The energy-optimal motion of a vibration-driven robot in a resistive medium

The rectilinear motion of a two-mass system in a resistive medium is considered. The motion of the system as a whole occurs by longitudinal periodic motion of one body (the internal mass) relative to the other body (the shell). The problem consists of finding the periodic law of motion of the internal mass that ensures velocity-periodic motion of the shell at a specified average velocity and minimum energy consumption. The initial problem reduces to a variational problem with isoperimetric conditions in which the required function is the velocity of the shell. It is established that, with optimal motion, the shell velocity is a piecewise-constant time function taking two values (a positive value and a negative value). The magnitudes of these velocities and the overall size of the intervals in which they are taken are uniquely defined, while the optimal motion itself is non-uniquely defined. The simplest optimal motion, for which the period is divided into two sections – one with a positive velocity and the other with a negative velocity of motion of the shell – is investigated in detail. It is shown that, among all the optimal motions, this simplest motion is characterized by the maximum amplitude of oscillations of the internal mass relative to the shell. © Elsevier Ltd. All rights reserved.     

Optimal control of the rectilinear motion of a rigid body on a rough plane by means of the motion of two internal masses

The problem of the optimal control of a rigid body moving along a rough horizontal plane due to motion of two internal masses is solved. One of the masses moves horizontally parallel to the line of motion of the main body, while the other mass moves in the vertical direction. Such a mechanical system models a vibration-driven robot–a mobile device able to move in a resistive medium without special propellers (e.g., wheels, legs or caterpillars). Periodic motions are constructed for the internal masses to ensure velocity-periodic motion of the main body with maximum average velocity, provided that the period is fixed and the magnitudes of the accelerations of the internal masses relative to the main body do not exceed prescribed limits. Based on the optimal solution obtained for a fixed period without any constraints imposed on the amplitudes of vibration of the internal masses, a suboptimal solution that takes such constraints into account is constructed.     

The optimal quasi-stationary motion of a vibration-driven robot in a viscous medium

We consider a rectilinear quasi-stationary motion of a two-mass system in a viscous medium. The motion of the system as a whole occurs due to periodic movements of the internal mass relatively to the shell. The problem is to describe the law of motion of the internal mass that provides the minimum energy consumption with a specified average velocity of the shell. We propose an algorithm for solving the problem with any law of the resistance of the medium. We obtain the energy-optimal law of motion of a spherical shell in a viscous liquid.     

Approximate equations of rotational motion of a rigid body carrying a movable point mass

The dynamics of a compound system, consisting of a rigid body and a point mass, which moves in a specified way along a curve, rigidly attached to the body is investigated. The system performs free motion in a uniform gravity field. Differential equations are derived which describe the rotation of the body about its centre of mass. In two special cases, which allow of the introduction of a small parameter, an approximate system of equations of motion is obtained using asymptotic methods. The accuracy with which the solutions of the approximate system approach the solutions of the exact equations of motion is indicated. In one case, it is assumed that the point mass has a mass that is small compared with the mass of the body, and performs rapid motion with respect to the rigid body. It is shown that in this case the approximate system is integrable. A number of special motions of the body, described by the approximate system, are indicated, and their stability is investigated. In the second case, no limitations are imposed on the mass of the point mass, but it is assumed that the relative motion of the point is rapid and occurs near a specified point of the body. It is shown that, in the approximate system, the motion of the rigid body about its centre of mass is Euler–Poinsot motion.     

The self-propulsion of a body with moving internal masses in a viscous fluid

An investigation of the characteristics of motion of a rigid body with variable internal mass distribution in a viscous fluid is carried out on the basis of a joint numerical solution of the Navier — Stokes equations and equations of motion for a rigid body. A nonstationary three-dimensional solution to the problem is found. The motion of a sphere and a drop-shaped body in a viscous fluid in a gravitational field, which is caused by the motion of internal material points, is explored. The possibility of self-propulsion of a body in an arbitrary given direction is shown.     

