Drone maths

Euler angles, z then y then x

\[ p' = R_x(\alpha)\,R_y(\beta)\,R_z(\gamma)\,p \] \[ R_y(\beta) = \begin{pmatrix} \cos\beta & 0 & \sin\beta \\ 0 & 1 & 0 \\ -\sin\beta & 0 & \cos\beta \end{pmatrix} \]

Orthographic view: drop depth, scale \((x', y')\) by zoom \(s\), draw back to front by \(z'\). Released spin \(\omega_0\) decays over a slow idle turn:

\[ \dot\beta(t) = \omega_{\text{idle}} + \omega_0 e^{-kt}, \qquad \ddot\beta = -k\,\omega_0 e^{-kt} \] \[ \beta(t) = \beta_0 + \omega_{\text{idle}}\,t + \frac{\omega_0}{k}\left(1 - e^{-kt}\right) \]

Rotors spin at constant \(\dot\theta = \omega_r\); diagonal pairs counter-rotate so torques cancel.

Pitch α

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Yaw β (mod 360°)

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Angular velocity β̇

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Angular acceleration β̈ = −kω

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Easing, velocity, tilt

\[ \begin{aligned} s(t) &= L\,E(t/T) \\ v(t) &= \tfrac{L}{T}\,E'(t/T) \\ a(t) &= \tfrac{L}{T^2}\,E''(t/T) \end{aligned} \]

inOutSine: \(E(u) = \tfrac{1-\cos \pi u}{2}\), peak speed \(\tfrac{\pi}{2}\tfrac{L}{T}\) mid-leg. Tilt \(\theta = \arctan(a/g)\).

\[ \begin{aligned} w_{\perp} &= 2h\tan\tfrac{\varphi}{2}, \quad w_{\parallel} = \tfrac34 w_{\perp} \\ \Delta &= w_{\parallel}(1-o_f) \\ S &= w_{\perp}(1-o_s) \end{aligned} \]

Altitude h

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Ground speed v

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Acceleration a

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Body tilt θ

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Leg progress u

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Peak v this ease

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Footprint w⊥ × w∥

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Trigger spacing Δ

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Lane spacing S

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Photos taken

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