Engineering Guides · Module 11

Choosing process and measurement noise

The filter trusts a measurement more when Q is large relative to R. Choose R from the sensor and tune Q against the motion model’s real error.

Intermediate
Module 11: Sensor Fusion & State Estimation
Written for ROS 2 Jazzy where ROS 2 is used
Sources checked 2026-11-06

Assumptions: Numbers in the examples are illustrative and the code is executed by the site’s tests. Facts taken from a cited source are marked; other statements are general engineering knowledge and must be confirmed on the datasheet of the part you use.

1. Engineering problem

A localisation filter blends a motion model with measurements. Q says how much the motion model can be wrong between steps; R says how noisy each measurement is. Wrong values give a filter that lags (Q too small) or jitters (Q too large).

2. When is this needed?

Setting up robot_localization or a custom filter.
When the pose lags the truth or jitters.
After changing a sensor.

3. Selection and design decisions

The robot_localization documentation (read 2026-11-06) states that the process noise covariance Q “can be difficult to tune” and that “the larger the value for Q relative to the variance for a given variable in an input message, the faster the filter will converge to the value in the measurement”; it can be given as a full matrix or just the diagonal (off-diagonal values are then zero). It also offers dynamic_process_noise_covariance, which scales Q with velocity so that the covariance stops growing when the robot is stationary (default false), and initial_estimate_covariance (P), whose small values make the filter slow to trust early measurements.

Overview
Parameter	Role	Source of the value
R (measurement covariance)	Noise of each sensor measurement	Sensor datasheet or measured variance; the message covariance
Q (process noise)	Model error per step	Tune from the difference between predicted and measured motion
P0 (initial covariance)	Initial confidence	Large unless the start state is known
Gating threshold	Rejects outliers (see the outlier guide)	Mahalanobis distance

4. Advantages and disadvantages

Larger Q follows measurements closely and passes their noise; smaller Q smooths and lags.
Honest R from the sensor beats a tuned fudge factor; if a driver reports a covariance of zero, supply one.
Dynamic Q avoids covariance growth at rest and needs a velocity-dependent model.

5. Requirements and constraints

R reflects each sensor’s real noise at its operating conditions.
Q is tuned on logged data, not by feel in one manoeuvre.
Covariances in messages are consistent with the units used by the filter.

6. Architecture

motion model (Q) ──predict──▶ state, P ◀──update── measurement (R) ; gain K = P / (P + R)

7. Implementation

Set R from the sensor’s datasheet or a stationary recording.
Start with a small diagonal Q and a large initial P.
Record a manoeuvre and compare predicted and measured motion.
Raise Q until the lag is acceptable and the jitter is not.
Check the filter’s covariance against the actual error.

def steady_gain(q, r, steps=500):
    p = 1.0
    for _ in range(steps):
        p = p + q                      # predict: variance grows by process noise
        k = p / (p + r)                # Kalman gain
        p = (1 - k) * p                # update
    return k
r = 0.04                               # measurement variance (0.2 m std)
for q in (0.0001, 0.001, 0.01, 0.1):
    print(f"Q {q:<7} R {r}: steady gain {steady_gain(q, r):.3f}")
x, p, q = 0.0, 1.0, 0.01
for z in (1.0, 1.0, 1.0):
    p += q; k = p / (p + r); x += k * (z - x); p *= (1 - k)
print("estimate after three measurements of 1.0:", round(x, 3))

Printed (checked by the site’s tests):

Q 0.0001  R 0.04: steady gain 0.049
Q 0.001   R 0.04: steady gain 0.146
Q 0.01    R 0.04: steady gain 0.390
Q 0.1     R 0.04: steady gain 0.766
estimate after three measurements of 1.0: 0.99

Reading it: with a measurement variance R of 0.04 the steady-state gain rises from 0.049 (Q = 0.0001) to 0.766 (Q = 0.1): a larger Q makes the filter follow the measurements. After three measurements of 1.0 with Q = 0.01 the estimate has moved from 0 to 0.99. Values are illustrative; real filters have more states and the gain is a matrix.

8. Real examples

AMR with wheel odometry and a LiDAR pose: Q tuned so that the filter follows the LiDAR pose within one second after a slip.

9. Failure scenarios

A covariance of zero in a message treated as perfect.
Q so small that the filter ignores real pose corrections for minutes.
Tuning on a single straight run.

10. Troubleshooting

If the pose lags, increase Q or check the sensor delay.
If it jitters, decrease Q or increase R.
If the covariance shrinks to zero while the error does not, Q is too small.

11. Validation

The example prints the lines shown (executed by the site tests).
On a logged run the filter’s error stays within its reported covariance (consistent) most of the time.

13. Further references

robot_localization documentation (ros2 branch): state_estimation_nodes.rst and params/ekf.yaml, read 2026-11-06.