8.1: Introduction to Stochastic Simulation

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Page ID: 49305

Franz S. Hover & Michael S. Triantafyllou
Massachusetts Institute of Technology via MIT OpenCourseWare

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Whereas in optimization we seek a set of parameters \(\vec{x}\) to minimize a cost, or to maximize a reward function \(J(\vec{x})\), here we pose a related but different question. Given a system \(S\), it is desired to understand how variations in the defining parameters \(\vec{x}\) lead to variations in the system output. We will focus on the case where \(\vec{x}\) is a set of random variables, that can be considered unchanging - they are static. In the context of robotic systems, these unknown parameters could be masses, stiffness, or geometric attributes. How does the system behavior depend on variations in these physical parameters? Such a calculation is immensely useful because real systems have to be robust against modeling errors.

At the core of this question, the random parameters \(x_i\) in our discussion are described by distributions; for example, each could have a pdf \(p(x_i)\). If the variable is known to be normal or uniformly distributed, then of course it suffices to specify the mean and variance. However, in the general case more information may be needed.