Differentiable Boustrophedon Paths That Enable Optimization Via Gradient Descent

TL;DR

This work addresses the non-differentiability of traditional boustrophedon path optimization by introducing a differentiable representation for path plans within convex polygons, enabling gradient-based optimization. It defines a differentiable polygon containment via sigmoid-encoded half-spaces, approximates transect lengths through a length integral, and constructs a differentiable fitness function with gradients computable for transect angle and $x$-offset using auto-differentiation. Experiments show the differentiable scores closely match the discrete baseline with fidelity improving as temperature and sampling increase, but gradient descent alone struggles due to severe non-convexity, suggesting a hybrid grid search plus refinement approach. The findings highlight both the potential and limits of gradient-based methods for path planning, offering insights into the optimization landscape and enabling integration with differentiable systems in robotics, while outlining future work on closed-form integrals and concave-domain extensions.

Abstract

This paper introduces a differentiable representation for the optimization of boustrophedon path plans in convex polygons, explores an additional parameter of these path plans that can be optimized, discusses the properties of this representation that can be leveraged during the optimization process and shows that the previously published attempt at optimization of these path plans was too coarse to be practically useful. Experiments were conducted to show that this differentiable representation can reproduce scores from traditional discrete representations of boustrophedon path plans with high fidelity. Finally, optimization via gradient descent was attempted but found to fail because the search space is far more non-convex than was previously considered in the literature. The wide range of applications for boustrophedon path plans means that this work has the potential to improve path planning efficiency in numerous areas of robotics, including mapping and search tasks using uncrewed aerial systems, environmental sampling tasks using uncrewed marine vehicles, and agricultural tasks using ground vehicles, among numerous others applications.

Figures (5)

• Figure 1: Transects discretely generated within a parallelogram. Transects covering the polygon are shown in yellow. Some may consider the transects in (a) superior to (b) as the same space is covered using a smaller count of uniform-length transects.
• Figure 2: Transect lines passing through the parallelogram from Figure \ref{['discrete_transects']} using the differentiable formulation described. The z-axis corresponds to an indicator function stating if a particular point is contained within the parallelogram. Transect lengths are approximated based on the integral of these lines. Left shows Temperature=17, right shows Temperature=100.
• Figure 3: The score surface of the differentiable formulation based on the parallelogram and transects, shown in Figure \ref{['discrete_transects']}. Here, the angle varies, and the x-offset is fixed at 0.0 and Temperature at 10000.
• Figure 4: The 2D score surface is described by the differentiable formulation based on the parallelogram and transects, shown in Figure \ref{['discrete_transects']}. Here, the angle and the x-offset are varied. The slice where x-offset is equal to 0.0 is displayed in Figure \ref{['surface_slice']}.
• Figure 5: The 2D score surface as described by the differentiable formulation based on the parallelogram and transects, shown in Figure \ref{['discrete_transects']}. Here, the angle and Temperature are varied. While this figure only shows temperatures less than 200, the slice where the Temperature is equal to 10000 is displayed in Figure \ref{['surface_slice']}.