PhD Progress Report VI: Extending Probabilistic Roadmaps for Unknown Obstacles

1.1 Review of MDP model formulation The major theme of my research since last autumn has been in the development of the MDP formulation of PRM route graphs with uncertain edges. In the last progress report (Kneebone 2007, section 2) details were given on how a PRM graph with uncertain edges (where the agent does not know a priori if they are traversable) could be represented as either an MDP or POMDP (Kaelbling et al. 1996) model. Each modelled the agent's belief about edge traversability as a probability distribution over possible combinations of occluded edges. The exact form of this distribution depended upon the model. As was noted in the previous report, the POMDP formulation has little practical value due to the time complexity of current methods of solving them. Attempting to solve (produce policies for) anything except trivial examples created POMDP models that are intractable on current hardware. The method itself does not scale well with graph size irrespective of hardware. The other main idea introduced was that of 'clustering' edges in the PRM graph to allow the probability distribution over specific graph edges to become dependent on each other. Briefly, if two edges are close the same obstacle in the environment then it is reasonable to assume there is some amount of dependence between them. Exploiting this allows to the agent to infer information about edges it hasn't directly received an observation of. If the agent receives a " blocked " observation for one of a pair of edges that it knows are perfectly dependent, then it can infer that the second edge is blocked. As was discussed, the hindrance to this approach is that the number of elements required to represent the probability distribution over dependent edges is an exponential function of the number of edges present (specifically 2 n). This is due to each additional uncertain edge doubling the number of possible worlds that could be the true state. If two or more edges are far away from each other, or near different obstacles in the graph, then it becomes unlikely they will be dependent on each other. By clustering these edges into different groups, their probabilities are calculated independently. Considering the probabilities over different clusters independently greatly reduces the number of variables required to record the agent's entire belief state-the set of probability distributions over all uncertain edges in the graph. For a graph of …