High Order Control Barrier Functions

Abstract

The control barrier functions that we introduced in Definition 2.3 can only be used to find safe controllers if the control explicitly shows up in Eq. (2.13). Formally, this means that the function capturing the safety requirement needs to have relative degree one with respect to the dynamics of the system. In this chapter, we relax this assumption, and define a more general notion of a CBF. To setup the problem, we start with High Order Barrier Functions (HOBFs) in Sect. 3.1. We then continue with High Order Control Barrier Functions (HOCBFs) in Sect. 3.2 which lead to sufficient conditions guaranteeing safety expressed using functions of arbitrary relative degree. In Sect. 3.4, we introduce integral HOCBFs (iHOCBFs) for systems with multiple control inputs further generalizing the use of CBFs. We also include two simulation examples, one that illustrates the use of HOCBFs in Sect. 3.3 and another for the use of iHOCBFs in Sect. 3.5.