On the Trajectory Generation of the Hydrodynamic Chaplygin Sleigh

In this letter we consider the asymptotic behaviour and the trajectory generation problem for the Chaplygin sleigh interacting with a potential fluid. We investigate which trajectories can be obtained, at least asymptotically as t tents to infinity, by controlling some of the coordinates (shape-control variables) and using the theory of reconstruction. Moreover we support our conclusions via numerical simulations

Purcell magneto-elastic swimmer controlled by an external magnetic field

International audienceThis paper focuses on the mechanism of propulsion of a Purcell swimmer whose segments are magnetized and react to an external magnetic field applied into the fluid. By an asymptotic analysis, we prove that it is possible to steer the swimmer along a chosen direction when the control functions are prescribed as an oscillating field. Moreover, we discuss what are the main obstructions to overcome in order to get classical controllability result for this system

On the Trajectory Generation of the Hydrodynamic Chaplygin Sleigh

In this letter we consider the asymptotic behaviour and the trajectory generation problem for the Chaplygin sleigh interacting with a potential fluid. We investigate which trajectories can be obtained, at least asymptotically as t tents to infinity, by controlling some of the coordinates (shape-control variables) and using the theory of reconstruction. Moreover we support our conclusions via numerical simulations

Control of locomotion systems and dynamics in relative periodic orbits

The connection between the dynamics in relative periodic orbits of vector fields with noncompact symmetry groups and periodic control for the class of control systems on Lie groups known as `(robotic) locomotion systems' is well known, and has led to the identification of (geometric) phases. We take an approach which is complementary to the existing ones, advocating the relevance|for trajectory generation in these control systems|of the quali-tative properties of the dynamics in relative periodic orbits. There are two particularly important features. One is that motions in relative periodic orbits of noncompact groups can only be of two types: Either they are quasi-periodic, or they leave any compact set as t →±∞ (`drifting motions'). Moreover, in a given group, one of the two behaviours may be predominant. The second is that motions in a relative periodic orbit exhibit `spiralling', `meandering' behaviours, which are routinely detected in numerical integrations. Since a quantitative description of meandering behaviours for drifting motions appears to be missing, we provide it here for a class of Lie groups that includes those of interest in locomotion (semidirect products of a compact group and a normal vector space). We illustrate these ideas on some examples (a kinematic car robot, a planar swimmer)

Distributed Utility Estimation with Heterogeneous Relative Information

In this letter, we consider a scenario where a set of agents, interconnected by a network topology, aim at computing an estimate of their own utility, importance or value, based on pairwise relative information having heterogeneous nature. In more detail, the agents are able to measure the difference between their value and the value of some their neighbors, or have an estimate of the ratio between their value and the value the remaining neighbors. This setting may find application in problems involving information provided by heterogeneous sensors (e.g., differences and ratios), as well as in scenarios where estimations provided by humans have to be merged with sensor measurements. Specifically, we develop a distributed algorithm that lets each agent asymptotically compute a utility value. To this end, we first characterize the task at hand in terms of a least-squares minimum problem, providing a necessary and sufficient condition for the existence of a unique global minimum, and then we show that the proposed algorithm asymptotically converges to a global minimum. This letter is concluded by numerical analyses that corroborate the theoretical findings

ELEMENTARY MECHANICS OF THE MITRAL VALVE

We illustrate a bare-bones mathematical model that is able to account for the elementary mechanics of the mitral valve when the leaflets of the valve close under the systolic pressure. The mechanical model exploits the aspect ratio of the valve leaflets that are represented as inextensible rods, subject to the blood pressure, with one fixed endpoint (on the endocardium) and an attached wire anchored to the papillary muscle. Force and torque balance equations are obtained exploiting the principle of virtual work, where the first contact point between rods is identified by the Weierstrass-Erdmann condition of variational nature. The chordae tendineae are modeled as a force applied to the free endpoint of the flaps. Different possible boundary conditions are investigated at the mitral annulus, and, by an asymptotic analysis, we demonstrate that in the pressure regime of interest generic boundary conditions generate a tensional boundary layer. Conversely, a specific choice of the boundary condition inhibits the generation of high tensional gradients in a small layer

Modeling and steering magneto-elastic micro-swimmers inspired by the motility of sperm cells

Controlling artificial devices that mimic the motion of real microorganisms, is attracting increasing interest, both from the mathematical point of view and applications. A model for a magnetically driven slender micro-swimmer, mimicking a sperm cell is presented, supported by two examples showing how to steer it. Using the Resistive Force Theory (RTF) approach [J. Gray and J. Hancock, J. Exp. Biol. 32, 802 (1955)] to describe the hydrodynamic forces, the micro-swimmer can be described by a driftless affine control system where the control is an external magnetic field. Moreover we discuss through at first an asymptotic analysis and then by numerical simulations how to realize different kinds of paths

Equilibrium of Two Rods in Contact Under Pressure

We study the equilibrium of a mechanical system composed by two rods that bend under the action of a pressure difference; they have one fixed endpoint and are partially in contact. This system can be viewed as a bi-valve made by two smooth leaflets that lean on each other. We obtain the balance equations of the mechanical system exploiting the principle of virtual work and the contact point is identified by a jump condition. The problem can be simplified exploiting a first integral. In the case of quadratic energy, another first integral exists: its peculiarity is discussed and a further reduction of the equations is carried out. Numerical integration of the differential system shows how the shape of the beams and the position of the contact point depend on the applied pressure. For small pressure, an asymptotic expansion in a small parameter allows us to find an approximate solutions of polynomial form which is in surprisingly good agreement with the solution of the original system of equations, even beyond the expected range of validity. Finally, the asymptotics predicts a value of the pressure that separates the contact from the no-contact regime of the beams that compares very well with the one numerically evaluated

Optimal Motion of a Scallop: Some Case Studies

In this letter, we analyze two optimal control problems for the scallop: a two-link swimmer that is able to self-propel changing dynamics between two fluids regimes. We address and solve explicitly the minimum time problem and the minimum quadratic one, computing the cost needed to move the swimmer between two fixed positions using a periodic control. We focus on the case of only one switching in the dynamics and exploiting the structure of the equation of motion we are able to split the problem into simpler ones. We solve explicitly each sub-problem obtaining a discontinuous global solution. Then we approximate it through a suitable sequence of continuous functions