Optimal control of vibrationally excited locomotion systems

Optimal controls are constructed for two types of mobile systems propelling themselves due to relative oscillatory motions of their parts. The system of the first type is modelled by a rigid body (main body) to which two links are attached by revolute joints. All three bodies interact with the environment with the forces depending on the velocity of motion of these bodies relative to the environment. The system is controlled by high-frequency periodic angular oscillations of the links relative to the main body. The system of the other type consists of two bodies, one of which (the main body) interacts with the environment and with the other body (internal body), which interacts with the main body but does not interact with the environment. The system is controlled by periodic oscillations of the internal body relative to the main body. For both systems, the motions with the main body moving along a horizontal straight line are considered. Optimal control laws that maximize the average velocity of the main body are found.     

Optimal control of the rotation of a system of two bodies connected by an elastic rod

Optimal controls of the rotation of a mechanical system consisting of two rigid bodies, joined by an elastic rod, through a specified angle about an axis passing through the centre of mass of one of the bodies are constructed. The problem of the optimal control of the rotation of the system through a given angle with complete suppression of the oscillations of the elastic rod at the minimum of the energy functional of the control moment and the problem of time-optimality for a specified constraint on the energy functional of the control moment are solved.     

Parametric controllability of certain systems of rigid bodies

The concept of parametric controllability as applied to systems of rigid bodies is discussed. The topic of discussion is Langrangian systems for which “unfreezing” of the parameters is possible such as, for example, the refinement of a model by taking account of the small variability of the links assumed by rigid bodies in the first approximation. As has been shown, just taking account of a small change in the parameters can ensure the controllability of a mechanism which was not controllable assuming rigidity absolute of the links. Certain sufficient conditions are proposed for parametric controllability in invariant manifolds for objects with cyclic coordinates. A two-link pendulum in the horizontal plane under the action of an internal moment (from the first link to the second) is considered as an example. The effect of its mass inertial parameters on the controllability is investigated. The parametric controllability of such an object in a manifold of zero angular momentum, due to the elastic pliability of the second link or the oscillations of an additional mass on a spring along the second link, is demonstrated. An example of a parametrically controllable planetary mechanism with slippage of discs is also considered.     

Rigidity of multi-graphs. I. Linking rigid bodies in n-space

A linkage of rigid bodies in n-space consists of a set of n-dimensional rigid bodies in n-space, certain pairs of which are linked together by one or more rigid bars using universal (ball) joints. Such a structure is canonically associated (bodies to vertices, bars to edges) with a multi-graph. We show that a multi-graph can be realized as a rigid linkage of rigid bodies in n-space if and only if it contains $\text{n(n + 1)}\text{2}$ edge disjoint spanning trees. Some techniques for generating rigid linkages are also discussed.     

An approximate method of calculating the oscillations of a liquid in inclined elastic cavities and channels

Small oscillations of an ideal incompressible liquid, which partially fills an inclined elastic container (a mobile cavity or channel) of arbitrary form and having a longitudinal plane of symmetry, are considered. The integral condition of continuity of the liquid is obtained by integrating the differential condition of incompressibility and exact satisfaction of the kinematic boundary conditions on the wetted side walls. Using this equation, systems of coordinate functions are constructed which represent the kinematically possible displacements of the liquid, for calculating the oscillations by the Ritz method and the finite-element method. The basic unknown functions, which describe the displacements of the liquid in cross-sections, are approximated by power functions and Legendre functions. Transverse layers of liquid, within the limits of the thickness of which a linear approximation for the unknown functions can be used, are considered as finite elements.     

