Real-time Safety Index Adaptation for Parameter-varying Systems via Determinant Gradient Ascend

TL;DR

The paper addresses the challenge of maintaining safety guarantees for systems whose dynamics vary in real time by proposing Safety Index Adaptation (SIA) based on Determinant Gradient Ascend (DGA). By reinterpreting the SIS feasibility constraint as a determinant-based condition on a PSD matrix $Q(\theta,\bm{p},\rho)$ and leveraging Sylvester's criterion, the approach yields closed-form, real-time updates to safety-index parameters, warm-started from the previous feasible solution. A 2-DOF planar robot arm with parameter variations demonstrates that SIA preserves forward invariance (FI) and finite-time convergence (FTC) while significantly reducing computation time compared to full SIS. The methodology enables robust, real-time safety maintenance for dynamic, real-world systems with varying dynamics and control bounds, with practical implications for autonomous robotics and safety-critical control.

Abstract

Safety Index Synthesis (SIS) is critical for deriving safe control laws. Recent works propose to synthesize a safety index (SI) via nonlinear programming and derive a safe control law such that the system 1) achieves forward invariant (FI) with some safe set and 2) guarantees finite time convergence (FTC) to that safe set. However, real-world system dynamics can vary during run-time, making the control law infeasible and invalidating the initial SI. Since the full SIS nonlinear programming is computationally expensive, it is infeasible to re-synthesize the SI each time the dynamics are perturbed. To address that, this paper proposes an efficient approach to adapting the SI to varying system dynamics and maintaining the feasibility of the safe control law. The proposed method leverages determinant gradient ascend and derives a closed-form update to safety index parameters, enabling real-time adaptation performance. A numerical study validates the effectiveness of our approach.

Figures (4)

• Figure 1: Illustration of safety index adaptation. After the drone picks up a package whose weight is not known in advance, its dynamics change. The safe control law is adapted to the new dynamics and continues to guarantee safety, e.g., collision avoidance.
• Figure 2: 2-DOF Robot Arm.
• Figure 3: Arm end-effector tracking without and with safety index adaptation. Each goal is marked with the same color as the corresponding tracking trajectory. The robot is initialized with a feasible safety index with respect to the initial system dynamics and starts to track the first goal in blue. Every time a goal is reached, the system dynamics change. (a) Without adaptation, when tracking the second goal in orange, the arm runs into a state (marked by a cross) where it is approaching the wall quickly and no safe control can be found within the control limits. (b) With adaptation, the safety index is updated upon changes to the dynamics. That keeps the safe control law always feasible. As a result, the arm decelerates in advance when approaching the wall and safely tracks each of the goals.
• Figure 4: Feasibility rate of adapted safety indices, safety index parameters $\theta'=[k']$ and adaptation time under different system parameters $\rho'$. The first point ($\rho: c_1=c_2=1.0$) corresponds to the nominal system. $b_1$ and $b_2$ are always $0$.

Theorems & Definitions (4)

• Remark 2.1
• Remark 2.2
• Remark 3.1
• Remark