Fossen 6-DOF Hydrodynamics

• Fossen-based 6-DOF hydrodynamics are defined by a unified Newton–Euler framework that captures full rigid-body and hydrodynamic forces in a 6×6 matrix formulation.
• They integrate added-mass effects, Coriolis–centripetal coupling, and multi-term damping to accurately simulate underwater maneuvering and fluid interactions.
• The approach extends to coupling with spatial rigid-body solvers and internal actuators, with experimental validations achieving sub-degree attitude RMSE.

Fossen-based 6-DOF hydrodynamics refer to the class of maneuvering models for underwater vehicles that encapsulate six degrees of freedom (DOF) in a unified Newton–Euler framework, originally formulated by Thor I. Fossen. These models precisely capture rigid-body dynamics, hydrodynamic added-mass, Coriolis–centripetal coupling, multi-term damping, and nonlinear hydrostatics, making them essential for high-fidelity simulation, control, and experimental validation of underwater vehicles and their extensions such as manipulators and internal actuators. By structuring all dynamic terms as 6×6 spatial matrices (or higher when internal actuation is present), the Fossen approach underpins robust simulation toolchains and facilitates systematic experimental benchmarking (Kolano et al., 2022, Rambech et al., 29 Oct 2025).

1. Canonical 6-DOF Equations of Motion

The Fossen formulation for underwater vehicle motion is most succinctly written in the body-fixed frame as

$$M(\nu)\,\dot\nu + C(\nu)\,\nu + D(\nu)\,\nu + g(\eta) = \tau\,,$$

where:

• $\nu \in \mathbb{R}^6$ is the velocity vector $[u, v, w, p, q, r]^{\top}$ (surge, sway, heave, roll, pitch, yaw rates),
• $\eta \in \mathbb{R}^6$ denotes the pose (earth-fixed position and orientation),
• $M$ is the total inertia matrix,
• $C$ the Coriolis–centripetal matrix,
• $D$ the hydrodynamic damping,
• $g$ the restoring vector (gravity and buoyancy),
• $\tau$ the 6-DOF input wrench (forces and moments).

Each term decomposes further:

• $M = M_{RB} + M_A$, with $M_{RB}$ the rigid-body inertia (explicitly partitioned via mass, center of mass, and spatial inertia) and $M_A$ the hydrodynamic added-mass matrix.
• $C(\nu) = C_{RB}(\nu) + C_A(\nu)$, each constructed to maintain overall skew-symmetry.
• $D(\nu)\nu = D_L\nu + D_Q(\nu)|\nu|$, with $D_L$ linear and $D_Q$ quadratic drag, typically both diagonal.
• $g(\eta)$ accounts for translational and rotational restoring, with separate centers of mass ($r_G$) and buoyancy ($r_B$).

Explicit matrix forms: $$M_{RB} = \begin{bmatrix} m I_3 & -m S(r_G) \ m S(r_G) & I_O \end{bmatrix}$$

$$M_A = \operatorname{diag}\{X_{\dot u}, Y_{\dot v}, Z_{\dot w}, K_{\dot p}, M_{\dot q}, N_{\dot r}\}$$

The skew operator $S(\cdot)$ encodes cross products.

Coriolis–centripetal matrices exploit the structure

$$C_{RB}(\nu) = \begin{bmatrix} 0 & -mS(v) \ -mS(v) & -S(I_O\omega) \end{bmatrix}\,,\qquad C_A(\nu) = \begin{bmatrix} 0 & -S(M_A v) \ -S(M_A v) & -S(M_A \omega) \end{bmatrix}$$

with $v = [u, v, w]^{\top}$, $\omega=[p, q, r]^{\top}$.

The restoring force $g(\eta)$ is constructed via

$$g(\eta) =\begin{bmatrix} R(\eta)^\top (\rho V g - m g) e_3 \ S(r_B)R(\eta)^\top \rho V g e_3 - S(r_G)R(\eta)^\top m g e_3 \end{bmatrix}$$

where $R(\eta)$ is the rotation matrix, $\rho V g$ is the buoyant force. For details of these definitions and their experimental context, see (Kolano et al., 2022, Rambech et al., 29 Oct 2025).

2. Parameterization and Hydrodynamic Coefficient Assignment

Hydrodynamic coefficients—added-mass and drag terms—are critical to model fidelity and are typically assigned from curated empirical tables, CFD, or dedicated experiments. In the validation study of a BlueROV2 vehicle (Kolano et al., 2022), for instance:

• $M_A$ diagonals: $X_{\dot u}\simeq -0.8\,m$, $Y_{\dot v}\simeq -1.2\,m$, $Z_{\dot w}\simeq -1.4\,m$, $K_{\dot p}\simeq -0.02\,I_x$, etc.
• $D_L = \operatorname{diag}\{X_u, Y_v, \ldots, N_r\}$
• $$D_Q = \operatorname{diag}\{X_{|u|u}, Y_{|v|v}, \ldots\}$$

Manipulator links are often treated as pure rigid bodies with negligible hydrodynamics, unless the manipulation speeds or link cross-sections justify augmented models. For small, slow manipulation (as in UVMS studies), link-level added mass and drag are omitted as a modeling simplification (Kolano et al., 2022).

