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Design and Attitude Control of an Underwater Quadruped Robot
Davide Molinaroli, Mohit Singh, Kostas Alexis
TL;DR
Underwater quadruped locomotion is promising but limited by actuator waterproofing and hydrodynamic modeling. The paper presents a reproducible robot, simplified drag-based dynamics, and an SO(3) closed-loop attitude controller, which successfully transfers to hardware for roll, pitch, and coupled maneuvers.
Problem
Underwater quadruped attitude control is challenged by reliable actuator sealing and the complexity of complete hydrodynamic modeling.
Method
The paper combines a reproducible waterproof quadruped, a simplified drag-based floating-base model, and an SO(3) closed-loop controller.
Results
The simplified model was sufficient to design a controller that transferred to the real system for roll, pitch, and coupled reorientation.
Takeaways & Limitations
Drag-based actuation through spherical end effectors can achieve underwater quadruped attitude control with a reproducible low-cost platform.
Takeaways & Limitations
Performance is limited by uncoordinated leg recovery phases and recovery strokes covering roughly half the available workspace.
Abstract
from arXiv · showhide
Legged robots are versatile on land, but their use in underwater environments remains limited. Extending quadruped locomotion to water enables amphibious mobility with applications in inspection, environmental monitoring and disaster response. This paper presents the design, modeling, and experimental validation of a reproducible underwater quadruped robot. The robot is built around custom waterproof motor housings machined from polyoxymethylene plastic, which use off-the-shelf O-rings and dynamic shaft seals. A simplified model is derived to describe the dynamics of this underwater legged system, capturing how drag forces on spherical end effectors transmit torque to the floating base. Building on this model, a closed-loop attitude controller is developed using an error formulation defined on the special orthogonal group SO(3). The controller is evaluated both in simulation and experimentally in a water tank, where the robot tracks desired orientation setpoints in roll, pitch and yaw.
1 Introduction
The paper targets underwater attitude control for quadruped robots, addressing actuator sealing and hydrodynamic-modeling challenges with a reproducible platform, simplified dynamics, and closed-loop validation.
- Underwater quadruped locomotion could support amphibious operation for inspection, environmental monitoring, and partially flooded environments.
- Reliable underwater operation is constrained by waterproof actuator design and the complexity of identifying complete hydrodynamic models.
- The paper develops an open-source, low-cost, reproducible robot using off-the-shelf components and custom waterproof housings.
- A simplified model captures drag- and buoyancy-generated forces and torques, while an SO(3) PID controller maps desired body torques into foot and motor commands.
- The complete closed-loop system is experimentally verified during underwater attitude maneuvers.
2 Related work
Prior amphibious legged robots use bio-inspired trajectories, central pattern generators, or underwater locomotion strategies, whereas this work exploits hydrodynamic drag on submerged end effectors for attitude control.
- Prior amphibious legged robots have mimicked turtles or salamanders, using recorded limb trajectories or central pattern generators for locomotion.
- A sea turtle robot used motion-capture trajectories and a PD joint-space controller for reef-based vision following.
- A separate quadruped combined custom gearboxes and waterproof enclosures with seafloor walking and obstacle-avoidance swimming.
- The present approach differs by using hydrodynamic drag on submerged end effectors for quadruped attitude reorientation.
3 Robot Design
The robot combines a waterproof central enclosure, four three-servo legs, spherical drag-producing end effectors, and sealed POM motor housings with onboard sensing and control electronics.
- The robot uses a central waterproof electronics enclosure, mirrored four-leg architecture, and a cylindrical battery housing beneath the body.
- Each leg has three waterproofed servos: one hip actuator and two lateral actuators driving a planar five-bar linkage.
- POM motor casings use bolted halves, static O-ring seals, dynamic radial lip seals, bearings, retainers, and cable penetrators to limit water ingress.
- Spherical end effectors simplify hydrodynamic modeling through isotropic drag, while perforations increase drag and reduce buoyancy.
- The robot runs ROS1 at 100 Hz, combining ADC feedback, IMU orientation, inverse kinematics, and PWM servo commands.
4 Mathematical Modeling
The robot is modeled as a floating base with four mirrored kinematic chains, using spherical end effectors as the basis for forward and inverse leg kinematics.
