Rigid solid dynamics in practice

1. Initial condition
- - \(p(t_0)\), \(v(t_0)\), \(\mathrm{R}(t_0)\), \(\omega(t_0)\) given as initial condition
- - Precompute \(I_0=I(t_0)\)
- - Compute \(L(t_0) = I_0\;\omega(t_0)\)

2. Temporal Evolution

Iterate over time \(t_k\)
- - Compute total force \(F(t_k)\) and torque \(\tau(t_k)\)
- - Compute \(I(t_k)=\mathrm{R}(t_k)\,I_0\,\mathrm{R}^T(t_k)\)
- - Compute \(\omega(t_k)= I(t_k)^{-1}\,L(t_k)\)

- - Numerical integration updating state vector
- - \(\rightarrow\) \((p(t_{k+1}), P(t_{k+1}), \mathrm{R}(t_{k+1}), L(t_{k+1}))\)

- - Handle collision

\(\displaystyle \left(\begin{array}{c}p'(t) \\ P'(t) \\ \mathrm{R}'(t) \\ L'(t) \end{array}\right)=\left(\begin{array}{c}v(t) \\ F(t) \\ \hat \omega(t)\,\,\mathrm{R}(t) \\ \tau(t) \end{array}\right)\)

\(I(t)=\mathrm{R}(t)\,I_0\,\mathrm{R}^T(t)\)
\(L(t)=I(t)\,\omega(t)\)
\(P(t)=m\,v(t)\)

pictures/cereals.png