Life at Low Reynolds Number

The manner in which quadrupeds change their locomotive patterns—walking, trotting, and galloping—with changing speed is poorly understood. In this paper, we provide evidence for interlimb coordination during gait transitions using a quadruped robot for which coordination between the legs can be self-organized through a simple “central pattern generator” (CPG) model. We demonstrate spontaneous gait transitions between energy-efficient patterns by changing only the parameter related to speed. Interlimb coordination was achieved with the use of local load sensing only without any preprogrammed patterns. Our model exploits physical communication through the body, suggesting that knowledge of physical communication is required to understand the leg coordination mechanism in legged animals and to establish design principles for legged robots that can reproduce flexible and efficient locomotion.

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A Quadruped Robot Exhibiting Spontaneous Gait Transitions from Walking to Trotting to Galloping

Internal models and prediction of visual gravitational motion

We consider a two coupled Duffing–van der Pol oscillators with a nonlinear coupling. The method of multiple scales is used to obtain a set of autonomous equations for the amplitudes and phases of the response of the system. The stability boundaries of fixed points of the approximate equations are obtained using Routh–Hurwitz criterion. The stability boundaries are drawn in various parameters space. The influence of nonlinear damping, detuning, nonlinearity and the coupling strength on the response dynamics is studied. Jump phenomenon is found to occur for a range of values of the parameters of the system. The results obtained from the approximate equations are verified by numerically solving the original system and good agreement is obtained.

Resonance behaviour and jump phenomenon in a two coupled Duffing–van der Pol oscillators

Air levitation is the process by which an object is lifted without mechanical support in a stable position, by providing an upward force that counteracts the gravitational force exerted on the object. This work presents a low-cost lab implementation of an air levitation system, based on open solutions. The rapid dynamics makes it especially suitable for a control remote lab. Due to the system's nature, the design can be optimized and, with some precision trade-off, kept affordable both in cost and construction effort. It was designed to be easily adopted to be used as both a remote lab and as a hands-on lab.

https://www.mdpi.com/1424-8220/17/10/2321/pdf

Design of a Low-Cost Air Levitation System for Teaching Control Engineering

We review the problem of a vertically thrown ball, with a drag force which is either linear or quadratic in the speed. It is stressed from the outset that these two types of drag correspond to specific ranges of the Reynolds number (Re<1 and 103<Re<2×105, respectively) and do not hold outside these intervals. We also include the buoyant force in our treatment of the problem. The equations of motion are solved analytically and several true-to-life examples are discussed. The calculations are somewhat harder than for the well-known case without drag force, but no highbrow mathematics is required and the extra effort is amply compensated by the gain in realism and surprise value. © 1999 American Association of Physics Teachers.

/pdf/on-the-rise-and-fall-of-a-ball-with-linear-or-quadratic-drag-2sgdah5k18.pdf

On the rise and fall of a ball with linear or quadratic drag

This paper is about mode interaction in systems of coupled nonlinear oscillators. The main ideas are demonstrated by means of a model consisting of two coupled, parametrically driven pendulums. On the basis of this we also discuss mode interaction in the Faraday experiment (as observed by Ciliberto and Gollub) and in running animals. In all these systems the interaction between two modes is seen to take place via a third mode: This interaction mode is a common daughter, born by means of a symmetry breaking bifurcation, of the two interacting modes. Thus, not just any two modes can interact with each other, but only those that are linked (in the system's group-theoretical hierarchy) by a common daughter mode. This is the quintessence of mode interaction. In many cases of interest, the interaction mode is seen to undergo further bifurcations, and this can eventually lead to chaos. These stages correspond to lower and lower levels of symmetry, and the constraints imposed by group theory become less and less restrictive. Indeed, the precise sequence of events during these later stages is determined not so much by group-theoretical stipulations as by the accidental values of the nonlinear terms in the equations of motion.

https://www.math.upatras.gr/~weele/files/Oscillators%20%26%20Horses%20(AmJPhys%2069,%202001).pdf

Jacobus P. van der Weele

Papers

On the rise and fall of a ball with linear or quadratic drag

Mode interaction in horses, tea, and other nonlinear oscillators: the universal role of symmetry