I am trying to implement 'belief space' planning for a robot that has a camera as its main sensor. Similar to SLAM, the robot has a map of 3D points, and it localizes by performing 2D-3D matching with the environment at every step. For the purpose of this question, I am assuming the map does not change.
As part of belief space planning, I want to plan paths for the robot that take it from start to goal, but in a way that its localization accuracy is always maximized. Hence, I would have to sample possible states of the robot, without actually moving there, and the observations the robot would make if it were at those states, which together (correct me if I am wrong) form the 'belief' of the robot, subsequently encoding its localization uncertainty at those points. And then my planner would try to connect the nodes which give me the least uncertainty (covariance).
As my localization uncertainty for this camera-based robot depends entirely on things like how many feature points are visible from a given locations, the heading angle of the robot etc.: I need an estimate of how 'bad' my localization at a certain sample would be, to determine if I should discard it. To get there, how do I define the measurement model for this, would it be the camera's measurement model or would it be something relating to the position of the robot? How do I 'guess' my measurements beforehand, and how do I compute the covariance of the robot through those guessed measurements?
EDIT: The main reference for me is the idea of Rapidly exploring Random Belief Trees, which is an extension of the method Belief Road Maps. Another relevant paper uses RRBTs for constrained planning. In this paper, states are sampled similar to conventional RRTs, represented as vertices as a graph, but when the vertices are to be connected, the algorithm propagates the belief from the current vertex to the new, (PROPAGATE function in section V of 1), and here is where I am stuck: I don't fully understand how I can propagate the belief along an edge without actually traversing it and obtaining new measurements, thereby new covariances from the localization. The RRBT paper says "the covariance prediction and cost expectation equations are implemented in the PROPAGATE function": but if only the prediction is used, how does it know, say, whether there are enough features at the future position that could enhance/degrade the localization accuracy?