Soft-Minimum and Soft-Maximum Barrier Functions for Safety with Actuation Constraints

Pedram Rabiee1, Jesse B. Hoagg1
1University of Kentucky
Paper arXiv

Abstract

This paper presents two new control approaches for guaranteed safety (remaining in a safe set) subject to actuator constraints (the control is in a convex polytope). The control signals are computed using real-time optimization, including linear and quadratic programs subject to affine constraints, which are shown to be feasible. The first control method relies on a soft-minimum barrier function that is constructed using a finite-time-horizon prediction of the system trajectories under a known backup control. The main result shows that the control is continuous and satisfies the actuator constraints, and a subset of the safe set is forward invariant under the control. Next, we extend this method to allow from multiple backup controls. This second approach relies on a combined soft-maximum/soft-minimum barrier function, and it has properties similar to the first. We demonstrate these controls on numerical simulations of an inverted pendulum and a nonholonomic ground robot.

Algorithm for Safe Control with Single Backup Control

  • The soft-minimum barrier function is constructed using a finite-time-horizon prediction of system trajectories under a known backup control. The set \( \mathcal{S} = \{x \in \mathbb{R}^n : h(x) \geq 0 \} \) is a subset of the set from which the finite-time trajectory under the backup control remains in the safe set \( \mathcal{S}_{\rm s} \) and reaches the backup set \( \mathcal{S}_{\rm b} \) in time \( N T_{\rm s} \).
  • This soft-minimum barrier function serves as a constraint in a minimum intervention quadratic program (QP). The QP generates a safe control input and is guaranteed to be feasible when executed.
  • A mixture law uses the feasibility metric and barrier function value \( h \) as switching parameters. It combines the backup policy with the QP solution \( u_* \), ensuring a smooth transition to the backup policy when leaving \( \mathcal{S} \) or the feasible set \( \mathcal{B} \). This approach guarantees both safety and control continuity.
Algorithm Flowchart Single Backup
Algorithm Single Backup

Algorithm for Safe Control with Multiple Backup Control

Algorithm Flowchart Multiple Backup
Algorithm Multiple Backup

BibTeX

@misc{rabiee2024softminimum,
      title={Soft-Minimum and Soft-Maximum Barrier Functions for Safety with Actuation Constraints}, 
      author={Pedram Rabiee and Jesse B. Hoagg},
      year={2024},
      eprint={2305.10620},
      archivePrefix={arXiv},
      primaryClass={eess.SY}
}