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Swim-and-Breach at Palm Scale: A Rudder-Steered Two-Propeller Underwater Robot Platform with Differential-Thrust Pitch Control
Daehyun Choi, Ian Bergerson, Hengjia Zhu, Tianjun Lan, Saad Bhamla
TL;DR
Underwater barriers and dry gaps can stop robots in streams and other settings, motivating fish-inspired swim-and-breach movement. This palm-scale platform combines propeller-driven swimming and pitch control with rudder-stabilized exits, and demonstrates a cruise-leap-cruise sequence.
Problem
Method
Results
13.9 BL/s swimming speed and 209 deg/s turning accompanied a tank cruise-leap-cruise sequence reaching 1.6 BL high and 3.7 BL long.
Source: §V · pp. 6–7Takeaways & Limitations
The platform shows the possibility of underwater robots crossing barriers and dry gaps for sensing, inspection, and monitoring.
Source: §V · p. 7
Abstract
from arXiv · showhide
We present a palm-scale (65 mm, 34 g) swim-and-breach robot platform. Two vertically stacked propellers provide both propulsion and differential-thrust pitch control under a proportional-integral-derivative (PID) loop, and a tail rudder adds yaw control. The hull, evaluated by flow simulation, reduces the drag five-fold relative to an equivalent cuboid, and the propellers are optimized using B-series modeling validated by dynamometer measurements. The current robot swims at 13.9 body lengths per second and turns at 209 deg per second, corresponding to the upper limits reported for underwater robots. In free swimming, the pitch loop turns the body to any commanded nose-up pitch angle, and, with the rudder stabilizing the exit, the current robot leaps 1.6 body lengths high and 3.7 long in a seamless cruise-leap-cruise sequence. The platform can be used to build small-scale robots that cross barriers and dry gaps between pools for inspection in streams, flooded structures, and industrial systems.
I. INTRODUCTION
Underwater robots’ agility has largely been reported in horizontal-plane turning, while a water exit requires burst thrust and precise vertical-plane pitch and yaw control. This paper introduces a fish-inspired platform that swims and leaps from the water.
I. INTRODUCTION
- Fish leave the water by pitching nose-up, accelerating toward the surface with a thrust burst, and holding pitch through the exit.The paper notes that fish leap for purposes including predator evasion, migration, parasite shedding, prey capture, and escape from disturbance.Source: §I · p. 1
- The platform uses vertically stacked propellers for propulsion and differential-thrust pitch control, with a rudder correcting yaw deviations.The propellers are chosen over a flapping tail because they are reported to deliver at least an order of magnitude greater static thrust per unit weight.Source: §I · p. 1
- 13.9 BL/s swimming speed and a 2.8 BL vertical leap place the platform among the fastest reported robotic swimmers and the highest-leaping fish-shaped or jumping robots.Its swimming and turning performance are also reported among the fastest systems compared, while its pitching rate is comparable to sharks.Source: §I · p. 1
II. SYSTEM DESIGN OF THE ROBOT PLATFORM
The platform combines a compact, passively roll-stable hull with a streamlined shape designed to reduce drag. Simulations show substantially lower drag than for an equivalent cuboid.
1) Hydrodynamics of the hull and rudder:
- The robot is 65 mm long and 15 mm wide, with a total mass of 34 g and neutral buoyancy.The design figure depicts the sealed hull, stacked motors, custom electronics, geared rudder, body axes, and assembled prototype.Source: 1) Hydrodynamics of the hull and rudder · p. 2, Figure 3 · p. 2
- The hull uses a truncated NACA four-digit airfoil form following the Kammback principle to keep drag and body size small.The front-view section places the center of buoyancy 1.7 mm above and 0.7 mm forward of the center of mass, supporting passive roll stability.Source: 1) Hydrodynamics of the hull and rudder · p. 2
- The drag comparison uses URANS computational-fluid-dynamics simulations at a free-stream speed of 0.5 m/s against an equivalent cuboid with matching projected areas.The current design sheds a narrow vortex train, whereas the cuboid sheds larger vortices from its leading edges onward.Source: 1) Hydrodynamics of the hull and rudder · p. 2
- At zero angle of attack, the hull’s drag coefficient is ≈0.18 versus ≈0.9 for the equivalent cuboid, a roughly five-fold reduction.Across −30 to 30 deg, the hull’s Cd stays at ≈0.2–0.35, while the cuboid’s stays at ≈0.9–1.3.Source: 1) Hydrodynamics of the hull and rudder · p. 2
2) Vertically stacked thruster pair and tail rudder:
The vertically stacked propellers provide propulsion and pitch torque, while a tail rudder supplies yaw authority with little pitch or roll coupling.
