Sensors & How They Work

A gyro/accelerometer reading is the true value plus a slowly-varying bias $b$ (itself a random walk, i.e. bias instability), a scale-factor error, and white noise.

Integrating rate to get angle (or acceleration to get position) integrates the bias too, so a constant bias becomes a linear drift in angle and a quadratic drift in position — this is why raw IMU dead-reckoning is unbounded without an external reference.

Pinhole projection: $x = f X / Z$, $y = f Y / Z$ — a 3D point projects to the image plane scaled by focal length and inversely by depth.

Lens distortion (radial/tangential) bends this ideal mapping near the edges of the frame; intrinsic calibration (focal length, principal point, distortion coefficients) corrects it, typically fit from checkerboard images.

$Z = \dfrac{fB}{d}$, where $d$ is the disparity (pixel shift of the same point between the two cameras).

Depth resolution degrades quadratically with distance ($\partial Z/\partial d \propto Z^2$), so stereo is precise up close and gets rapidly noisier far away.

Intrinsic: a sensor's internal geometry (camera focal length/distortion, LiDAR beam angles). Extrinsic: the rigid transform between sensors' frames (or sensor to robot body). Temporal: aligning timestamps across sensors that sample at different rates or with different latency — an uncorrected time offset shows up as apparent extrinsic error at any speed.

GPS gives roughly meter-level absolute position from satellite ranging but degrades badly with multipath (reflections off buildings) and urban-canyon signal blockage; RTK correction data can push this to centimeter level.

Wheel encoders give cheap, high-rate relative motion via dead reckoning, but wheel slip means the wheel can rotate without the robot actually translating that far, so position error still grows unbounded over distance without correction.

Datasheets report noise as zero-mean Gaussian with a stated $\sigma$ (or an Allan deviation plot for IMUs), which separates the noise into regions dominated by white noise (short averaging times, deviation falls as $1/\sqrt{\tau}$) versus bias instability (flattens out) versus rate random walk (rises again at long $\tau$) — the minimum of the curve is the sensor's best achievable stability.