What alpha = 0.98 Really Sets

Ada re-derives this chapter’s own numbers step by step, at full precision

ADA · CALCULATION AUDIT

What alpha = 0.98 Really Sets

The chapter tunes a complementary filter at alpha=0.98 (98% gyro weight, 2% accelerometer) running at dt=0.02 (50 Hz update rate), and reports that after 10 min the gyro alone drifts 5.1° while the fused estimate drifts only 0.3°. The 98/2 split reads like a minor accelerometer contribution. This audit asks the question that framing hides: what does alpha = 0.98 actually set in terms of a time constant, and is the “2% accelerometer” weight really as small an effect as it sounds?

Companion to the chapter Production Sensor Fusion and Validation — every number here comes from that chapter.

Ada: The recommended alpha = 0.98 is presented as a 98/2 weight split, which is true but hides the more useful number. With the filter’s own sample step dt = 0.02 s (50 Hz), alpha also fixes the crossover time constant – the horizon over which the accelerometer correction blends in. The standard complementary-filter result is tau = alpha x dt / (1 - alpha):

• Weights: gyro 0.98 -> 98%, accelerometer 1 - 0.98 = 0.02 -> 2%
• Time constant: tau = 0.98 x 0.02 / (1 - 0.98) = 0.0196 / 0.02 = 0.98 s
• Crossover frequency: f_c = 1 / (2 x pi x tau) = 1 / (2 x pi x 0.98) = 0.162 Hz

So the gyro owns everything faster than about 0.16 Hz and the accelerometer quietly corrects everything slower. That is why the raw gyro drift spec of 0.5 deg/min is survivable: over the table’s 10-minute test the uncorrected drift would reach 0.5 deg/min x 10 min = 5.0 deg (matching the measured ~5.1 deg gyro-only figure), yet the accelerometer’s low-frequency authority holds the fused drift near a fraction of a degree.

The design-meaningful reading is that the “2% accelerometer” weight is not a small effect: spread across a roughly 1-second time constant it is exactly the low-frequency anchor that keeps the integrated gyro from walking away, so tuning alpha is really choosing where fast trust ends and slow correction begins.

Every number above is taken from the chapter’s own material and re-derived step by step.