On the Numerical Computation of Second Order Control Barrier Functions

In recent years, the development of Control Barrier Functions (CBF) has allowed safety guarantees to be placed on nonlinear control affine systems. While powerful as a mathematical tool, CBF implementations on systems with high relative degree constraints can become too computationally intensive for real-time control. Such deployments typically rely on the analysis of a system's symbolic equations of motion, leading to large, platform-specific control programs that do not generalize well. To address this, a more generalized framework is needed. This thesis provides a formulation for second-order CBFs for rigid open kinematic chains. An algorithm for numerically computing the safe control input of a CBF is then introduced based on this formulation. It is shown that this algorithm can be used on a broad category of systems, with specific examples shown for convoy platooning, drone obstacle avoidance, and robotic arms with large degrees of freedom. These examples show up to three-times performance improvements in computation time as well as 2-3 orders of magnitude in the reduction in program size.