  In this paper, we focus on non-conservative obstacle avoidance between robots
with control affine dynamics with strictly convex and polytopic shapes. The
core challenge for this obstacle avoidance problem is that the minimum distance
between strictly convex regions or polytopes is generally implicit and
non-smooth, such that distance constraints cannot be enforced directly in the
optimization problem. To handle this challenge, we employ non-smooth control
barrier functions to reformulate the avoidance problem in the dual space, with
the positivity of the minimum distance between robots equivalently expressed
using a quadratic program. Our approach is proven to guarantee system safety.
We theoretically analyze the smoothness properties of the minimum distance
quadratic program and its KKT conditions. We validate our approach by
demonstrating computationally-efficient obstacle avoidance for multi-agent
robotic systems with strictly convex and polytopic shapes. To our best
knowledge, this is the first time a real-time QP problem can be formulated for
general non-conservative avoidance between strictly convex shapes and
polytopes.
