John B. Ferris

Development of Proper Models of Hybrid Systems

A method to improve the ability of design engineers to generate proper dynamic models of systems from sets ofcomponent models is developed. This two stage method, called Eigenvalue MODA, is a new model deduction procedure for developing proper models of hybrid systems. The first stage in Eigenvalue MODA consists of using a previously published Model Order Deduction Algorithm (MODA) to systematically increment the complexity of each component model in the system. The first stage continues until a set of Critical System Eigenvalues (CSE) has been defined. During the second stage, the complexity component models is incremented based on the convergence of the CSE. The second stage continues until each CSE has converged to within a user specified tolerance. Component models may be represented by first-order (state space) or second-order equations and may be modal expansion or finite segment models. An example shows that the deduction of the proper system model depends on the interactions between the component model representations and the model deduction method. Eigenvalue MODA is a model deduction method that facilitates the generation of models of sufficient accuracy with physically meaningful parameters and states. This makes the models useful for system design and, as such, Eigenvalue MODA would be a useful tool in an automated modeling environment for design engineers.

Characterising road profiles as Markov Chains

Load data representing severe customer usage is needed throughout a chassis development programme. It is necessary to understand the excitation, the road profile, in order to set the target chassis loads. Defining a large set of roads is impractical, so roads with similar characteristics must be grouped. Rough roads used for chassis development are not easily characterised however. In this work, roads are characterised via a Markov Chain. Any realisation generated from this process represents all profiles in the set. Realisations of any length can be generated, allowing efficient simulation. Some definitions and elementary properties of road profiles and Markov Chains are reviewed before introducing a transition matrix as a tool to characterise a measured road profile. A statistical test is introduced to assess whether a given road profile could be a realisation of a given transition matrix. An example demonstrates the limitations, benefits, and applicability of this representation.

Preliminary results for model identification in characterizing 2-D topographic road profiles

Load data representing severe customer usage is needed throughout a chassis development program; the majority of these chassis loads originate with the excitation from the road. These chassis loads are increasingly derived from vehicle simulations. Simulating a vehicle traversing long roads is simply impractical, however, and a greatly reduced set of characteristic roads must be found. In order to characterize a road, certain modeling assumptions must be made. Several models have been proposed making various assumptions about the properties that road profiles possess. The literature in this field is reviewed before focusing on two modeling assumptions of particular interest: the stationarity of the signal (homogeneity of the road) and the corresponding interval over which previous data points are correlated to the current data point. In this work, 2-D topographic road profiles are considered to be signals that are realizations of a stochastic process. The objective of this work is to investigate the stationarity assumption and the interval of influence for several carefully controlled sections of highway pavement in the United States. Two statistical techniques are used in analyzing these data: the autocorrelation and the partial autocorrelation. It is shown that the road profile signals in their original form are not stationary and have an extremely long interval of influence on the order of 25m. By differencing the data, however, it is often possible to generate stationary residuals and a very short interval of influence on the order of 250mm. By examining the autocorrelation and the partial autocorrelation, various versions of ARIMA models appear to be appropriate for further modeling. Implications to modeling the signals as Markov Chains are also discussed. In this way, roads can be characterized by the model architecture and the particular parameterization of the model. Any synthetic road realized from a particular model represents all profiles in this set. Realizations of any length can be generated, allowing efficient simulation and timely information about the chassis loads that can be used for design decisions. This work provides insights for future development in the modeling and characterization of 2-D topographic road profiles.

Characterizing 2D road profiles using ARIMA modeling techniques

The principal excitation to a vehicles chassis system is the road profile. Simulating a vehicle traversing long roads is impractical and a method to produce short roads with given characteristics must be developed. There are many methods currently available to characterize roads when they are assumed to be homogeneous. This work develops a method of characterizing non-stationary road profile data using ARIMA (Autoregressive Integrated Moving Average) modeling techniques. The first step is to consider the road to be a realization of an underlying stochastic process. Previous work has demonstrated that an ARIMA model can be fit to non-stationary road profile data and the remaining residual process is uncorrelated. This work continues the examination of the residual process of such an ARIMA model. Statistical techniques are developed and used to examine the distribution of the residual process and the preliminary results are demonstrated. The use of the ARIMA model parameters and residual distributions in classifying road profiles is also discussed. By classifying various road profiles according to given model parameters, any synthetic road realized from a given class of model parameters will represent all roads in that set, resulting in a timely and efficient simulation of a vehicle traversing any given type of road.

