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Sensors & Noise

How robots perceive — encoders, IMUs, LIDAR, and cameras, plus sampling and the noise every reading carries.

Every estimate a robot makes starts with a sensor, and every sensor is imperfect: it samples reality at discrete instants, with limited range and resolution, and adds noise. Understanding what each sensor measures — and how it lies — is the foundation under localization, SLAM, and control.

The visualizer sweeps a 2D LIDAR around a room. Each beam returns the distance to the nearest surface, and the returns form a point cloud. Drag the noise slider: at zero the points sit exactly on the walls; raise it and they scatter, more so at longer range — just like real hardware.

measurement noise σ = 4

Speed

0/120 beams

sensor beam range return surface

range 156 (true 150)

A LIDAR measures distance by timing reflected light. Each beam returns the range to the nearest surface, and the fan of returns forms a point cloud the robot uses to perceive its surroundings. With noise at zero the points sit exactly on the walls; raise it and the returns scatter around the true surface — and the scatter grows with range, just like a real sensor. Encoders, IMUs, and cameras each carry their own noise model in the same way.

The common sensors

Encoders — count wheel or joint rotation for odometry. Cheap and precise per step, but errors integrate, so position drifts over distance.
IMU (inertial measurement unit) — accelerometers and gyroscopes report acceleration and angular rate. Great short-term, but integrating twice to get position lets bias drift quickly.
LIDAR — fires laser beams and times the reflections to measure range, producing the point cloud above. Accurate geometry; struggles with glass and rain.
Cameras — dense, rich, cheap pixels, but they need heavy computation to recover geometry and falter in poor light.

These split into proprioceptive (about the robot itself: encoders, IMU) and exteroceptive (about the world: LIDAR, cameras).

Sampling

Sensors are discrete. A reading arrives every $\Delta t = 1/f$ seconds at sample rate $f$ , and a scanner has finite angular resolution. Sample too slowly and you miss fast events — the Nyquist limit (Math track) says you must sample faster than twice the highest frequency you care about. Resolution and rate bound everything downstream.

Noise

A real reading is the true value plus error. The standard model is additive Gaussian noise,

z = z^{\star} + \varepsilon, \qquad \varepsilon \sim \mathcal{N}(0, \sigma^2),

where $z^{\star}$ is the true measurement and $\sigma$ is the noise standard deviation — the slider in the demo. Larger $\sigma$ means a wider spread of returns. Often $\sigma$ itself grows with range (more distant points are noisier), which is why the far walls scatter more than the near ones.

Beyond Gaussian noise, sensors also suffer bias (a constant offset), outliers (spurious returns off reflections), and quantization from finite resolution. This is exactly why robots fuse many sensors and track uncertainty explicitly — a filter weights each reading by how much, given its $\sigma$ , it should be trusted.

Takeaways

Sensors are proprioceptive (encoders, IMU) or exteroceptive (LIDAR, cameras); each has distinct strengths and failure modes.
Sampling is discrete in time and resolution; the rate bounds what you can observe.
Real readings carry noise (often Gaussian, growing with range) plus bias and outliers, which downstream estimators model and fuse.

References

Thrun, Burgard & Fox, Probabilistic Robotics — Ch. 6 (sensor models, including beam and range models).
Siegwart, Nourbakhsh & Scaramuzza, Introduction to Autonomous Mobile Robots — Ch. 4 (perception and sensors).