Multi-Obstacle Muscle Wrapping Based on a Discrete Variational Principle

Biomechanical simulations with dynamic muscle paths on NURBS surfaces

“…In the given example, the evolution of the muscle length and orientation shows the same qualitative behaviour as in widely used multi obstacle wrapping methods, see e.g. [4,5]. The major difference is that the wrapping obstacle is modeled as a single closed surface and no calculation of G1-continuous transitions is necessary.…”

“…variational principle δE = 0 yields that for a stationary point γ of E, the corresponding Euler-Lagrange equation 1has to hold, where Φ(γ) = ∂φ(γ) /∂γ ∈ R 1×3 is the surface constraint Jacobian and γ is the derivative of the geodesic curve with respect to s. Given a nonsingular differentiable parametrization γ = F (ν) in terms of surface coordinates ν ∈ R 2 , the Jacobian ∂F (ν) /∂ν ∈ R 3×2 can be used to project equation (1) into the tangent space of the manifold defined by the surface constraint. Thus, differential equation 2is equivalent to equation (1) and the Jacobian ∂F (ν) /∂ν plays the role of a null space matrix [1,3,4]. From the point of view of classical mechanics, the solution of equations (1) and (2) is the trajectories of a free particle on a constraint manifold.…”

“…Moreover, the evolution of the muscle length is shown for two different wrapping formulations. On the left hand side, the bone geometry is approximated via a NURBS surface, while the muscles in the centre wrap around several cylinders and spheres with G1-continuous transitions, see [4].…”

“…Typically, muscle paths cannot be adequately represented as straight lines because the anatomical structure of the human body forces the muscles to wrap around bones and adjacent tissue. Assuming that the muscles and tendons are always under tension, their path can be modelled as a locally length minimizing curve that wraps smoothly over adjacent obstacles [2,4,5].…”

“…Left: Muscle path of the biceps and the triceps around NURBS bone geometry approximation 2 . Center: Muscle path of the biceps and the triceps around multiple obstacles with G1-continuous transitions from[4]. Right: Length evolution of the biceps and triceps for two different wrapping surfaces.…”