3. Computational Realization and Integration with Rigid Body Solvers

Fossen-based hydrodynamics can be integrated seamlessly with spatial rigid-body algorithms such as Featherstone's ABA (Articulated Body Algorithm). The experimental work in Julia’s RigidBodyDynamics.jl implements the following scheme (Kolano et al., 2022):

• Each body's spatial inertia is augmented by its corresponding added-mass,
• Gravity is set to zero within the rigid-body solver, since all restoring forces and buoyancy are handled as explicit external wrenches,
• At each time step, hydrodynamic effects (drag, buoyancy, gravity) are assembled into wrenches and superimposed.

This architecture allows open kinematic chains (floating base + manipulator) and leverages efficient inverse/forward dynamics. All hydrodynamic loads appear as spatial wrenches on the respective bodies, compatible with recursive rigid-body pipelines.

Exemplary code snippet (Kolano et al., 2022):

for body in mechanism.bodies I_rigid = body.rigid_inertia I_added = body.added_mass_inertia set_inertia!(body, I_rigid + I_added) end set_gravity!(mechanism, SpatialAcceleration(Zero,Zero))

External loads are assembled per time step, summed, and provided to the dynamics integrator.

4. Model Extensions: Internal Moving Masses and Additional DOF

Fossen's formalism is extensible to include actuators beyond classical external wrenches, such as internal moving mass actuators. This modifies the system by enlarging the state (with internal mass position and velocity), updating the inertia matrices, and augmenting Coriolis and restoring formulations (Rambech et al., 29 Oct 2025).

Key extensions:

• Mass–inertia matrix is expanded to incorporate contributions from internal moving masses, with additional skew and block structure,
• Coriolis–centripetal terms are systematically derived using Kirchhoff's energy-based approach, ensuring overall skew-symmetry and passivity,
• Restoring vector acquires additive terms from gravitational moments and forces exerted by the motion of the internal mass,
• Drag is not influenced by enclosed internal masses.

The combined equations, written in block form, allow for arbitrary placement and control of internal actuators, as demonstrated in explicit Newton–Euler and Lagrangian derivations (Rambech et al., 29 Oct 2025).

5. Experimental Validation, Accuracy, and Limitations

Rigorous experimental validation is critical to substantiate Fossen-based 6-DOF models. In a comprehensive study with a 10-DOF UVMS (BlueROV2 base and Reach Alpha arm), 50 randomized manipulator trajectories were executed in pooled water, and vehicle attitude (roll/pitch) was recorded via onboard IMU and motion capture (Kolano et al., 2022). Simulated responses were compared against this ground truth. Reported results:

• Roll RMSE $=1.27^\circ$, $\sigma=0.69^\circ$,
• Pitch RMSE $=1.73^\circ$, $\sigma=0.82^\circ$,
• After correcting a $\sim$1.6 s simulation lag, RMSE reduced to $0.86^\circ$ (roll), $1.03^\circ$ (pitch).

Model strengths:

• Accurately captures qualitative and steady-state system response,
• Enables fully-coupled vehicle-manipulator simulation,
• Integrates transparently with high-performance spatial rigid-body solvers,
• Open parameterization and reproducibility in Julia.

Model weaknesses:

• Notable simulation lag, attributed to missing actuator dynamics and possible unmodeled hydrodynamics,
• Simplified, isotropic added-mass may under-represent anisotropic effects,
• Neglects unsteady phenomena (vortex shedding, wave-induced loads) and manipulator link drag,
• Environmental disturbances treated at most as constant drift.

A plausible implication is that achieving sub-degree accuracy across all axes and time lags will require more detailed hydrodynamic parameterization, actuator modeling, and unsteady flow effects.

6. Potential Extensions and Research Directions

Advancing the state-of-the-art in Fossen-based hydrodynamics includes several promising directions:

• Explicit incorporation of the full $6\times 6$ added-mass on all links (panel-method hydrodynamics as in Stonefish),
• Modeling finite actuator dynamics (motor inertia, friction, torque limits),
• Extension to unsteady hydrodynamics employing memory kernels or Morison’s equation to capture wave effects,
• Inclusion of stochastic environmental disturbances (e.g., spatiotemporal current profiles, irregular wave spectra),
• Application of the fully coupled models to advanced control algorithms for both external- and internal-actuated underwater systems (Kolano et al., 2022, Rambech et al., 29 Oct 2025).

These directions are directly suggested by the reported strengths and identified discrepancies in experimental benchmarking, and by the ability of the Fossen approach to generalize via its systematic matrix framework. Adopting more granular geometric, CFD-based, or data-driven methods for hydrodynamic coefficients is also a critical trajectory for future work.

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References

• (Kolano et al., 2022) "The Coupling Effect: Experimental Validation of the Fusion of Fossen and Featherstone to Simulate UVMS Dynamics in Julia"
• (Rambech et al., 29 Oct 2025) "Combining Moving Mass Actuators and Manoeuvring Models for Underwater Vehicles: A Lagrangian Approach"