robotics//navigation//dead reckoning
Dead reckoning is the estimation of a vehicle's position and attitude by integrating its measured velocities and accelerations from a known starting point, without any external reference, and it is the backbone of every inertial navigation system, from a ship's log to the attitude of a drone between two GPS fixes. Its appeal is that it needs nothing from the outside world: it works in a tunnel, under jamming, indoors, and it delivers a smooth estimate at the full rate of the sensor with no latency. Its flaw is that every error integrated stays in the sum.
Dead reckoning is the estimation of a vehicle's position and attitude by integrating its measured velocities and accelerations from a known starting point, without any external reference, and it is the backbone of every inertial navigation system, from a ship's log to the attitude of a drone between two GPS fixes. Its appeal is that it needs nothing from the outside world: it works in a tunnel, under jamming, indoors, and it delivers a smooth estimate at the full rate of the sensor with no latency. Its flaw is that every error integrated stays in the sum.
Integrate a gyroscope with a constant bias bbb and white noise of coefficient NNN (the angle random walk of the datasheet) and the angle error is
δθ(t)=b t+∫0tn(s) ds,σθ(t)=Nt.\delta\theta(t) = b\,t + \int_0^t n(s)\,ds,\qquad \sigma_\theta(t) = N\sqrt{t}.δθ(t)=bt+∫0tn(s)ds,σθ(t)=Nt.
The bias grows as a straight line; the noise opens as t\sqrt tt, a random walk. The accelerometer is worse because it is integrated twice: a bias bab_aba gives a position error 12bat2\tfrac12 b_a t^221bat2. A bias of 10 mg, normal for an uncalibrated cheap MEMS part, is 5 m after 10 seconds and about 180 m after a minute. And an attitude error leaks gravity into the horizontal: a tilt error of 1 mrad (0.06°) projects gravity as a 1 mg horizontal bias, so attitude and position errors feed each other.
mean error at 60 s1.64° largest at 60 s1.77° RMS in theory at 60 s1.67° dominant at 60 sbias A gyroscope at rest integrated for 120 s, twenty realizations: white noise of 0.50 °/√h, a bias of 100 °/h and an instability of 10.0 °/h. At 60 s the angle error is 1.67° RMS, most of it from the bias.
Start with white noise alone and watch the fan of twenty runs open as t\sqrt tt; add the constant bias and the whole fan tilts, switch on calibration at rest and the tilt disappears, then raise the bias instability, the part that changes after calibrating and that no calibration removes.
Integrating sensors accumulates error without limit.
Dead reckoning is excellent over seconds and useless over hours, so it always needs an external reference (GNSS, a camera, a map) to pull it back, and its job in a fused estimator is to carry the vehicle between those references.
Grade buys time, never immunity: a navigation-grade gyroscope drifts four orders of magnitude slower than a consumer one, which stretches minutes into hours (IMU, Allan variance).
The fusion that combines it with slow, drift-free references is sensor fusion, and for attitude alone the complementary filter; the wider problem is navigation.