[SP] Properties of Curve Subdivision in SE(2)

[datahaki]

Background

GLC and RRT* based motion planning for car-like robots output Dubins paths, which are the concatenation of straight lines and circle segments with constant curvature each. When the vehicle follows such a trajectory rigorously, the steering wheel angle makes a rotation at every transition point to adjust for the sudden change in curvature. Any passenger is likely to experience a jerky lateral acceleration.

Curve subdivision in SE(2) applied to a Dubins path produces a trajectory with continuous curvature. Numerical stability and convergence of the smoothing method is already proven.

There are several opportunities for future work:

Task description

• Motion planning accounts for the maximum curvature that the vehicle can achieve. Generally, the maximum curvature of the smoothed trajectory is slightly higher than that of the original Dubins path. The task is to quantify/bound the increase in curvature produced by curve subdivision in SE(2).
• Collision avoidance asserts that a Dubins path is sufficiently far away from obstacles. Generally, the refined trajectory deviates slightly from the original path. Here also, a quantification of the maximum displacement would be useful to derive.
• The outcome of curve subdivision in SE(2) depends on the underlying linear scheme. The choice of the underlying linear scheme determines, for instance, if the control points are interpolated or just approximated. Many of such linear schemes exists. The methods are up for experimentation and investigation in SE(2). One advanced example is the 6-point interpolatory scheme by Deslauriers/Dubuc.

Illustrations

Dubins path with discontinuous curvature

Modified trajectory (blue) with continuous curvature

References

2014 Dyn/Sharon Manifold-valued subdivision schemes based on geodesic inductive averaging
2018 Hakenberg Curve Subdivision in SE(2)
2018 video on Curve Subdivision in SE(2)