Featherstone's algorithm

Definition

Let a multibody system have bodies $1, \ldots, n$ and a fixed base $0$, whose joint graph is a rooted tree. Let body $i$ be joined to its parent $\lambda(i)$ by joint $i$, with joint coordinates $q_i$, velocities $\dot q_i$ and applied joint forces $\tau_i$. Featherstone's algorithm computes the joint accelerations $\ddot q_1, \ldots, \ddot q_n$ in three passes over the tree.

1. Outward pass: compute the velocity of each body from that of its parent and $\dot q_i$, and its velocity-dependent acceleration terms.
2. Inward pass: compute each body's articulated-body inertia and bias force from its own inertia and the values of its children.
3. Outward pass: compute $\ddot q_i$ from the parent's acceleration, $\tau_i$ and the articulated quantities of body $i$, then the acceleration of body $i$.

The outward passes go from the base to the leaves, and the inward pass from the leaves to the base. The articulated-body inertia and bias force describe how body $i$ with all its descendants responds to a force on body $i$.