Terrain Roughness Standards for Mobility and Ultra-Reliability Prediction

The U.S. Army uses the root mean squared of elevation, or the RMSE standard for characterizing road/off-road roughness descriptions. This standard has often appeared in contracts as a performance requirement for the vehicle system. One important application of the standard is describing the testing environment for the vehicle. A physical test, which uses the standard, is the 30,000 mile endurance test. More recently, another metric has been used, the power spectral density (PSD) of road roughness. The international standard for road roughness is known as the International Roughness Index (IRI), and all road construction projects in the U.S. are based on this, as well as Department of Transportation analyses. This paper will analyze the different standards by comparing and contrasting the various aspects of each. Depending on the standard and metrics chosen, the simulation results will have different correlations with actual test. The goal is to better understand each standards limitations and how it affects the correlations.

Establishing chassis reliability testing targets based on road roughness

A process for establishing chassis reliability testing targets, based on road roughness, is developed in this work. The testing target is determined from road profile and traffic measurements. The process is applicable to a variety of road types (from highways to gravel roads) and locations. Roughness levels are calculated via the International Roughness Index from each road profile. These roughness levels are combined with corresponding traffic volume measurements to develop the target roughness level. This target roughness level is used in selecting the target roads. These target roads are then the testing targets for durability test development and reliability predictions. The work establishes a new target development process that does not rely on specific response data from a vehicle. Chassis durability tests can be developed from a database of target roads, without costly acquisition of vehicle-specific response data.

Review of current developments in terrain characterization and modeling

As computational power builds to meet the needs of ground vehicle designers, the focus has begun to shift from laboratory testing of prototype parts and subsystems to computational simulations of the vehicle. In the automotive and defense industries, large strides have been made in simulating full vehicle responses, such as durability. These simulations are most meaningful when excited by proper mathematical models that accurately characterize the terrain. It is important to understand the roughness indices that are used to judge the terrain profiles. The state-of-the-art in terrain characterization and modeling is reviewed in this work for models including Power Spectral Density (PSD), Markov Chains, Autoregressive Integrated Moving Average (ARIMA), Parametric Road Spectrum (PRS), Shifted Spatial Range Spectrum (SSR), Direct Spectrum Estimation (DSE) and Transformed Direct Spectrum Estimation (TrDSE). The applicability, limitations, and benefits of these models are assessed based on their effectiveness in capturing the stochastic nature of the terrain being characterized. A discussion of terrain characterization usage to advance reliability testing concludes this work as an example of the applicability of this technology.

A polynomial chaos approach to ARIMA modeling and terrain characterization

During the vehicle design process, excitation loads are needed to correctly model the system response. The main source of excitation to this dynamic system comes from the terrain. Characteristic models of terrain topology, therefore, would allow for more accurate models and simulations of the system response. Terrain topology can be characterized as a realization of an underlying stochastic process. It has been demonstrated that ARIMA modeling can be used to characterize non-stationary road profiles. In this work it is suggested that ARIMA models of terrain topology can be further developed by characterizing the previously deterministic autoregressive coefficients as random variables. In this way uncertainty is introduced into the system parameters and propagated through the process to yield a distribution of terrain topology. This distribution is then dependent on the distribution of the residuals as well as the distribution of the ARIMA coefficients. The use of random variables to classify road types is discussed as possible future work.

A planar quasi-static constraint mode tyre model

The fast-paced, iterative, vehicle design environment demands efficiency when simulating suspension loads. Towards that end, a computationally efficient, linear, planar, quasi-static tyre model is developed in this work that accurately predicts a tyres lower frequency, reasonably large amplitude, nonlinear stiffness relationship. The axisymmetric, circumferentially isotropic, stiffness equation is discretised into segments, then parameterised by a single stiffness parameter and two shape parameters. The tyres deformed shape is independent of the overall tyre stiffness and the forces acting on the tyre. Constraint modes capture enveloping and bridging properties and a recursive method yields the set of active constraints at the tyre–road interface. The nonlinear stiffness of a tyre is captured by enforcing unidirectional geometric boundary conditions. The model parameters are identified semi-empirically; simulated cleat test loads match experiments within 7% including nonlinear stiffness when simulating a flat plate test and a discontinuous stiffness when simulating a cleat test.

Developing stable autoregressive models of terrain topology

It is desirable to evaluate vehicle models over a wide range of terrain types, but it is computationally impractical to simulate over measured terrain. A method to characterise terrain topology is developed in this work to group terrain into meaningful sets with similar physical characteristics. Specifically, measured terrain profiles are considered realisations of an underlying stochastic process; an autoregressive model provides the mathematical framework to describe this process. The autocorrelation of the spatial derivative of the terrain profile is examined to determine the form of the model. The required order for the model is determined from the partial autocorrelation of the spatial derivative of the profile. The stability of the model is evaluated and enforced by transforming the AR difference equation into an Infinite Impulse Response filter representation. Finally, the method is applied to a set of USA highway data and an optimal model order is determined for this application.