Optimal control of the rectilinear motion of a two-body system in a resistive medium

The rectilinear motion of a two-body system is considered. One of the bodies (the main body) interacts with a resistive environment, while the other body (the internal body) interacts with the main body but does not interact with the environment. The force applied to the internal body leads to a reaction that acts on the main body and produces a change in its velocity, which causes a change in the resistance of the environment to the motion of the main body. Thus, by controlling the motion of the internal body, one can control the external force acting on the main body and, as a consequence, the motion of the entire system. A periodic motion of the internal body relative to the main body, which generates the motion of the main body with periodically changing velocity and the maximum displacement for the period, is constructed for a wide class of laws of resistance of the environment to the motion of the main body.The principle of motion considered is appropriate for mobile mini- and micro-robots. The body (housing) of such robots can be hemetically sealed and smooth, without protruding parts, which enables these robots to be used for the non-destructive inspection of miniature engineering structures such as thin pipe-lines, as well as in medicine. Problems of optimizing the control modes for such systems are of interest both to researchers in the field of optimal control and to specialists in applied mechanics and robotics.     

Optimal motion of a multilink system in a resistive medium

We consider the motion in a resistive medium of a mechanical system consisting of a main body and one or two links attached to it by means of cylindrical joints. The motion is controlled through high-frequency periodic oscillations of the links. For this system, an equation of motion is deduced and the average velocity of locomotion is estimated under certain assumptions. This velocity is positive if the angular velocity of diverting the attached links is less than the angular velocity of bringing them to the axis of the body. An optimal control problem of maximizing the average velocity is formulated and solved. An example is given.     

Quadratic integrals and the reducibility of the equations of motion of a complex mechanical system in a central field

A mechanical system, consisting of a non-variable rigid body (a carrier) and a subsystem, the configuration and composition of which may vary with time (the motion of its elements with respect to the carrier is specified), is considered. The system moves in a central force field at a distance from its centre which considerably exceeds the dimensions of the system. The effect of the system motion about the centre of mass on the motion of the centre of mass, which is assumed to be known, is ignored (the analogue of the limited problem [1] for a rigid body). The necessary and sufficient conditions for a quadratic integral of the motion around the centre of mass to exist are obtained in the case when there is no dynamic symmetry. It is shown that, for a quadratic integral to exist, it is necessary that the trajectory of the motion of the centre of mass should be on the surface of a certain circular cone, fixed in inertial space, with its vertex at the centre of the force field. If the trajectory does not lie on the generatrix of the cone, only one non-trivial quadratic integral can exist and the initial system, in the presence of this quadratic integral, reduces to autonomous form. For the motion of the centre of mass along the generatrix or the motion of the system around a fixed centre of mass, the necessary and sufficient conditions for a non-trivial quadratic integral to exist are obtained, which are generalizations of the energy integral, the de Brun integral [2] and the integral of the projection of the kinetic moment. When three non-trivial quadratic integrals exist, the condition for reduction to an autonomous system describing the rotation of the rigid body around the centre of mass and integrable in quadratures are indicated [3, 4].     

Analysis and optimization of the motion of a body controlled by means of a movable internal mass

The controlled horizontal motion of a body in the presence of dry friction forces is investigated. Control is accomplished by means of a movable mass that can move within the body in a bounded range. Some simple modes of periodic relative motions of the movable mass, under which the entire system moves as a whole, are investigated. Constraints are imposed on the relative displacement, velocity and acceleration of the movable mass. The optimum parameters of this relative motion, under which the maximum mean velocity of the body is reached, are determined.     

Optimal control of the motion of a multilink system in a resistive medium

The optimal control of the motion of a system consisting of a main body and one or two links joined to it by cylindrical joints in a resistive medium is investigated. The resistance force of the medium acting on the moving body is assumed to depend on their velocity. The control is accomplished through high-frequency angular oscillations of the links. The equations of motion are analysed, and the mean velocity of translational motion of the system is estimated under certain assumptions. Optimal control problems are formulated and solved, and the laws of control of the oscillations of the links for which the maximum mean velocity of motion is obtained are found as a result. The data obtained are in qualitative agreement with observations of the swimming of fish and animals. The results of this study can be used in developing mobile robots that move in a liquid.     

