Real robots are called embodied. And this embodyment makes a big difference to simulated robots. A simulated robot is still a program, living so to say inside the computer, also if hardware is in the loop that is calculating based on simulated input data. A real robot is receiving real signals with no ground truth readable in some memory cells of a computer. Information about a real state is only coming from better sensor, and quantum mechanics is telling us that also a better sensor has fundamental limits. Whilst in a computer program one can work with semantically true for logical program outputs, the semantics (i.e. meaning) in the embodied world needs an underlying truth to which the robot has no access to. Instead the robot has to build a model, the model is implemented as computer program, and there: The robot can decide on its own for a semantic truth. This decision has no direct effect on the semantic truth in the real world. How does such model inside the robot look like? In its most general form it is a set of equations describing a random dynamical system. Also the Designer has to work with models for the semantic truths. It seems to be logically necessary that the models in the (to be designed) robot and in the Design Space Exploration phase are assumed to be the same. Whilst the robot works with data, in the design phase we do neither have a robot nor data. Hence, we need a method to produce as much data as necessary inserting our model knowledge into an adequate sampling method. In this Book, we choose the Polynomial Chaos Expansion (PCE) as a data generation method. From an analytical point of view, we transform the equations describing the random dynamical system into a sequence of samples, but instead of performing the sampling, we only further investigate the syntactical shape of the transformation equations, and we are extracting from these equations the moments of the pdfs these equations are describing. We find, that we can successively formulate the random dynamical system as a cascade of increasing detailedness with each layer in this hierarchical system of decision-making driven by Gaussian noise. We further link to analytical tools to describe and further elaborate on the Gaussian noise driven decision-making scenario: Stochastic Optimal Control, Koopman Operator, Hidden Markov Model and its analytical pendant (the Integrated Nested Laplace Approximation).