Simulation of Ancient Mechanisms for Hero’s Automata

By William Machado (2164749)

Supervisor: Dr Euan McGookin

Introduction Hero of Alexandria , a Greek mathematician and engineer, designed

self-propelling automata. These were put in motion by a falling

counterweight; as the weight fell, it would unwrap a cord from the axle of

the drive wheels, causing the wheels to turn. He described four unique

wheel configurations, which would allow for different paths to be driven. A

fifth design, inspired by his work, was also analyzed. A mathematical

model was derived for each of the five automata, and then simulated in

Matlab, in order to determine the performance capabilities and limitations

of each automata.

Dimensions

Base length, [m] 0.533 Thickness of base wall, [m] 0.022

Base width, [m] 0.355 Central column radius[m] 0.133

Base height, [m] 0.266 Square axle side length, [m] 0.013

Column height, [m] 0.888 Nominal drive wheel radius, rw [m] 0.1185

Total height, [m] 1.863 Coefficient of friction, μ 0.0153

Drum radius, ra[m] 0.011 Length of cord, [m] 0.833

Total mass, mb[kg] 23 Second drive wheel radius of RDA, [m] 0.073

Automata Types Linearly Driving Automata (LDA) • 1 drive axle • 2 identical drive wheels • 1 counterweight Circularly Driving Automata (CDA) • 1 drive axle • 2 drive wheels of different radii • 1 counterweight Differential Drive Automata (DDA) • 2 drive axles • 2 identical drive wheels • 1 counterweight Rectangular Drive Automata (RDA) • 2 drive axles, perpendicular to each other • 2 sets of drive wheels, each set a different radius • 1 counterweight Multi-Weight Automata (MWA) • 2 drive axles • 2 identical drive wheels • 1 counterweight per axle

LDA wheel configuration

RDA wheel configuration

DDA/MWA wheel configuration

CDA wheel configuration

Conclusions After simulating the five different automata types, various relationships

between the performance and the geometry and mass were observed,

including those for velocities, distances, and turning radii.

DDA

• Left and right turns are symmetrical, but forward and backward driven

turns are not

• When turning, surge and sway velocities do not achieve steady state,

but the yaw velocity does achieve steady state

-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

X,[m]

Y,[

m]

Differential Drive Turning Paths

Forward-Left

Backward-Left

Forward-Right

Backward-Right

0 0.5 1 1.5 2 2.5 3-5

-4

-3

-2

-1

0

1

2103.5 kg; u v r

t, [s]

[m/s

] or

[rad/s

]

u

v

r

RDA

• Progresses as an LDA, but in surge and sway directions.

• Cannot turn, due to single drive axle per direction.

-1 -0.5 0 0.5 1 1.5 2 2.5-1

-0.5

0

0.5

1

1.5

2

2.5

X, [m]

Y,

[m]

Animation of automata, powered by 103.5 kg weight

0 0.5 1 1.5 2 2.5-3

-2

-1

0

1

2

3103.5 kg; u v r

t, [s]

[m/s

] or

[rad/s

]

u

v

r

Theory from: T. Fossen, Marine Control Systems: Guidance, Navigation, and Control of Ships, Rigs and Underwater Vehicles, Marine Cybernetics (2002)

K. J. Worrall, Guidance and Search Algorithms for Mobile Robots: Application and Analysis within the Context of Urban Search and Rescue, Diss. U of Glasgow (2008).

Sketches from: Hero of Alexandria, Herons von Alexandria Druckwerke und Automatentheater = Pneumatica et Automata, Trans W. Schmidt, Illus H Querfurth, B. G. Teubner (1899)

Theory The mathematical model is a non-linear state

space model, based on the Newton-Euler

Formulation and transformations between

frames of reference. The dynamic component

of the model expresses the forces and

moments acting on and produced by the

automata, while the kinematic component expresses the position and

velocities. The model equation is:

η ν

=𝐽 η . ν

𝑀−1[τ − 𝐶 ν . ν − 𝐷 ν . ν − 𝑔 η ]

where η is the Earth-fixed position vector, ν is the body-fixed velocity

vector, J(η) is the transformation matrix, M is the mass and inertia

matrix, C(ν) is the Coriolis-centripetal matrix, D(ν) is the dampening

matrix, g(η) is the gravitational vector, and τ is the input vector.

𝑌𝐵

𝑌𝐸

Earth-fixed Frame

𝑦 𝜃

𝑧

𝜓

𝑥 𝜙

𝑤

𝑢

𝑝

𝑣

𝑞

𝑟

Body-fixed Frame

𝑋𝐸

𝑍𝐸

𝑋𝐵

𝑍𝐵

DoF Axis Term Type of Motion

Forces and Moments (τ)

Velocities (ν)

Position and Angles (η)

1 XE/XB Surge Linear X, [N] u, [m/s] x, [m] 2 YE/YB Sway Linear Y, [N] v, [m/s] y, [m] 3 ZE/ZB Heave Linear Z, [N] w, [m/s] z, [m] 4 XE/XB Roll Rotation K, [Nm] p, [rad/s] ϕ, [rad] 5 YE/YB Pitch Rotation M, [Nm] q, [rad/s] θ, [rad] 6 ZE/ZB Yaw Rotation N, [Nm] r, [rad/s] ψ, [rad]

CDA and MWA • Increasing wheel radii ratio or

mass ratio decreases the turning

radius.

• A wheel radii ratio of 1: 𝑛 has the

same turning radius as a mass

ratio of 1:n

• CDA total distance traveled (𝐷𝑖) is

a function of linear distance (D)

and wheel radii: 𝐷𝑖 = 𝐷 ∗𝑟𝑤𝑟+𝑟𝑤𝑙

2∗𝑟𝑤𝑟

• Increasing the wheel radii ratio by 0.25 allows the CDA to progress

another 0.75π radians along the drive path

• For MWA, the wheel with the larger mass will unwind faster. The

length of cord unwound from each drum is: 𝑚𝑐𝑤𝑟

𝑚𝑐𝑤𝑙=

𝐿𝑟

𝐿𝑙

-4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

X,[m]

Y,[

m]

Drive paths for MWA and CDA

MWA 1:1

MWA 1:1.5

MWA 1:2

MWA 1:2.5

MWA 1:4

CDA 1:1

CDA 1:sqrt(1.5)

CDA 1:sqrt(2)

CDA 1:sqrt(2.5)

CDA 1:2

LDA

• Increasing counterweight mass to 𝑛 ∗ 𝑚𝑏 increases steady state surge

velocity (U):

𝑈 = 2.009 ∗ 𝑒0.0147∗𝑛 − 0.9282 ∗ 𝑒−0.8952∗𝑛

• For a given cord length (L) and length of cord per coil 𝐿𝑐 , increasing

drive wheel radius to 𝑛 ∗ 𝑟𝑤 increases drivable distance (x):

𝑥𝑓𝑖𝑛𝑎𝑙 = (𝐿 ∗ 2 ∗ 𝜋 ∗ 𝑛 ∗ 𝑟𝑤)/𝐿𝑐

• LDA drives same distance, forward or backward

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

time,[s]

u,[

m/s

]

Comparing U over various counterweight masses

34.5 kg

46 kg

57.5 kg

69 kg

80.5 kg

92 kg

103.5 kg

0.2 0.4 0.6 0.8 1 1.2 1.4 2

3

4

5

6

7

8

9

10

11

12

1.6 Wheel radius ratio

Fin

al X

positio

n,

[m]