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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]
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