Logarithmic Spiral Trajectory Research Articles

Analytical solution to logarithmic spiral trajectories with circumferential thrust and mission applications

This study made use of a shape-based method to analyze the orbital dynamics of a spacecraft subject to a continuous propulsive acceleration acting along the circumferential direction. Under the assumption of a logarithmic spiral trajectory, an exact solution to the equations of motion exists, which allows the spacecraft state variables and flight time to be expressed as a function of the angular coordinate. There is also a case characterized by specific initial conditions in which the time evolution of the state variables may be analytically determined. In this context, the presented solution is used to analyze circle-to-circle trajectories, where the combination of two impulsive maneuvers and a logarithmic spiral path are used to accomplish the transfer. The determined results are then applied to the achievement of the Earth—Mars and the Earth—Venus transfers using actual data from a recent thruster developed by NASA.

Maximum Entropy (Most Likely) Double Helical and Double Logarithmic Spiral Trajectories in Space-Time

The ubiquity of double helical and logarithmic spirals in nature is well observed, but no explanation is ever offered for their prevalence. DNA and the Milky Way galaxy are examples of such structures, whose geometric entropy we study using an information-theoretic (Shannon entropy) complex-vector analysis to calculate, respectively, the Gibbs free energy difference between B-DNA and P-DNA, and the galactic virial mass. Both of these analytic calculations (without any free parameters) are consistent with observation to within the experimental uncertainties. We define conjugate hyperbolic space and entropic momentum co-ordinates to describe these spiral structures in Minkowski space-time, enabling a consistent and holographic Hamiltonian-Lagrangian system that is completely isomorphic and complementary to that of conventional kinematics. Such double spirals therefore obey a maximum-entropy path-integral variational calculus (“the principle of least exertion”, entirely comparable to the principle of least action), thereby making them the most likely geometry (also with maximal structural stability) to be adopted by any such system in space-time. These simple analytical calculations are quantitative examples of the application of the Second Law of Thermodynamics as expressed in geometric entropy terms. They are underpinned by a comprehensive entropic action (“exertion”) principle based upon Boltzmann’s constant as the quantum of exertion.

Control of nonlinear systems exhibiting chaos to desired periodic or quasi-periodic motions

A novel method in the design of controllers to drive general nonlinear systems to desired periodic or quasi-periodic motions is presented in this paper. The viability of the approach is demonstrated by controlling chaotic systems to desired motions. The proposed control system consists of a combination of a nonlinear feedforward controller and a linear feedback controller. The control gains for the feedback controller are determined by performing the stability analysis of the closed-loop systems that contain periodic or quasi-periodic coefficients. For the case of periodic coefficients, stability is determined using the well-known Floquet theory. Since there is no rigorous mathematical theory for the analysis of quasi-periodic systems, an approximate technique has been applied to determine the stability where the coefficients turned out to be quasi-periodic. In this approach, a quasi-periodic system is replaced by an approximate periodic system with an appropriate large principal period such that Floquet theory can be applied. State transition matrices for these closed-loop systems are computed symbolically in terms of the unknown control gains using shifted Chebyshev polynomials and Picard iterations. Stability diagrams obtained from the symbolic approach are used to select control gains to guarantee asymptotic stability of the feedback systems. Three examples of chaotic systems are studied in order to show applicability to a diverse class of problems. In the first case, the chaotic motion of a forced Duffing oscillator is driven to a two-frequency quasi-periodic square wave and a fixed point. In the control to a fixed point, the system is constrained to follow a logarithmic spiral trajectory. The second case involves the control of the chaotic attractor of a parametrically forced Lorenz system to a periodic orbit whose frequency is irrationally related to the parametric excitation frequency. In the last case, the chaotic behavior of a Mathieu–Duffing oscillator is successfully driven to a two-frequency quasi-periodic motion.

