Finneas the Undulator

Introduction and Project Background

Persian carpet flatworms (Pseudobiceros bedfordi), also described as a “magic carpet,” evolved to swim by undulating the ruffled margins of their thin bodies. They also typically crawl on the seafloor using such undulating motion. When tasked with creating a mechanically-driven amphibious vehicle, the team decided to take on this bio-inspired approach.

Persian carpet flatworm (Pseudobiceros bedfordi) swimming by undulating its ruffled body margins. — Source: roundglasssustain.com/species/marine-flatworms

We are simplifying the traditional undulating fin robot design by utilizing a central camshaft instead of a large number of servos. Below are two examples of previous undulating fin robots that utilize more robust motor control rather than pure mechanical power transmission.

Servo-driven undulating fin robot. — Source: academic.oup.com/jom/article/doi/10.1093/jom/ufae037/7823757

Pliant Energy Systems amphibious robot. — Source: pliantenergy.com/robotics

We follow a similar structure to the figure shown below, where we chose to utilize a common CAM shaft that drives each part of the fin.

Cam-mechanism undulating fin robot diagram. — Source: mdpi.com/2077-1312/10/9/1327

Requirements and Specifications

Qualitative

Moves in water and on land
- This is the main purpose of our project
- The fins allow for both thrust in water and rolling friction on land
Waterproof
- Electronics need a dry environment
Floats
- We want a surface-operating robot that doesn't need to adjust buoyancy
Looks swag and fishy

Quantitative

Tip of fins move 2 inches peak to peak
- Realistic distance we need the fin to move to generate enough thrust in water and enough distance to reach over obstacles on land
- The original goal was 4 inches but had to pivot due to geometric constraints. The fin rod pivot point must be at the hull/water interface and the hull must be a certain width for stability.
CAM follower moves 1 inch peak to peak
Camshaft rotates at 60rpm
- Creates a realistic and effective undulating frequency for water and land

Design Features of Main Subsystems

Significant Design Decisions

Cam shaft
- We opted for a camshaft to create a simplified version of the more common servo-based undulating fin robot. Using a servo for each fin rod adds control complexity and unnecessary tuning for a mechanical design project. A single central camshaft leads to a more mechanically robust and simple robot.
Cam geometry
- An eccentric circle CAM geometry was chosen to create the sinusoidal motion for the fins.
Fin rod pivot geometry
- We designed a slotted hole for the pivot at the hull/water interface that restricts the vertical and horizontal translation of the fin rods while allowing them to pitch back and forth. As shown in the actual physical system, additional waterproofing measures were taken on top of what is shown in the SolidWorks model.
Follower to fin rod connection
- A halved pivot joint was machined to connect the fin rods to the followers.
ESP for WiFi control
- A simple ESP32 microcontroller was implemented so that the vehicle can move autonomously over WiFi, taking out the need for long wires to be attached to a separate microprocessor. As shown below, we use an L298N motor driver to drive the 24 VDC gear motor. An LM2596 DC-DC buck converter steps down a 22.2 V 6S LiPo (3300 mAh) battery to a safe input voltage for the ESP32, preventing the on-board regulator from overheating.
Fin rod pivot point location
- Due to size constraints the maximum amplitude of the follower motion was highly limited. To achieve our quantitative specification of 2 inches of fin rod tip motion, the pivot point of the fin rod was strategically placed to create mechanical leverage.

Evaluation of Design

Analysis

CAM MATLAB

An eccentric cam was chosen as the cam shape to produce a sinusoidal follower motion. The total amplitude of the follower is given by two times the shaft hole offset. The sinusoidal motion of the followers is translated directly to the motion of the fin rods which can be seen in the sinusoidal fin motion. The cams are press fit onto a ⅜″ D-shaft to restrict both axial motion and rotational motion.

Fastener

In order to demonstrate the feasibility of our fastener choice, we calculate the failure mode for the bolts.

CAM Torque analysis with motor

Finneas’ drive shaft carries 8 eccentric circular CAMs that lift 8 spring-loaded followers. To verify our design before production, we had to verify two things:

Kinematics: the follower travels the distance at the speed and acceleration that is expected.
Drive torque: the motor at 60 RPM can supply the average and peak torque needed to spin all 8 of the CAMs against the spring force and friction.

Free Body Diagram

Inputs

Symbol	Value
$R$	1.25 in
$e$	0.40 in
$s = 2e$	0.80 in
$n$	60 RPM
$\omega = 2\pi n/60$	6.28 rad/s
$N_\text{cam}$	8
$\Delta\phi$	90°
$m$	0.084 lbm (38 g)
$k$	20 lbf/in (3.5 N/mm)
$x_0$	0.10 in
$\mu$	0.25
$g$	386.09 in/s²

Kinematics

$$x(\theta) = e(1-\cos\theta)$$

$$v(\theta) = \dot{x} = e\,\omega\sin\theta$$

$$a(\theta) = \ddot{x} = e\,\omega^{2}\cos\theta$$

$$x_{\max} = 2e = 0.80\ \text{in}$$

$$v_{\max} = e\omega = 2.51\ \text{in/s}$$

$$a_{\max} = e\omega^{2} = 15.79\ \text{in/s}^{2}$$

$$\tan\phi(\theta) = \frac{e\sin\theta}{\sqrt{R^{2}-e^{2}\sin^{2}\theta}}$$

$$\tan\phi_{\max} = \frac{e}{\sqrt{R^{2}-e^{2}}} = \frac{0.40}{1.184} = 0.338$$

$$\phi_{\max} = 18.7^{\circ}$$

Force analysis

$$N\cos\phi + mg - F_{s} = m\,a, \qquad F_{s} = k(x_{0}+x)$$

$$N\cos\phi(\theta) = m\,e\,\omega^{2}\cos\theta + k\bigl(x_{0}+e(1-\cos\theta)\bigr) - mg$$

$$N_{\max} = k(x_{0}+2e) - mg - m\,e\,\omega^{2} = 20(0.10+0.80) - 0.084 - 0.003 = 17.9\ \text{lbf}\;(79.6\ \text{N})$$

Drive torque

$$T\,\omega = (N\cos\phi)\,\dot{x}$$

$$T_\text{cam}(\theta) = \bigl[m\,e\,\omega^{2}\cos\theta + k(x_{0}+e(1-\cos\theta)) - mg\bigr]\,e\sin\theta$$

$$T_\text{fric}(\theta) = \mu\,N(\theta)\,(R-e\cos\theta)$$

$$T_\text{cam,peak} \approx 4.8\ \text{lbf}\cdot\text{in}\quad (\theta \approx 120^{\circ})$$

$$\bar{N} \approx \frac{N_\text{min}+N_\text{max}}{2} = \frac{1.9 + 17.9}{2} \approx 10\ \text{lbf}$$

$$\bar{T}_\text{fric} \approx \mu\,\bar{N}\,R = 0.25 \times 10 \times 1.25 \approx 3.1\ \text{lbf}\cdot\text{in}$$

$$T_\text{shaft} \approx 8\,\bar{T}_\text{fric} \approx 25\ \text{lbf}\cdot\text{in}\;(2.8\ \text{N}\cdot\text{m})$$

$$P = T_\text{shaft}\,\omega \approx 18\ \text{W}$$

Future Work

Two camshafts to allow for steering
More machined components to allow for higher torque
Sensors for more robust control and various tasks
Fin rod adaptations or additional tilt motors that allow it to work at different angles (such as pointing straight down for land movement or optimizing movement based on environment)
Non-PLA hull for better waterproofing
Fully enclosed hull to allow diving