer verification of hyperbolicity is provided based on statistical analysis of the angles of intersection of stable and unstable manifolds for mechanical systems with hyperbolic attractors of Smale–Williams type: (i) a particle sliding on a plane under periodic kicks, (ii) interacting particles moving on two alternately rotating disks, and (iii) a string with parametric excitation of standing-wave patterns by a modulated pump. The examples are of interest as contributing to filling the hyperbolic theory of dynamical systems with physical content.     

A simple characterization of disjoint covering systems

A simple necessary and sufficient condition for a system of arithmetic sequences to cover the set of all integers is given.     

Water resources planning based on complex system dynamics: A case study of Tianjin city

A complex system dynamic (SD) model focusing on water resources, termed as TianjinSD, is developed for the integrated and scientific management of the water resources of Tianjin, which contains information feedback that governs interactions in the system and is capable of synthesizing component-level knowledge into system behavior simulation at an integrated level, thus presenting reasonable predictive results for policy-making on water resources allocation and management. As for the Tianjin city, interactions among 96 components for 12 years are explored and four planning alternatives are chosen, one of which is based on the conventional mode assuming that the existing pattern of human activities will be prevailed, while the others are alternative planning designs based on the interaction of local authorities and planning researchers. Optimal mode is therefore obtained according to different scenarios when compared the simulation results for evaluation of different decisions and dynamic consequences.     

Parsimonious representation of nonlinear dynamical systems through manifold learning: A chemotaxis case study

Nonlinear manifold learning algorithms, such as diffusion maps, have been fruitfully applied in recent years to the analysis of large and complex data sets. However, such algorithms still encounter challenges when faced with real data. One such challenge is the existence of “repeated eigendirections,” which obscures the detection of the true dimensionality of the underlying manifold and arises when several embedding coordinates parametrize the same direction in the intrinsic geometry of the data set. We propose an algorithm, based on local linear regression, to automatically detect coordinates corresponding to repeated eigendirections. We construct a more parsimonious embedding using only the eigenvectors corresponding to unique eigendirections, and we show that this reduced diffusion maps embedding induces a metric which is equivalent to the standard diffusion distance. We first demonstrate the utility and flexibility of our approach on synthetic data sets. We then apply our algorithm to data collected from a stochastic model of cellular chemotaxis, where our approach for factoring out repeated eigendirections allows us to detect changes in dynamical behavior and the underlying intrinsic system dimensionality directly from data.     

A numerical study of the dynamics of elastic articulated systems in a fluid

We propose a method and an algorithm for computing the dynamics of elastic structures of articulated form in a fluid flow taking account of the weakening in certain structural elements. In describing the motion we use two sets of radius-vectors, which are approximated in the computations by parametric local splines of first degree. The possibilities of the proposed method are illustrated using the example of the study of the dynamics of transition processes in an articulated anchor-buoy structure, which arise when there is an abrupt change in the direction of the fluid flow velocity. We determine the kinematic and force characteristics of the structure under various changes in the direction of the flow velocity. We determine the structural elements in which the weakening occurs. Three figures. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 128–134.