Finneas the Undulator

Hugo Guckert · Tay Han · Henry Heathwood · Will Sedo

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$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

Conclusions