M3-T12: Observability and consistency

Learning objectives

After this lesson, you should be able to:

• Define observability and compute the observability matrix of a linear system.
• Identify the unobservable directions of visual-inertial navigation (global position and yaw) and the effect of bias states.
• Explain how spurious information in an unobservable direction makes a filter inconsistent.
• Choose measurements or motion that make a state observable.

Conceptual explanation

A state is observable if it can be determined from the measurements (and controls) over some time. For a linear system $x_{k+1}=F\,x_{k}$, $z_{k}=H\,x_{k}$, the system is observable if the observability matrix has full column rank. Unobservable directions are those along which the state can change without altering any measurement.

Engineering intuition

Shift the whole world a metre east and rotate it about the vertical: every image of the environment and every IMU measurement (relative to gravity) is unchanged. Position and yaw are therefore unobservable in VIO: four degrees of freedom, the “gauge”. The true uncertainty in those directions grows with time (drift). A filter whose linearisation points happen to depend on its current estimate can, through numerical accident, appear to learn about yaw from a measurement that contains none: its covariance in yaw shrinks when it should not. That is inconsistency, and the cure is to build the Jacobians so the unobservable directions stay in their null space.

Mathematical formulation

For the linear system, the observability matrix and its rank test:

Example: position–velocity with an accelerometer bias (state [$p$, $v$, $b$]); acceleration measured as a $=u+b$. With only a velocity or position measurement the bias is observable if the vehicle accelerates (the bias then changes the prediction), but with no measurement of position or velocity the whole state is unobservable. For the nonlinear VIO case, observability is analysed with Lie derivatives (Hesch et al., Kelly and Sukhatme): the result is four unobservable directions (3 translation, 1 yaw), plus scale becoming unobservable in a monocular system without sufficient excitation.

Consistency requires that the filter’s linearised model have the same unobservable subspace as the true nonlinear system. Observability-constrained EKF (OC-EKF) and first-estimate Jacobians enforce this: they choose the linearisation points so that $\operatorname{null} (O)$ contains the true gauge directions.

$O$

observability matrix (linear case) or observability Gramian.

$\operatorname{null} (O)$

directions that cannot be inferred from the data.

Architecture and data flow

The practical consequence for system design: provide a measurement that observes what you need. GNSS observes position and (via motion) heading, a magnetometer observes yaw, a map or loop closure observes the gauge directions in a drifting VIO. A navigation manager should know which states are observable in each mode and report a growing covariance in the others (lesson M2-T16).

Real-world UAV applications

• Deciding whether VIO needs a heading source for long flights.
• Checking that a filter keeps yaw uncertainty growing without GNSS.
• Choosing manoeuvres that excite scale or bias for calibration.
• Explaining overconfident covariance in a research estimator.

Implementation example

This runs as shown: observability ranks for a 1-D inertial navigation model with accelerometer bias and for a planar vehicle with yaw, and the observability Gramian eigenvalues for a constant-heading case.

import numpy as np

def obs_matrix(F, H, n):
    blocks, M = [], np.eye(n)
    for _ in range(n): blocks.append(H @ M); M = F @ M
    return np.vstack(blocks)
rank = lambda O: np.linalg.matrix_rank(O, tol=1e-9)

dt = 0.1
# 1-D inertial navigation: state [p, v, b_a]; the accelerometer measures a + b_a, which we treat as a known input u (so the bias drives velocity)
F = np.array([[1, dt, 0], [0, 1, -dt], [0, 0, 1.0]])
assert rank(obs_matrix(F, np.array([[1.0, 0, 0]]), 3)) == 3         # position measurements make velocity and bias observable
assert rank(obs_matrix(F, np.array([[0, 1.0, 0]]), 3)) == 2         # velocity alone: absolute position stays unobservable
assert rank(obs_matrix(F, np.array([[0, 0, 0.0]]), 3)) == 0         # no measurement: nothing is observable

# planar vehicle, state [x, y, yaw], heading-referenced position increments only observe relative motion
F3 = np.eye(3); H_rel = np.array([[0.0, 0, 0]])
assert rank(obs_matrix(F3, H_rel, 3)) == 0                          # odometry alone: no absolute pose is observable (gauge)
H_pos = np.array([[1.0, 0, 0], [0, 1.0, 0]])
assert rank(obs_matrix(F3, H_pos, 3)) == 2                          # absolute position observes x and y, yaw remains unobservable
H_all = np.vstack([H_pos, [0, 0, 1.0]])
assert rank(obs_matrix(F3, H_all, 3)) == 3                          # adding a heading source (magnetometer, GNSS course) observes yaw

# observability Gramian eigenvalues: the unobservable direction has (numerically) zero information
G = sum(np.linalg.matrix_power(F3.T, k) @ H_pos.T @ H_pos @ np.linalg.matrix_power(F3, k) for k in range(10))
lam = np.linalg.eigvalsh(G); print("Gramian eigenvalues (x, y observed):", np.round(lam, 3))
assert lam[0] < 1e-12 and lam[-1] > 5

Common engineering mistakes

Failure modes

Debugging approach

1. Compute the rank and the null space of the observability matrix of your model.
2. Run NEES on sub-blocks: position, yaw, biases separately (lesson M3-T15).
3. Check that yaw variance grows monotonically without a heading source.
4. Design excitation manoeuvres and check convergence of calibration states.

Practical exercises

1. Compute the observability matrix of a 2-D constant-velocity model measuring position only.
2. Show that adding a magnetometer makes yaw observable in the planar example.
3. Explain why scale is unobservable for a monocular camera flying at constant velocity.
4. Find the dimension of the unobservable subspace of VIO with unknown camera–IMU extrinsics (see the references).

Interview questions

Write an answer in your own words, then reveal the model answer. Nothing is sent anywhere.

References and further reading

Primary papers, official documentation and standards. Each claim above that is specific to a tool or version points here.

• J. A. Hesch, D. G. Kottas, S. L. Bowman, S. I. Roumeliotis, “Camera-IMU-based localization: observability analysis and consistency improvement”, IJRR, 2014
• J. Kelly, G. S. Sukhatme, “Visual-inertial sensor fusion: localization, mapping and sensor-to-sensor self-calibration”, IJRR, 2011
• $T$. $D$. Barfoot, State Estimation for Robotics, Cambridge University Press, 2017, chapter 3.
• $C$. $T$. Chen, Linear System Theory and Design, 3rd ed., Oxford University Press, 1999.

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