Manatee
Hydrodynamic & Biomimetic
Analysis

Mechanatee at a Glance

The full visual output of the Mechanatee project — anatomical reconstruction, swimming kinematics, hydrodynamic flow, and design-space comparison — in one place. Mathematical derivations, force balances, and the engineering details for each of these figures follow in the sections below.

Anatomical Layover — Five-Panel Exploded View

Five progressive layers from outer envelope to skeleton, modeled after the classic medical-illustration reference figure. The envelope (panel 1) is regenerated at runtime from the current muscle and skeletal reconstruction. The nervous system is routed using comparative cetacean and sirenian neuroanatomy (Reep, Morgane, Marshall, Pabst); musculature follows the dugong-derived Domning (1977) map; the skeleton is the real manatee STL.

Generated by dimensional_analysis/manatee_anatomy_layover.py. Hybrid sirenian model — see the Anatomy Sources callout in the Propulsion section.

Swim cycle. Dorso-ventral undulation across one full stroke at design cruise (0.8 m/s, f ≈ 0.42 Hz). Subcarangiform gait derived from K&F07 kinematic regressions.

CFD flow. Velocity field, streamlines, and reverse Kármán vortex street at cruise speed. Panel-method + boundary layer + Theodorsen unsteady-lift prediction (cfd_hydrodynamic_prediction.py).

Mechanatee in the swimmer design space. Operating at St ≈ 0.14 — below the canonical 0.20–0.40 efficiency band. The low-Strouhal niche of slow, quiet, vegetated-water survey.

Textured beauty render. Dragon Skin silicone outer body with Sharklet-inspired antifouling microtexture. Visual indistinguishability target: juvenile manatee at 5 m in turbid water.

Eight-frame swim sequence. Body posture sampled across one full undulation cycle. Travelling-wave kinematics with quadratic amplitude envelope.

Orthographic multi-view. Side, top, front, and 3/4 perspectives. Visual indistinguishability target: juvenile manatee at 5 m in turbid water.

▾ The mathematical derivations for every figure above — geometry, kinematics, Reynolds & Strouhal, Lighthill EBT, Froude efficiency, CPG oscillator network, and cost of transport — follow in the sections below. ▾

Manatee Anatomy & Biomechanics

The Mechanatee's engineering is rooted in the detailed musculoskeletal anatomy of Trichechus manatus. Understanding how biological manatees generate thrust through muscle-skeleton interaction is the foundation for every actuator placement, spine segment, and kinematic parameter.

Musculoskeletal architecture of the manatee caudal region showing muscle insertion points, vertebral geometry, and force vectors.

From Biology to Engineering

Aquatic biomimetic robots emerge from the synthesis of ichthyology, marine biology, hydrodynamics, electronics, mechanics, controls, and computer science. The manatee offers a unique biomimetic platform: quiet, efficient locomotion in shallow coastal environments using low-frequency dorso-ventral undulation.

The manatee axial skeleton is segmented into four functional regions — cervical, thoracic, lumbar, and caudal. The caudal vertebrae (24–29 segments) are the primary propulsive elements, with progressive simplification and flattening toward the fluke. Key muscle groups include:

Longissimus (LS) and Multifidus/Semispinalis (MS) — extend from the lumbar region to the fluke root, providing the primary dorso-ventral bending force. The fluke itself contains no muscles — it is a passive hydrofoil of connective tissue that pitches with the local spine tangent angle, generating thrust via dynamic angle of attack.

Vibrissae — specialized tactile hairs distributed across the body enable hydrodynamic sensing of water currents and pressure gradients, analogous to the lateral line system in fish. This inspires the Mechanatee's distributed sensor network.

The Florida West Indian Manatee

Target organism: a 6.5 ft (2.0 m) juvenile Trichechus manatus latirostris. Kinematic parameters from Kojeszewski & Fish (2007) for adults (mean body length 3.34 m, mass 857.8 kg), geometrically scaled to the juvenile form factor using length ratio L_r = 2.0/3.34 = 0.60.

