Chapter 9 essentially took a break from planning by indicating how to make a single decision in the presence of uncertainty. In this chapter, we return to planning by formulating a sequence of decision problems. This is achieved by extending the discrete planning concepts from Chapter 2 to incorporate the effects of multiple decision makers. The most important new decision maker is nature, which causes unpredictable outcomes when actions are applied during the execution of a plan. State spaces and state transition equations reappear in this chapter; however, in contrast to Chapter 2, additional decision makers interfere with the state transitions. As a result of this effect, a plan needs to incorporate state feedback, which enables it to choose an action based on the current state. When the plan is determined, it is not known what future states will arise. Therefore, feedback is required, as opposed to specifying a plan as a sequence of actions, which sufficed in Chapter 2. This was only possible because actions were predictable.
Keep in mind throughout this chapter that the current state is always known. The only uncertainty that exists is with respect to predicting future states. Chapters 11 and 12 will address the important and challenging case in which the current state is not known. This requires defining sensing models that attempt to measure the state. The main result is that planning occurs in an information space, as opposed to the state space. Most of the ideas of this chapter extend into information spaces when uncertainties in prediction and in the current state exist together.
The problems considered in this chapter have a wide range of applicability. Most of the ideas were developed in the context of stochastic control theory [93,564,567]. The concepts can be useful for modeling problems in mobile robotics because future states are usually unpredictable and can sometimes be modeled probabilistically  or using worst-case analysis . Many other applications exist throughout engineering, operations research, and economics. Examples include process scheduling, gambling strategies, and investment planning.
As usual, the focus here is mainly on arriving in a goal state. Both nondeterministic and probabilistic forms of uncertainty will be considered. In the nondeterministic case, the task is to find plans that are guaranteed to work in spite of nature. In some cases, a plan can be computed that has optimal worst-case performance while achieving the goal. In the probabilistic case, the task is to find a plan that yields optimal expected-case performance. Even though the outcome is not predictable in a single-plan execution, the idea is to reduce the average cost, if the plan is executed numerous times on the same problem.