Safety-Critical Control and Planning for Obstacle Avoidance between Polytopes with Control Barrier Functions

Abstract

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.

Lie Algebraic Cost Function Design for Control on Lie Groups

“…From the point of view of computational complexity, the worst-case for arbitrary obstacle configuration still results in SDPs of computationally intractable size. Thus, a convex representation of free space [15], obstacles [30,67,68], or risk-bounded contours of [69][70][71] should be explored.…”

Handling Disjunctions in Signal Temporal Logic Based Control Through Nonsmooth Barrier Functions

“…Therefore, we note that xpt k `tq " ϕpt; xpt k q, u ˚p¨; t k qq " x ˚pt; t k q and that (23a) subject to (23b-e) is equal to H T pxpt k qq as defined in (9). By using (16) from the proof of Theorem 3 we derive that H T pxpt k`1 qq " H T pϕp∆t; xpt k q, u ˚p¨; t k qqq (13), (16) ě min…”

Gaussian Control Barrier Functions: Non-Parametric Paradigm to Safety

“…is the partial derivative of (15) with respect to x. For the case when the query state is noisy, x q , we use the predictive mean and variance (21) as described in Section IV-D. Similarly, the corresponding partial derivatives are used to compute the Jacobians and Hessians from ( 22)- (23) to compute the Lie derivatives for the noisy query state.…”

“…Adaptive CBFs were formulated to handle time-varying control bounds and noise in the system dynamics [20]. CBFs have been combined with model predictive control methods for safe motion and path planning [21], [22]. For stochastic dynamical systems, stochastic CBFs are developed [23].…”