The Case for Manatee-Based Biomimetics

Aquatic organisms have been evolving through natural selection for millions of years, endowing them with morphological and structural adaptations for highly efficient locomotion. Propellers — the primary method of AUV propulsion — are inherently inefficient due to perpendicular vortex formation that directly increases power consumption.

MIT's RoboTuna (1995) was 16% slower than predicted, hydrodynamically unstable vertically, and under-predicted maneuverability by a factor of two. Rather than merely aiming for superficial resemblance, the Mechanatee emphasizes biological fidelity across four domains: bionic morphology, kinematics, hydrodynamics, and neural control.

The manatee's fusiform body, spatulate tail, and dorso-ventral oscillation plane produce a wake with significantly less turbidity and sediment disruption than propeller-driven vehicles — critical for wildlife observation and environmental monitoring in shallow coastal waters.

Subcarangiform Dorso-Ventral Low Acoustic Coastal/Riverine 10 Prototypes

Biological T. manatus alongside the Mechanatee digital twin geometry.

Biological Kinematic Parameters

Body Morphology & Cross-Section Distribution

25,200-point triangulated mesh (Mechanatee BIAUV STL) aligned to a canonical 2.5 m body axis with 60 cross-section stations computed via convex hull slicing.

Dorsal, lateral, and cross-sectional views. Principal body axis aligned, nose at origin, centered Y/Z.

Fusiform Body Form

The manatee body follows a fusiform planform — maximum girth near 34% body length (x ≈ 0.85 m), tapering anteriorly to a blunt rostrum and posteriorly to a dorso-ventrally flattened caudal peduncle. The raw spine extends 1.425 m, scaled to the canonical body using a uniform factor:

Geometric Scale Factor

λ = L / L_raw = 2.500 / 1.425 = 1.7544

Dynamic similarity maintained: Re at cruise (~1.9×10⁶) remains turbulent, consistent with adult range.

Dimension	Value
Max cross-section area	0.578 m²
Max width	0.912 m
Max height	0.814 m
Wetted surface area	2.200 m²
Fluke span	0.650 m
Fluke chord	0.280 m
Mesh facets	25,200

Cross-section area, width, and height across 60 axial stations.

Axial Area Distribution & Added Mass

The area distribution defines the added-mass profile critical for Lighthill's elongated-body theory. Peak area at mid-body drives the bulk of inertial coupling with the fluid. The rapid taper through the caudal peduncle minimizes recoil and concentrates momentum transfer at the fluke.

Added mass per unit length follows the elliptical cross-section model:

Added Mass (Elliptical)

m_a(x) = ρ · π · b(x)²

b(x) = local half-height at axial position x. Varies from ~0.25 m mid-body to ~0.003 m at the thin fluke trailing edge.

The caudal region (x > 1.46 m) encompasses 26 articulated segments with inter-segment spacing of 40 mm, each individually actuated by antagonistic muscle pairs in the dorso-ventral plane.

Principal body dimensions and regional boundaries (cervical, thoracic, lumbar, caudal).

Pectoral flipper planform. Used for pitch/yaw stabilization and low-speed maneuvering.

Swimming Kinematics

Manatee swimming is classified as subcarangiform: dorso-ventral undulation propagating as a traveling wave from peduncle to fluke tip, distinct from the lateral undulation of most fish.

Progressive Wave Model

The tail displacement at each point along the caudal spine follows a traveling sinusoidal wave. The amplitude grows quadratically from the tail base to the fluke tip:

Tail Displacement

y(x, t) = A(x) · sin(2πft − 2πx/λ)

Amplitude Envelope (Quadratic Growth)

A(s) = A_tip · (s / L_tail)²

A/BL = 0.22 peak-to-peak (K&F07); A_tip = 0.275 m single-side amplitude. L_tail ≈ 1.04 m (26 segments × 40 mm). Quadratic envelope matches biological amplitude growth.