2) Vertically stacked thruster pair and tail rudder:
- A thrust difference between the vertically offset motors creates pitching torque, while their thrust sum sets net propulsive force.Mirrored propellers contra-rotate so reaction torque about the roll axis cancels when both motors run at the same speed.Source: 2) Vertically stacked thruster pair… · p. 2
- The hydrodynamic evaluation maps drag against angle of attack and yaw, pitch, and roll moment coefficients against rudder angle.It also depicts wake structures, pressure contours, and flow with the rudder neutral or deflected.Source: Figure 4 · p. 3
- Rudder deflection creates yaw moment that grows with angle over ±60 deg, while pitch and roll moment coefficients remain negligible.The rudder steers in the horizontal plane without disturbing the pitch PID control; gyro-integrated yaw error drives its motor.Source: 2) Vertically stacked thruster pair… · p. 3
3) Propeller optimization:
The propeller design uses a Wageningen B-series model and motor–propeller torque matching to select among printed variants. Dynamometer tests identify a three-blade 20 mm propeller as the best tested option.
3) Propeller optimization:
- The B-series model leaves blade count and diameter as design variables after setting P/D = 0.6 and AE/A0 = 0.40 at their lower bounds.At zero advance ratio, decreasing pitch ratio increases KT/KQ, while smaller blade area reduces torque and mass.Source: 3) Propeller optimization · p. 3
- For each diameter, the motor–propeller torque intersection sets rotation rate, which is used to predict bollard thrust.The figure compares torque-matched predictions with dynamometer measurements for two- and three-blade variants.Source: 3) Propeller optimization · pp. 3–4, Figure 5 · p. 3
- 78.9 gf at 2.95 A is the peak sustained thrust of the selected three-blade 20 mm propeller.The two-blade 20 mm variant produces 73.7 gf; with both thrusters, the static thrust-to-weight ratio is approximately 4.7.Source: Figure 5 · p. 3, 3) Propeller optimization · p. 4
- At 15 mm, measured thrust falls below model predictions because printed blade thickness exceeds the B-series specification at the printer’s resolution limit.At 30 mm, thrust drops as the larger propeller moves the operating point toward motor stall.Source: 3) Propeller optimization · p. 4
B. Electronics and Firmware
A compact custom board integrates sensing, motor control, and power regulation. BLE supports setup and telemetry, while runs execute onboard and are logged for later transfer.
B. Electronics and Firmware
- The 32 × 25 mm, approximately 5 g board integrates an ESP32-C3, MPU-6050 IMU, MOSFET thruster switches, and an H-bridge for the rudder gearmotor.A buck converter steps tether power down to 5 V, and the MCU module regulates 3.3 V for logic and the IMU.Source: B. Electronics and Firmware · p. 4
- BLE connects the host PC for parameterization and approximately 70 Hz telemetry of IMU, motor PWM, sequence, pitch, and controller data.The telemetry is 14-field; the onboard log records raw IMU, motor and rudder PWM, angles, and targets at about 25 Hz.Source: B. Electronics and Firmware · p. 4
- After a run starts, the sequence executes without BLE, and its onboard RAM log is streamed to the host afterward, preventing dropped links during a run from losing data.The log holds up to about 120 s.Source: B. Electronics and Firmware · p. 4
C. Closed-Loop Pitch Control
The robot estimates pitch onboard and uses a PID controller to adjust the differential thrust of its two motors. In tank tests, it tracks pitch commands and recovers from imposed taps, with high-frequency tracking limited by actuator saturation and phase lag.
C. Closed-Loop Pitch Control
- A time-based complementary filter estimates pitch by fusing gyro rate with the accelerometer gravity reference.The accelerometer term is rejected when measured specific force differs from 1 g by more than 0.15 g; gyro bias adapts only while idle.Source: C. Closed-Loop Pitch Control · p. 4
- The PID acts on the error between estimated and commanded pitch, using saturated trim to redistribute top- and bottom-motor duties while base duty sets net thrust.Tank-test gains were Kp = 10, Ki = 0, and Kd = 0.5, held fixed throughout.Source: C. Closed-Loop Pitch Control · p. 4
- For 10–30 deg commands at 0.2–0.33 Hz, tracking RMSE stays within 11% of commanded amplitude; it grows to 30% at 1 Hz, where actuator saturation begins, due to phase lag rather than amplitude error.Source: C. Closed-Loop Pitch Control · p. 5
- With PID control at 0 deg, tap-induced swings stay within ±3 deg and recover in about 1 s, versus ±12 deg and slow ring-down without control.During sinusoidal tracking, taps produce 6–8 deg instantaneous error that is absorbed within about 0.3 s, leaving overall RMSE indistinguishable from undisturbed runs.Source: C. Closed-Loop Pitch Control · p. 5, Figure 7 · p. 5
A. Sinusoidal Pitch Tracking
In free swimming, the pitch loop tracks sinusoidal commands and turns the robot from level to steep nose-up angles while retaining forward speed. These results demonstrate pitch control across tracking and pitch-up maneuvers.