Logarithmic spiral trajectories generated by Solar sails

Analytic solutions to continuous thrust-propelled trajectories are available in a few cases only. An interesting case is offered by the logarithmic spiral, that is, a trajectory characterized by a constant flight path angle and a fixed thrust vector direction in an orbital reference frame. The logarithmic spiral is important from a practical point of view, because it may be passively maintained by a Solar sail-based spacecraft. The aim of this paper is to provide a systematic study concerning the possibility of inserting a Solar sail-based spacecraft into a heliocentric logarithmic spiral trajectory without using any impulsive maneuver. The required conditions to be met by the sail in terms of attitude angle, propulsive performance, parking orbit characteristics, and initial position are thoroughly investigated. The closed-form variations of the osculating orbital parameters are analyzed, and the obtained analytical results are used for investigating the phasing maneuver of a Solar sail along an elliptic heliocentric orbit. In this mission scenario, the phasing orbit is composed of two symmetric logarithmic spiral trajectories connected with a coasting arc.

An analytical model of a typhoon wind field based on spiral trajectory

In typhoon risk assessment and warning, a critical component is a good representation of the typhoon wind field model. In this study, a new analytical model based on the logarithmic spiral trajectory model is developed to simulate the surface wind speed distribution of a typhoon. The logarithmic spiral trajectory model could overcome the limitation of the parametric gradient wind model. A slab surface layer of constant depth is used to solving the tangential equilibrium equation, and the frictional drag at the upper boundary of the surface layer is considered correctly. Consequently, the theoretical method for determining the Holland β parameter is derived from the logarithmic spiral trajectory model. It is concluded that β increases with the surface layer depth or decreases with the radius to maximum winds. By analyzing the change in kinetic energy of the air particle, the interpretation of the relationship between β and the influencing factors is provided. The models are applied to the 17th typhoon NESAT and the 19th typhoon NALGAE of 2011. Through comparisons between the observed wind records and the simulation results, the logarithmic spiral trajectory model proposed in this study could accurately simulate the wind speeds.

Measurement of Cutting Speed of Mechanical Excavator

Only when mining at reasonable cutting speed, can a mechanical excavator achieve high efficient and low consumption. The same cutting-angle logarithmic spiral trajectory is initially confirmed for the model of a reasonable excavating trajectory to reduce digging resistance. An analytical expression between hoisting and pushing speed is derived from the mechanism kinematics theory and the constraint equation method. This expression accurately describes the cooperative relationship between these speeds. Virtual prototype kinematics of a large mechanical excavator are simulated and verified through ADAMS. The simulation result is consistent with the theoretical result, and thus validates the analytical expression between hoisting and pushing speed.

A bio-inspired grasp optimization algorithm for an anthropomorphic robotic hand

A fundamental requirement for assistive robots is to guarantee a safe and human-like way to perform their tasks. In particular, the ability to realize smooth movements and obtain a stable grasp is of primary importance. In this perspective, this paper aims at studying human grasping and developing a bio-inspired method for power-grip posture prediction and finger trajectory planning for a robotic hand. The developed method is based on neuroscientific assumptions and experimental evidence coming from the observation of the human behavior during power grip. It is based on the minimization of a suitably defined function to identify the optimal grasp configuration and the choice of a logarithmic spiral trajectory for moving the fingers. The behavior of ten different subjects during the grasping action has been analyzed with the CyberGlove motion capture data glove. A common thumb posture has been observed and has been introduced in the grasping algorithm. The algorithm performance has been tested on an anthropomorphic robotic hand by means of simulation trials. The results demonstrate the effectiveness of the approach and pave the way for the implementation on a real robotic hand.

Design of a logarithmic k-space spiral trajectory.

Spiral acquisitions are used in fast cardiac imaging because they traverse k-space efficiently and minimize flow artifacts. A variable pitch logarithmic spiral trajectory is designed to critically sample the low-frequency region in k-space and gradually undersample the high-frequency region. An approximate analytical expression for the trajectory provides a fast means to calculate the gradient waveforms and the sampled data points. A numerical method is introduced based on the trajectory curvature and the rate of change in the gradient magnitude with time for the composite Archimedean-logarithmic trajectory. The pulse sequence is implemented and images are acquired on phantoms and human hearts. The images show improved image resolution and some improvement in image quality as a result of increased extent in k-space and reduction in aliasing artifacts, respectively.