Frequency–Speed Relationship

f(U) = 0.24 + 0.22U   [Hz]

Cycle period range: 1.7–3.3 s. Two reference speeds are used throughout this analysis: biological cruise = 0.5 m/s (mean adult manatee swimming speed, Kojeszewski & Fish 2007) → f = 0.35 Hz; and design cruise = 0.8 m/s (Mechanatee target operating point near peak η_p) → f = 0.416 Hz, T = 2.40 s. All Strouhal, drag, and thrust numbers below are reported at the design cruise unless noted.

Wave Speed

V_wave = 0.51 + 1.09U   [m/s]

Always exceeds body speed, maintaining thrust-generating phase lag.

Segmental Peak Acceleration

a(x, t) = −A(x) · ω² · sin(kx − ωt)

Drives inertial loads on the actuator system; ω = 2πf, k = 2π/λ.

Traveling wave propagation through one complete swim cycle.

Deformation envelope showing quadratic amplitude growth along caudal spine.

Inter-vertebral angles, peak deflection, and angular velocity across 26 joints at multiple speeds.

Inter-Vertebral Articulation

Each vertebral disk joint allows ±15° dorso-ventral and ±10° lateral articulation via ball-and-socket or flexible silicone joints. Angular displacement increases distally while segment mass decreases, creating the characteristic whip-like tail action.

Region	Max Angle	Max ω
Proximal (0–7)	1.5–1.9°	3.8–4.9 °/s
Mid-caudal (8–16)	2.3–5.9°	5.9–7.0 °/s
Distal (17–24)	2.9–7.8°	6.9–11.1 °/s

Joint Angle

θ_j(t) = arctan[(Δz_j+1 − Δz_j) / Δx_j]

Force Requirements Per Segment

For a single segment of mass m_seg ≈ 0.3 kg at f = 0.5 Hz, peak inertial force F_inertia = m_seg · A · ω² ≈ 0.05 N. However, hydrodynamic drag dominates at the fluke tip (v ≈ 0.69 m/s): F_drag = ½ρC_DA · v² ≈ 2.9 N. Net actuator force: 5–15 N at distal segments, 1–3 N proximal.

Hydrodynamic Force Analysis

Thrust, drag, and propulsive efficiency derived from Buckingham Pi analysis, elongated-body theory (Lighthill, 1971), and Strouhal optimization.

Dimensional Analysis — Key Dimensionless Groups

Buckingham Pi theorem identifies three groups governing the Mechanatee's hydrodynamic regime:

Reynolds Number

Re = ρUL / μ

ρ = 1025 kg/m³, μ = 1.08×10⁻³ Pa·s (seawater @ 25°C).
Re = 2.37×10⁵ (0.1 m/s) → 3.57×10⁶ (1.5 m/s). Turbulent regime confirmed.

Strouhal Number

St = fA_single / U

At cruise (0.8 m/s): St = 0.416 × 0.275 / 0.8 = 0.143. Below the 0.20–0.40 optimal band (Taylor, Nudds & Thomas, 2003), consistent with manatees being low-St swimmers. K&F07 report mean St = 0.28 for adults at ~0.5 m/s; efficiency remains high (η_p = 0.73–0.82).

Reduced Frequency

k = πfc / U

Quantifies unsteady effects on the fluke; c = 0.28 m (fluke chord).

Strouhal sweep showing propulsive efficiency across St range. Mechanatee operates at St ≈ 0.14 at cruise.

Annotated flow field with thrust, drag, and reactive force components.

Thrust, Drag & Navier-Stokes

The underwater vehicle dynamics are described by revised Newton-Euler equations incorporating variable buoyancy effects. The system inertia matrix M combines rigid-body inertia with hydrodynamic added-mass terms; the Coriolis matrix C(ν) accounts for translational-rotational coupling; and the damping matrix D(ν) captures linear and quadratic hydrodynamic damping.

Friction Drag (ITTC 1957)

F_D = ½ρU²S_wetC_D   where   C_D = 0.075 / (log₁₀Re − 2)²

C_D ≈ 0.0065 at cruise (Re = 1.9×10⁶, Hoerner form factor k = 1.59); S_wet = 2.2 m². F_D ≈ 4.7 N.

