Dynamic Skinning

The formula, in plain words

Read κ(t) as the response of the skin to a sudden kick: A sets how far it swings, sin(2πf t) makes it go back and forth f times per second, and exp(−t/τ) makes the swing fade out after about τ seconds.

1. Measure. For each joint, its angular velocity Ω_j is measured every 1/60 s and kept in a short history.
2. Filter. Every change of speed (a start, a stop, a change of direction) launches a damped wobble shaped by κ, and the wobbles add up. The drag term D Ω_j adds a lag proportional to the current speed: alone, it is Velocity Skinning, and it vanishes as soon as the joint stops, while the wobbles keep going.
3. Bend. Each vertex is rotated around the joint, against the motion, by an angle that grows with its distance to the joint. The weight β_ij sums the skinning weights of the joint and of all its descendants: turning the shoulder bends the whole arm.

The algorithm

// Precomputation, from the skinning weights α
for each vertex i, joint j:
  βij = sum of αik over k = j and its descendants

// Each frame, after standard skinning (LBS)
for each joint j:
  Ωj = angular velocity of j
  append Ωj to its history Hj      // step Δt = 1/60 s
  Fj = D Ωj                         // drag
     + Σk κ(kΔt) · ΔHj(t − kΔt)   // oscillations

for each vertex i at position pi:
  di = 0
  for each joint j with βij > 0:
    θ = −γi ‖Fj × (pi − qj)‖
    p' = rotate pi around joint qj,
         axis Fj, angle θ
    di += βij (p' − pi)
  pi = pi + di

ΔH_j is the change of angular velocity between two consecutive samples, and γ_i the (paintable) bend weight of the vertex.