General Uncertainty Concepts

There are generally two different ways in which uncertainty is introduced into the motion strategy problem:

• Uncertainty in Prediction: This form is uncertainty generally arises from our inability to predict a future situation. When unpredictability is present, it is generally inappropriate to assume that a precomputed path from an initial configuration to a goal configuration will solve the problem. If unpredictable changes occur, the robot might have to change its course of action during execution. In general, the robot will have to use a motion strategy that includes feedback, allowing its actions to depend on the current configuration or state. Typical problems that involve uncertainty in predictability include: indoor mobile robots, planetary exploration vehicles, robot manipulators, underwater robots, floating blimp robots, target-tracking robots, and an automated car in traffic.
• Uncertainty in Sensing: This form of uncertainty occurs if we are unable to determine the current situation. When this form is present, a motion strategy often cannot be described directly in terms of configuration or state. The robot must choose actions based on information that is gathered from sensing. Examples include mobile robots, robots with vision systems, and robots in highly-unstructured and partially-known environments.

With either of these forms of uncertainty, there are several possible things that can be uncertain, such as the configuration, the state, the model of the world, or even the model of the robot.

For a given motion motion strategy problem, it might be appropriate to model either one of these uncertainties, or both.

A game against nature A vast set of motion strategy problems that involve uncertainty can be formulated as a game that is played between the robot and ``nature.'' The robot and nature each has a set of actions to choose from. The nature player will generally choose actions that are unknown or unpredictable to the robot. Thus, a robot faced with uncertainties must make decisions that overcome an ``opponent's'' actions. The robot has a loss or cost function, L, which serves the same purpose as in the forward search method for nonholonomic planning.

To quickly identify the main concepts, consider the toy problem illustrated below. The robot would like to move from the initial position to the goal position, but along the shortest route lies a train crossing. The robot can choose either to go around the train crossing (over a bridge), or it can take the shortest route. There is a chance, however, that the robot would have to wait for a while if the train is coming.

The robot has two actions: go straight or take the longer route. Nature has two actions: it will either prevent a train from coming, or allow a train to come. Since each player has two possible actions, there are four possible outcomes. If the robot goes straight and there is no train, then L = 1, but if the train comes, then L=4. If the robot takes the longer route, then L=3, regardless of whether a train comes. This information can be encoded in a matrix,

$$\pmatrix{ 1 & 4 \cr 3 & 3 \cr },$$

in which the rows correspond to the robot's actions and the columns correspond to nature's actions.

There are two general ways to model nature: nondeterministic and probabilistic.

Nondeterministic modeling A nondeterministic model means that we have no idea what nature will choose. This usually leads to worst-case analysis when designing a motion strategy.

In the example, the robot should take the longer route, because in the worst-case, the train will come, implying L=4. By taking the longer route, L=3, which is always preferred.

Probabilistic modeling A probabilistic model means that nature's actions will be selected according to a probability distribution. This generally allows more information to be exploiting when designing a motion strategy, but in many applications it may be impossible to determine statistically-valid probabilities. The probabilistic model usually leads to average-case or expected-case analysis.

In the example, suppose that a train will arrive with probability $\frac{1}{4}$ (thus, there will not be a train with probability $\frac{3}{4}$). If the robot choose to go straight, the expected loss is $1 (\frac{3}{4}) + 4 (\frac{1}{4}) = 1.75$. If the robot takes the longer route, the expected loss is clearly 3. In this case, the robot should go straight. If the probability that a train would arrive is higher, such as $\frac{3}{4}$, then the expected-case analysis would lead the robot to select the longer route, as one would intuitively expect.