Probabilistic Robots

1. Uncertainty

After calibration the robot should on average return to the desired location, but scatter due to uncontrollable factors. These are zero mean errors, occurring incrementally, the size of the distribution growing with distance travelled. This can be modelled as a *gaussian distribution.

In reality, we have uncertain terms , and with a zero-mean and a Gaussian distribution:

Simple sensing / actions procedures are locally effective but limited in complex problems in the real-world. We can build incrementally updating probabilistic models to estimate the position of our robot on the map.

Every action & sensor measurement is uncertain. When estimating a robot's state, we use these, so the state estimate is also uncertain. Usually, we take an uncertain measurement and then take action to receive new information. This then updates the estimate.

2. Probabilistic Inference

Prior knowledge combined with new measurements is generally modelled as a Bayesian Network - a series of weighted combinations of old & new information.

Sensor fusion is the general process of combining multiple uncertain measurements to produce a better estimate.

2.1 Bayesian Probabilistic Inference

Baysian is the measure of subjective belief. Probabilties describe our state of knowledge. The Bayes' Rule relates probabilities of discrete statements:

Here, is the prior probability of before observing . is the likelihood of observing given . is the marginal likelihood. is the posterior probability of after observing .

We use Bayes' rule to incrementally digest new information from sensors about a robot's state.

2.2 Probability Distributions

Discrete probabilistic inference generalizes to large numbers of possible states, so we can use a continuous probability density function: .

A guassian distribution often represents uncertainty in measurements well , where is the mean and is the standard deviation.

The prior (wide Gaussian) is updated with a likelihood (narrow Gaussian) to produce a posterior (narrower Gaussian).

2.3 Particles

Here, a probability distirbution is represented by a finite set of weighted samples of the state , where . These are simple and can represent multi-modal distributions.