Obstacle Slalom

Problem Statement

The nominal controller flies dead straight at the goal and has never heard of the obstacles. Everything keeping the vehicle out of them is the barrier — which is the point: the safety argument does not depend on the controller being any good.

Model and Formulation

$h=\Vert p-c{\Vert }^2-(r+\delta {)}^2$ for each obstacle, with relative degree two under acceleration control, so the high-order condition applies. Unlike the pairwise case the obstacle does not manoeuvre, so the whole avoidance burden is the vehicle's and each constraint touches one block of the decision vector.

Tuning and Failure Modes

• Offset the obstacles by less than a radius from the direct path. At exactly one radius the straight line is tangent, and the unfiltered run looks almost safe by accident — the first version of this scene grazed at +0.001 m instead of penetrating.
• Clearance is a separate parameter from radius so the margin can be tuned without moving the obstacle.

Evidence

run	min clearance	goal error
filtered	+1.189 m	0.00 m
unfiltered	−1.997 m	0.00 m

Required clearance 1.200 m; the filtered run sits 1 % inside it, which is discretisation overshoot on a condition that holds in continuous time.

References

• Ames et al., CBF-based Quadratic Programs for Safety Critical Systems (2017)

Related Algorithms

• Position Exchange
• Potential Field 3D