Lighthill Thrust (Elongated Body Theory)

T = ½m_a(L) · w(L,t)² − recoil terms

m_a(L) = added mass at trailing edge = ρπb² = 340 kg/m; w(L,t) = dorso-ventral velocity of fluke tip. Computed in dimensional_analysis/biauv_swimming_suite.py (Lighthill EBT with recoil correction) and cross-checked against the panel-method + Theodorsen unsteady-lift solver in hydrodynamics/cfd_hydrodynamic_prediction.py. At design cruise (0.8 m/s): net thrust T ≈ 24.1 N vs. drag F_D ≈ 4.7 N (full speed sweep in the COT table below). Kinematic Froude efficiency η_F = U/V_wave = 0.579; this is distinct from the biological propulsive efficiency η_p reported by K&F07 (see η_p column in the COT table).

Force Vector Analysis

Distributed reactive & resistive forces during tail undulation. Thrust results from the phase offset between dorso-ventral velocity and body curvature.

Per-segment force decomposition: inertial, pressure, and net thrust. Distal segments contribute disproportionately due to higher oscillation velocity.

Lift generated dynamically through active modulation of fluke pitch angle — not static control surfaces. SMA-driven deformation enables real-time pitch adjustments.

Joint reaction forces and moment arms along the caudal spine. Proximal joints bear 62% inertial, 38% hydrodynamic loading.

Total bending moment decomposed into inertial and hydrodynamic components across speed range.

Bending Moment Distribution

Each caudal segment must overcome the cumulative inertial acceleration of all downstream mass plus hydrodynamic reaction forces. Moments are computed from the tip inward using the added-mass distribution from the cross-section analysis:

Total Bending Moment at Joint j

M_total,j = ∑_k=j+1^N (m_k + m_a,k) · a_k · r_jk

Peak M_total ≈ 122.4 N·m at the proximal caudal joint (x = 1.456 m). Moment arms range 15–40 mm depending on segment.
Load path: ~30% reacted by active actuators (HASEL torque = 2 × 70 N × 40 mm = 5.6 N·m per segment), ~70% borne by passive spine stiffness (PA12 nylon / UHMWPE vertebral disks and silicone inter-disk elastomers). The proximal joint acts primarily as a structural anchor; active bending authority increases distally where moments are lower and actuator moment arms are more favorable.

Fluke Stiffness Optimization

Thrust scales as T ∝ A_tip². Optimal fluke stiffness balances flexibility and resistance — low stiffness increases amplitude but risks uncontrolled deformation; high stiffness improves force transmission but reduces amplitude. The design target is a fluke with graded stiffness: compliant at the trailing edge, stiffer at the attachment point.

Artificial Muscle Propulsion

The Mechanatee tail is driven by artificial muscles placed at biologically accurate attachment sites across 25 caudal segments. Each muscle bundle replicates the pennation angle, fiber direction, and force vector of its biological counterpart. The muscles are angled and attached to the facet joints on each end of the vertebrae for realistic thrust, efficiency, and controlled undulation.

Functional Muscle Groups — Sirenian Tail (Dugong-Derived Map)

The propulsive muscles below are organized by function. Each group attaches to specific vertebral processes and fires in coordinated sequence via the CPG controller. Artificial muscles are substituted at each biological attachment site, preserving the original fiber angles and force vectors. Muscle nomenclature follows Domning (1977) on the dugong — see the caveat above on the dugong-muscle / manatee-spine hybrid.

Biomimetic Design Principles — Aquatic Tail & Spine (Garman)

The following design principles, derived from the Aquatic Tail & Spine research paper (Garman), establish the biomechanical framework that governs how biological muscle architecture is translated into the Mechanatee actuator system. These principles bridge evolutionary biology, functional anatomy, and mechanical engineering.

Functional Muscle Groups (Garman Classification)

The propulsive muscles are classified into three functional color groups based on their stroke contribution, following the muscle fiber layout analysis from Domning (1977) as annotated in Garman’s research.

