Lambeth, J., Ferris, J., Reid, A., and Gorsich, D., "Developing a Compact Continuous-State Markov Chain for Terrain Road Profiles," SAE Technical Paper 2013-01-0629, 2013, doi:10.4271/2013-01-0629.
Accurate terrain models provide the chassis designer with a powerful tool to make informed design decisions early in the design process. It is beneficial to characterize the terrain as a stochastic process, allowing limitless amounts of synthetic terrain to be created from a small number of parameters. A continuous-state Markov chain is proposed as an alternative to the traditional discrete-state chain currently used in terrain modeling practice. For discrete-state chains, the profile transitions are quantized then characterized by a transition matrix (with many values). In contrast, the transition function of a continuous-state chain represents the probability density of transitioning between any two states in the continuum of terrain heights. The transition function developed in this work uses a location-scale distribution with polynomials modeling the parameters as functions of the current state. A maximum likelihood estimator is used to parameterize the coefficients of these polynomials. The logistic distribution is applied as a proof of concept, and other distributions are discussed as targets of future research.