Path Following for Underactuated Marine Vessels

Velocity and orientation control of underwater snake robots using absolute velocity feedback

“…Again, the function arguments in (14) and (15) have been omitted for compactness. The control law (14) yields the following dynamics of the error z 2 :…”

“…Proposition 2: Suppose that the robot described by (1) moves according to (3) on the manifold Γ 3 . The adaptive control law (14), (18) asymptotically stabilizes the set…”

“…Note that the last term on the right hand side of (22) cannot be compensated by the feedback linearising controller (23). This is due to the design of u λ in (14), which contains a term depending onφ 0 . Remark 6: Because of prioritizing the velocity control higher than the orientation control, the choice v ref > V c,max , and Ass.…”

“…However, the following property holds for the closed-loop system. Proposition 4: Suppose that the body shape, forward velocity, and orientation of an underwater snake robot described by (1) are controlled according to (3), (14), and (23). Then, the sideways velocity, v n , is uniformly bounded.…”

“…Other than methods such as central pattern generators, using VHCs to encode the robotic gait has the advantage that it makes the control design amenable to hierarchical synthesis, where the gait is enforced at the lowest level and velocity and orientation control are done at a higher level. To this end, we will employ the hierarchical control design methodology from [13], that was used for control design for ships in [14] and terrestrial snake robots in [15]. In order to make the feedback independent of the unknown relative velocities that enter the dynamic equations via the fluid drag forces, we will make use of adaptive backstepping control [16].…”