Geometry of the motion of robot manipulators

A p-parametric robot is a mapping g of R^p into the homogeneous space P=C₆×C₆/Diag(C₆×C₆) given by the formula g(u₁,...,u_p=exp u₁X₁..... exp u_pX_p, where C₆, is the Lie group of all congruences of E₃ and X₁,..., X_p are fixed vectors from the Lie algebra of C₆. We characterize the set g(R^p) locally by a system of PDE and give some geometrical properties of g as a p-dimensional motion for p<6. We also characterize the Frenet frame of g and show how to construct it for the robot manipulator given by its axes X₁,...,X_p.     

Stabilization of the motion of a spherical robot using feedbacks

The paper is concerned with the problem of stabilizing a spherical robot of combined type during its motion. The focus is on the application of feedback for stabilization of the robot which is an example of an underactuated system. The robot is set in motion by an internal wheeled platform with a rotor placed inside the sphere. The results of experimental investigations for a prototype of the spherical robot are presented.     

A method for controlling the motion of a robot snakeboarder

The problem of planning the motions of a robot snakeboarder consisting of a snakeboard and a flywheel mounted on it and capable of performing controlled rotation relative to the snakeboard crossbar is investigated. Programmed control of the wheel axes to provide an assigned path of motion of an arbitrary point on the crossbar is described. A control of the angular acceleration of the flywheel on the snakeboard that ensures a required course of variation of the velocity of the crossbar centre both on a horizontal plane and on an inclined plane is constructed. The problem of the maximum acceleration of the snakeboard along a “figure-of-eight” trajectory is solved.     

Wave Instabilities

A uniform train of periodic waves may be unstable to large scale variations so that some incoherence can develop. The analysis is presented for the case of gravity waves, and for a class of interaction problems typical of many physical systems.     

Experimental investigations of the controlled motion of a screwless underwater robot

In this paper we describe the results of experimental investigations of the motion of a screwless underwater robot controlled by rotating internal rotors. We present the results of comparison of the trajectories obtained with the results of numerical simulation using the model of an ideal fluid.     

Algorithmic and complexity issues of robot motion in an uncertain environment

This paper presents a survey of one approach to planning collision-free paths for an automaton operating in an environment with obstacles. Path planning is one of the central problems in robotics. Typically, the task is presented in the two- or three-dimensional space, with the automaton being either an autonomous vehicle or an arm manipulator with a fixed base. The multiplicity of approaches one finds in this area revolves around two basic models: in one, called path planning with complete information, perfect information about the geometry and positions of the robot and the obstacles is assumed, whereas in the other, called path planning with incomplete information, an element of uncertainty about the environment is present. The approach surveyed here, called dynamic path planning, has been developed in the last few years; it is based on the latter model and gives rise to algorithmic and computational issues very different from those in the former model. The approach produces provable (nonheuristic) path planning algorithms for an automaton operating in a highly unstructured environment where no knowledge about the obstacles is available beforehand and no constraints on the geometry of the obstacles are imposed.     

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.     

Reactive Infiltration Instabilities

When a fluid flow is imposed on a porous medium, the infiltrationflow may interact with the reaction-induced porosity variationswithin the medium and may lead to fingering instabilities. Anonlinear model of such interaction is developed and morphologicalinstability of a planar dissolution front is demonstrated usinga linear stability analysis of a moving-free-boundary problem.The fully nonlinear model is also examined numerically usingfinite-difference methods. The numerical simulations confirmthe predictions of linear stability theory and, more importantly,reveal the growth of dissolution fingers that emerge as a resultof these instabilities     

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.     

An approach to analytical construction of the equations of motion of robot manipulators

Lagrange equations of the second kind are used to develop a method for constructing the elements of the equations of motion of robot manipulators in explicit (analytical) form with minimum computational complexity. The equations of motion can be represented by multilevel decomposition. Algorithms are given for constructing the inertia matrix of the manipulator segments in a number of steps equal to the number of degrees of freedom of the robot manipulator.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 82–88, 1990.     

