Fractal dimensions of terrain profiles

Analysing terrain profiles of fields, roads, and other terrains, it was determined that terrain profiles are random and non-periodical. Mandelbrot has defined non-scaling, self-similar figures as fractals, and many investigators have tried to characterize natural forms and structures using fractal geometry. The work here investigates whether terrain profiles can be defined as fractals. Fractal dimensions of profiles were calculated. These were compared with a locus of Brownian motion further to investigate characteristics of terrain profiles. Fractals are defined to be self-similar and irregular. Measuring and analysing terrain profiles, it was established that the statistical characteristics of any part of a terrain profile are similar and that the statistical characteristics of profiles of any kind of terrain are similar irrespective of roughness. This means that terrain profiles are self-similar, and irregular. From these results, it was determined that terrain profiles are fractals. The fractal dimensions were calculated with a coarse-graining method and by Power Spectral Densities (PSD), and fractal dimensions by Scaling were between 1.1 and 1.8 and by PSD between 1.3 and 1.5. Using the locus of Brownian motion, fractal dimensions were 1.5 or slightly larger than those of the terrain profiles. Fractal dimensions for the locus of smoothed Brownian motion were nearly equal to terrain profiles. Therefore terrain profiles could be artificially generated from the locus of smoothed Brownian motion. It appears that terrain roughness is formed by random and non-periodical force.     

Calculating fractal parameters from low-resolution terrain profiles

Driver comfort on rough terrain is an important factor in the off-road performance of wheeled and tracked ground vehicles. The roughness of a terrain has typically been quantified by the U.S. Army as the root-mean-square elevation deviation (RMS) of the terrain profile. Although RMS is an important input into many mobility calculations, it is not scale invariant, making it difficult to estimate RMS from low resolution terrain profiles. Fractal parameters are another measure of roughness that are scale invariant, making them a convenient proxy for RMS. While previous work found an empirical relationship between fractal dimension and RMS, this work will show that, by including the cutoff length, an analytic relationship between fractal properties and RMS can be employed. The relationship has no free parameters and agrees very well with experimental data - thus providing a powerful predictive tool for future analyses and a reliable way to calculate surface roughness from low-resolution terrain data in a way that is scale invariant. In addition, we show that this method applies to both man-made ride courses and natural terrain profiles.     

Country road and field surface profiles acquisition,modelling and synthetic realisation for evaluating fatigue life of agricultural machinery

Fatigue life of agricultural machinery is strongly affected by the surfaces that these machines operate on. In the present paper a sensor-frame was developed to acquire road and field profiles in absolute geo-referenced coordinates. The sensor-frame was validated by measuring discrete trapezoidal bumps with known dimensions resulting in a root mean squared (RMS) error of 6–8 mm. Profiles were acquired from a country road and from a mowed grass field. Using the quarter-car vehicle model, the movement of an agricultural vehicle was simulated for various speeds. The resulted vertical loads were rainflow-counted and the accumulated fatigue pseudo damage was calculated using Palmgren–Miner linear rule. Based on the derived Power Spectrum Density (PSD), the profiles were classified according to ISO 8608 standard. Two methodologies were followed to model and create a number of synthetic realisations for each profile: Direct Spectrum Estimate (DSE) and ISO based modelling. Simulating the produced synthetic profiles with the quarter-car vehicle model, the corresponding pseudo damage was calculated. The accumulated damages from the DSE models were closer to the corresponding ones from the measured profiles. ISO based models could not model the profile irregularities, which proved to contribute the largest part of the accumulated fatigue damage.     

Measurement of mixed-mode crack profiles by holographic interferometry

Holographic interferometry was used to study the mixed-mode fracture characteristics of mortar. The nature of crack propagation in such quasibrittle materials and the theoretical model used to interpret the experimental results indicated that a highly sensitive measurement technique was required. The gradual curvature of the propagating crack at different sections of the specimen necessitated full-field observation capability. The nature of the problem made holographic interferometry the ideal technique for this application. To measure the in-plane components of the opening and sliding of the crack surfaces during propagation, a single holographic plate was placed very close to the specimen. This allowed four independent observations of any point on the specimen from the four corners of the plate without any need for additional optics of exposures. Double-exposure holograms were made at different crack-propagation stages. The developed plate was illuminated by an unexpanded reference beam to form a real image of the object and observe displacement fringes. Fringe data were interpreted by using computer software written for this research.     

