Stability in dynamical systems subject to some law of force is considered. This leads to a set of differential equations which govern the motion. (AIP)

Stability in dynamical systems

The aerodynamic performance of hovering insects is largely explained by the presence of a stably attached leading edge vortex (LEV) on top of their wings. Although LEVs have been visualized on real, physically modeled, and simulated insects, the physical mechanisms responsible for their stability are poorly understood. To gain fundamental insight into LEV stability on flapping fly wings we expressed the Navier–Stokes equations in a rotating frame of reference attached to the wing's surface. Using these equations we show that LEV dynamics on flapping wings are governed by three terms: angular, centripetal and Coriolis acceleration. Our analysis for hovering conditions shows that angular acceleration is proportional to the inverse of dimensionless stroke amplitude, whereas Coriolis and centripetal acceleration are proportional to the inverse of the Rossby number. Using a dynamically scaled robot model of a flapping fruit fly wing to systematically vary these dimensionless numbers, we determined which of the three accelerations mediate LEV stability. Our force measurements and flow visualizations indicate that the LEV is stabilized by the `quasi-steady' centripetal and Coriolis accelerations that are present at low Rossby number and result from the propeller-like sweep of the wing. In contrast, the unsteady angular acceleration that results from the back and forth motion of a flapping wing does not appear to play a role in the stable attachment of the LEV. Angular acceleration is, however, critical for LEV integrity as we found it can mediate LEV spiral bursting, a high Reynolds number effect. Our analysis and experiments further suggest that the mechanism responsible for LEV stability is not dependent on Reynolds number, at least over the range most relevant for insect flight (100<Re<14,000). LEVs are stable and continue to augment force even when they burst. These and similar findings for propellers and wind turbines at much higher Reynolds numbers suggest that even large flying animals could potentially exploit LEV-based force augmentation during slow hovering flight, take-offs or landing. We calculated the Rossby number from single-wing aspect ratios of over 300 insects, birds, bats, autorotating seeds, and pectoral fins of fish. We found that, on average, wings and fins have a Rossby number close to that of flies (Ro=3). Theoretically, many of these animals should therefore be able to generate a stable LEV, a prediction that is supported by recent findings for several insects, one bat, one bird and one fish. This suggests that force augmentation through stably attached (leading edge) vortices could represent a convergent solution for the generation of high fluid forces over a quite large range in size.

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Rotational accelerations stabilize leading edge vortices on revolving fly wings

In a rst part we discuss the well-known prob- lem of the motion of a star in a general non-axisymmetric 2D galactic potential by means of a very simple but al- most universal system: the pendulum model. It is shown that both loop and box families of orbits arise as a natural consequence of the dynamics of the pendulum. An approx- imate invariant of motion is derived. A critical value of the latter sharply separates the domains of loops and boxes and a very simple computation allows to get a clear pic- ture of the distribution of orbits on a given energy surface. Besides, a geometrical representation of the global phase space using the natural surface of section for the prob- lem, the 2D sphere, is presented. This provides a better visualization of the dynamics. In a second part we introduce a new indicator of the basic dynamics, the Mean Exponential Growth fac- tor of Nearby Orbits (MEGNO), that is suitable to inves- tigate the phase space structure associated to a general Hamiltonian. When applied to the 2D logarithmic poten- tial it is shown to be eective to obtain a picture of the global dynamics and, also, to derive good estimates of the largest Lyapunov characteristic number in realistic physi- cal times. Comparisons with other techniques reveal that the MEGNO provides more information about the dynam- ics in the phase space than other wide used tools. Finally, we discuss the structure of the phase space as- sociated to the 2D logarithmic potential for several values of the semiaxis ratio and energy. We focus our attention on the stability analysis of the principal periodic orbits and on the chaotic component. We obtain critical energy values for which connections between the main stochastic zones take place. In any case, the whole chaotic domain appears to be always conned to narrow laments, with a Lyapunov time about three characteristic periods.

/pdf/simple-tools-to-study-global-dynamics-in-non-axisymmetric-4460vhvkzi.pdf

Simple tools to study global dynamics in non-axisymmetric galactic potentials – I

This paper considers the vehicle navigation problem for an autonomous underwater vehicle (AUV) with six degrees of freedom. We approach this problem using an error state formulation of the Kalman filter. Integration of the vehicle's high-rate inertial measurement unit's (IMU's) accelerometers and gyros allow time propagation while other sensors provide measurement corrections. The low-rate aiding sensors include a Doppler velocity log (DVL), an acoustic long baseline (LBL) system that provides round-trip travel times from known locations, a pressure sensor for aiding depth, and an attitude sensor. Measurements correct the filter independently as they arrive, and as such, the filter is not dependent on the arrival of any particular measurement. We propose novel tightly coupled techniques for the incorporation of the LBL and DVL measurements. In particular, the LBL correction properly accounts for the error state throughout the measurement cycle via the state transition matrix. Alternate tightly coupled approaches ignore the error state, utilizing only the navigation state to account for the physical latencies in the measurement cycle. These approaches account for neither the uncertainty of vehicle trajectory between interrogation and reply, nor the error state at interrogation. The navigation system also estimates critical sensor calibration parameters to improve performance. The result is a robust navigation system. Simulation and experimental results are provided.

Autonomous Underwater Vehicle Navigation

This thesis is about modelling, design and control of Miniature Flying Robots (MFR) with a focus on Vertical Take-Off and Landing (VTOL) systems and specifically, micro quadrotors. It introduces a mathematical model for simulation and control of such systems. It then describes a design methodology for a miniature rotorcraft. The methodology is subsequently applied to design an autonomous quadrotor named OS4. Based on the mathematical model, linear and nonlinear control techniques are used to design and simulate various controllers along this work. The dynamic model and the simulator evolved from a simple set of equations, valid only for hovering, to a complex mathematical model with more realistic aerodynamic coefficients and sensor and actuator models. Two platforms were developed during this thesis. The first one is a quadrotorlike test-bench with off-board data processing and power supply. It was used to safely and easily test control strategies. The second one, OS4, is a highly integrated quadrotor with on-board data processing and power supply. It has all the necessary sensors for autonomous operation. Five different controllers were developed. The first one, based on Lyapunov theory, was applied for attitude control. The second and the third controllers are based on PID and LQ techniques. These were compared for attitude control. The fourth and the fifth approaches use backstepping and sliding-mode concepts. They are applied to control attitude. Finally, backstepping is augmented with integral action and proposed as a single tool to design attitude, altitude and position controllers. This approach is validated through various flight experiments conducted on the OS4.

T. B. Putsyata

Papers

Analytical dynamics. L. A. Pars