The dynamics of pyramidal bodies within the framework of the local interaction model

Two-dimensional inertial motion of pyramidal bodies in a medium is investigated, on the assumption that the force exerted by the medium on their surface is described by the local interaction model. Assuming unseparated flow around the bodies and small perturbations applied at the initial time to the parameters of rectilinear motion, an analytical solution is constructed of the problem of the two-dimensional motion of slender bodies with bases whose contour is a rhombus or a star consisting of four symmetrical cycles. It is shown that the solution provides the basis for a complete parameterc analysis of the dynamics of the body and for evaluating the forces and torques experienced by the body along its trajectory. A criterion for the stability of the body is found, using which, knowing the velocity, mass and position of the body's centre of gravity, one can determine the form of the perturbed motion of the pyramidal body. It is shown that the body shape is one of the most important factors affecting the stability of motion, and that, of all bodies with the same shape and position of the centre of mass, those with the least mass have the largest reserve of stability. The analytical results are confirmed by numerical solution of the Cauchy problem for the system of equations of motion obtained without the simplifying assumptions.     

Self-propulsion of a body with rigid surface and variable coefficient of lift in a perfect fluid

We study the system of a 2D rigid body moving in an unbounded volume of incompressible, vortex-free perfect fluid which is at rest at infinity. The body is equipped with a gyrostat and a so-called Flettner rotor. Due to the latter the body is subject to a lifting force (Magnus effect). The rotational velocities of the gyrostat and the rotor are assumed to be known functions of time (control inputs). The equations of motion are presented in the form of the Kirchhoff equations. The integrals of motion are given in the case of piecewise continuous control. Using these integrals we obtain a (reduced) system of first-order differential equations on the configuration space. Then an optimal control problem for several types of the inputs is solved using genetic algorithms.     

The optimal periodic motions of a two-mass system in a resistant medium

The rectilinear motions of a two-mass system, consisting of a container and an internal mass, in a medium with resistance, are considered. The displacement of the system as a whole occurs due to periodic motion of the internal mass with respect to the container. The optimal periodic motions of the system, corresponding to the greatest velocity of displacement of the system as a whole, averaged over a period, are constructed and investigated using a simple mechanical model. Different laws of resistance of the medium, including linear and quadratic resistance, isotropic and anisotropic, and also a resistance in the form of dry-friction forces obeying Coulomb's law, are considered.     

The motion in a perfect fluid of a body containing a moving point mass

The motion of a system (a rigid body, symmetrical about three mutually perpendicular planes, plus a point mass situated inside the body) in an unbounded volume of a perfect fluid, which executes vortex-free motion and is at rest at infinity, is considered. The motion of the body occurs due to displacement of the point mass with respect to the body. Two cases are investigated: (a) there are no external forces, and (b) the system moves in a uniform gravity field. An analytical investigation of the dynamic equations under conditions when the point performs a specified plane periodic motion inside the body showed that in case (a) the system can be displaced as far as desired from the initial position. In case (b) it is proved that, due to the permanent addition of energy of the corresponding relative motion of the point, the body may float upwards. On the other hand, if the velocity of relative motion of the point is limited, the body will sink. The results of numerical calculations, when the point mass performs random walks along the sides of a plane square grid rigidly connected with the body, are presented.     

Simulation of the Spatial Action of a Medium on a Body of Conical Form

We consider amathematical model of the spatial action of a medium on the axisymmetric rigid body whose external surface has a part that is a circular cone.We present a complete system of equations of motion under the quasistationary conditions. The dynamical part forms an independent system of the sixth order in which the independent subsystems of lower order are distinguished. We study the problem of stability with respect to the part of variables of the key regime—the spatial rectilinear translational deceleration of the body. For a particular class of bodies, we show the inertial mass characteristics under which the key regime is stable. For a plane analog of the problem, we obtain a family of phase portraits in the space of quasivelocities.     

