In this section, we will propose a formation control algorithm that can guarantee the stable schooling for multiple underactuated underwater vehicles whose models can be expressed as (1) or (5) with Assumption 1. It is easy to see that (5) is in a second-order nonlinear strict-feedback form, and we will solve the schooling problem using general backstepping method (Krstic et al., 1995).
Step 1. As aforementioned, the vehicles' schooling is conducted by the various potential functions introduced in the previous section. For this reason, in this step, we consider the following Lyapunov function candidate
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