02 · Perception, state estimation and time

Assumptions, implementation & acceptance

The scalar Kalman example assumes direct measurement , independent zero-mean noises and positive innovation variance. Covariances use squared position units.

Formula conditions, units and worked arithmetic reviewed. The implementation isolates one numerical mechanism; it does not train a policy or control real hardware.

Before running: predict a boundary case

Posterior lies between prior and measurement; posterior variance is nonnegative and smaller.

Minimal executable example

Save the example and linked modules in one folder. Use Node.js 22 or newer, then run node perception-and-state.mjs. No third-party runtime dependencies.

const solve=()=>{
const prior=1, P=4, measurement=3, R=1;
const K=P/(P+R);return [K,prior+K*(measurement-prior),(1-K)*P];
};
const actual=solve();
const expected=[0.8,2.6,0.8];
if(JSON.stringify(actual)===JSON.stringify(expected)){console.log('PASS',actual)}else{
const a=[actual].flat(), b=[expected].flat();
if(a.length!==b.length||a.some((v,i)=>!Number.isFinite(v)||Math.abs(v-b[i])>1e-10))throw Error('Numerical mismatch');
console.log('PASS within 1e-10',actual);
}

[
  0.8,
  2.6,
  0.8
]

Test and transfer

Posterior lies between prior and measurement; posterior variance is nonnegative and smaller.

Record input, code version, expected result and observed result in the notebook below. Then construct a counterexample that violates the assumption above and explain why the original conclusion no longer follows.