Section 29.2: Odometry and dead reckoning

"Dead reckoning is useful because it admits how fast it is becoming unsure."

A Loop Closure That Came Back With Receipts

Problem First

Figure 29.2.1 sets up the running theme of this section: a visual idea only becomes useful state estimation once it is tied to a state variable, an uncertainty model, and the next robot action.

Blindfold a person, spin them once, and ask them to walk back to where they started: they will stride out confidently and end up meters off, never doubting a single step. A robot running on odometry alone is that blindfolded walker, because wheel ticks, Inertial Measurement Unit (IMU) integration, and visual motion increments keep producing a pose even when no map feature is visible, and the danger is that small bias accumulates without an external correction.

For a differential-drive base, where two independently powered wheels on a common axle set both speed and heading by how much each wheel turns relative to the other, wheel angular increments become a forward displacement and heading change. The covariance must grow with each integration step because uncertainty compounds. Dead reckoning serves as a short-horizon prediction, not a long-term truth source. A robot that knows only its best guess, and not how wrong that guess might be, navigates on hope rather than geometry. The numbers make the stakes concrete. A 1% heading bias hides in a single step. After 100 meters it produces roughly a 1-radian heading error and a position error on the order of the route length itself. This phenomenon is called unbounded open-loop drift, and every production system corrects it before it compounds. Figure 29.2.2 below traces this pipeline end to end: wheel measurements feed the kinematic update, which advances the pose, while the covariance ellipse grows step by step until heading error dominates.

Formal Model

The belief that carries both drift and covariance forward is a chained motion posterior conditioned on wheel, leg, visual, or inertial increments. Its value is determined by how quickly uncertainty grows and by whether that growth is visible to the planner.

The equation below is the midpoint kinematic update introduced informally in Figure 29.2.2: it takes the forward displacement \(\Delta s\) and heading change \(\Delta\theta\) computed from wheel increments and advances the pose \((x_t, y_t, \theta_t)\) by one integration step, evaluating the direction of travel at the midpoint heading \(\theta_t + \Delta\theta/2\) rather than at the start or end heading, which is what makes the update accurate to second order instead of first order.

$$ x_{t+1}=x_t+\Delta s\cos(\theta_t+\Delta\theta/2),\quad y_{t+1}=y_t+\Delta s\sin(\theta_t+\Delta\theta/2),\quad \theta_{t+1}=\theta_t+\Delta\theta $$

The important evidence terms are wheel radius, slip, encoder quantization, IMU bias, integration timestep, and frame transform. Each term needs a residual or drift estimate because dead reckoning fails gradually before it fails obviously.

Covariance propagation matters because a robot tracking only its best-guess pose cannot decide when to trust that pose or ask for a correction. Without covariance, a planner never widens its safety margin as uncertainty grows, and a warehouse robot commits to a narrow aisle it no longer knows it can clear. A missing or frozen covariance is among the most common reasons a sensor-fusion stack silently accepts a stale pose.

The mechanism follows the linearized motion model. After each integration step, the update sets the new covariance \(\Sigma_{t+1} = J_f \Sigma_t J_f^T + Q\), where \(J_f\) is the Jacobian (the matrix of partial derivatives that gives the best linear approximation of the kinematic update near the current state) of the kinematic update with respect to the current state and \(Q\) is a diagonal matrix of per-wheel or per-axis noise. Heading uncertainty grows faster than position uncertainty because heading error rotates every subsequent translation increment off-axis. A concrete illustration shows the gap. A 0.1-degree per-step heading bias stays invisible in the covariance after one step. After 200 steps across a 10-meter route, that same bias will typically rotate the robot's frame of reference enough to push the lateral position error past roughly 35 cm, though the exact figure depends on the specific motion profile and noise model used. A 0.1-meter constant position offset, by contrast, shifts the whole path by exactly 0.1 meter and nothing more.

Worked Diagnostic

The covariance equations above tell you how drift should grow, but the only way to trust them is to run the kinematic update on a known increment and watch the heading error appear. Code Fragment 1 is the odometry unit test: integrate a known motion increment, inspect pose drift, then compare the hand result with the library or ROS node before trusting long routes.

# Integrate a differential-drive odometry increment.
# A small wheel mismatch creates a heading change that will compound.
import math

r = 0.05
baseline = 0.32
d_left = 1.00
d_right = 1.06
delta_s = r * (d_right + d_left) / 2.0
delta_theta = r * (d_right - d_left) / baseline
print(f"forward={delta_s:.3f} m")
print(f"heading_change={math.degrees(delta_theta):.2f} deg")

Expected output interpretation. The forward increment is only about 5 cm, but it already carries a nonzero yaw error. That is the important readout: if the robot repeats this biased update for hundreds of steps, the translational error will grow mostly because the heading estimate keeps rotating the future motion in the wrong direction.

Tool Workflow

Keep the dead-reckoning calculation small enough to audit by hand, then use robot_localization, Nav2 odometry sources, or a factor graph for deployment. The invariant is that motion increments and covariance grow in the expected frame.