Correlation between the properties of eigenfrequencies and eigenmodes in a chain of rigid bodies with torque connections

A chain of N rigid bodies with elastic connections transmitting a torque interaction is considered. Similar individual inertial elements of the chain have one degree of freedom. An exact analytical solution of the problem of oscillation eigenfrequencies and eigenmodes is constructed for an arbitrary value of N. The properties of frequencies and modes of such a chain are compared with those of a canonical Newton chain. It is established that the correlation between eigenfrequencies and the properties of the corresponding eigenmodes fundamentally differs from that for a canonical Newton chain. Lower eigenfrequencies of a chain of rigid bodies with inertialess torque connections correspond to modes with a larger number of nodes, while higher eigenfrequencies correspond to the smoother modes. The nontypical correlation between the oscillation eigenfrequencies and eigenmodes discovered based on the exact analytical solution to the problem of free oscillations of a chain of N rotating rigid bodies contradicts the ideas underlying theoretical studies in the field of solid state physics devoted to simulation of mechanical and thermal dynamic processes in crystalline lattices.     

Rigid–flexible interactive dynamics modelling approach

Dynamics modelling of multi-body systems composed of rigid and flexible elements is elaborated in this article. The control of such systems is highly complicated due to severe underactuated conditions caused by flexible elements and an inherent uneven non-linear dynamics. Therefore, developing a compact dynamics model with the requirement of limited computations is extremely useful for controller design, simulation studies for design improvement and also practical implementations. In this article, the rigid–flexible interactive dynamics modelling (RFIM) approach is proposed as a combination of Lagrange and Newton–Euler methods, in which the motion equations of rigid and flexible members are separately developed in an explicit closed form. These equations are then assembled and solved simultaneously at each time step by considering the mutual interaction and constraint forces. The proposed approach yields a compact model rather than a common accumulation approach that leads to a massive set of equations in which the dynamics of flexible elements is united with the dynamics equations of rigid members. The proposed RFIM approach is first detailed for multi-body systems with flexible joints, and then with flexible members. Then, to reveal the merits of this new approach, few case studies are presented. A flexible inverted pendulum is studied first as a simple template for lucid comparisons, and next a space free-flying robotic system that contains a rigid main body equipped with two manipulating arms and two flexible solar panels is considered. Modelling verification of this complicated system is vigorously performed using ANSYS and ADAMS programs. The obtained results reveal the outcome accuracy of the new proposed approach for explicit dynamics modelling of rigid–flexible multi-body systems such as mobile robotic systems, while its limited computations provide an efficient tool for controller design, simulation studies and also practical implementations of model-based algorithms.     

Low frequency resonances in the dynamic interaction of an elastic solid with a semi-bounded medium

The vibration of an elastic rod with a large point mass at one end to which a given harmonic load is applied is investigated. The other end of the rod is connected to a rigid punch in continuous contact with a semi-bounded elastic medium. Some lemmas are proved on systems with simplified boundary conditions with properties which majorize the characteristics of the initial system. The effect of the system parameters on the conditions under which an unbounded resonances occur is thereby determined. Analytic relations for determining the number of resonance frequences are obtained.     

Fluid-structure interaction in porous media for loaded filter pleats

Currently, the known simulation efforts with respect to fluid-structure interaction (FSI) are mainly restricted to the study of flow interacting with deformable solid bodies. On the other hand, there is extensive literature on simulation of flow through porous media. In particular, algorithms and software for CFD simulations of filtration processes in the case of rigid filtration medium were presented earlier by Fraunhofer ITWM, see, e.g., [1, 2]. In these papers the deformation of the solid skeleton is neglected. In many cases of water filtration, filtration for hydraulic applications, and even in certain air filtration regimes, the fluid pressure can be quite high, and the deformation of the pleats can be an issue. The deflection of pleats and its effect on the filtration process merits examination because under operating conditions (and especially after a partial loading of the media) the pleats often cannot be anymore considered as rigid porous media. Therefore, in this paper, the deflection is considered for clean media, as well as for partially loaded media. A new model describing the coupled flow and deformation, and corresponding numerical results are presented. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Integrable cases of the problem of the free motion of a gyrostat