基于地形图像特征与模糊决策的海底地形可导航性分析方法

基于多波束测深数据和图像匹配技术的水下地形辅助导航方法需要对实施区域的地形可导航性进行评估,其中的地形独特性计算需要提取图像特征,并利用各特征参数进行综合评判。地形图像的直方图包含实际地理区域的起伏程度信息,通过定义直方图复杂度对此信息进行了有效提取。均方根对比度和拉普拉斯模方和分别侧重描述了地形整体和局部的变化程度,图像的二维熵从信息论的角度反映了地理区域包含的信息量。此外,依据单一特征参数对于待评价地形的可导航性进行排名,其结果存在不一致性,利用模糊综合评判对各参数的信息进行了有效综合。通过实测多波束地形数据仿真分析,基于图像纹理特征的地形匹配方法,在模糊评判结果最优的地形区域能够将匹配误差限制在2 m以内,结果验证了该可导航性分析方法的有效性。     

Qualitative mathematical analysis of the Richards equation

The Richards equation is widely used as a model for the flow of water in unsaturated soils. For modelling one-dimensional flow in a homogeneous soil, this equation can be cast in the form of a specific nonlinear partial differential equation with a time derivative and one spatial derivative. This paper is a survey of recent progress in the pure mathematical analysis of this last equation. The emphasis is on the interpretation of the results of the analysis. These are explained in terms of the qualitative behaviour of the flow of water in an unsaturated soil which is described by the Richards equation.Nomenclature a coefficient in second-order diffusion term of equation - b coefficient in first-order advection term of equation - D soil-moisture diffusivity [L²T^-1] - h pressure head [L] - H quarter-plane domain for Cauchy-Dirichlet problem [L] x [T] - K hydraulic conductivity scalar [LT^–1] - K hydraulic conductivity tensor [LT^–1] - q soil-moisture flux scalar [LT^–1] - q soil-moisture flux vector [LT^–1] - r dummy variable - R rectangle [L] x [T] - s dummy variable - s* representative value of dummy variable - S half-plane domain for Cauchy problem [L] x [T] - t time [T] - u unknown solution of partial differential equation - u₀ initial-value function - v soil-moisture velocity scalar [LT^–1] - v soil-moisture velocity vector [LT^–1]     

A mathematical programming algorithm for limit analysis

This paper deals with the limit analyses of perfect rigid-plastic continua. Based on the kinematic theorem of the limit analysis theory, a mathematical programming finite element formula for determining the upper bound load multiplier has been established, and an iteration algorithm proposed accordingly. In this algorithm the plastic and rigid zones are distinguished for every iteration step, and the goal function is modified gradually. The difficulties caused by the nonsmoothness of the goal function are overcome. Some examples solved by this algorithm are presented. The project supported by National Natural Science Foundation of China.     

改进的地形熵算法在地形辅助导航中的应用

地形辅助导航是水下航行器导航技术的一个发展新方向,但它并不能够在任何地形区域都可以工作,比如在平坦区域的导航效果很差。通过计算不同区域的水深标准差,选择地形特征独特的区域作为适配区域。基于熵的算法对于地形复杂区域的匹配分析是快速有效的,但传统的地形熵算法匹配精度不高,本文引入了地形差异熵的概念并对其进行改进,在选定的地形区域使用MATLAB软件进行了仿真研究。仿真结果表明,改进的地形熵算法在选定的地形区域位置误差在250m左右,可以满足水下航行器的导航要求。     

Spectrum analysis of road roughness for earthmoving machinery

The power spectral density functions of roughness of road surfaces for earthmoving machines such as tractors, scrapers and dumpers were determined by use of an accelerometer mounted on a towed fifth wheel of which the frequency-response function had previously been known. From obtained data, a range of power spectral density curves of the road roughness for these earthmoving machines was derived and expressed mathematically.     

Characterizing terrain image classification difficulties through reduced-dimension class convex hull analysis

The classification of natural terrain ahead of an autonomous vehicle can help it make decisions regarding the traversability of various paths, but remains an open research problem. Despite the explosion in popularity of deep learning networks,there is still little work available on informed neural network design procedures for specific tasks such as terrain image classification, save through performance measures. A related problem is understanding features of a dataset that lead to difficulties in separating classes of images from one another. This research proposes an algorithm and accompanying analytical procedure to characterize such image classification difficulties; identifying what makes some images easily distinguishable as their class and what makes others readily confused with other classes. This is achieved by learning reduced-dimensionality representations of the input data, constructing a convex hull of class members in the reduced dimensionality representation, then examining between-class overlap within each space, incrementally increasing the dimensionality until overlap is eliminated. Summarizing the between-class overlap statistics reveals trends and anomalies that can be linked back to visual features, characteristic of the original data. Case studies are presented of insights identified through selected example analyses: relative intensities of terrain classes from images taken by Mars rovers, and the impact of color gradients in separating sand from bedrock in color images of terrain. Such insights are discussed as steps toward a more directed approach to designing neural networks for image classification.     