Source: Garman, H. (2024). “Florida Manatee Bio-Inspired Tail — Aquatic Spine.” Sirenia Systems Research. Muscle classification based on Domning, D.P. (1977), Smithsonian Contributions to Zoology, No. 226.

Sirenian Caudal Musculature — Dugong Dissection Reference

The muscle architecture of the Mechanatee is mapped directly from dissection studies of the closely related dugong (Dugong dugon). The following anatomical plates from Domning (1977) reveal the layered musculature of the posterior trunk and tail through progressive dissection — from superficial skin muscles down to the deepest vertebral attachments. Each layer informs actuator placement in the Mechanatee.

Figures 49 & 50 — Progressive lateral dissection of the dugong posterior trunk and tail. Fig. 49 (top): Superficial layer with cutaneus trunci (CuT) and outer rectus sheath removed, exposing iliocostalis thoracis (IlT), intercostales externi (IntE), longissimus dorsi (LnD), intertransversarii (Intr), obliquus abdominis externus (OAE), rectus abdominis (RA), coccygeus ventralis/dorsalis (CoVe/CoVu), ischiococcygeus (Isc), and sacrococcygeus ventralis lateralis (SVL). Fig. 50 (bottom): Second layer removed from ventral side. Deep intertransversarius bundles (Intr_d) and pelvic origin exposed. Obliquus abdominis internus (OAI) and transversus abdominis (TrA) visible. Sacrococcygeus ventralis lateralis triangular cross-section with aponeurosis (ap) marked.

Figures 51–54 — Deep musculature and pelvic attachments. Fig. 51 (top): Deep ventral muscles with longissimus dorsi (LnD) structure shown schematically. Sacrococcygeus ventralis lateralis (SVL) and medialis (SVM) exposed alongside flexor haemalis (FH) and chevron bones (ch₅–ch₁₂). Fig. 52 (middle): Pelvic and genital region. Deep intertransversarius bundles (Intr_d) originating on pelvis. Ischiococcygeus (Isc), rectus abdominis (RA), and coccygeus ventralis (CoVe) attachment geometry visible. Figs. 53–54 (bottom): Left pelvis bone showing muscle attachment sites in lateral (53) and medial (54) views — ischiococcygeus (Isc), rectus abdominis (RA), intertransversarius (Intr), transversus abdominis (TrA), coccygeus ventralis (CoVe), and retractor of the ischium (ReI).

Domning (1977) — Dugong Full-Body Profile. Figure 1 (top): Skin thickness along the body in mm — D = dorsal midline, L = lateral, V = ventral midline. Thickness varies from ~8 mm (mid-body lateral) to ~18 mm (dorsal head/neck). Figure 2 (middle): Lateral view of dermal muscles and superficial aponeuroses showing the full muscle architecture that informs the Mechanatee actuator map. Figure 3 (bottom): Dorsal view of superficial muscles on the right side; deeper layer on the left (splenius, longissimus capitis, shoulder muscles, and forelimb removed). This is the source anatomy for the dugong-derived muscle map used throughout the Mechanatee reconstruction.

Muscle Attachment to Vertebrae

The biological model has specific attachment sites on each vertebra. The muscles are angled and attached to the facet joints on each end of the vertebrae. By knowing where the muscles attach to the pelvic bone and throughout each vertebra, the design replicates the motion control of the biological tail.

Each caudal vertebra has attachment points for 6-8 muscle groups: dorsal extensors, ventral flexors, lateral sacrococcygeus pairs, and fine-control intertransversarii. The fiber angles match the biological pennation, with muscles angled from the facet joints to create both axial contraction and rotational torque at each segment.

The artificial muscles are substituted at each of these biological attachment sites, preserving the original fiber angles. This creates a force distribution that matches the biological system rather than using simplified dorsal/ventral pairs alone.

