Level set methods underpin modern safety techniques such as control barrier functions (CBFs), while also serving as implicit surface representations for geometric shapes via distance fields. Inspired by these two paradigms, we propose a unified framework where the implicit surface itself acts as a CBF. We leverage Gaussian process (GP) implicit surface (GPIS) to represent the safety boundaries, using safety samples which are derived from sensor measurements to condition the GP. The GP posterior mean defines the implicit safety surface (safety belief), while the posterior variance provides a robust safety margin. Although GPs have favorable properties such as uncertainty estimation and analytical tractability, they scale cubically with data. To alleviate this issue, we develop a sparse solution called sparse Gaussian CBFs. To the best of our knowledge, GPIS have not been explicitly used to synthesize CBFs. We validate the approach on collision avoidance tasks in two settings: a simulated 7-DOF manipulator operating around the Stanford bunny, and a quadrotor navigating in 3D around a physical chair. In both cases, Gaussian CBFs (with and without sparsity) enable safe interaction and collision-free execution of trajectories that would otherwise intersect the objects.

翻译：水平集方法构成了现代安全技术（如控制屏障函数CBFs）的基础，同时通过距离场作为几何形状的隐式曲面表示。受这两种范式的启发，我们提出了一个统一框架，其中隐式曲面本身充当CBF。我们利用高斯过程隐式曲面表示安全边界，通过源自传感器测量的安全样本对GP进行条件约束。GP后验均值定义了隐式安全曲面（安全置信度），而后验方差提供了鲁棒的安全裕度。尽管GP具有不确定性估计和解析可处理性等优良特性，但其计算复杂度随数据量呈立方增长。为缓解此问题，我们开发了一种称为稀疏高斯CBF的稀疏解决方案。据我们所知，GPIS尚未被明确用于合成CBF。我们在两种场景的避碰任务中验证了该方法：在斯坦福兔子周围操作的模拟7自由度机械臂，以及在实体椅子周围进行3D导航的四旋翼无人机。两种情况下，高斯CBF（含稀疏与不含稀疏版本）均实现了安全交互和无碰撞轨迹执行，而这些轨迹原本会与物体发生干涉。