- 4 Mathematical Modeling: The model treats the underwater quadruped as a floating base with four kinematic chains, neglecting leg dynamics and retaining end-effector-generated drag and buoyancy forces and torques.The dominant actuation mechanism is drag from spherical end effectors moving through water.
- 4 Mathematical Modeling: Each leg’s body-frame end-effector position is obtained by transforming its leg-frame position with a leg-specific mirroring matrix and origin offset.The mirroring accounts for the symmetric arrangement and differing frame orientations of the four legs.
- 4 Mathematical Modeling: Forward kinematics maps motor angles θ1, θ2, θ3 to end-effector position through the planar five-bar mechanism and hip rotation.The five-bar geometry uses equal actuated-link lengths, equal passive-link lengths, and motor spacing c.
- 4 Mathematical Modeling: Inverse kinematics recovers the joint angles from a desired end-effector position by first mapping that position into the leg frame and then solving the five-bar geometry.The procedure is identical across legs after applying the relevant frame transformation.
4.4 Leg Translational Jacobian
The leg translational Jacobian maps joint and floating-base velocities to end-effector velocity while accounting for the closed-chain five-bar mechanism and hip rotation.
- 4.4 Leg Translational Jacobian: The full translational Jacobian combines the five-bar closed-chain Jacobian with the hip-rotation contribution to map motor velocities into body-frame end-effector velocity.The five-bar terms are derived from loop closure and depend on the passive joint angles.
- 4.4 Leg Translational Jacobian: The generalized leg state includes joint angles and velocities together with floating-base linear and angular velocities.The base linear velocity is expressed in the world frame, while angular velocity is expressed in the body frame.
- 4.4 Leg Translational Jacobian: The floating-base Jacobian transforms base motion and leg motion into the world-frame velocity of each end effector.The transformation uses the body-to-world rotation extracted from the base orientation quaternion and the body-frame end-effector position.
4.6 Drag Force and Torque Transmission
Spherical end effectors generate drag and buoyancy forces that are transmitted through the leg Jacobians into forces and torques acting on the floating base.
- 4.6 Drag Force and Torque Transmission: Each spherical end effector experiences hydrodynamic drag determined by its velocity, fluid density, drag coefficient, and cross-sectional area.The spherical geometry supports an isotropic drag model based on paw speed and area.
- 4.6 Drag Force and Torque Transmission: The end-effector drag force maps into generalized base forces and torques through the transpose of the translational floating-base Jacobian.Because leg dynamics are neglected, only end-effector forces contribute to the floating-base dynamics.
- 4.6 Drag Force and Torque Transmission: The four legs’ drag contributions are summed to obtain the total force and torque acting on the floating base.The model explicitly aggregates the contributions across legs.
- 4.6 Drag Force and Torque Transmission: Each submerged spherical paw also produces a net vertical buoyancy force acting at its end-effector position, thereby contributing a body torque.The net force depends on paw mass, volume, fluid density, and gravitational acceleration.
4.8 Inertia-Based Body Drag Model
The body-drag model approximates the robot with an inertia-matched box and combines quadratic and viscous drag terms, while omitting added mass and lift.
- 4.8 Inertia-Based Body Drag Model: The simulation approximates the body as a rectangular box whose half-dimensions reproduce the robot’s diagonal inertia tensor.The equivalent box is parameterized from the body mass and principal moments of inertia.
- 4.8 Inertia-Based Body Drag Model: Body drag combines a quadratic term based on box geometry and velocity with a viscous term computed using Stokes’ law for an equivalent sphere.The equivalent sphere radius is the mean of the three box half-dimensions.
- 4.8 Inertia-Based Body Drag Model: The complete drag force is rotated into the world frame, whereas the drag torque remains expressed in the body frame.The quadratic and viscous force and torque components are first stacked into vector quantities.
- 4.8 Inertia-Based Body Drag Model: The model neglects added mass and lift, and computes its coefficients directly from robot geometry without experimental identification.This provides a simplified hydrodynamic treatment rather than a fully identified fluid model.
- 4.8 Inertia-Based Body Drag Model: The floating-base equations sum the contributions of all legs, body drag, and buoyancy as external forces and torques.Torso gravity and buoyancy are neglected because the robot is slightly negatively buoyant and suspended by a rope during experiments.