Actuator Candidates

Two solid-state technologies are under evaluation for these muscle sites:

Parameter	HASEL C-6519 (Artimus)	ThermoFlex MK-1 (Delta)
Technology	Electrostatic (dielectric pouch)	Nitinol SMA (phase transformation)
Force	70 N	TBD (bundled)
Stroke	5 mm	~3-8% contraction
Drive	6 kV	3.6 V / 15 A
Power per actuator	~4 W	~55 W
Frequency (submerged)	0-200 Hz	0.3-0.6 Hz
Form factor	Flat pouch / patch mount	McKibben sheath + integrated PCB
Biomimetic fidelity	Moderate (surface mount)	High (pennated bundles at bio sites)
Fluid system	None	None
TRL	5-6	4-5

Actuator force, stroke, and per-segment kinematic coverage across the caudal tail.

Degree-of-Freedom Configurations

Recommended: Config C (2-DOF, 52 actuators) is the minimum viable configuration. Manatees are subcarangiform swimmers — dorso-ventral oscillation is the dominant mode. Lateral steering is handled by differential amplitude, not separate lateral actuators.

Hazel Witch Actuators — Bundle Integration

Mechanatee's tail propulsion is built around bundles of Hazel Witch Actuators — nitinol-based artificial muscles strategically placed at biological attachment angles so they actuate the way the underlying organism does. Each bundle preserves the force load and fiber orientation of the muscle it replaces, giving the BIAUV a realistic load profile and undulating gait without conventional servos or rotary drives.

Design Concept

Bundles of Hazel Witch nitinol muscles are routed between each vertebra on the left and right of the spinous process. This placement is ideal for spine stabilization and lets the nitinol properties carry controlled, repeatable contraction across the full stroke cycle. Cooling is handled by a braided bundle attached to a hydraulic loop — the standard nitinol bundle can be grouped or elongated depending on the muscle being replicated.

Biological Performance Targets

Manatee reference specs (Kojeszewski, 2007): idling 0.56–0.83 m/s · cruising 0.84–1.94 m/s · sprinting 5–6.94 m/s · velocity range 0.08–0.38 BL/s · full stroke 130°.

Skeletal Allocation

The biological Florida manatee carries 16–19 thoracic vertebrae, 1–3 lumbar, 23–27 caudal, and 9–13 chevron bones. Mechanatee's tail subset incorporates 2 lumbar + 23 caudal vertebrae — matching the dorsoventral oscillation region where Hazel Witch bundles deliver the dominant propulsive stroke.

Scaled Muscle Bundle Dimensions

Hazel Witch bundles are sized to a 5-ft body-length juvenile manatee target. Original biological lengths are scaled to a working bundle envelope while preserving function and attachment topology.

Central Pattern Generator — Spinal Drive Architecture

Mechanatee's tail is not driven by an open-loop sine wave. The traveling-wave kinematics emerge from a network of coupled neural oscillators — a central pattern generator (CPG) — that mirrors the spinal motor circuits found in real swimming vertebrates. This is the same control architecture validated by Ijspeert et al. on the Salamandra robotica salamander robot, adapted here for sirenian dorso-ventral undulation.

Why a CPG, not a Lookup Table?

Vertebrate locomotion is generated bottom-up: a small descending drive signal from the brainstem (the “mesencephalic locomotor region”) sets a single scalar — intent — and a chain of segmental oscillators in the spinal cord produces the rhythmic muscle activation pattern, complete with the correct phase lag from one segment to the next. Speed, gait, and turning all emerge from modulating that single drive signal plus a few asymmetry terms.

For Mechanatee this matters for three concrete reasons:

• Robustness. A CPG re-entrains to perturbations (waves, contact, actuator faults) within one cycle — an open-loop sine wave does not.
• Smooth gait transitions. Cruise → sprint → station-hold → reverse are all expressed as drive-signal changes, not as separate motion files.
• Biological fidelity. The same equations describe lamprey, salamander, dogfish, and (by homology) sirenian spinal locomotor networks. This is the missing biomechanical layer between “anatomy” and “hydrodynamics.”