The free spatial motion of a gyrostat in the form of a carrier body with a triaxial ellipsoid of inertia and an axisymmetric rotor is considered. The bodies have a common axis of rotation, which coincides with one of the principal axes of inertia of the carrier. In the Andoyer–Deprit variables the equations of motion reduce to a system with one degree of freedom. Stationary solutions of this system are found, and their stability is analysed. Cases in which the longitudinal moment of inertia of the carrier is greater than the largest of the transverse moments of inertia of the system of bodies, is smaller than the smallest or belongs to a range composed of the moments of inertia indicated, are investigated. General analytical solutions that describe the motion on separatrices and in regions corresponding to oscillations and rotation on the phase portrait are obtained for each case. The results can be interpreted as a development of the Euler case of the motion of a rigid body about a fixed point when one degree of freedom, namely, relative rotation of the bodies, is added.     

Gyroscopic control and stabilization

Summary In this paper, we consider the geometry of gyroscopic systems with symmetry, starting from an intrinsic Lagrangian viewpoint. We note that natural mechanical systems with exogenous forces can be transformed into gyroscopic systems, when the forces are determined by a suitable class of feedback laws. To assess the stability of relative equilibria in the resultant feedback systems, we extend the energy-momentum block-diagonalization theorem of Simo, Lewis, Posbergh, and Marsden to gyroscopic systems with symmetry. We illustrate the main ideas by a key example of two coupled rigid bodies with internal rotors. The energy-momentum method yields computationally tractable stability criteria in this and other examples.     

Rigid Body Dynamics Using Clifford Algebra

In this paper the dynamics of rigid bodies is recast into a Clifford algebra formalism. Specifically, the algebra Cℓ(0, 6, 2), is used and it is shown how velocities, momenta and inertias can be represented by elements of this algebra. The equations of motion for a rigid body are simply derived by differentiating the momentum of the body.     

Research about transmission of the mechanical vibration done by machine-tools on the group of bones shoulder-neck-head

Different researches have shown that long–term exposure to whole–body vibration can induce different injuries like back pain, injuries of the different part of the body, disturbing the physical and intellectual activities. The resulted diseases could be occasional or could be for ever. Bearing in mind that vibration is applied on the hand or feet, the health rick can be assessed if the forces transmitted in the shoulder during vibration are known. To estimate the forces a biomechanical model has been developed in which the shoulder, neck and head are represented by rigid bodies. The bodies are connected by visco–elastic joint elements. The applied forces are resulted from experimental measurements. To assess the health risk the forces must be divised in two components, an ascendant one and a descendent one. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Forced oscillations with a sliding regime range of a two-mass system interacting with a fixed stop : PMM vol. 37, n6, 1973, pp. 999–1006

We investigate, by the method developed in [1]. the forced oscillations with a sliding regime range of a two-mass system with elastic connection between the elements, impacting a fixed stop. The system being considered is a dynamic model for a number of vibrational mechanisms. Forced oscillations with a sliding regime range of a system with shock interactions are periodic motions accompanied by a period of an infinite succession of instantaneous collisions of two fixed elements of the model [2]. Within the framework of conditions of roughness of the parameter space [3], in this paper we study by the method of [1] periodic motions with a sliding regime range of a two-mass system with a stop. This problem was posed because in real systems the velocity recovery factor R changes from shock to shock, mainly taking small values (0, 0.2). At the same time, the regions of realizability of one-impact oscillations, in practice the most essential ones among motions with a finite number of interactions over a period, narrow down sharply as R decreases and becomes very small even for R < 0.6 [4]. Thus, the stability of the given operation can be ensured by a law of motion which is independent or weakly dependent on R (^*) (see footnote on the next page). By virtue of what has been said above, finite-impact periodic modes are little suitable for this purpose. Regions, delineated in the parameter space of the model being considered, of existence of stable periodic motions with a sliding regime range have proved to be sufficiently broad. By virtue of the adopted approximation of the sliding regime, the dynamic characteristics of these motions do not depend upon R. The circumstances mentioned confirm the practical value of motions with a sliding regime range in dynamic systems with impact interactions.     

