Math/Physics

Cross product

calculator

Law of Cosines

$c=\sqrt{a^2+b^2-2ab\cdot \cos \gamma }$

Small angle approximations:

$\sin \theta \approx \theta ,\tan \theta \approx \theta ,\cos \theta \approx 1-\frac{{\theta }^2}{2}$ $\sin (u)\sin (v)=\frac{1}{2}[\cos (u-v)-\cos (u+v)]$ $\cos (u)\cos (v)=\frac{1}{2}[\cos (u-v)+\cos (u+v)]$

Physics prequisite

force to torque:

$\tau =r\times F=rF\sin \theta$

friction torque(viscous) >Wiki:

$\tau =-k\dot{\theta }$

弹性力（k 为弹簧的劲度系数，x 为振子偏离平衡位置的位移）： $F=-kx$ 阻尼力（c 为阻尼系数，v 为振子速度）： $F=-cv$

Potential energy

Springs: $\frac{1}{2}k(x-x_0{)}^2$

Gravity: $mgh$

Kinetic energy

Linear motion: $\frac{1}{2}m{v}^{T}v$

Angular motion: $\frac{1}{2}{\omega }^{T}I\omega$

Inertia

$I$ is always symmetric. Diagonal if CoM aligns with primary axes.

For cylinder: $I_z=\frac{1}{2}M{R}^2$, $I_x=I_y=\frac{1}{4}M{R}^2+\frac{1}{12}M{L}^2$