Coupled-Oscillator Equations

Segmental Phase Oscillator (Ijspeert 2008 form)

θ̇_i = 2πν + Σ_j w_ij r_j sin(θ_j − θ_i − φ_ij)     r̈_i = a (a/4 (R_i − r_i) − ṙ_i)

θ_i = phase of oscillator at vertebra i; r_i = amplitude; ν = global drive (sets frequency); R_i = target amplitude (sets envelope); φ_ij = inter-segmental phase bias (sets wavelength); w_ij = nearest-neighbor coupling weight.

Phase → Joint Angle (Antagonistic Pair)

θ_joint,i(t) = r_i(t) · cos(θ_i(t)) · A_env(s_i/L_tail)

A_env is the quadratic amplitude envelope defined in the kinematics section. The dorsal extensor (epaxial) fires for θ_i ∈ [−π/2, π/2]; the ventral flexor (hypaxial) fires for the antagonistic half-cycle. This produces the alternating epaxial/hypaxial bursts seen in sirenian EMG-equivalent dissection studies.

Drive-Signal → Behavior Mapping

A single descending drive vector d = (ν₀, Δν_L/R, ΔR_U/D) produces the full behavioral repertoire:

Behavior	Drive Signal	Resulting Kinematics
Station-hold	ν0 = 0	Limit-cycle collapses; tail goes flaccid + neutrally trimmed
Biological cruise (0.5 m/s)	ν0 = 0.35 Hz	Symmetric DV oscillation, λ ≈ 3.3 m
Design cruise (0.8 m/s)	ν0 = 0.42 Hz	Symmetric DV oscillation, peak ηp ≈ 0.79
Slow turn	ΔνL/R ≠ 0	Asymmetric segmental amplitude; ~3 m turning radius
Surface / dive	ΔRU/D ≠ 0	DC offset on dorsal vs. ventral antagonist amplitude — bias the mean tail angle
Reverse (escape)	φij → −φij	Wave propagates head-ward instead of tail-ward

Why this layer was missing — and why it matters

Most fish-inspired AUVs script their tail motion as prescribed sinusoids. That works in a tank and fails in surge. A spinal CPG is what lets a real animal swim through chop without re-planning every cycle — it's a closed-loop oscillator stabilized by sensory feedback, not a clock. Mechanatee adopts this architecture so the same controller spans calm-water survey, surge-loaded coastal transit, and contact-rich seagrass navigation without modal switching.

The phase-lagged joint-angle profile across 26 caudal segments — the kinematic output of the coupled-oscillator network. Inter-segment phase bias φ_ij determines wavelength; drive frequency ν determines stride.

References: Ijspeert, A.J. (2008). “Central pattern generators for locomotion control in animals and robots: a review.” Neural Networks 21(4):642–653. · Ijspeert et al. (2007). “From swimming to walking with a salamander robot driven by a spinal cord model.” Science 315(5817):1416–1420. · Crespi & Ijspeert (2008). “Online optimization of swimming and crawling in an amphibious snake robot.” IEEE Trans. Robotics 24(1):75–87.

Efficiency, Cost of Transport & Power Budget

How the Mechanatee's undulatory kinematics translate to real-world swimming performance, benchmarked against biological manatees and conventional propeller AUVs.

Froude Efficiency & COT

Froude Efficiency

η_F = U / V_wave = 0.8 / 1.382 ≈ 0.58

V_wave must exceed U for thrust generation — this is fundamental to undulatory propulsion. K&F07 report biological propulsive efficiency η_p = 0.67–0.82 across the speed range.

Cost of Transport

COT = P_total / (mgU)   [J/(m·kg)]

Speed	Thrust	Power	ηprop	COT
0.3 m/s	13.2 N	0.15 W	0.72	0.0010
0.5 m/s	17.3 N	0.64 W	0.75	0.0026
0.8 m/s	24.1 N	2.49 W	0.79	0.0065
1.0 m/s	29.5 N	4.25 W	0.81	0.0100
1.3 m/s	39.0 N	9.22 W	0.76	0.0173
1.5 m/s	45.9 N	13.8 W	0.73	0.0231

Froude efficiency and cost of transport across the full operating speed range.

