Andrei Andreyevich Markov proposed in 1889 the problem (solved by Dubins in 1957) of finding the twice continuously differentiable (arc length parameterized) curve with bounded curvature, of minimum length, connecting two unit vectors at two arbitrary points in the plane. In this note we consider the following variant, which we call the dynamic Markov–Dubins problem (dM-D): to find the time-optimal $C^2$ trajectory connecting two velocity vectors having possibly different norms. The control is given by a force whose norm is bounded. The acceleration may have a tangential component, and corners are allowed, provided the velocity vanishes there. We show that for almost all the two vectors boundary value conditions, the optimization problem has a smooth solution. We suggest some research directions for the dM-D problem on Riemannian manifolds, in particular we would like to know what happens if the underlying geodesic problem is completely integrable. Path planning in robotics and aviation should be the usual applications, and we suggest a pursuit problem in biolocomotion. Finally, we suggest a somewhat unexpected application to "dynamic imaging science". Short time processes (in medicine and biology, in environment sciences, geophysics, even social sciences?) can be thought as tangent vectors. The time needed to connect two processes via a dynamic Markov–Dubins problem provides a notion of distance. Statistical methods could then be employed for classification purposes using a training set.

We describe a model for the dynamic interaction of a sphere with uniform density and a system of coaxial circular vortex rings in an ideal fluid of equal density. At regular intervals in time, a constraint is imposed that requires the velocity of the fluid relative to the sphere to have no component transverse to a particular circular contour on the sphere. In order to enforce this constraint, new vortex rings are introduced in a manner that conserves the total momentum in the system. This models the shedding of rings from a sharp physical ridge on the sphere coincident with the circular contour. If the position of the contour is fixed on the sphere, vortex shedding is a source of drag. If the position of the contour varies periodically, propulsive rings may be shed in a manner that mimics the locomotion of certain jellyfish. We present simulations representing both cases.

We consider the problem of motion of axisymmetric vortex rings in an ideal incompressible fluid. Using the topological approach, we present a method for complete qualitative analysis of the dynamics of a system of two vortex rings. In particular, we completely solve the problem of describing the conditions for the onset of leapfrogging motion of vortex rings. In addition, for the system of two vortex rings we find new families of motions where the relative distances remain finite (we call them pseudo-leapfrogging). We also find solutions for the problem of three vortex rings, which describe both the regular and chaotic leapfrogging motion of vortex rings.

This article concerns microrobots for solid and liquid environments. A short overview of microrobotics, suitable actuators and energy systems is given. The principles of terrestrial and aquatic locomotion are discussed and illustrated with examples from the literature on robotics. The state of the art with a focus on piezo microrobots for solid and liquid environments is presented.
Furthermore, we report an amphibious prototype, which can move on flat solid ground and on the free surface of water. The design, characteristic parameters and experiments on locomotion are described. The robot is characterized by a light and simple design and can perform twodimensional locomotion in different environments with a speed up to 30 mm/s. An analytical model to predict the maximum carrying capacity of the robot on water is solved numerically.

Magnetic actuation of elasto-magnetic devices has long been proposed as a simple way to propel fluid and achieve locomotion in environments dominated by viscous forces. Under the action of an oscillating magnetic field, a permanent magnet, when attached to an elastic tail, is able to generate bending waves and sufficient thrust for propulsion. We study the hydrodynamical effects of the magnetic head geometry using a geometrically exact formulation for the elastic tail elastohydrodynamics.We show that the spherical head geometry fails to take full advantage of the propulsive potential from the flexible tail. Nevertheless, while elongated prolate spheroids demonstrate a superior swimming performance, this is still regulated by the nature of the imposed magnetic field. Interestingly, the highest swimming speed is observed when the magnitude of the magnetic field is weak due to delays between the orientation of the magnetic moment and the oscillating magnetic field. This allows the stored elastic energy from the deformed tail to relax during the phase lag between the imposed magnetic field and the swimmer’s magnetic moment, favouring in this way the net propulsion. In particular, this result suggests the existence of optimal magnetic actuations that are non-smooth, and even discontinuous in time, in order to fully explore the propulsive potential associated with the relaxation dynamics of periodically deformed elastic filaments.

