Fast Differential-Geometric Methods for Continuous Muscle Wrapping Over Multiple General Surfaces
Musculoskeletal simulation has become an essential tool for understanding human locomotion and movement disorders. Muscle-actuated simulations require methods that continuously compute musculotendon paths, their lengths, and their rates of length change to determine muscle forces, moment arms, and the resulting body and joint loads. Musculotendon paths are often modeled as locally length-minimizing curves that wrap frictionlessly over moving obstacle surfaces representing bone and tissue. Biologically accurate wrapping surfaces are complex and a single muscle path may wrap around multiple obstacles. However, state-of-the-art muscle wrapping methods are either limited to analytical results for a pair of simple surfaces, or they are computationally expensive. This thesis describes a new method for the fast and accurate computation of a massless musculotendon’s shortest path that wraps frictionlessly across an arbitrary number of general smooth wrapping surfaces. Furthermore, an explicit formula for the path’s exact rate of length change is presented, as well as an algorithm for simulating path lift-off and touchdown. The total path is regarded as a concatenation of straight-line segments between local surface geodesics, where each geodesic is naturally parameterized by its start point, direction, and length. The shortest path is computed by finding the root of a vector-valued global path-error constraint equation that enforces that the geodesics connect collinearly with adjacent straight-line segments. High computational efficiency is achieved using Newton’s method to zero the path error with an explicit, banded Jacobian that maps natural variations of the geodesic parameters to path-error variations. Simulation benchmarks demonstrate that the proposed method computes high-precision solutions for path length and rate of length change, and that it allows for wrapping over biologically accurate surfaces that can be described either parametrically or implicitly. By using the explicit path-error Jacobian, the proposed method is very efficient and thus allows for simulating muscle paths over hundreds of surfaces in real time.