利用系统对称性加速安全控制函数的计算
Leveraging Equivariances and Symmetries in the Control Barrier Function Synthesis
- 通过动态对称性和约束对称性，推导出控制屏障函数的对称结构
- 仅需计算部分区域即可推断全空间的函数值，大幅降低计算量
- 适用于非对称约束场景，可复用已有函数构建新安全约束
控制屏障函数（CBF）的合成通常计算成本高或构造繁琐。本文研究系统动力学与约束结构中的等变性（一种广义对称性）如何缓解这一问题。尽管CBF本身通常不具备对称性，我们证明：在动力学具有等变性、约束具有对称性的情况下，通过可达性分析导出的CBF会继承这些对称性。这一发现使我们只需在子域计算CBF值，即可推断整个定义域的取值，带来显著计算节省。有趣的是，即使约束不对称，仍可通过已知部分CBF与等变性结合，构造适用于多种新约束的CBF。论文通过多个实例验证理论结果，并通过数值实验评估了引入等变性带来的计算优势。
原文摘要 · Abstract (English)
The synthesis of Control Barrier Functions (CBFs) often involves demanding computations or a meticulous construction. However, structural properties of the system dynamics and constraints have the potential to mitigate these challenges. In this paper, we explore how equivariances in the dynamics, loosely speaking a form of symmetry, can be leveraged in the CBF synthesis. Although CBFs are generally not inherently symmetric, we show how equivariances in the dynamics and symmetries in the constraints induce symmetries in CBFs derived through reachability analysis. This insight allows us to infer their CBF values across the entire domain from their values on a subset, leading to significant computational savings. Interestingly, equivariances can be even leveraged to the CBF synthesis for non-symmetric constraints. Specifically, we show how a partially known CBF can be leveraged together with equivariances to construct a CBF for various new constraints. Throughout the paper, we provide examples illustrating the theoretical findings. Furthermore, a numerical study investigates the computational gains from invoking equivariances into the CBF synthesis.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。