RL Robotics LabSearch ↗
Lesson / authored

Week 11 · Experiment: Derivative is change per time

Session 3 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 derivative is change per time.

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

Derivative is change per time

A derivative term reacts to the error slope, D = (e − e_previous)/Δt. It can damp rapidly changing responses, but small measurement noise divided by a short interval can produce a large correction. Differentiating the measured speed rather than the reference error avoids a sudden derivative kick when the target changes; the sign is then negative. At the first sample there is no previous observation, so set derivative to zero. Filtering reduces noise but adds lag. Derivative is optional; PI may suffice for a slow wheel-speed task.

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)

Use your own recorded data or named test fixtures instead of the illustrative inputs. Save expected and actual values side by side. 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. Add derivative on measured speed with a zero first-sample contribution.
  2. Compare clean measurements with the same alternating ±0.005 m/s noise sequence for all controllers.
  3. Run P, PI and PID with the same sample interval and output limits.
  4. Plot command jitter as well as speed so a smoother-looking speed trace cannot hide excessive effort.

Experiment

Hold plant and target fixed while varying measurement noise. Quantify maximum command change and mean absolute error for P/PI/PID.

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

Comparable P/PI/PID traces with measurement-derivative convention and noise effects visible.

Save notebook/w11-s3.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.

← Previous sessionCourse roadmapNext session →