Dynamics modelling and Hybrid Suppression Control of space robots performing cooperative object manipulation

Dynamics modelling and control of multi-body space robotic systems composed of rigid and flexible elements is elaborated here. Control of such systems is highly complicated due to severe under-actuated condition caused by flexible elements, and an inherent uneven nonlinear dynamics. Therefore, developing a compact dynamics model with the requirement of limited computations is extremely useful for controller design, also to develop simulation studies in support of design improvement, and finally for practical implementations. In this paper, the Rigid–Flexible Interactive dynamics Modelling (RFIM) approach is introduced as a combination of Lagrange and Newton–Euler methods, in which the motion equations of rigid and flexible members are separately developed in an explicit closed form. These equations are then assembled and solved simultaneously at each time step by considering the mutual interaction and constraint forces. The proposed approach yields a compact model rather than common accumulation approach that leads to a massive set of equations in which the dynamics of flexible elements is united with the dynamics equations of rigid members. To reveal such merits of this new approach, a Hybrid Suppression Control (HSC) for a cooperative object manipulation task will be proposed, and applied to usual space systems. A Wheeled Mobile Robotic (WMR) system with flexible appendages as a typical space rover is considered which contains a rigid main body equipped with two manipulating arms and two flexible solar panels, and next a Space Free Flying Robotic system (SFFR) with flexible members is studied. Modelling verification of these complicated systems is vigorously performed using ANSYS and ADAMS programs, while the limited computations of RFIM approach provides an efficient tool for the proposed controller design. Furthermore, it will be shown that the vibrations of the flexible solar panels results in disturbing forces on the base which may produce undesirable errors and perturb the object manipulation task. So, it is shown that these effects can be significantly eliminated by the proposed Hybrid Suppression Control algorithm.     

On some variational principles in mechanics of continuous media : PMM vol. 37, n6, 1973, pp. 963–973

Some problems of the analytical mechanics of continuous media are considered. The kinematc constraints, restricting the motion of elements of a continuous medium are separated into internal and external constraints; the internal surface stresses in a continuous medium are treated as reactive forces of the internal constraints. Since the work of these latter on possible displacements of elements of the continuous medium is not zero in the general case, then the internal constraints in a continuous medium should be referred to the category of nonideal constraints. The general equation of the dynamics of a continuous medium expressing the d'Alembert-Lagrange variational principle and including all the dynamic laws is examined. The work of the internal surface stresses in this equation can be given by using the first and second laws of thermodynamics, whereupon the general equation of the dynamics of continuous media can be represented in two other forms. Furthermore, an extension of the Gauss and Chetaev principles to continuous media is given.     

The small free oscillations of a strip

The refined equations of the free oscillations of a rod-strip, constructed previously in a first approximation by reducing the two-dimensional equations to one-dimensional equations by using trigonometric basis functions and satisfying the static boundary conditions on the boundary surfaces are analysed. These equations, the solutions of which are obtained for the case of hinge-supported end sections of the rod, are split into two independent systems of equations. The first of these describe non-classical fixed longitudinal-transverse forms of free oscillations, which are accompanied by a distortion of the plane form of the cross section. It is shown that the oscillation frequencies corresponding to them depend considerably on Poisson's ratio and the modulus of elasticity in the transverse direction, while for a rod of average thickness for the same value of the frequency parameter (the tone) they may be considerably lower than the frequencies corresponding to the classical longitudinal forms of free oscillations, which are performed while preserving the plane form of the cross sections. The second system of equations describes transverse flexural-shear forms of free oscillations, whose frequencies decrease as the transverse shear modulus decreases. They are practically equivalent in quality and content to the similar equations of well-known versions of the refined theories, but, unlike them, when the number of the tone increases and the relative thickness parameter decreases they lead to the solutions of the classical theory of rods.