Finneas the Undulator

Hugo Guckert · Tay Han · Henry Heathwood · Will Sedo

First deployment

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.

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

Quantitative

Design Features of Main Subsystems

Significant Design Decisions

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.

MATLAB plot of follower position profile.
Cam profile dimensioned drawing.

Fastener

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

Fastener free-body diagram.
Fastener shear-failure calculations.

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:

  1. Kinematics: the follower travels the distance at the speed and acceleration that is expected.
  2. 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
SymbolValue
$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$2.8 lbf/in (0.49 N/mm)
$x_0$0.10 in
$\mu$0.25
$g$386.09 in/s²

Sign convention. $x$ is the follower’s lift, measured downward from the base-circle (minimum-lift) position. The cam sits above the follower and pushes it down; the spring sits below and pushes it back up. Gravity acts in $+x$ (downward).

Kinematics

For an eccentric circular cam with an in-line follower (and $e \ll R$ so the contact stays close to the follower’s centerline), the lift is the standard simple-harmonic profile:

$$x(\theta) = e(1-\cos\theta)$$
$$v(\theta) = \dot{x} = e\,\omega\sin\theta$$
$$a(\theta) = \ddot{x} = e\,\omega^{2}\cos\theta$$

Peak values:

$$x_{\max} = 2e = 0.80\ \text{in}$$
$$v_{\max} = e\omega = 2.51\ \text{in/s}$$
$$a_{\max} = e\omega^{2} = 15.79\ \text{in/s}^{2}$$

Pressure angle (angle between the cam-surface normal at the contact point and the follower’s direction of motion):

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

This is largest at $\theta = 90^{\circ}$:

$$\tan\phi_{\max} = \frac{e}{\sqrt{R^{2}-e^{2}}} = \frac{0.40}{1.184} = 0.338$$
$$\phi_{\max} = 18.7^{\circ}$$

Standard rule of thumb: keep $\phi \le 30^{\circ}$ for translating roller / knife-edge followers. 18.7° is well within bounds — no risk of jamming or excessive side-load on the follower guide.

Force analysis

Newton’s second law on the follower, along the line of motion ($+x$ downward):

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

Solving for the cam normal force projected onto the motion line:

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

$N\cos\phi$ peaks at $\theta = 180^{\circ}$ (max lift, max spring compression, $\cos\phi = 1$):

$$N_{\max} = k(x_{0}+2e) - mg - m\,e\,\omega^{2} = 2.8(0.10+0.80) - 0.084 - 0.003 \approx 2.43\ \text{lbf}\;(10.8\ \text{N})$$

The minimum occurs at $\theta = 0$ (base circle, $x = 0$, $\cos\phi = 1$):

$$N_{\min} = m\,e\,\omega^{2} + k\,x_{0} - mg = 0.003 + 0.28 - 0.084 \approx 0.20\ \text{lbf}\;(0.89\ \text{N})$$

$N_{\max}$ is the design load for the cam-follower contact stress (Hertzian contact, sized separately) and the bolt-shear analysis on the halved pivot joint.

Drive torque

For each cam, instantaneous shaft torque from the contact force comes from instantaneous power balance, $T\omega = (N\cos\phi)\,\dot{x}$:

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

Plus a Coulomb friction term at the cam–follower contact (always opposing shaft rotation), with moment arm equal to the contact-point radius $R - e\cos\theta$:

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

The first term is conservative: the spring stores energy on the down-stroke and returns it on the up-stroke. Its average over one revolution is zero. Its peak is:

$$T_\text{cam,peak} \approx 0.65\ \text{lbf}\cdot\text{in}\quad (\theta \approx 118^{\circ})$$

The friction term is dissipative (always positive). Using the mean normal force from the swing between minimum and maximum:

$$\bar{N} \approx \frac{N_\text{min}+N_\text{max}}{2} = \frac{0.20 + 2.43}{2} \approx 1.3\ \text{lbf}$$
$$\bar{T}_\text{fric} \approx \mu\,\bar{N}\,R = 0.25 \times 1.3 \times 1.25 \approx 0.41\ \text{lbf}\cdot\text{in}$$
8-cam total

With 90° phasing between adjacent cams (4 distinct phases × 2 cams each), the first and second harmonics of the conservative torque cancel when summed across the 4 phase positions. The net shaft torque is dominated by friction:

$$T_\text{shaft} \approx 8\,\bar{T}_\text{fric} \approx 3.3\ \text{lbf}\cdot\text{in}\;(0.37\ \text{N}\cdot\text{m})$$

with only small higher-harmonic ripple on top.

Drive power
$$P = T_\text{shaft}\,\omega \approx 2.3\ \text{W}$$

Motor sizing. A 24 V geared DC motor delivering ≥ 5 lbf·in (0.6 N·m) of continuous torque at the output shaft (after gear reduction to 60 RPM) leaves comfortable margin for startup inertia, breakaway friction, and conservative-torque ripple. The IG32-1:27 motor at 20 V already covers this with ample margin.

Future Work

Conclusions

Finneas the Undulator met the desired requirements and specifications. Finneas moves in water and on land, is waterproof, floats, and looks swag and fishy. Not only this, but the tips of the fins move ~2 inches peak to peak with a corresponding CAM follower movement of ~1 inch peak to peak. The camshaft rotates at differing speeds based on a quickly configurable, web-browser PWM controller, also allowing it to reverse and move backwards. A closed-loop PID controller was also implemented to maintain the desired RPM. The team achieved their learning goals, especially gaining valuable experience maintaining CAD models, using an unconventional locomotion mechanism, manufacturing different components in modular, reasonable ways, waterproofing, and performing detailed analysis to verify outputs.

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Team

Hugo Guckert
Hugo Guckert
Tay Han
Tay Han
Henry Heathwood
Henry Heathwood
Will Sedo
Will Sedo