5 Attitude Controller Design
The controller converts orientation error on SO(3) into drag-based foot motions that generate corrective body torque. It combines PID control, torque allocation, and practical stroke-cycle handling for implementation on the robot.
- 5.1 Orientation Error: The controller computes SO(3) orientation error, applies a PID torque law with feedforward compensation, and allocates the resulting torque to foot velocity commands.The allocation distributes saturated torque across the legs and computes drag-force directions and speeds from foot geometry.
- 5.2 Control Law: Leg dynamics are treated as virtually instantaneous relative to torso dynamics, allowing the controller to use rigid-body attitude error dynamics for torque control.This assumption neglects multi-body leg dynamics in the control design.
- 5.2 Control Law: Torque saturation is set to τmax = 2 Nm, and saturation feedback through Kaw provides anti-windup during commanded-torque limiting.The torque limit was estimated from preliminary trials using measured angular and joint velocities.
- 5.4 Power and Recovery Stroke Cycle: Diagonal leg pairs alternate power and recovery phases to approximate continuous torque generation despite finite workspace and limited servo bandwidth.A half-duration initial recovery phase offsets the pair timing, while power trajectories are integrated and then retraced during recovery.
- 5.5 Simulation: Simulation evaluated simultaneous roll, pitch, and yaw setpoints from 15 to 75 degrees in 30-degree increments using the closed-loop controller.The simulation used fourth-order Runge-Kutta integration with Δt = 0.01 s and trec = 1.5 s.
6 Experiments
Water-tank experiments evaluated single-axis roll and pitch regulation, yaw correction from varied initial conditions, and coupled roll-pitch commands. The controller stabilized the tested coupled setpoints, while larger pitch commands exposed stroke-phasing and representation effects.
- 6.1 Experimental Setup: The experiments commanded six roll and six pitch setpoints from 15 to 90 degrees in 15-degree increments, plus three coupled roll-pitch trials.Yaw began from different initial conditions ranging from -78 to +123 degrees and was regulated toward zero.
- 6.1 Experimental Setup: Reported metrics use centered 3 s moving averages, so they characterize sustained average orientation across stroke cycles rather than instantaneous oscillations.The table includes rise time, settling time, steady-state error, overshoot, initial geodesic error, and yaw performance.
- 6.3 Pitch Regulation: Pitch experiments showed increasing rise and settling times through 60 degrees, while 75- and 90-degree trials required trec = 1.0 s for stabilization.Temporary pitch loss occurred when leg phasing became inefficient for torque generation; workspace asymmetry later restored more favorable synchronization.
- 6.4 Coupled Roll and Pitch Regulation: The robot stabilized coupled roll-pitch commands at 30, 45, and 60 degrees while correcting yaw from varied initial conditions.The 60-degree case used trec = 1.0 s and had shorter rise and settling times than the 45-degree case.
7 Discussion
The results show that spherical-end-effector drag can control an underwater quadruped’s attitude, and that the simplified model transfers sufficiently to hardware. Performance remains limited by discontinuous torque generation during leg recovery.
- 7 Discussion: Drag-based actuation through spherical end effectors achieved attitude control of the underwater quadruped, with the simplified model transferring from design to the real system.The authors identify modeling servo and leg dynamics, added mass, and restoring forces as ways to reduce simulation-to-hardware discrepancy.
- 7 Discussion: Performance is primarily limited by speed because uncoordinated recovery phases can create intermittent torque gaps and longer settling times.Retracting legs toward a fixed workspace-center position also limits each power stroke to roughly half the available range.
8 Conclusion
The paper presents and validates a low-cost, reproducible underwater quadruped for attitude control, combining open-source hardware, simplified dynamics, and a closed-loop SO(3) controller.
- The robot combines off-the-shelf components with POM-machined casings to provide a reproducible and cost-effective waterproofing solution.The design uses servo motors, electronics, sealing components, and custom-machined housings.
- A simplified drag-based dynamics model supports closed-loop SO(3) control for underwater reorientation.The model is derived from floating-base kinematics and drag-based swimming.
- The controller was deployed on the physical platform for roll, pitch, and coupled reorientation maneuvers.