Week 11 · Learn: Accumulating error

Session 1 of 4 · Integral and derivative · Phase 3

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

When does remembering error help, and when does it create a problem? This session focuses on accumulating error.

Before you start

The previous week’s recorded baseline and Week 10, 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

Accumulating error

An integral term remembers persistent error. In sampled form, I_next = I + e Δt. If e is m/s, the accumulated quantity has units metres. Ki converts it to output units. A small speed deficit that remains for many samples keeps increasing the correction, potentially overcoming steady drag. The accumulated state must be reset when starting a new trial so the controller does not inherit yesterday's error. Omitting Δt changes the effective gain when sampling changes. An integral term cannot distinguish genuine error from a biased sensor.

Worked example — illustrative values

At error 0.02 m/s for 0.1 s, accumulated error increases by 0.002 m. A speed measurement jump from 0.08 to 0.10 m/s over 0.1 s has slope 0.2 m/s². With measurement derivative gain 0.1, its damping contribution is −0.02 normalized output units.

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/w11.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.
target, speed, integral, previous = 0.1, 0.0, 0.0, None
dt, kp, ki, kd = 0.1, 3.0, 2.0, 0.05
for step in range(80):
    error = target - speed
    integral = max(-0.2, min(0.2, integral + error * dt))
    derivative = 0.0 if previous is None else -(speed - previous) / dt
    raw = kp * error + ki * integral + kd * derivative
    command = max(0.0, min(1.0, raw))
    previous = speed
    speed += 0.2 * (0.3 * command - speed)
    print(step, speed, integral, raw, command)

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 integral accumulator adds error times 0.1 s and is clipped to ±0.2 in the model’s accumulated-error units. The derivative uses the negative change in measured speed, so a sudden target change does not directly create a derivative kick. previous starts as None, making the first derivative zero rather than dividing an undefined change. Output is then limited to 0–1. Integral clipping limits accumulation but is not a perfect anti-windup design for every actuator. Compare P, PI and PID using the same disturbance/noise traces, and reset accumulator/history whenever the experiment restarts.

Practical instructions

1. Trace four constant-error samples by hand using e×dt.
2. Compare accumulation at dt 0.1 and 0.2 over the same total time.
3. Explain how a sensor bias can produce an integral correction even when true speed is acceptable.
4. Set Ki and Kd to zero in the example to reproduce the P baseline.

Experiment

Integrate identical errors over equal total times at two sample rates. The accumulated physical quantity should agree apart from endpoint convention.

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

Unit-correct integral/slope calculations and an explanation of sensor bias.

Save notebook/w11-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 accumulating error in your own words using this session’s example and its units/assumptions.
• Produce the specific evidence above: Unit-correct integral/slope calculations and an explanation of sensor bias.
• 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: MathWorks: Understanding PID, Part 1. Study task: Sketch a constant error and predict what its accumulated integral will do.
• Video/lecture option: MathWorks: What is PID control?. Study task: Draw the closed-loop signal path and label the measured quantity. 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.