Velocity and acceleration profiles — peak propulsive efficiency at 0.95 m/s matches biological data.

System Power Budget (HASEL Config C)

No pump, no solenoid valves, no Tesla coils required. The HASEL path trades higher HV driver power for radical mechanical simplification — 312 fluid connections eliminated.

Turning radius and depth authority vs. speed. Variable buoyancy mimics the manatee's use of lung volume for passive depth regulation.

Maneuverability & 6-DOF Dynamics

Turning is achieved through asymmetric undulation amplitude and pectoral flipper deflection. The 6-DOF dynamics include:

M (system inertia = rigid body + added mass) · C(ν) (Coriolis coupling) · D(ν) (linear + quadratic damping) · τ (propulsive + gravitational/buoyancy + environmental forces).

A dynamic ballast system replicates the manatee's use of lungs as variable buoyancy organs, enabling passive depth regulation without energy-intensive thrusters.

Swimming profile showing body posture across one complete undulation cycle at cruise speed.

Presentation-quality profile view with dimensional annotations and regional boundaries.

Dual-view profile: dorsal (top) and lateral (side) projections with cross-section overlay.

Hydrodynamic design space: thrust, drag, and net propulsive force as functions of speed and frequency.

Per-segment force vector decomposition showing inertial, hydrodynamic, and net thrust contributions.

Combined geometry and kinematic parameter summary — body morphology mapped to propulsive mechanics.

3D Model & Visualization

Interactive exploration of the full Mechanatee geometry — the bridge between biological morphology and engineered form.

Musculoskeletal System — Hybrid Sirenian Model

Complete musculoskeletal model: 50 shaped manatee vertebrae across 5 spinal regions with 34 dugong-derived muscle groups mapped from Domning (1977). See the Anatomy Sources caveat in the Propulsion section for the dugong-muscle / manatee-spine hybrid disclosure. Use preset buttons to isolate upstroke vs. downstroke muscle groups.

Hydrodynamic force vectors during dorso-ventral undulation — thrust generation at each stroke phase.

Lift and drag decomposition along the tail during one full oscillation period.

Inter-vertebral joint angles across the caudal spine — amplitude increases progressively toward the fluke.

Bending torque distribution — peak at the lumbar-caudal transition where muscle cross-section is greatest.

Mesh & Spine Registration

The digital twin is built from a 25,200-point triangulated exterior mesh with an embedded 60-point spine centerline extracted from biological CT data. Spine curvature defines the neutral axis for all bending moment calculations and serves as the reference frame for kinematic deformation.

Each of the 26 caudal vertebral disks is modeled as an elliptical cross-section (~3.5 cm × 2.5 cm) in PA12 nylon or UHMWPE, with ball-and-socket joints allowing ±15° dorso-ventral and ±10° lateral articulation. Actuators mount to 6 attachment points per disk (2 dorsal, 2 ventral, 2 lateral) via stainless steel clevis pins or embedded Kevlar tendons.

The mesh-to-spine binding uses cubic spline interpolation along the body axis, with the fluke region deforming via local tangent-angle rotation to replicate the passive pitching behavior of biological manatee connective tissue.

Hydrodynamic Infographic

CFD framework uses ANSYS Fluent or OpenFOAM for co-simulation with the CPG control system: CPG generates joint angles → set boundary motion in CFD → extract forces → feed to RL reward for parameter optimization.

Spine curvature analysis — segment lengths inform inter-disk spacing and actuator mounting geometry.

3D scatter of 60-point spine within the mesh hull. Virtual mass coefficients estimated from slender-body theory vary along this axis.

Conservation, Marine-Veterinary & Coastal Science Applications

Mechanatee is designed for the operating envelope where conventional propeller AUVs fail or cause unacceptable disturbance: shallow, vegetated, animal-rich water. Its low Strouhal number (St ≈ 0.14), absence of cavitation, and biomimetic visual profile open mission classes that propeller vehicles cannot safely or ethically perform.