Turn observations into estimates.

A sensor does not hand you the current state

A robot’s state might include position and velocity. A camera can provide evidence about position, while a proprioceptive sensor measures something about the robot’s own motion. Both measurements can contain noise, systematic error or delay. This lab uses fictional scalar position and velocity readings; it does not process images or model a particular physical sensor.

Every reading has a measurement time and an arrival time. If a camera reports where the cart was half a second ago, the number may be accurate for that earlier time and still be poor evidence of its current position. The lab displays both sensor timestamps separately. Its first reading is reused during initial delay; it never invents measurements from before the experiment.

Make a prediction before revealing truth

Start with the raw readings. Reduce the weight on each new reading and predict whether the curve becomes smoother. Then reveal the simulator truth diagnostic. Does smoother also mean closer to the current true position?

The truth curve is available because we authored the simulation. It is used for evaluation, not by the estimator. On a real robot, a trustworthy reference measurement would require additional equipment or a carefully designed experiment. A filter cannot normally measure its own true error just by inspecting its output.

Explain the weighted average

At a weight of 0.2, the next estimate takes 80% of the previous estimate and 20% of the delivered reading. For example, a previous estimate of 1 metre and a reading of 2 metres produce 0.8 × 1 + 0.2 × 2 = 1.2 metres. The weights add to one, so the result lies between its two inputs.

Giving older information more influence can suppress rapid noise changes. It also delays the response to real motion. With no noise at all, smoothing can introduce error that the raw current reading did not have. At weight 1, there is no smoothing: the estimate equals the delivered reading, including any delay or bias.

Position and velocity are filtered separately here. They are not combined to predict each other. That makes this a simple baseline, not a Kalman filter or a complete sensor-fusion system. The next lesson will add a motion model and a prediction/correction cycle.

Uncertainty needs assumptions

A position reading with bounded noise of 0.15 metres is within that distance of its measurement-time truth, if the noise bound is correct and there is no bias. To use it at the current time, we must also allow for movement during its age. At a declared maximum speed of 0.8 m/s, a reading 0.5 seconds old needs another 0.4 metres of allowance.

The filter’s previous estimate also grows stale between updates. Its position-error allowance grows by maximum speed × elapsed time before being combined with the new reading’s allowance. Velocity uses a maximum-acceleration assumption in the same way. These worst-case envelopes can be wide. They are neither probabilities nor statistical confidence intervals.

The error-bound arithmetic

If a previous estimate has error at most B, after one time step its error against current position is at most B + maximum speed × time step. Let R be the delivered reading’s current-time error bound and a the weight. The next estimate’s error is at most (1 − a) × (B + maximum speed × time step) + a × R.

This follows because the magnitude of a sum cannot exceed the sum of the magnitudes: the triangle inequality. It does not require the errors to be independent. The first estimate starts from the first reading and its bound. The same argument applies to velocity using maximum acceleration. The lab’s authored sinusoidal motion satisfies the declared limits of 0.8 m/s and 0.8 m/s².

Deliberately break an assumption

Set position noise to zero, weight to one and camera bias to 0.4 metres. The estimator receives a consistent offset but is not told to compensate for it. Its zero-bias envelope can miss the truth even though the curve looks clean.

A biased trace might sometimes fit inside a wide envelope by coincidence. That does not restore the validity of a broken assumption. Likewise, changing settings reuses the same underlying motion and random offsets; these comparisons are repeated development practice, not independent evaluations.

Produce an observation audit

For guided practice, capture a raw-reading baseline, a strongly smoothed trace and a delayed or biased trace. Explain the difference between noise reduction and current-state accuracy.

For implementation practice, independently expand the filter into a weighted sum of all delivered readings. Verify the timestamp indexing and prove the error-envelope recurrence. For a decision brief, identify which measurements, timing guarantees and motion bounds would need verification before relying on this estimator.

The diagnostic RMSE is the square root of the average squared difference from current simulator truth. Report position and velocity separately because their units differ. These software experiments do not establish independent implementation proficiency or physical deployment readiness. Human review and learner pilots remain pending.