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LAB / TWO SMALL EXPERIMENTS

DON'T JUST READ
THE MODEL.
CHANGE IT.

A practical way to explore the mathematics behind my interests. These are interactive teaching models, not measured data from hardware or validated physical simulations.

EXPERIMENT 01 / GEOMETRIC CONTROL

THE ACTUATOR
ENVELOPE.

Command a normalized chassis velocity. Each wheel has a unit speed limit. The feasible region changes as yaw rate changes.

Command inputs

Choose body-frame velocity components and angular velocity. The example uses a symmetric, normalized four-wheel holonomic mapping with unit wheel bounds.

FEASIBILITY CHECK● FEASIBLE

All four wheel commands are within the normalized speed bounds.

Wheel commands / normalized

W₁0.00
W₂0.00
W₃0.00
W₄0.00
WORKSPACE / vₓ, vᵧω = +0.15
BLUE POLYGON = FEASIBLE COMMANDS AT THE CURRENT YAW RATE · WHITE/RED POINT = SELECTED COMMAND
w = A·[vₓ, vᵧ, ω]ᵀ,   A = [[1, −1, −1], [1, 1, 1], [1, 1, −1], [1, −1, 1]],   |wᵢ| ≤ 1

Interpretation: This is a deliberately normalized teaching model. It is not a calibration of an actual VEX X-drive, and its coefficients do not assert any particular wheel geometry, wheel radius, motor response or friction condition. The shape is the intersection of linear speed-limit inequalities.

EXPERIMENT 02 / TOPOLOGICAL DATA ANALYSIS

CONNECTIVITY
UNDER A LENS.

Start with a fixed point cloud. Increase the connection radius and watch the graph's components merge and cycles appear.

Filtration parameter

Connect any two points whose distance is at most ε. This builds the 1-skeleton of a Vietoris–Rips complex.

GRAPH SUMMARY9 NODES

The indicators below describe this graph at the selected threshold.

CONNECTED COMPONENTS—
EDGES—
GRAPH CYCLE RANK—
VERTICES9
β₀ = C,  rank(G) = E − V + C

β₀ is the number of connected components. The graph cycle rank is not the persistent H₁ barcode of the full Vietoris–Rips complex; filling in triangles changes the homology.

POINT CLOUD / DISTANCE GRAPHADJUST ε TO FILTER
CIRCLES = DATA POINTS · LINES = PAIRS WITH DISTANCE ≤ ε · COLOURS INDICATE CONNECTED COMPONENTS

This simplified interactive graph illustrates the first step toward a filtration and helps visualize the kind of structure used in topological data analysis. It is independent of the financial-modeling manuscript's specific analytical examples.

CONTINUE / FROM EXPERIMENT TO EXPLANATION

INTERESTED IN THE
ACTUAL RESEARCH?

The full archive describes my mathematical approach, assumptions, and what still needs empirical or physical validation.

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