IMU Cheat-Sheet

Kalman

Pros Optimal if noise modeled; explicit uncertainty.

Cons Needs Q/R tuning; heavier math.

$$\theta_k = \theta_{k-1} + \omega_k\Delta t + K(\theta_{\text{acc}}-\theta_{k-1})$$

Suitability (balancing robot): Strong if you model gyro bias and process/measurement noise; most accurate but more complex to tune and compute.

Madgwick

Pros Lightweight, fast convergence.

Cons β tuning critical.

$$\dot{q} = \tfrac12 q \otimes \omega - \beta \frac{\nabla f(q)}{\|\nabla f(q)\|}$$

Suitability (balancing robot): Good for full 3D attitude; for 1‑axis pitch control it works, but β must track sample rate and noise—often more than needed.

Mahony

Pros Simple PI feedback; integral cancels bias.

Cons Needs Kp/Ki tuning.

$$\dot{q} = \tfrac12 q \otimes (\omega + K_p e + K_i \int e\,dt)$$

Suitability (balancing robot): Excellent for pitch estimation—low compute, handles gyro bias via integral term; widely used on MCUs.

Complementary

Pros Very simple; tunable cutoff; low compute.

Cons Fixed-frequency assumption; phase lag around cutoff.

$$\theta_k = \alpha\,(\theta_{k-1} + \omega_k\,\Delta t)\; +\; (1-\alpha)\,\theta_{\text{acc}}$$

With time-constant \(\tau\) and sample time \(\Delta t\): \(\alpha = \tfrac{\tau}{\tau + \Delta t}\). High-pass gyro, low-pass accel.

Suitability (balancing robot): Ideal baseline for 1‑axis tilt—robust and trivial to implement; choose cutoff ≈ 1–5 Hz for typical robots.