*2.2. Task and Motion Planning under Uncertainty*

There are some situations in which a robot has incomplete information about its manipulation environment; therefore, it needs to plan under uncertainty. Task planning under uncertainty is a well-established field in Artificial Intelligence. Conditional-based task planners can provide conditional plan to cope with uncertain information when either the initial state is not completely known, or the result of actions are nondeterministic. There are various classes of planning in this field like conformant, contingent, or probabilistic planning.

Conformant planning looks for plans under given uncertainty concerning the start state and the effects of symbolic actions, assuming no sensing capabilities during the execution of the plan. The plan should be successful regardless of which is the start state. Contingent planning also considers uncertainty regarding the start state and the effects of actions. However, it can provide some sort of observation over a conditional plan in execution. Probabilistic planning does planning under probabilistic uncertainty regarding the start state and the effects of actions.

More details on some approaches following conditional-based task planning are commented next as we are interested in this type of planner due to its feature of providing observation over a conditional plan. Some conditional task planners are *Contingent-FF* [1], *POND* [20], and *PKS* [21]. They plan in the belief space and compute conditional plans in the offline mode, which are guided by the result of sensing actions. On the other hand, there are some conditional task planners like *K-Planner* [22], *SDR* [23], and *HCP* [24] solving conditional plans online. Although these planners can prune some branches by considering online sensing actions, satisfying the goal of task may not be possible and the planners may face with dead-end even if there is a solution.

The concept of contingent-based task and motion planning has also emerged. For instance, the *Planning with Knowledge and Sensing (PKS)* planner considers incomplete information and performs contingent planning [25] in two main scenarios, using *force sensing* and *visual sensing*. In a similar direction, offline-based hybrid conditional task and motion planning has been proposed [26], i.e., task planning is foremost performed, and then geometric evaluation is considered by incorporating low-level feasibility checks inside conditional planning (assuming that actuation actions are deterministic). On the contrary, the approach proposed here interweaves simultaneously efficient geometric reasoning inside the task planning process to provide geometrically feasible plans. The approach also copes with collaboration between the mobile robot and a human operator to perform a manipulation task.

#### **3. A Proposal for Contingent Task and Motion Planning**

This section first presents a brief overview of the original *Contingent-FF* task planning, and the modifications introduced in the present proposal to compute geometrically feasible manipulation conditional plans.

#### *3.1. Contingent-FF Overview*

The *Contingent-FF* task planner [1] handles uncertainty in the initial state and in the result of actions. The task planner has two main components which are heuristic computation and search space. For the heuristic computation, the planner uses a modified version of the *Relaxed Planning Graph (RPG)* used in the *Fast-Forward (FF)* planner [27]. The relaxed plan including a number of relaxed actions is computed from the *RPG*, and the heuristic value is the length of this relaxed plan. Also, promising actions (called helpful actions in *FF*) are extracted from the relaxed plan as a pruning technique in the search space, as discussed in *FF*. The *Contingent-FF* planner extends the *RPG* process, called *CRPG*, by adding unknown facts in an additional layer in the heuristic phase. Known facts are basically those which do not have uncertainty and unknown facts are the ones which could be the result of nondeterministic actions or uncertain in the initial state. It introduces reasoning about unknown facts that allows such facts to become known in the *RPG* process. Once *CRPG* is successfully built, the relaxed plan is extracted.

In *Contingent-FF*, belief states including known and unknown facts are considered. The search space starts from the initial belief state and applies an *And-Or* search. The search space progress is guided by the heuristic value and helpful actions. The result of planning provides conditional plans that may involve a variety of sensing actions whose outcome causes different plan branches.