Generalized limit analysis in poroplasticity by mathematical programming

Classical limit analysis of structures by the statical approach computationally means maximization of a load multiplier under equilibrium and yield condition constraints, namely convex mathematical programming. In elastoplasticity, generalizations of limit analysis had been proposed in order to achieve, still by load factor constrained optimization, the safety factor with respect to plastic collapse. This paper presents similar generalization in two-phase poroelastoplasticity. A method is here developed (and validated by numerical application to a masonry dam) apt to assess the safety factor of a structure interpretable as a poroplastic system, with respect to both plastic collapse and critical thresholds on deformations, by solving a nonconvex nonsmooth constrained optimization problem usually referred to in the literature as “mathematical program under equilibrium constraints”. Piece-wise linearization of yield surfaces and reduction of yield planes by a “sifting” procedure are adopted to reduce computing efforts.     

Structure design and traction trafficability analysis of multi-posture wheel-legs bionic walking wheels for sand terrain

Nowadays, the existing walking wheels still have problems with the wheel-legs structure and the traction trafficability on the loose sand. It is commonly believed that African ostrich (Struthio camelus) is a kind of bipedal species with superior running performance on the sandy environment. Being enlightened by this, four bionic walking wheels (herringbone wheel, in-line wheel, V-shaped wheel and combination wheel) were designed and tested by imitating the structure and posture of ostrich’s feet travelling on sand. The results showed that when the wheel load was 20, 30 and 50N respectively and the slip ratio was less than 35%, the herringbone wheel had better traction trafficability than that of other wheels. When the wheel load was 30, 50 and 70N and the slip ratio was more than 35%, the in-line wheel had better performance than that of other wheels. It was shown in this thesis that the bionic walking wheels designed with the multi-posture wheel-legs and the simple structure could reduce the soil resistance and the disturbance to sand, thereby achieving a superior performance of traveling on sand. In addition, a new idea and research method for designing of walking mechanism on soft terrain has been provided in this thesis.     

Experimental modal analysis: From transducers to frequency analysis to mathematical models

Experimental Techniques -     

Photometric measurement of terrain roughness

Information on the roughness of terrain is an important factor in designing the rolling gear of cross-country vehicles. In view of a number of advantages—especially the great information density—of the photometric measuring of terrain roughnesses, this study report discusses the applicability of this technique as a new method to complement those already in use. The report demonstrates that the resolution limit of approximately ± 5 cm in a scale 1:2000 reproduction is sufficient, for all practical purposes, because the resulting total measuring error will be distributed over the entire profile-frequency range.A comparison between an accurate ground-based technique and photometric measurement is presented to demonstrate that, in the range of the profile wavelength of 1 m, the roughness root-mean-square-values differ only by a few mm, so that these differences may be neglected.It is suggested that, for assessing terrain roughness, range-pair classification be used in addition to the usually applied power spectral density, because range-pair classification describes the terrain amplitudes occuring in terms of their absolute magnitude.The report indicates evaluation errors that may result from non-stationary roughness profiles, and it shows the possibility of conducting the photometric terrain measurement in wider and less expensive steps, and of complementing this measurement by superimposing a suitable random noise signal on the measurement curve.     

Method of analysis of mathematical models of media under complex loading

A method of analysis of constitutive models of media with complex loading trajectories is proposed. It is based on a comparison of data from laboratory experiments and corresponding results of numerical calculations. In previous laboratory experiments, nearly homogeneous complex loading with continuous rotation of the principal axes of the strain tensor and loading with broken strain trajectories was performed. Numerical calculations for the types of loading corresponding to the experiments performed are based on the constitutive equations of the model. The numerical results obtained and data of the laboratory experiments are compared. The hypoplastic model of a geomedium is analyzed. Analysis shows that the model is a satisfactory qualitative and quantitative approximation of data from laboratory experiments on complex loading of geomaterials. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 133–142, September–October, 1999.     

Dynamic analysis of mathematical model of ethanol fermentation with gas stripping

In this paper, a mathematical model for ethanol fermentation with gas stripping is investigated. Firstly, the model with continuous substrate input is taken. We study the existence and local stability of two equilibrium points. According to Poincare–Bendixson Theorem, the sufficient condition for the globally asymptotical stability of positive equilibrium point is obtained, which implies that we can get stable ethanol product. Secondly, we study the model with impulsive substrate input and obtain the sufficient condition for the local stability of cell-free periodic solution by using the Floquet’s theory of impulsive differential equation and small-amplitude perturbation skills. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical (subcritical) bifurcation. Finally, our results are confirmed by means of numerical simulation.