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Week 17 · Improve: Residuals explain the model

Session 4 of 4 · Estimate position · Phase 5

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

How accurately can wheel rotation tell you where the robot is? This session focuses on residuals explain the model.

Before you start

The previous week’s recorded baseline and Week 16, 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

Residuals explain the model

A residual is measured truth minus prediction. Signed x and y residuals retain direction; a Euclidean error sqrt(dx² + dy²) records magnitude. Plot both. Repeated residual directions suggest a biased parameter or starting pose. Variable directions may reflect slip or inconsistent setup. Replaying the same raw encoder data with a changed calibration isolates the mathematical effect, but cannot prove future physical improvement. Validate a chosen correction on fresh travel and retain the original trace as a baseline.

Worked example — illustrative values

If dL = 0.09 m, dR = 0.11 m and b = 0.10 m, ds = 0.10 m and dθ = 0.20 rad. Starting at (0,0,0), midpoint integration gives x ≈ 0.0995 m, y ≈ 0.0100 m. These are calculated values, not measurements. A 0.05 rad heading error over 1 m produces approximately 5 cm sideways error.

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/w17.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 cos, sin

def update(pose, left_m, right_m, track_m):
    if track_m <= 0:
        raise ValueError("track_m must be positive")
    x, y, heading = pose
    ds = (left_m + right_m) / 2
    turn = (right_m - left_m) / track_m
    return (x + ds * cos(heading + turn / 2),
            y + ds * sin(heading + turn / 2), heading + turn)

pose = (0.0, 0.0, 0.0)
for left, right in [(0.1, 0.1), (0.09, 0.11), (0.1, 0.1)]:
    pose = update(pose, left, right, 0.1)
    print(pose)

Compare the unchanged baseline and your proposed change on the same inputs. Keep the original files so another reader can reproduce the comparison. 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

Each function call consumes new wheel-distance increments and returns a new pose. Equal 0.1 m increments add 0.1 m along the present heading. Unequal increments also rotate the heading; the midpoint orientation places the small translation approximately along the arc. Repeated identical cumulative counts must produce zero increments, not another copy of the entire distance. Save count pairs and elapsed time before deriving pose. A true-world comparison at stopped checkpoints reveals drift that the function cannot observe internally. A heading error also changes later x/y travel even if the distance scale is correct.

Practical instructions

  1. Choose one evidence-supported correction: scale, effective track width or starting alignment.
  2. Replay the same counts with old and changed parameters before moving again.
  3. Validate the selected correction on three fresh short routes.
  4. Write an odometry limitations paragraph and carry the calibrated configuration forward.

Experiment

Compare old/new parameters on the same trace, then collect three fresh validation runs with unchanged test geometry.

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

A justified calibration decision validated separately and an explicit statement that odometry drifts.

Save notebook/w17-s4.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

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

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.

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