A. Sinusoidal Pitch Tracking
- Free-swimming sinusoidal pitch tracking has RMSE from 1.47 ± 0.17 deg at 10 deg amplitude and 0.5 Hz to 4.15 ± 1.29 deg at 10 deg amplitude and 1 Hz.Tests also included 20 deg commands at 0.5 Hz and 10 deg commands at 0.67 Hz.Source: A. Sinusoidal Pitch Tracking · p. 5, Figure 8 · p. 5
- Forward speed peaks at 4–6 BL/s at 24% PWM duty during sinusoidal steering, which uses the thruster pair without dedicated dive planes or a buoyancy controller.These runs used the earlier rectangular hull; the current hull is expected to alter drag and added mass, but not the loop.Source: A. Sinusoidal Pitch Tracking · p. 5
B. Pitch-Up
- The robot reaches commanded nose-up angles of 0, 30, 60, or 90 deg within 0.75 s while retaining forward speed.The reported range spans level to vertical, covering the range needed for a water exit.Source: B. Pitch-Up · p. 5
IV. WATER EXIT PERFORMANCE
Rudder control stabilizes the robot’s water exit against yaw while pitch control holds the commanded launch angle. The exit test quantifies the yaw reduction and explains how the rudder counters yaw generated by differential-thrust pitch corrections.
A. Exit Sequence
- The pitch-up trials command target angles of 0, 30, 60, and 90 deg and track pitch and forward speed through the maneuver.The figure shows the straight approach before the pitch-up step at t = 0.Source: Figure 9 · p. 6
- About 3 deg of exit yaw with the yaw loop engaged, versus about 22 deg with it disengaged, as the robot exits at 0.78 m/s and rises 1.3 BL.With the loop engaged, the body breaks through at the 65 deg launch angle.Source: A. Exit Sequence · pp. 5–6
- The rudder opposes yaw from unequal propeller duties, keeping the robot in the launch plane despite exit speeds of O(10) BL/s.The pitch PID’s continual thrust corrections leave the contra-rotating propellers’ reaction torques unbalanced; the rudder supplies an opposing yaw moment with negligible pitch coupling.Source: A. Exit Sequence · p. 6
B. Complete Leap
The robot completes an on-device cruise-to-leap-to-cruise sequence, clears the tank obstacle in all three trials, and also exits the water in a turbulent natural stream with rudder control.
B. Complete Leap
- In all three tank trials, the robot cleared the obstacle, rising 1.6 BL above the surface and landing 3.7 BL horizontally from takeoff.The five phases are cruise, pitch-up, burst, leap, and re-entry; both thrusters are off during the ballistic leap, and cruise resumes after re-entry.Source: B. Complete Leap · p. 6
A. Exit Sequence
- Fig. 10 compares exit trajectories and yaw angles with and without rudder control, showing five trials per condition.The vertical-plane trajectories are plotted as z against x, and the yaw-angle distribution is shown as a polar histogram.Source: Figure 10 · p. 6
C. Field Demonstration
- With rudder control, the robot exited in Boulder Creek’s slow, turbulent current and reached 2.8 BL above the surface.Without rudder control, it yawed away from the exit plane and struck the surface far off along the yaw direction.Source: C. Field Demonstration · p. 6
V. CONCLUSIONS
The paper presents a palm-scale robot that combines swimming with controlled water exits and obstacle-clearing leaps. It reports fast swimming and turning, a rudder-stabilized exit in a natural stream, and a tank leap; untethered operation and further maneuvering and hop-cycle studies remain future directions.
Platform
- The 65 mm, 34 g platform uses vertically stacked propellers for propulsion and differential-thrust pitch control, with a tail rudder for yaw control.The propellers are mounted vertically in the body’s side-view center plane.Source: §V · p. 6
Performance
- The robot swims at 13.9 BL/s and turns at 209 deg/s, while the Kamm-cut NACA hull’s simulated drag is five-fold lower than an equivalent cuboid’s.The pitch loop tracks sinusoidal commands with amplitude ratios of 0.88–0.96; the hull comparison is from URANS simulation.Source: §V · p. 6
Results
- The complete cruise-leap-cruise sequence cleared a tank obstacle, reaching 1.6 BL above the surface and spanning 3.7 BL; a rudder-controlled stream exit reached 2.8 BL.The leap rises and spans the distances described in the figure captions; in the stream, the result supports robustness to external flow.Source: §V · p. 7, Figure 11 · p. 7, Figure 12 · p. 7
Future Directions
- Future work includes untethered operation, quantitative horizontal-plane maneuvering analysis, repeated hop cycles with controlled re-entry and air attitude control, and systematic field trials.Roll is unregulated during the proposed air-attitude-control work.Source: §V · p. 7
Potential Use
- The authors identify crossing weirs, rock sills, pipe lips, and dry gaps between pools as possible applications in streams, flooded structures, and industrial systems.The platform is also described as modular for environmental sensing, inspection, and monitoring.Source: §V · p. 7