Experimental study of vector and scalar properties of 9Kh2 steel in deformation along flat curvilinear trajectories. Part 1. Logarithmic-spiral trajectories

Experimental study of vector and scalar properties of 9Kh2 steel in deformation along flat curvilinear trajectories. Part 1. Logarithmic-spiral trajectories

Superstrings, Cantorian-fractal Spacetime and Quantum-like Chaos in Atmospheric Flows

Atmospheric flows exhibit long-range spatiotemporal correlations manifested as the fractal geometry to the global cloud cover pattern concomitant with the inverse power law form for spectra of temporal fluctuations. Such non-local connections are ubiquitous to dynamical systems in nature and are identified as signatures of self-organized criticality. A recently developed cell dynamical system model for atmospheric flows predicts the observed self-organized criticality as a natural consequence of quantum-like mechanics governing flow dynamics. The model is based on the concept that spatial integration of enclosed small scale fluctuations results in the formation of large eddy circulations. The model predicts the following: (a) The flow structure consists of an overall logarithmic spiral trajectory with the quasiperiodic Penrose tiling pattern for the internal structure. (b) Conventional power spectrum analysis will resolve such spiral trajectories as a continuum of eddies with progressive increase in phase. (c) Increments in phase are concomitant with increases in period length and also represent the variance, a characteristic of quantum systems identified as Berrys phase. (d) The universal algorithm for self-organized criticality is expressed in terms of the universal Feigenbaum constants, a and d, as 2 a 2= πd, where the fractional volume intermittency of occurrence πd contributes to the total variance 2 a 2 of fractal structures. (e) The Feigenbaum constants are expressed as functions of the golden mean. ( f) The quantum mechanical constants fine structure constant and ratio of proton mass to electron mass, which are pure numbers and are obtained by experimental observations only, are now derived in terms of the Feigenbaum constant, a. (g) Atmospheric flow structure follows Keplers third law of planetary motion. Therefore, Newtons inverse square law for gravitation also applies to eddy masses. The centripetal acceleration representing the inertial masses (of eddies) are equivalent to gravitational masses. The fractal-Cantorian structure of spacetime can also be visualized as a nested continuum of vortex (eddy) circulations, whose inertial masses obey Newtons inverse square law of gravitation. The model concept resembles a superstring model for subatomic dynamics which incorporates gravitational forces.

Orbital Motion Under Continuous Tangential Thrust

The effect of continuous tangential thrust on the orbital motion and mass loss of a vehicle initially in a circular orbit is investigated analytically. It is shown that, for a thrust-to-weight ratio of greater than 0.16175, escape speed will eventually be reached along an unwinding spiral trajectory. For lower thrust-to-weight ratios, escape speed is never attained, and the flight path oscillates around a logarithmic spiral trajectory. Formulas are obtained for the approximate orbital motion and time of flight along each type of trajectory and for mass loss due to expenditure of rocket propellant.

Shear-stress trajectories produced by ballistic impact

Plastic distribution of stresses in a metal plate under ballistic impact can result in slip lines which can be visibly seen as two orthogonal families of logarithmic spiral trajectories. These spirals can be produced dynamically in oxide surface coatings of steel plate ballistically impacted. The visible spirals in the oxide coating can be described mathematically by assuming a Mohr-Coulomb yield surface with a friction angle of zero. The mutally orthogonal trajectories are continuous lines without breaks, and when the elastic steel body is strained beyond the elastic limit the failure lines follow these trajectories. This is the first time that Lüder bands have been measured under the dynamic conditions described so that not only theories of ideal plasticity are at issue, but large deformation in dynamic plasticity as well.

The Geometry of the Shear Zone in Metal Cutting

The choice of a suitable trajectory shape and slip-line field for the shear zone must be influenced by the degree of work hardening and thermal softening, and is necessarily difficult. Although probably incorrect, the geometry of a polar slip-line field is described in terms of the properties of circular and logarithmic spiral trajectories, as this affords a suitable starting point. It is then assumed that the fundamental quantity is the trajectory shape, and it is shown that a slip-line field may be determined which corresponds to any given set of spiral trajectories. The choice of spirals is limited by the condition of volume continuity. The results of Kececioglu (1960) [4] are reexamined in the light of more recent theories of the shear zone and the experimentally determined strain-rate values are shown to be incorrectly derived. A suitable trajectory shape must first be adopted before calculating the value of the strain rate in terms of the width of the zone.