M09 · optional contextual support
Connect dynamics estimates and constraints
Step a feedback system and separate numerical accuracy from stability and measurement assumptions.
Start with arithmetic, percentages and basic algebra. Open any section directly. Practice and completion here do not change any AI course requirement or the original five-lesson mathematics pilot.
1. Try a concrete case
A unit-mass cart has x=1,v=0 and force −2 N for 0.1 s. Velocity-first integration gives v′=−0.2 m/s, then x′=0.98 m. Position-first integration would give a different finite-step result.
2. Predict before changing a value
Change stiffness and damping, then the number of integration steps. Separately inspect the scalar correction with fixed prior variance 4 m² and sensor variance 1 m². Explain why a small reported variance cannot rule out bias. Name the quantity you expect to change and one you expect to stay fixed before using the controls.
3. Experiment
Change one quantity, follow every calculation
- Final position
- 0.30712
- Final velocity
- -0.90061
- Scalar Kalman gain
- 0.80000
- Corrected mean for measurement 2
- 1.80000
- Corrected variance
- 0.80000
Vertical axis: Position metres. Fixed scales; 0 points clipped for display. Lines join sampled values to aid reading; they do not create extra observations. Exact numerical values are below.
| Time seconds | Position metres |
|---|---|
| 0.00000 | 1.00000 |
| 0.10000 | 0.98000 |
| 0.20000 | 0.94240 |
| 0.30000 | 0.88971 |
| 0.40000 | 0.82450 |
| 0.50000 | 0.74932 |
| 0.60000 | 0.66667 |
| 0.70000 | 0.57895 |
| 0.80000 | 0.48842 |
| 0.90000 | 0.39718 |
| 1.00000 | 0.30712 |
Unit mass, velocity-first integration, u=clip(−k x−d v,±2 N). Smaller steps change numerical error; gain changes can change stability. The separate scalar correction uses prior variance 4 m² and fixed measurement variance 1 m², independently of the damping control. It assumes independent zero-mean sensor noise; it does not detect an unmodeled bias. Clipping is a constraint, not proof of safe trajectories.
Change stiffness and damping, then the number of integration steps. Separately inspect the scalar correction with fixed prior variance 4 m² and sensor variance 1 m². Explain why a small reported variance cannot rule out bias.
Controls save locally as practice. No timers, remote services or GPU are required. Maximum numerical work is a bounded small table or 256 bootstrap resamples.
4. Understand the calculation
Start with a calculation you can inspect
A unit-mass cart has x=1,v=0 and force −2 N for 0.1 s. Velocity-first integration gives v′=−0.2 m/s, then x′=0.98 m. Position-first integration would give a different finite-step result.
Step a feedback system and separate numerical accuracy from stability and measurement assumptions. The experiment changes one named quantity at a time and shows the intermediate values, rather than asking you to accept a final score. Before changing a control, write down which value you expect to move and why.
Read the formula in words and symbols
v′=v+Δt u/m; x′=x+Δt v′; K=P/(P+R); corrected mean=prior mean+K(measurement−prior mean)
Δt is seconds per step, u force, m mass, P prior variance and R measurement-noise variance. K is dimensionless. The simulation and separate scalar sensor example are not one fitted physical system.
A subscript names a coordinate, case or time index; it is not multiplication. A summation sign means add the terms for the stated indices. Parentheses identify which calculation happens first. Carry units through arithmetic: adding incompatible quantities or confusing a rate with a count changes the question.
Connect the visual and the numbers
Change stiffness and damping, then the number of integration steps. Separately inspect the scalar correction with fixed prior variance 4 m² and sensor variance 1 m². Explain why a small reported variance cannot rule out bias. The plot uses declared fixed axes, so a new setting does not silently rescale the apparent change. Any clipped values are counted and remain available in the numerical table. Do not infer an exact curve or a new observation from a line drawn between points.
Use it in an AI experiment
These ideas support C11, C12, C13. Use the return link when you entered from an AI lesson; its original section and experiment remain separate. A mathematical support calculation is a way to inspect an assumption, not evidence that a trained model generalizes.
Optional derivation and implementation depth
The scalar posterior variance is (1−K)P under independent zero-mean noise. Matrix filters propagate cross-covariance; numerically robust updates preserve symmetry and nonnegative variance. Feedback stability depends on dynamics, delay and discretization. Projecting a control into bounds solves only that declared constraint; full trajectory safety needs independent checks.
Recompute the worked example with a separate piece of arithmetic or a short program. State the domain and test a boundary case before using the formula elsewhere. Guided numerical practice, independent implementation and human review are different outcomes.
Optional implementation and research depth
The scalar posterior variance is (1−K)P under independent zero-mean noise. Matrix filters propagate cross-covariance; numerically robust updates preserve symmetry and nonnegative variance. Feedback stability depends on dynamics, delay and discretization. Projecting a control into bounds solves only that declared constraint; full trajectory safety needs independent checks.
Recompute one row independently, then test a boundary case and state which assumption it probes. Passing the supplied practice case does not establish independent implementation proficiency.
5. Apply it to a different case
Practice not yet passed; independent mastery and human review are not claimed.
6. Continue when useful
Completion not recorded. Skipping retains settings and practice; it never awards completion.
Used in C11, C12, C13. Browse every support topic and course connection.