  Even if path planning can be solved using standard techniques from dynamic
programming and control, the problem can also be approached using probabilistic
inference. The algorithms that emerge using the latter framework bear some
appealing characteristics that qualify the probabilistic approach as a powerful
alternative to the more traditional control formulations. The idea of using
estimation on stochastic models to solve control problems is not new and the
inference approach considered here falls under the rubric of Active Inference
(AI) and Control as Inference (CAI). In this work, we look at the specific
recursions that arise from various cost functions that, although they may
appear similar in scope, bear noticeable differences, at least when applied to
typical path planning problems. We start by posing the path planning problem on
a probabilistic factor graph, and show how the various algorithms translate
into specific message composition rules. We then show how this unified
approach, presented both in probability space and in log space, provides a very
general framework that includes the Sum-product, the Max-product, Dynamic
programming and mixed Reward/Entropy criteria-based algorithms. The framework
also expands algorithmic design options for smoother or sharper policy
distributions, including generalized Sum/Max-product algorithm, a Smooth
Dynamic programming algorithm and modified versions of the Reward/Entropy
recursions. We provide a comprehensive table of recursions and a comparison
through simulations, first on a synthetic small grid with a single goal with
obstacles, and then on a grid extrapolated from a real-world scene with
multiple goals and a semantic map.
