Dead Reckoning & Position Integration
Dead Reckoning & Position Integration
Before GPS, ships and aircraft navigated by dead reckoning (DR): start at a known position, measure speed and heading, and integrate forward through time. DR remains the core of every inertial navigation system on submarines and satellites — it fails gracefully when external fixes are unavailable but accumulates error with time.
Each step in DR is simple: $x(t+\Delta t) = x(t) + V\cos(\theta)\Delta t$, $y(t+\Delta t) = y(t) + V\sin(\theta)\Delta t$, where $V$ is ground speed and $\theta$ is track angle (measured CCW from east in math convention, or clockwise from north in navigation). Errors in speed, heading, or timing all accumulate linearly — so a 1% speed error becomes a 600 m error in 60 km.
We practise stepping DR by hand and analyse how error grows with distance.
$$\Delta x = V\cos\theta\,\Delta t, \qquad \Delta y = V\sin\theta\,\Delta t$$
Heading is where the vehicle points; track is the actual ground path. In wind or current, they differ by a drift angle. Drift correction: $\sin(\text{drift}) = V_{\text{wind}}\sin(\alpha)/V_{\text{ground}}$.
Start at origin. Fly at $V = 120\,\text{kt}$ on heading $60°$ (compass) for 30 minutes. Find the new position.
Compass bearing 60° = math angle $90° - 60° = 30°$ from east. Distance = $120 \cdot 0.5 = 60\,\text{nmi}$.
$\Delta x = 60\cos 30° = 51.96\,\text{nmi}$ east, $\Delta y = 60\sin 30° = 30\,\text{nmi}$ north.
From Example 1 end point, turn to heading $150°$ (ESE-S) and fly 20 nmi. Find new position.
Math angle = $90° - 150° = -60°$. $\Delta x = 20\cos(-60°) = 10$ east; $\Delta y = 20\sin(-60°) = -17.32$ north.
Final: $(51.96 + 10,\, 30 - 17.32) = (61.96,\, 12.68)\,\text{nmi}$.
A vehicle traverses 100 km. Speed sensor has $\sigma_V = 1\%$, heading sensor has $\sigma_\theta = 1°$. Estimate the 1-σ position error.
Along-track: $\sigma_{\text{along}} \approx d \cdot 0.01 = 1\,\text{km}$.
Cross-track from heading: $\sigma_{\text{cross}} \approx d \cdot \sin(1°) \approx 100\cdot 0.01745 = 1.745\,\text{km}$.
Total 1-σ: $\sqrt{1^2 + 1.745^2} \approx 2.01\,\text{km}$.
Practice Problems
Show Answer Key
1. $\Delta x = V\cos\theta\,\Delta t$, $\Delta y = V\sin\theta\,\Delta t$.
2. $90° - 270° = -180°$, equivalent to $+180°$ — due west.
3. $300$ nmi.
4. Small per-step errors don't average out because heading bias is systematic; they accumulate.
5. Angle between heading and actual ground track caused by wind or current.
6. Inertial navigation systems (aircraft, submarines), wheel-encoder odometry on robots.