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The best results of each planner on the static world of the Arena2 map are shown in Table 5.3. RTD solves all planning problems. LSS-LRTA, MOCART and variations of MOCART are not scalable in this benchmark. LSS-LRTA is more scalable than MOCART and its variations. MOCART-CAS performs significantly better than all other planners with respect to time per search. This shows the MOCART-CAS planner has the fastest action selection mechanism as compared to the other planners. The better performance of MOCART-CAS is due to the use of the corridors of the actions. To discuss the sub-optimality of the Monte-Carlo planners on Arena2, we selected a problem from Arena2 as an example. We call this problem P 1. The initial and the goal locations of this problem are shown on the Arena2 map in Figure 5.5.

Planner Depth timelimit nc Avg Ts Avg Sub Scale MOCART 15 200 500 0.0172 11.54 273/300 MOCART-RAS 15 200 500 0.005 12.34 271/300 MOCART-CAS 3 30 1500 0.0004 7.04 284/300 RTD 15 - - 0.083 3.63 300/300 LSS-LRTA 15 - - 0.002 2.96 296/300 Table 5.3: Time per search, Sub-optimality and Scale of five planners on Arena2 (Avg means Average).

There is a long wall between the initial and goal locations. The wall is not straight

Figure 5.5: The planning problem P 1 on Arena2 map.

and creates a large room of a rectangular shape as shown in Figure 5.5. The prob- lem is challenging for any local search planner based on admissible heuristic search because the admissible heuristic prefers the path towards the large room and there is no path to the goal location via this room. The room can be considered as a local

maximum (or minimum) for the local search planners.

The path found by MOCART for this problem is shown in Figure 5.6. MOCART keeps moving inside the large room for a long time. The planner visits every empty

Figure 5.6: The path followed by MOCART to solve P 1 on Arena2.

space in this room for a few times and then it learns to avoid moving inside the room. Learning to avoid moving towards a previously visited state is guaranteed by a limit on the number of times an action is executed at a state. The process of learning to escape the local maximum is very slow in MOCART because this process depends on the limit on the action execution at a state and every state in the big room has more than one applicable action such that execution of any one of these actions keeps the planning agent moving inside the room (local maximum).

The solution of MOCART-CAS for the same problem is shown in Figure 5.7. MOCART- CAS follows the same path (as that followed by MOCART) to solve P 1 but it learns quicker than MOCART to get out of local maximum. MOCART-CAS learns quicker

Figure 5.7: The path followed by MOCART-CAS to solve P 1 on Arena2. (to escape the local maximum) than MOCART because of the use of the corridor of actions in a rollout. In MOCART-CAS, if an action at a state has exceeded the limit of execution then it is excluded from the corridor and hence it is not sampled in the look-ahead search. But the look-ahead search in MOCART does not impose this restriction on an action. Therefore, MOCART-CAS has fewer actions which are executed for the given limit and it learns more quickly to avoid moving to the same state again.

The path found by LSS-LRTA to solve P 1 is shown in Figure 5.8. LSS-LRTA could not reach the goal location within pathlimit. By comparing the performance of LSS- LRTA with MOCART (Figure 5.6) and MOCART-CAS (Figure 5.7) for the same problem i.e. P 1, it is notable that LSS-LRTA also keeps traveling in the big room for a long time but it also moves to areas other than the room (for example towards the second wall of the Arena2 map). It escapes the local minimum (i.e. room) for a few times but again moves into the room. LSS-LRTA learns to avoid moving in

Figure 5.8: Path followed by LSS-LRTA for the planning problem P 1.

the room by updating the heuristic values of the states it sees during the look-ahead search. However, if the change in the heuristic values (of the locations inside the room) are inaccurate with respect to the heuristic values of the neighbouring loca- tions (outside the room), it can again move inside the room (even after moving out of the room). The inaccurate heuristic values create the heuristic depression (also defined and discussed in section 2.1.4 of Chapter 2) for LSS-LRTA. The area inside the room is a heuristically depressed region for LSS-LRTA. It can be deduced that the poor performance of LSS-LRTA in Arena2 is due to the presence of heuristically depressed areas in those planning problems which are unsolved by LSS-LRTA. RTD is efficient in Arena2 with respect to sub-optimality but it is still poor in terms of time per search. The RTD planner successfully solves all planning problems on this map under the given pathlimit. The problem P 1 is also used to analyse the performance of RTD with respect to sub-optimality on Arena2 more closely. The path followed by RTD to solve P 1 problem is shown in Figure 5.9. RTD explores

Figure 5.9: The solution of P 1 by RTD.

the boundaries of the wall to find the exit from the local minimum and eventually finds a way to move to the goal location. This success of RTD is due to the global search i.e. D*-Lite. The backward search guides the planning agent to move towards the corners of the walls. RTD does not behave like LSS-LRTA or MOCART (and its variations) because the other side of wall was visible to the global search algorithm. This example highlights the importance of the use of the global backward search in combination with a local forward search algorithm to solve path planning problems in a partially visible grid world. The Arena2 results show that RTD is not sensitive to the initial heuristic error and it performs well if the obstacle ratio is small.