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.

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.

The idea of a model-based control of rehabilitation for hemiparetic patients requires efficient models of human walking, healthy walking as well as hemiparetic walking. Such models are presented in this paper. They include 42 degrees of freedom and allow especially the evaluation of kinetic magnitudes with the goal to evaluate measures for the hardness of hemiparesis. As far as feasible, the simulations have been compared successfully with measurements, thus improving the confidence level for an application in clinical practice. The paper is mainly based on the dissertation [19].

The paper deals with the dynamics of a spherical rolling robot actuated by internal rotors that are placed on orthogonal axes. The driving principle for such a robot exploits nonholonomic constraints to propel the rolling carrier. A full mathematical model as well as its reduced version are derived, and the inverse dynamics are addressed. It is shown that if the rotors are mounted on three orthogonal axes, any feasible kinematic trajectory of the rolling robot is dynamically realizable. For the case of only two rotors the conditions of controllability and dynamic realizability are established. It is shown that in moving the robot by tracing straight lines and circles in the contact plane the dynamically realizable trajectories are not represented by the circles on the sphere, which is a feature of the kinematic model of pure rolling. The implication of this fact to motion planning is explored under a case study. It is shown there that in maneuvering the robot by tracing circles on the sphere the dynamically realizable trajectories are essentially different from those resulted from kinematic models. The dynamic motion planning problem is then formulated in the optimal control settings, and properties of the optimal trajectories are illustrated under simulation.

In our earlier paper [3] we examined the problem of control of a balanced dynamically nonsymmetric sphere with rotors with no-slip condition at the point of contact. In this paper we investigate the controllability of a ball in the presence of friction. We also study the problem of the existence and stability of singular dissipation-free periodic solutions for a free ball in the presence of friction forces. The issues of constructive realization of the proposed algorithms are discussed.

We consider a mechanical system inside a rolling ball and show that if the constraints have spherical symmetry, the equations of motion have Lagrangian form. Without symmetry, this is not true.

We consider two types of trajectories found in a wide range of mechanical systems, viz. box orbits and loop orbits. We elucidate the dynamics of these orbits in the simple context of a perturbed harmonic oscillator in two dimensions. We then examine the small-amplitude motion of a rigid body, the rock’n’roller, a sphere with eccentric distribution of mass. The equations of motion are expressed in quaternionic form and a complete analytical solution is obtained. Both types of orbit, boxes and loops, are found, the particular form depending on the initial conditions. We interpret the motion in terms of epi-elliptic orbits. The phenomenon of recession, or reversal of precession, is associated with box orbits. The small-amplitude solutions for the symmetric case, or Routh sphere, are expressed explicitly in terms of epicycles; there is no recession in this case.

The dynamical behavior of a heavy circular cylinder and a point vortex in an unbounded volume of ideal liquid is considered. The liquid is assumed to be irrotational and at rest at infinity. The circulation about the cylinder is different from zero. The governing equations are Hamiltonian and admit an evident autonomous integral of motion — the horizontal component of the linear momentum. Using the integral we reduce the order and thereby obtain a system with two degrees of freedom. The stability of equilibrium solutions is investigated and some remarkable types of partial solutions of the system are presented.

Point vortices have been extensively studied in vortex dynamics. The generalization to higher singularities, starting with vortex dipoles, is not so well understood.We obtain a family of equations of motion for inviscid vortex dipoles and discuss limitations of the concept.We then investigate viscous vortex dipoles, using two different formulations to obtain their propagation velocity. We also derive an integro-differential for the motion of a viscous vortex dipole parallel to a straight boundary.