Fig. 3.

Sparsity structures of the Jacobians ∂f/∂x, ∂f/∂ẋ, and ∂f/∂u of the implicit dynamics equation f (x, ẋ, u) = 0 for the 2D gait model with 9 degrees of freedom and 16 muscles [24]. Columns correspond to the state variables defined in (12). The dense blocks in rows 10–18 are due to multibody dynamics (equation 1). The 3×3 holes in these blocks are due to the limbs, each of which has three degrees of freedom, being decoupled in the implicit formulation. The stair stepping pattern in these rows is due to muscle-skeleton coupling, which only produces non-zero Jacobian elements when the muscle actually crosses the joint. Explicit formulations will have fully dense blocks in those parts of the Jacobian, due to dynamic coupling which causes each force generating element to affect each generalized acceleration in the system.