Auto-oscillations during deceleration of a rigid body in a resisting medium

Some qualitative analysis is carried out of the rectilinear and spatial problems concerning the motion of a rigid body in a resisting medium.Anonlinearmodel is constructed of impact of the mediumon the rigid body, which takes into account the dependence of the arm of force on the reduced angular velocity of the body. Moreover, the moment of this force itself is also a function of the angle of attack. As was shown by the processing the experimental data on the motion of homogeneous circular cylinders in water, these circumstances should be taken into account in the simulation. The analysis of the plane and spatial models of the interaction of a rigid body with a medium reveal the sufficient conditions of stability of the key regime of motion, i.e., the translational rectilinear deceleration. It is also shown that, under certain conditions, both stable or unstable auto-oscillating regimes can be presented in the system.     

Vibrations of a cylinder in a concentric vessel filled with a two-layer fluid

A hybrid vibrational system containing a solid (a cylinder) with an elastic connection to a coaxial cylindrical cavity, completely filled with a heavy ideal stably stratified two-layer fluid, is considered. The combined self-consistent vibrations of the body and the fluid (of the internal waves) are studied. An explicit solution of the internal boundary value problem of an inhomogeneous liquid in an annular domain for a specified motion of the body is obtained. An integrodifferential equation of the Newton type is constructed on the basis of this. This equation describes the self-consistent oscillations of the cylinder. In the case of weak coupling of the interaction between the solid and the medium, an approximate solution is obtained using asymptotic methods and an analysis is carried out. Qualitative effects of the mutual effect of the motions of the cylinder and the fluid are found.     

Controlled motion of a rigid body with internal mechanisms in an ideal incompressible fluid

We consider the controlled motion in an ideal incompressible fluid of a rigid body with moving internal masses and an internal rotor in the presence of circulation of the fluid velocity around the body. The controllability of motion (according to the Rashevskii–Chow theorem) is proved for various combinations of control elements. In the case of zero circulation, we construct explicit controls (gaits) that ensure rotation and rectilinear (on average) motion. In the case of nonzero circulation, we examine the problem of stabilizing the body (compensating the drift) at the end point of the trajectory. We show that the drift can be compensated for if the body is inside a circular domain whose size is defined by the geometry of the body and the value of circulation.     

On the minimum and maximum averaged resistance problem of moving bodies

A body moving in a rarefied medium of point fixed particles is considered. The center of mass of the body moves with a constant speed, and, moreover, the body executes slow rotational motions. The resistance force of the medium to the body’s motion is defined. The concept of a rough body is introduced, and it is proved that the ratio between the resistance of the rough body and that of the smooth body corresponding to it lies in the limits from 0.969 up to 2. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 36, Suzdal Conference-2004, Part 2, 2005.     

Development of a new method for online parameter identification in seismically excited smart building structures using virtual synchronization and adaptive control design

In this paper a new method for online parameter identification and damage detection in smart building structures that are subjected to arbitrary seismic excitation is proposed. It uses real-time measurements of a structure's motion to identify its unknown constant or piecewise constant parameters such as stiffness, damping and mass over the time. The method is based on elements of system synchronization and adaptive control theories. First, a computational system, called the virtual system, is defined. Next, by using properly designed controller and estimations for the unknown parameters, the state of the virtual system is forced to follow the measured motion of the real structure. The mentioned estimations are computed from a proposed update law which depends on the measured motion of the real structure and the virtual system’s state. A major theoretical novelty of this paper is a proposed convergence condition which is applicable in case of arbitrary external forces or ground acceleration. It is shown that upon the satisfaction of that condition, as the synchronization completes, the computed estimation function converges to the true value of the vector of unknown parameters. In addition, an important practical contribution presented in this study is the introduction of a technique called scale factors. It helps to use available initial guesses of the unknown parameters to improve the speed of online identification. Numerical examples show that the proposed method is promising and has a good performance in both online identification and online damage detection problems.