Week 7 · Learn: Differential-drive geometry

Session 1 of 4 · Turning and kinematics · Phase 2

Plan about 15 minutes for explanation, 30 minutes for practical work and 10–15 minutes for documentation. A longer build may continue into the next session: stop safely, commit the current state and record the next check. Desktop simulations count as software evidence; label them clearly and record physical validation separately.

Engineering challenge

Can wheel travel predict a turn, and where does the rolling model break down? This session focuses on differential-drive geometry.

Before you start

The previous week’s recorded baseline and Week 6, Improve. For later sessions this week, retain the preceding session’s files and predictions.

Equipment: Desktop Python, editor, paper and ruler; for physical work, the configured 3pi+ 2040, clear floor mat and hardware checklist. Week 3 additionally uses the separate low-voltage LED circuit described in its procedure.

For any motion, verify the stop button, short time limit and clear floor area first. Keep the wheels raised for a new device program until its commands and stop behaviour are checked. A hazard or uncertain input is a reason to stop and document, not to force the trial to finish.

Theory and mathematics

Differential-drive geometry

A two-wheel robot turns by giving its wheels different travel. Let dL and dR be left and right ground distances and b the distance between the wheel contact tracks. The centre advances approximately ds = (dR + dL)/2. Heading changes by dθ = (dR − dL)/b radians. Equal travel makes dθ zero; opposite equal travel makes ds zero, a spin in place. These equations assume rolling without sideways slip and a rigid wheel separation. Effective b may differ slightly from a ruler measurement because of contact and scrub.

Worked example — illustrative values

For b = 0.080 m, dL = 0.050 m and dR = 0.070 m: ds = 0.060 m; dθ = 0.020/0.080 = 0.25 rad ≈14.3°. An in-place 90° turn requires ±0.080π/4 ≈±0.0628 m at the wheels.

Write the calculation in your notebook before running code. State which values you measured, which you assumed and which the program calculates. A correct numerical calculation cannot rescue an incorrect physical assumption.

Run and explain the model

The following is desktop Python, not a ready-to-run motor program. Download this week’s example, save it in your student repository and run python3 code/w07.py from the repository root. The same small model is reused across the week so you can learn it, build with it, test it and revise it.

# Desktop Python teaching example. Numerical inputs are illustrative.
from math import pi, degrees
track_m = 0.080
left_m, right_m = 0.050, 0.070
print("centre_m", (left_m + right_m) / 2)
print("heading_deg", degrees((right_m - left_m) / track_m))
print("90_degree_wheel_m", track_m * pi / 4)

Predict the example’s output by hand. Mark the inputs, units and assumptions; explain where this model could fail. If an exception appears, read its final line, identify the input or assumption that caused it and make the smallest explained correction. Do not delete validation merely to obtain output.

Understanding the model and its limits

The model averages wheel travel for centre displacement and divides the travel difference by track width for heading change. degrees converts the internal radian result for display. For an ideal in-place 90° turn, each wheel travels bπ/4 in opposite directions, because the centre does not translate and total travel difference is bπ/2. The effective track width may differ from a ruler measurement owing to tyre contact and slip. Fit it from modest turns, then check a different angle. Equal wheel travel should produce zero heading change; that is a useful sign/convention test.

Practical instructions

1. Measure wheel track width between contact centres, not overall body diameter.
2. Draw equal-forward, equal-reverse and opposite-travel cases.
3. Calculate ds and dθ for three signed pairs using the code.
4. Convert 45° and 90° to radians and label your positive heading convention.

Experiment

Calculate expected heading for three wheel-distance pairs, including a spin and straight travel. Verify results with the desktop code.

Before testing, record your prediction, changed factor, measured response, fixed conditions and stopping rule. Save every attempted run, including failures, with a condition and source version. If hardware is unavailable, use an explicitly labelled synthetic/replay dataset and list the physical question it cannot answer. Do not invent completed trials.

Deliverable

Correct signed centre travel and radian heading calculations; physical track width recorded.

Save notebook/w07-s1.md, the relevant code revision, raw CSV or test-case records, and one labelled diagram/plot/table. Link the files relatively from your notebook. Use the entry template and report guide.

Completion criteria

• Explain differential-drive geometry in your own words using this session’s example and its units/assumptions.
• Produce the specific evidence above: Correct signed centre travel and radian heading calculations; physical track width recorded.
• Keep predictions and raw outcomes, distinguish observations from interpretation, and explain one limitation or unresolved failure.
• Review the Git diff, commit the session’s intended files and state the next experiment or safe continuation point.

A documented failed prediction can meet the learning criteria. A missing physical trial must remain marked untested; software success alone does not validate the robot.

Reading and video

• Focused reading: Robotics Lab: units and trigonometry primer. Study task: Draw the wheel distances and derive the signs of ds and dθ.
• Video/lecture option: CS50 Python: loops. Study task: Pause on a loop and predict its output before continuing. Watch a relevant 5–10 minute excerpt or use the linked notes if video is inaccessible. This is supporting conceptual material; hardware in a demonstration may differ from yours.
• Practical reference: Engineering handbook and hardware setup. Manufacturer/API references and video metadata were checked on 2026-10-09; recheck the actual firmware before transferring code.

Reflection and next step

Which assumption most affected your result? Point to one observation that supports your explanation and one alternative explanation the evidence has not ruled out. Write a specific next test with a changed factor and measurable outcome, then proceed through the week’s Learn → Build → Experiment → Improve cycle.