  Planning safe paths is a major building block in robot autonomy. It has been
an active field of research for several decades, with a plethora of planning
methods. Planners can be generally categorised as either trajectory optimisers
or sampling-based planners. The latter is the predominant planning paradigm for
occupancy maps. Trajectory optimisation entails major algorithmic changes to
tackle contextual information gaps caused by incomplete sensor coverage of the
map. However, the benefits are substantial, as trajectory optimisers can reason
on the trade-off between path safety and efficiency.
  In this work, we improve our previous work on stochastic functional gradient
planners. We introduce a novel expressive path representation based on kernel
approximation, that allows cost effective model updates based on stochastic
samples. The main drawback of the previous stochastic functional gradient
planner was the cubic cost, stemming from its non-parametric path
representation. Our novel approximate kernel based model, on the other hand,
has a fixed linear cost that depends solely on the number of features used to
represent the path. We show that the stochasticity of the samples is crucial
for the planner and present comparisons to other state-of-the-art planning
methods in both simulation and with real occupancy data. The experiments
demonstrate the advantages of the stochastic approximate kernel method for path
planning in occupancy maps.
