In this chapter, we have investigated an asymptotic schooling scheme for multiple underactuated underwater vehicles. For each vehicle, there are only two control inputs -surge force and yaw moment available for its three DOF motion in the horizontal plane. The main difficulty in the tracking of this kind of vehicle is how to properly handle the vehicle's sway dynamics. To deal with this problem, in this chapter, we have introduced a certain polar coordinates transformation, through which the vehicle's dynamics can be reduced to a two-inputs strict-feedback form. The vehicles schooling has been conducted by properly selected smooth potential function, which consists of three different parts: one is for the interaction between vehicles, another is for group navigation, and the third one is for obstacle avoidance. The proposed formation algorithm guarantees the vehicles asymptotic schooling and velocity and heading matching while keeping obstacle avoidance. Proposed schooling scheme has been derived under the condition of u(t) > umln > 0, which inversely can be guaranteed by proposed formation control laws being combined with some suitable initial conditions. Therefore, the proposed schooling method only can guarantee the local stability. Moreover, it is notable that the following issues should be considered in our future works.

• Finite cut-off (b in Definition 1) of potential function, which was applied in the previous works (Leonard and Fiorelli, 2001; Olfati-Saber, 2006; Do, 2007), also plays an important role in the vehicles schooling in this chapter. However, since b < +<», it is easy to verify that dfp(Z,a,b)/dZ = 0 if Z> b . For this reason, the proposed schooling scheme only guarantees certain local minimum. It is of interest to upgrade the present result to the one where the global minimum can be guaranteed in our future works.

• Another practical concern is for the robustness of proposed schooling scheme. In practice, there various uncertainty terms have to be faced, such as vehicle's modelling error, measurement noise, and disturbance, etc. All of these terms should be considered in our future practical applications.

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This first volume will guide you through the basics of Photoshop. Well start at the beginning and slowly be working our way through to the more advanced stuff but dont worry its all aimed at the total newbie.

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