Underwater and Marine Vehicle Dynamics

Underwater, a robot floats or sinks depending on whether it weighs more than the water it pushes aside. Water drag grows with the square of speed, and the robot must push water along with it as it accelerates, so it feels heavier than it is. Currents push it off course the whole time.

Ballast (weight minus buoyancy) Forward thrust Vertical thruster Water current Depth hold controller

Forces under water

Along each axis, a vehicle of mass $m$ with added mass $m_a$ moving at velocity $\mathbf{v}$ relative to water flowing at $\mathbf{v}_c$ feels

$$(m + m_a)\,\dot{\mathbf{v}} = \mathbf{F}_{\text{thrust}} + (B - W)\,\hat{\mathbf{z}} - \tfrac{1}{2}\rho\,C_d A\,\lvert\mathbf{v}_r\rvert\,\mathbf{v}_r, \qquad \mathbf{v}_r = \mathbf{v} - \mathbf{v}_c$$

$B$ is buoyancy, the weight of the displaced water, and $W$ is the robot's weight. Added mass $m_a$ is the water that must be accelerated along with the robot, often close to the robot's own mass.

What to notice

With ballast near zero, the robot is neutrally buoyant and hangs at any depth, which is what operators aim for. A little extra weight and it slowly sinks, so it needs a vertical thruster or a depth controller. Drag grows with the square of speed, so doubling thrust gives only about 40 percent more speed. A current pushes the robot backward, and a weak robot pointed into a strong current can end up drifting backward. There is also no GPS under water, so the robot cannot easily tell that it is drifting, which is why underwater robots use acoustic navigation.