Estimating uncertain parameters

One of the current problems for model-based prediction with uncertain parameters is establishing the uncertainty on the parameters in the first place! There are a few ways, some better than others, to establish some sort of probabilistic bounds on parameters, and I’ll discuss a few here. Before I delve into techniques for estimating the parameters, I should honestly reintroduce a topic that I’ve discussed on this blog in the past. Although the notation is somewhat academic, I believe the concepts of aleatory and epistemic uncertainty do help clarify certain concepts in stochastic/uncertain system. Epistemic uncertainty relates to uncertainty introduced by lack of model fidelity. Imagine a system that operates as a sine wave, but we model it as just a linear curve. At low (absolute) values, the curves look nearly identical, but at higher magnitude values a divergence occurs. In the case of our model, we introduce higher uncertainty than actually exists in the system. Aleatory uncertainty, on the other hand, remains nearly impossible to eliminate. Statistical variations in the environment and setup contribute to the uncertainty in the system.

 Now that we have some concepts clarified, I’ll discuss more detailed problems with trying to quantify these said uncertain parameters. If we have an experimental setup in a controlled environment, we can easily run many experiments and statistically quantify the range of values of a parameter. However, if we’re operating a robot in a complex environment, the parameters that we encounter might vary according to time or position. Experimentally determined values would ignore the change in model operation. To update the estimate of the parameter, we can use a real-time state/parameter observer. The Wikipedia article on observers is actually quite excellent, and I recommend it to anyone interested in learning more! Essentially, we can update our estimate for the parameter in real time as the robot traverses terrain. However, this only reveals the instantaneous value of the parameter. For prediction we want to use historical data to more accurately assess a future operational status.

The quick example that I was working on today is a simple mass-spring-damper system with uncertain damping. An extended Luemberger observer estimates the current value of the damping parameter given some uncertainty. Using this estimator, we’re able to estimate the values of the damping coefficient (even if bimodally distributed) as shown in the figure above.