Feasible command strategies for the control of motion of a simple mobile robot

The concept of feasible command strategies is introduced and its applicability is demonstrated by solving a guidance and control problem. This problem concerns the motion of a system which is composed of a rolling disk and a controlled slender rod that is pivoted, through its endpoint, about the disk center. The motion of the disk-rod system is subjected to state and control constraints, and it serves as a model for the motion of a simple mobile robot. In addition, the concept of path controllability is introduced and a condition is derived for the system motion path controllability. The derivation of this condition enables one to design closed-loop control laws for the system motion.     

Influence of vortex structures on the controlled motion of an above-water screwless robot

This paper is devoted to an experimental investigation of the motion of a rigid body set in motion by rotating two unbalanced internal masses. The results of experiments confirming the possibility of motion by this method are presented. The dependence of the parameters of motion on the rotational velocity of internal masses is analyzed. The velocity field of the fluid around the moving body is examined.     

Effect of the deformability of structural links on the motion of a two-legged walking robot

The problem of the dynamics of parametric optimization of the motion of a walking robot is solved within the framework of a seven-link model with the elasticity of the structural components taken into account. The effect of the elasticity of the links on the kinematic, dynamic, and energy characteristics of the locomotion of the walking mechanism is studied.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 76–79, 1987.     

Turing Instabilities at Hopf Bifurcation

Turing–Hopf instabilities for reaction-diffusion systems provide spatially inhomogeneous time-periodic patterns of chemical concentrations. In this paper we suggest a way for deriving asymptotic expansions to the limit cycle solutions due to a Hopf bifurcation in two-dimensional reaction systems and we use them to build convenient normal modes for the analysis of Turing instabilities of the limit cycle. They extend the Fourier modes for the steady state in the classical Turing approach, as they include time-periodic fluctuations induced by the limit cycle. Diffusive instabilities can be properly considered because of the non-catastrophic loss of stability that the steady state shows while the limit cycle appears. Moreover, we shall see that instabilities may appear even though the diffusion coefficients are equal. The obtained normal modes suggest that there are two possible ways, one weak and the other strong, in which the limit cycle generates oscillatory Turing instabilities near a Turing–Hopf bifurcation point. In the first case slight oscillations superpose over a dominant steady inhomogeneous pattern. In the second, the unstable modes show an intermittent switching between complementary spatial patterns, producing the effect known as twinkling patterns.     

Instabilities of Waves on a Free Surface

This paper is concerned with instabilities which develop on a class of parallel shear flows in the presence of a free surface. Implicit in this type of study is the existence of three dimensional instabilities on slightly perturbed two dimensional spatially periodic shear flows.     

On Nonlinear Rayleigh-Taylor Instabilities

Some of the mathematical properties of the interface between two incompressible inviscid and immiscible fluids with different densities under the influence of a constant gravity field 9 are investigated. The purpose of this paper is to prove that linearly unstable modes for Rayleigh-Taylor instabilities give birth to nonlinear instabilities for the full nonlinear system. The main ingredient is a general instability theorem in an analytic framework which enables us to go from linear to nonlinear instabilities.     

Rolling motion dynamics of a spherical robot with a pendulum actuator controlled by the Bilimovich servo-constraint

Theoretical and Mathematical Physics - We discuss the problem of rolling without slipping for a spherical shell with a pendulum actuator (spherical robot) installed in the geometric center of the...     

Convective Instabilities in Anisotropic Porous Media

This paper addresses the problem of the onset of Rayleigh-Bénard convection in a porous layer using Brinkman's equation and anisotropic permeability. The critical Rayleigh number and wave number at marginal stabilities are calculated for both free and rigid boundaries. In both cases, it is noted that there exist ranges for which the stability criteria is intermediate to the low porosity Darcy approximation and to high porosity single viscous fluid. The permeability anisotropy is found to select the mode of instability.