A geometrical approach to the motion planning problem for a submerged rigid body

The main focus of this article is the motion planning problem for a deeply submerged rigid body. The equations of motion are formulated and presented by use of the framework of differential geometry and these equations incorporate external dissipative and restoring forces. We consider a kinematic reduction of the affine connection control system for the rigid body submerged in an ideal fluid, and present an extension of this reduction to the forced affine connection control system for the rigid body submerged in a viscous fluid. The motion planning strategy is based on kinematic motions; the integral curves of rank one kinematic reductions. This method is of particular interest to autonomous underwater vehicles which cannot directly control all six degrees of freedom (such as torpedo-shaped autonomous underwater vehicles) or in case of actuator failure (i.e. under-actuated scenario). A practical example is included to illustrate our technique.