# Safety Index Synthesis via Sum-of-Squares Programming

arXiv: https://arxiv.org/abs/2209.09134 · HTML: https://mlsanity.com/paper/2209.09134/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Sep 2022.

## Built on by (most cited first, 12 of 12)

- [State-wise Safe Reinforcement Learning: A Survey](https://mlsanity.com/paper/2302.03122/) (2023)
- [Exact Verification of ReLU Neural Control Barrier Functions](https://mlsanity.com/paper/2310.09360/) (2023)
- [GUARD: A Safe Reinforcement Learning Benchmark](https://mlsanity.com/paper/2305.13681/) (2023)
- [State-wise Constrained Policy Optimization](https://mlsanity.com/paper/2306.12594/) (2023)
- [Verifiable Safety Q-Filters via Hamilton-Jacobi Reachability and Multiplicative Q-Networks](https://mlsanity.com/paper/2506.15693/) (2025)
- [Proactive Human-Robot Co-Assembly: Leveraging Human Intention Prediction and Robust Safe Control](https://mlsanity.com/paper/2306.11862/) (2023)
- [Implicit Safe Set Algorithm for Provably Safe Reinforcement Learning](https://mlsanity.com/paper/2405.02754/) (2024)
- [Absolute Policy Optimization](https://mlsanity.com/paper/2310.13230/) (2023)
- [The Feasibility Theory of Constrained Reinforcement Learning: A Tutorial Study](https://mlsanity.com/paper/2404.10064/) (2024)
- [Demonstrating ViSafe: Vision-enabled Safety for High-speed Detect and Avoid](https://mlsanity.com/paper/2505.03694/) (2025)
- [Bridging Control with Neural Network Verifier alpha-beta-CROWN: A Tutorial](https://mlsanity.com/paper/2605.26577/) (2026)
- [Absolute State-wise Constrained Policy Optimization: High-Probability State-wise Constraints Satisfaction](https://mlsanity.com/paper/2410.01212/) (2024)

## Abstract (arXiv)

Control systems often need to satisfy strict safety requirements. Safety index provides a handy way to evaluate the safety level of the system and derive the resulting safe control policies. However, designing safety index functions under control limits is difficult and requires a great amount of expert knowledge. This paper proposes a framework for synthesizing the safety index for general control systems using sum-of-squares programming. Our approach is to show that ensuring the non-emptiness of safe control on the safe set boundary is equivalent to a local manifold positiveness problem. We then prove that this problem is equivalent to sum-of-squares programming via the Positivstellensatz of algebraic geometry. We validate the proposed method on robot arms with different degrees of freedom and ground vehicles. The results show that the synthesized safety index guarantees safety and our method is effective even in high-dimensional robot systems.