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.     

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.     

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 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 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.     

The motion of a solid with large deformations

We study the motion of a solid with large deformations. The solid may be loaded on its surface by needles, rods, beams, plates… Therefore it is wise to choose a third-gradient theory for the body. Stretch matrix of the polar decomposition has to be symmetric. This is an internal constraint which introduces a reaction stress in the Piola–Kirchhoff–Boussinesq stress. We prove that there exists a motion that satisfies the complete equations of Mechanics in a convenient variational framework. This motion is local-in-time because it may be interrupted by crushing, resulting in a discontinuity of velocity with respect to time, i.e., an internal collision.     

The 3D motion of a solid with large deformations

We study in dimension 3 the motion of a solid with large deformations. The solid may be loaded on its surface by needles, rods, beams, shells, etc. Therefore, it is wise to choose a third gradient theory for the body. It is known that the stretch matrix of the polar decomposition has to be symmetric. This is an internal constraint, which introduces a reaction stress in the Piola–Kirchhoff–Boussinesq stress. We prove that there exists a motion that satisfies the complete equations of Mechanics in a convenient variational framework. This motion is local-in-time for it may be interrupted by a crushing, which entails a discontinuity of velocity with respect to time, i.e., an internal collision.     

Control of motion of an inhomogeneous cylinder with internal movable masses along a horizontal plane

The controlled motion of a rigid inhomogeneous cylinder over a rough horizontal plane is considered. The control is provided by controlled motion of internal masses. Mathematical models are constructed that correspond to rolling without loss of contact or slippage. The conditions for the physical implementability of such a motion are derived. The case where the internal moving masses from a rigid flywheel the centre of inertia of which lies on the axis of the cylinder is investigated in detail. A near-time-optimal feedback control that enables the total energy to be changed in a required way is constructed on the basis of an asymptotic approach. The main operating modes are simulated, namely, swinging up of the cylinder to a large angular amplitude, rotation with a prescribed energy, deceleration of rolling to a complete stop, and oscillations and rotations in the neighbourhood of the separatrix.     

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.     

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.     

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.     

Optimum velocity for a person moving under rain

The total rain received by a moving body has previously been modelled by defining a wetness function. Several cases such as one-dimensional motion of an inclined plane, two-dimensional motion of an inclined plane, motion with time-varying velocity, inclined rain for an inclined plane, rain on a cylindrical surface and three-dimensional motion of a convex body were treated in detail. One of the major conclusions was that for a fixed distance, assuming vertical rain, the body should travel as fast as possible since the total wetness decreases with increasing velocity. The wetness function was shown to decrease asymptotically to a constant value as the velocity increases and in the high-speed range, increasing the velocity does not decrease the wetness substantially. One might think that the excess amount of energy required for a higher speed does not compensate for the small fraction of decrease in wetness. In this work, criteria are developed for a critical speed (optimum speed), for which the wetness is small enough for a reasonable energy consumption. Three cases are investigated: (1) vertical rain; (2) rain inclined towards the body; (3) rain inclined away from the body. In the first two cases, there is no absolute minimum for the wetness function and the optimum velocity is determined by special criteria. The third case is somewhat different, however, and if the inclination angle is higher than a critical value, an absolute minimum for wetness is obtained and the optimum velocity for this case is defined to be the velocity corresponding to this absolute minimum. Therefore the definition of optimum velocity is qualitatively different from the first two cases.     

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.     

Dynamics of a two-degrees-of-freedom system with friction and heat generation

A novel thermomechanical model of frictional self-excited stick-slip vibrations is proposed. A mechanical system consisting of two masses which are coupled by an elastic spring and moving vertically between two walls is considered. It is assumed that between masses and walls a Coulomb friction occurs, and stick-slip motion of the system is studied. The applied friction force depends on a relative velocity of the sliding bodies. Stability of stationary solutions is considered. A computation of contact parameters during heating of the bodies is performed. The possibility of existence of frictional auto-vibrations is illustrated and discussed.     

Optimization of the motion in a resistant medium of a body with a movable internal mass

The motion, in a resistant medium, of a system consisting of a rigid body and movable internal mass is considered. The external medium acts on the body by a force that piecewise linearly depends on its speed. The class of periodic motions of the internal mass for which the speed of this mass relative to the body is piecewise constant is studied. It is shown that, under certain conditions, the forward movement of the whole system in the medium is possible. The average speed of this movement over a period is determined. Optimal parameters of the motion of the internal mass for which the average speed of the system movement is maximal are found.     

Characteristic features of the high-velocity motion of bodies in dense media

The characteristic features of the high-velocity motion of conical and pyramidal bodies are investigated when the force acting on their surface is described by a local interaction model. It is assumed that the pressure on the body surface is represented by a binomial formula that is quadratic in the velocity. Three friction models are used to represent the tangential stresses: constant friction, friction that is proportional to the pressure and mixed friction. Analytical solutions of problems of the plane inertial motion of slender bodies with a base contour in the form of a circle, a rhombus or a star consisting of four cycles are constructed for an unseparated flow past the bodies and small perturbations imposed on the parameters of the linear motion at the initial instant of time. A criterion for the stability of the motion is found that enables the perturbed motion of the body to be determined when the medium parameters and the velocity, mass and shape of the body are known. The analytical results are validated by a numerical solution of the Cauchy problem for a system of equations of motion obtained without simplifying assumptions.     

The optimum rectilinear motion of a two-mass system

The forward rectilinear motion of a system of two rigid bodies along a horizontal plane is considered. Forces of dry friction act between the bodies and the plane, and the motion is controlled by internal forces of interaction between the bodies. A periodic motion in which the system moves along a straight line is constructed. The optimum parameters of the system and a control law are found corresponding to the maximum mean velocity of motion of the system as a whole.     

Control of the motion of a triaxial ellipsoid in a fluid using rotors

The motion of a body shaped as a triaxial ellipsoid and controlled by the rotation of three internal rotors is studied. It is proved that the motion is controllable with the exception of a few particular cases. Partial solutions whose combinations enable an unbounded motion in any arbitrary direction are constructed.     

The local boundedness of the perturbed motions of a gyroscope in gimbals with dissipative and accelerating forces

The motion of an unbalanced gyroscope in gimbals in a central Newtonian field of forces is considered, taking the masses of the suspension rings into account. It is assumed that there is a moment of forces of viscous friction acting on the axis of rotation of one of the rings, and there is an accelerating (electromagnetic) moment applied to the axis of rotation axis of the other ring. The equations of motion have a partial solution such that the mean velocity of the outer ring is perpendicular to the direction from the centre of gravitation S to the stationary point O, the middle plane of the inner ring contains this direction, and the gyroscope rotates about SO with an arbitrary constant angular velocity.     

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.